Quadrant Model of Reality: Quadrant Model of Reality Book 14 Philosophy and Game Theory

Philosophy chapter

QMRTheory of truth[edit]

See also: Two Truths

Wilber believes that the mystical traditions of the world provide access to, and knowledge of, a transcendental reality which is perennial, being the same throughout all times and cultures. This proposition underlies the whole of his conceptual edifice, and is an unquestioned assumption.[note 4] Wilber juxtaposites this generalisation to plain materialism, presenting this as the main paradigma of regular science.[19][quote 1]

Interior Exterior

Individual Standard: Truthfulness

(1st person)

(sincerity, integrity, trustworthiness) Standard: Truth

(3rd person)

(correspondence,

representation, propositional)

Collective Standard: Justness

(2nd person)

(cultural fit, rightness,

mutual understanding) Standard: Functional fit

(3rd person)

(systems theory web,

Structural functionalism,

social systems mesh)

In his later works, Wilber argues that manifest reality is composed of four domains, and that each domain, or "quadrant", has its own truth-standard, or test for validity:[20]

"Interior individual/1st person": the subjective world, the individual subjective sphere;[21]

"Interior collective/2nd person": the intersubjective space, the cultural background;[21]

"Exterior individual/3rd person": the objective state of affairs;[21]

"Exterior collective/3rd person": the functional fit, "how entities fit together in a system".[21

QMRAlthough not present in every New Age group,[123] a core belief of the movement is in channeling.[124] This is the idea that humans beings, sometimes (although not always) in a state of trance, can act "as a channel of information from sources other than their normal selves".[125] These sources are varyingly described as being God, gods and goddesses, ascended masters, spirit guides, extraterrestrials, angels, devas, historical figures, the collective unconscious, elementals, or nature spirits.[125] Hanegraaff described channeling as a form of "articulated revelation",[126] and identified four forms: trance channeling, automatisms, clairaudient channeling, and open channeling.[127]

QMRFurther, Dennett (1987, p. 52) argues that, based on our fixed personal views of what all humans ought to believe, desire and do, we predict (or explain) the beliefs, desires and actions of others “by calculating in a normative system”;[10] and, driven by the reasonable assumption that all humans are rational beings — who do have specific beliefs and desires and do act on the basis of those beliefs and desires in order to get what they want — these predictions/explanations are based on four simple rules:

The agent’s beliefs are those a rational individual ought to have (i.e., given their “perceptual capacities”, “epistemic needs” and “biography”);[11]

In general, these beliefs “are both true and relevant to [their] life;[12]

The agent’s desires are those a rational individual ought to have (i.e., given their “biological needs”, and “the most practicable means of satisfying them”) in order to further their “survival” and “procreation” needs;[13] and

The agent’s behaviour will be composed of those acts a rational individual holding those beliefs (and having those desires) ought to perform.

QMRDennett's three levels[edit]

The core idea is that, when understanding, explaining and/or predicting the behavior of an object, we can choose to view it at varying levels of abstraction. The more concrete the level, the more accurate in principle our predictions are; the more abstract, the greater the computational power we gain by zooming out and skipping over the irrelevant details.

Dennett defines three levels of abstraction, attained by adopting one of three entirely different “stances”, or intellectual strategies: the physical stance; the design stance; and the intentional stance:[14]

The most concrete is the physical stance, the domain of physics and chemistry, which makes predictions from knowledge of the physical constitution of the system and the physical laws that govern its operation; and thus, given a particular set of physical laws and initial conditions, and a particular configuration, a specific future state is predicted (this could also be called the “structure stance”).[15] At this level, we are concerned with such things as mass, energy, velocity, and chemical composition. When we predict where a ball is going to land based on its current trajectory, we are taking the physical stance. Another example of this stance comes when we look at a strip made up of two types of metal bonded together and predict how it will bend as the temperature changes, based on the physical properties of the two metals.

Somewhat more abstract is the design stance, the domain of biology and engineering, which requires no knowledge of the physical constitution or the physical laws that govern a system's operation. Based on an implicit assumption that there is no malfunction in the system, predictions are made from knowledge of the purpose of the system’s design (this could also be called the “teleological stance”).[16] At this level, we are concerned with such things as purpose, function and design. When we predict that a bird will fly when it flaps its wings on the basis that wings are made for flying, we are taking the design stance. Likewise, we can understand the bimetallic strip as a particular type of thermometer, not concerning ourselves with the details of how this type of thermometer happens to work. We can also recognize the purpose that this thermometer serves inside a thermostat and even generalize to other kinds of thermostats that might use a different sort of thermometer. We can even explain the thermostat in terms of what it's good for, saying that it keeps track of the temperature and turns on the heater whenever it gets below a minimum, turning it off once it reaches a maximum.

Most abstract is the intentional stance, the domain of software and minds, which requires no knowledge of either structure or design,[17] and “[clarifies] the logic of mentalistic explanations of behaviour, their predictive power, and their relation to other forms of explanation” (Bolton & Hill, 1996, p. 24). Predictions are made on the basis of explanations expressed in terms of meaningful mental states; and, given the task of predicting or explaining the behaviour of a specific agent (a person, animal, corporation, artifact, nation, etc.), it is implicitly assumed that the agent will always act on the basis of its beliefs and desires in order to get precisely what it wants (this could also be called the “folk psychology stance”).[18] At this level, we are concerned with such things as belief, thinking and intent. When we predict that the bird will fly away because it knows the cat is coming and is afraid of getting eaten, we are taking the intentional stance. Another example would be when we predict that Mary will leave the theater and drive to the restaurant because she sees that the movie is over and is hungry.

In 1971, Dennett also postulated that, whilst “the intentional stance presupposes neither lower stance”, there may well be a fourth, higher level: a “truly moral stance toward the system” — the “personal stance” — which not only “presupposes the intentional stance” (viz., treats the system as rational) but also “views it as a person” (1971/1978, p. 240).

The fourth square is always different

QMRThe Koch snowflake (also known as the Koch curve, star, or island[1]) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire) by the Swedish mathematician Helge von Koch.

The koch snowflake is a one dimensional double tetrahedron star of David Merkaba in its second itteration

QMRWhen tetrahedrons are recursively alternated in a pattern similar to the Koch Snowflake, the first symmetrical polyhedron to emerge is the stella octangula (or stellated octahedron), made of 8 tetrahedrons. In the third iteration, a cuboctahedron frame develops around the stellated octahedron and consists of 64 tetrahedrons (8^2). From here, the shape of a cube begins to emerge, wherein the fourth iteration forms 512 stacked tetrahedrons (8^3), with 2 octaves of cube octahedron geometry, and the fifth holds 4096 tetrahedrons (8^4). Upon further cycles of recursion, the resultant form approaches a perfect cube ad infinitum. See First Five Iterations of the 3D Kochcube

QMRVariants of the Koch curve[edit]

Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (quadratic), other angles (Cesàro), circles and polyhedra and their extensions to higher dimensions (Sphereflake and Kochcube, respectively)

Variant Illustration Construction

1D, 85° angle

Cesàro fractal

The Cesàro fractal is a variant of the Koch curve with an angle between 60° and 90° (here 85°).

1D, 90° angle

Quadratic type 1 curve

The first 2 iterations

1D, 90° angle

Quadratic type 2 curve

The first 2 iterations. Its fractal dimension equals 1.5 and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.

1D, ln 3/ln (√5)

Quadratic flake

The first 2 iterations. Its fractal dimension equals ln 3/ln (√5)=1.37.

1D, ln 3.33/ln (√5)

Quadratic Cross

Another variation. Its fractal dimension equals ln 3.33/ln (√5)=1.49.

2D, triangles

von Koch surface

The first 3 iterations of a natural extension of the Koch curve in 2 dimensions

2D, 90° angle

Quadratic type 1 surface

Extension of the quadratic type 1 curve. The illustration at left shows the fractal after the second iteration

Animation quadratic surface

2D, 90° angle

Quadratic type 2 surface

Extension of the quadratic type 2 curve. The illustration at left shows the fractal after the first iteration.

3D, spheres

Closeup of Haines sphereflake

Eric Haines has developed the sphereflake fractal, which is a three-dimensional version of the Koch snowflake, using spheres.

3D, tetrahedra into cube See Stellated Octahedron, the second recursion of the Kochcube When tetrahedrons are recursively alternated in a pattern similar to the Koch Snowflake, the first symmetrical polyhedron to emerge is the stella octangula (or stellated octahedron), made of 8 tetrahedrons. In the third iteration, a cuboctahedron frame develops around the stellated octahedron and consists of 64 tetrahedrons (8^2). From here, the shape of a cube begins to emerge, wherein the fourth iteration forms 512 stacked tetrahedrons (8^3), with 2 octaves of cube octahedron geometry, and the fifth holds 4096 tetrahedrons (8^4). Upon further cycles of recursion, the resultant form approaches a perfect cube ad infinitum. See First Five Iterations of the 3D Kochcube

Koch curve in 3D

A three-dimensional fractal constructed from Koch curves. The shape can be considered a three-dimensional extension of the curve in the same sense that the Sierpiński pyramid and Menger sponge can be considered as extensions of the Sierpinski triangle and Sierpinski carpet. The version of the curve used for the shape uses 85-degree angles.

Tartan registration[edit]

Depending upon how "different tartan" is defined, it has been estimated that there are about 3,500[56] to 7,000[57] different tartans, with around 150 new designs being created every year.[57] With four ways of presenting the hues in the tartan — "modern", "ancient", "weathered", and "muted" colours — there are thus about 14,000 recognised tartan variations from which to choose. The 7,000 figure above includes many of these variations counted as though they were different tartans.[citation needed]

Coat of arms of the now-defunct Scottish Tartans Society.

Until the late 20th century, instead of a central, official tartan registry, several independent organisations located in Scotland, Canada, and the United States documented and recorded tartan.[58] In the 1960s, a Scottish society called the Scottish Tartans Society (now defunct) was created to record and preserve all known tartan designs.[59] The society's register, the Register of All Publicly Known Tartans (RAPKT), contains about 2,700 different designs of tartan.[56] The society, however, ran into financial troubles in about the year 2000, and folded.[60] Former members of the society then formed two new Scottish-based organisations — the Scottish Tartans Authority (STA) and the Scottish Tartans World Register (STWR). Both of these societies initially based their databases on the RAPKT. The STA's database, the International Tartan Index (ITI) consists of about 3,500 different tartans (with over 7,000, counting variants), as of 2004.[56] The STWR's self-titled Scottish Tartans World Register database is made up of about 3,000 different designs as of 2004.[56] Both organisations are registered Scottish charities and record new tartans (free in the case of STS and for a fee in the case of STWR) on request.[61][62] The STA's ITI is larger, in part, because it has absorbed the entries recorded in the TartanArt database formerly maintained by the merged International Association of Tartan Studies and Tartan Educational and Cultural Association (IATS/TECA), based in the United States, and with whom the STA is directly affiliated.[citation needed]

The Scottish Register of Tartans (SRT) is Scotland's official tartan register. The SRT is maintained and administrated by the National Archives of Scotland (NAS), a statutory body based in Edinburgh.[63] The aim of the Register is to provide a definitive and accessible resource to promote and preserve tartan. It also aims to be the definitive source for the registration of new tartans (that pass NAS criteria for inclusion). The register itself is made up of the existing registers of the STA and the STWR as they were at the time of the SRT's launch, and new registrations from 5 February 2009 onward. On the Register's website users can register new tartans (for a fee), search for and request the threadcounts of existing tartans and receive notifications of newly registered tartans.[64][65] One criticism of the SRT and NAS's management of it is that its exclusivity, in both cost and criteria, necessarily mean that it cannot actually achieve its goals of definiteness, preservation and open access. The current version of the STA's ITI, for example, already contains a large number of tartans that do not appear in the SRT, and the gulf will only widen under current policy.[66]

Tartan etiquette[edit]

Since the Victorian era, authorities on tartan have stated that there is an etiquette to wearing tartan, specifically tartan attributed to clans or families. This concept of the entitlement to certain tartans has led to the term of universal tartan, or free tartan, which describes tartan which can be worn by anyone. Traditional examples of such are the Black Watch (also known as Government, Universal, and Campbell), Caledonian, Hunting Stewart, and Jacobite tartans.[67] In the same line of opinion, some tartan attributed to the British Royal Family are claimed by some to be "off limits" to non-royals.[68] Even so, there are no rules on who can, or cannot, wear a particular tartan. Note that some modern tartans are protected by trademark law, and the trademark proprietor can, in certain circumstances, prevent others from selling that tartan.[2] An example of one such tartan is the Burberry Check.[note 14]

Many books on Scottish clans list such rules and guidelines.[2] One such opinion is that people not bearing a clan surname, or surname claimed as a sept of a clan, should not wear the tartan of their mother's clan.[71] This opinion is enforced by the fact that in the Scottish clan system, the Lord Lyon states that membership to a clan technically passes through the surname. This means that children who bear their father's surname belong to the father's clan (if any), and that children who bear their mother's surname (her maiden name) belong to their mother's clan (if any).[72] Also, the Lord Lyon states that a clan tartan should only be worn by those who profess allegiance to that clan's chief.[73] Some clan societies even claim that certain tartans are the personal property of a chief or chieftain, and in some cases they allow their clansfolk "permission" to wear a tartan.[note 15] According to the Scottish Tartans Authority — which is closely associated with the Scottish tartan industry — the Balmoral tartan should not be worn by anyone who is not part of the British Royal Family. Even so, some weavers outside of the United Kingdom ignore the "longstanding convention" of the British Royal Family's "right" to this tartan. The society also claims that non-royals who wear this tartan are treated with "great disdain" by the Scottish tartan industry.[75][note 16] Generally, a more liberal attitude is taken by those in the business of selling tartan, stressing that anyone may wear any tartan they like. These "rules" are mere conventions; there are no laws regarding the wearing of any tartan.

"Ye principal clovris of ye clanne Stewart" which appeared in the Sobieski Stuarts's forgery Vestiarium Scoticum of 1842[38]

Wilkie's idealised depiction of George IV, in full Highland dress, during the visit to Scotland in 1822[note 4]

John Campbell of the Bank, 1749. The present official Clan Campbell tartans are predominantly blue, green and black.[20

The earliest image of Scottish soldiers wearing tartan, from a woodcut c.1631[12][note 2]

QMRRacing flags[1] are traditionally used in auto racing and similar motorsports to indicate track condition and to communicate important messages to drivers. Typically, the starter, sometimes the grand marshal of a race, waves the flags atop a flag stand near the start/finish line. Track marshals are also stationed at observation posts along the race track in order to communicate both local and course-wide conditions to drivers. Alternatively, some race tracks employ lights to supplement the primary flag at the start/finish line.

They are checkered flags

QMRMadras is a lightweight cotton fabric with typically patterned texture and plaid design, used primarily for summer clothing such as pants, shorts, dresses, and jackets. The fabric takes its name from the former name of the city of Chennai, Tamil Nadu, India. This cloth also was identified by the colloquial name, "Madrasi checks."

It is composed of quadrants

QMRIn geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex.

Conway calls it a quadrille.

QMRTribes have lived on the Great Plains for thousands of years. Early Plains cultures are commonly divided into four periods: Paleoindian (at least c. 10,000–4000 BCE), Plains Archaic (c. 4000–250 BCE), Plains Woodland (c. 250 BCE–950 CE), Plains Village (c. 950-1850 CE).[20] The oldest known painted object in North American was found in the southern plains, the Cooper Bison Skull, found in Oklahoma and dated 10,900-10,200 BCE. It's painted with a red zizzag.

QMRThe simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3)-torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values (sequence A002863 in OEIS) are given in the following table.

QMRIn knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot. The figure-eight knot is a prime knot.

QMRBy way of example, the unknot has crossing number zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.

QMRFour half-twists

(stevedore knot)

QMRIn mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

QMRIn parkourhe base technique to quadrupedal movement should have you assuming a position where:

> Your hands are placed shoulder width apart directly underneath your shoulders.

> Your back should be parallel to the ground

> Your shins parallel to the ground.

> Knee’s off the ground, toes in contact with the ground.

> Next you should start to move forward!

> When moving you should move alternate arms and legs.

> When the right hand goes forward the left leg should move forward at the same time.

> When the left hand moves forward the right leg should move as well.

> Keep the knees at approximately the same distance from the ground at all times, keep the back parallel with the ground.

Try not to stretch your self out too far, crowd yourself by bringing the knees in too close to the body or stick your backside into the air. Avoid resting the knees on the ground, if you wish to rest then assume a crouched position or stick your backside in the air.

The reason for this particular movement pattern is that it forces the mind to coordinate the body in such a way that it increases your overall body control and awareness; it is also more balanced in a physiological sense.

PROGRESSION

Once you have a firm grounding in the base technique of quadrupedal start to experiment with making it more difficult. Varying the shape and alignment of the body when doing it. Move up and down stairs, on rails, sideways, backwards, get down really low to the ground, do it on your elbows and knees like soldiers under barbed wire, use your imagination. There are many different ways to move on all fours. This is an opportunity to do some of the things that initially should be avoided; sticking the backside in the air, stretching yourself out or crowding yourself. But you only want to do this once you have mastered the base technique.

QMREver since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. As of May 2008, all prime knots up to 16 crossings have been tabulated. 16 is the squares of the quadrant number

QMRJim Hoste, Jeff Weeks, and Morwen Thistlethwaite used computer searches to count all knots with 16 or fewer crossings. This research was performed separately using two different algorithms on different computers, lending support to the correctness of its results. Both counts found 1701936 prime knots (including the unknot) with up to 16 crossings.[1]

Starting with three crossings (the minimum for any nontrivial knot), the number of prime knots for each number of crossings is

1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ..

QMRCeithri cathracha i r-robadar Tuatha De Danand ("[The four jewels of the Tuatha Dé Danann|The Four Jewels of the Tuatha Dé Danann

QMRIn statistics, a contingency table is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term contingency table was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation",[1] part of the Drapers' Company Research Memoirs Biometric Series I published in 1904.

A crucial problem of multivariate statistics is finding (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can use information theory concepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies.

contingency tables sort of look like quadrants

QMRIn the field of machine learning, a confusion matrix, also known as a contingency table or an error matrix [3] , is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one (in unsupervised learning it is usually called a matching matrix). Each column of the matrix represents the instances in a predicted class while each row represents the instances in an actual class (or vice-versa).[2] The name stems from the fact that it makes it easy to see if the system is confusing two classes (i.e. commonly mislabeling one as another).

Contents [hide]

1 Example

2 Table of confusion

3 See also

4 References

5 External links

Example[edit]

If a classification system has been trained to distinguish between cats, dogs and rabbits, a confusion matrix will summarize the results of testing the algorithm for further inspection. Assuming a sample of 27 animals — 8 cats, 6 dogs, and 13 rabbits, the resulting confusion matrix could look like the table below:

Predicted

Cat Dog Rabbit

Actual

class Cat 5 3 0

Dog 2 3 1

Rabbit 0 2 11

In this confusion matrix, of the 8 actual cats

QMRIn predictive analytics, a table of confusion (sometimes also called a confusion matrix), is a table with two rows and two columns that reports the number of false positives, false negatives, true positives, and true negatives. This allows more detailed analysis than mere proportion of correct guesses (accuracy). Accuracy is not a reliable metric for the real performance of a classifier, because it will yield misleading results if the data set is unbalanced (that is, when the number of samples in different classes vary greatly). For example, if there were 95 cats and only 5 dogs in the data set, the classifier could easily be biased into classifying all the samples as cats. The overall accuracy would be 95%, but in practice the classifier would have a 100% recognition rate for the cat class but a 0% recognition rate for the dog class.

Assuming the confusion matrix above, its corresponding table of confusion, for the cat class, would be:

5 true positives

(actual cats that were

correctly classified as cats) 2 false positives

(dogs that were

incorrectly labeled as cats)

3 false negatives

(cats that were

incorrectly marked as dogs) 17 true negatives

(all the remaining animals,

correctly classified as non-cats)

The final table of confusion would contain the average values for all classes combined.

Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

True condition

Total population Condition positive Condition negative Prevalence =

Σ Condition positive

Σ Total population

Predicted

condition Predicted condition

positive True positive False positive

(Type I error) Positive predictive value (PPV), Precision =

Σ True positive

Σ Test outcome positive

False discovery rate (FDR) =

Σ False positive

Σ Test outcome positive

Predicted condition

negative False negative

(Type II error) True negative False omission rate (FOR) =

Σ False negative

Σ Test outcome negative

Negative predictive value (NPV) =

Σ True negative

Σ Test outcome negative

Accuracy (ACC) =

Σ True positive + Σ True negative

Σ Total population

True positive rate (TPR), Sensitivity, Recall =

Σ True positive

Σ Condition positive

False positive rate (FPR), Fall-out =

Σ False positive

Σ Condition negative

Positive likelihood ratio (LR+) =

TPR

FPR

Diagnostic odds ratio (DOR) =

LR+

LR−

False negative rate (FNR), Miss rate =

Σ False negative

Σ Condition positive

True negative rate (TNR), Specificity (SPC) =

Σ True negative

Σ Condition negative

Negative likelihood ratio (LR−) =

FNR

TNR

See

The confusion table has four squares based off of two dualities

predicted condition positive and condition positive is a true positive

predicted condition negative and condition positive is a false negative type two error

predicted condition positive condition negative false positive type 1 error

predicted condition negative condition negative true negative

QMRDetection theory or signal detection theory is a means to quantify the ability to discern between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (called noise, consisting of background stimuli and random activity of the detection machine and of the nervous system of the operator). In the field of electronics, the separation of such patterns from a disguising background is referred to as signal recovery.[1]

Signal detection theory (SDT) is used when psychologists want to measure the way we make decisions under conditions of uncertainty, such as how we would perceive distances in foggy conditions. SDT assumes that the decision maker is not a passive receiver of information, but an active decision-maker who makes difficult perceptual judgments under conditions of uncertainty. In foggy circumstances, we are forced to decide how far away from us an object is, based solely upon visual stimulus which is impaired by the fog. Since the brightness of the object, such as a traffic light, is used by the brain to discriminate the distance of an object, and the fog reduces the brightness of objects, we perceive the object to be much farther away than it actually is (see also decision theory).

To apply signal detection theory to a data set where stimuli were either present or absent, and the observer categorized each trial as having the stimulus present or absent, the trials are sorted into one of four categories:

Respond "Absent" Respond "Present"

Stimulus Present Miss Hit

Stimulus Absent Correct Rejection False Alarm

square 1: stimulus present respond absent miss

square 2: stimulus absent respond absent correct rejection

square 3: stimulus present respond present hit

square 4: stimulus absent respond present false alarm

Based on the proportions of these types of trials, numerical estimates of sensitivity can be obtained with statistics like the sensitivity index d' and A',[7] and response bias can be estimated with statistics like c and β.[7]

Signal detection theory can also be applied to memory experiments, where items are presented on a study list for later testing. A test list is created by combining these 'old' items with novel, 'new' items that did not appear on the study list. On each test trial the subject will respond 'yes, this was on the study list' or 'no, this was not on the study list'. Items presented on the study list are called Targets, and new items are called Distractors. Saying 'Yes' to a target constitutes a Hit, while saying 'Yes' to a distractor constitutes a False Alarm.

Respond "No" Respond "Yes"

Target Miss Hit

Distractor Correct Rejection False Alarm

Bayes Criterion[edit]

In some cases, it is far more important to respond appropriately to H1 than it is to respond appropriately to H2. For example, if an alarm goes off, indicating H1 (an incoming bomber is carrying a nuclear weapon), it is much more important to shoot down the bomber if H1 = TRUE, than it is to send a fighter squadron to inspect a false alarm (i.e., H1 = FALSE, H2 = TRUE) (assuming a large supply of fighter squadrons). The Bayes criterion is an approach suitable for such cases.[8]

Here a utility is associated with each of four situations:

U_{11}: One responds with behavior appropriate to H1 and H1 is true: fighters destroy bomber, incurring fuel, maintenance, and weapons costs, take risk of some being shot down;

U_{12}: One responds with behavior appropriate to H1 and H2 is true: fighters sent out, incurring fuel and maintenance costs, bomber location remains unknown;

U_{21}: One responds with behavior appropriate to H2 and H1 is true: city destroyed;

U_{22}: One responds with behavior appropriate to H2 and H2 is true: fighters stay home, bomber location remains unknown;

QMREvaluation of binary classifiers[edit]

Main article: Evaluation of binary classifiers

From the contingency table you can derive four basic ratios

There are many metrics that can be used to measure the performance of a classifier or predictor; different fields have different preferences for specific metrics due to different goals. For example, in medicine sensitivity and specificity are often used, while in information retrieval precision and recall are preferred. An important distinction is between metrics that are independent on the prevalence (how often each category occurs in the population), and metrics that depend on the prevalence – both types are useful, but they have very different properties.

Given a classification of a specific data set, there are four basic data: the number of true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN). These can be arranged into a 2×2 contingency table, with columns corresponding to actual value – condition positive (CP) or condition negative (CN) – and rows corresponding to classification value – test outcome positive or test outcome negative. There are eight basic ratios that one can compute from this table, which come in four complementary pairs (each pair summing to 1). These are obtained by dividing each of the four numbers by the sum of its row or column, yielding eight numbers, which can be referred to generically in the form "true positive row ratio" or "false negative column ratio", though there are conventional terms. There are thus two pairs of column ratios and two pairs of row ratios, and one can summarize these with four numbers by choosing one ratio from each pair – the other four numbers are the complements.

The column ratios are True Positive Rate (TPR, aka Sensitivity or recall), with complement the False Negative Rate (FNR); and True Negative Rate (TNR, aka Specificity, SPC), with complement False Positive Rate (FPR). These are the proportion of the population with the condition (resp., without the condition) for which the test is correct (or, complementarily, for which the test is incorrect); these are independent of prevalence.

The row ratios are Positive Predictive Value (PPV, aka precision), with complement the False Discovery Rate (FDR); and Negative Predictive Value (NPV), with complement the False Omission Rate (FOR). These are the proportion of the population with a given test result for which the test is correct (or, complementarily, for which the test is incorrect); these depend on prevalence.

In diagnostic testing, the main ratios used are the true column ratios – True Positive Rate and True Negative Rate – where they are known as sensitivity and specificity. In informational retrieval, the main ratios are the true positive ratios (row and column) – Positive Predictive Value and True Positive Rate – where they are known as precision and recall.

One can take ratios of a complementary pair of ratios, yielding four likelihood ratios (two column ratio of ratios, two row ratio of ratios). This is primarily done for the column (condition) ratios, yielding likelihood ratios in diagnostic testing. Taking the ratio of one of these groups of ratios yields a final ratio, the diagnostic odds ratio (DOR). This can also be defined directly as (TP×TN)/(FP×FN) = (TP/FN)/(FP/TN); this has a useful interpretation – as an odds ratio – and is prevalence-independent.

There are a number of other metrics, most simply the accuracy or Fraction Correct (FC), which measures the fraction of all instances that are correctly categorized; the complement is the Fraction Incorrect (FiC). The F-score combines precision and recall into one number via a choice of weighing, most simply equal weighing, as the balanced F-score (F1 score). Some metrics come from regression coefficients: the markedness and the informedness, and their geometric mean, the Matthews correlation coefficient. Other metrics include Youden's J statistic, the uncertainty coefficient, the Phi coefficient, and Cohen's kappa.

QMRSensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function:

Sensitivity (also called the true positive rate, or the recall in some fields) measures the proportion of positives that are correctly identified as such (e.g., the percentage of sick people who are correctly identified as having the condition).

Specificity (also called the true negative rate) measures the proportion of negatives that are correctly identified as such (e.g., the percentage of healthy people who are correctly identified as not having the condition).

Thus sensitivity quantifies the avoiding of false negatives, as specificity does for false positives.

For any test, there is usually a trade-off between the measures. For instance, in an airport security setting in which one is testing for potential threats to safety, scanners may be set to trigger on low-risk items like belt buckles and keys (low specificity), in order to reduce the risk of missing objects that do pose a threat to the aircraft and those aboard (high sensitivity). This trade-off can be represented graphically as a receiver operating characteristic curve.

A perfect predictor would be described as 100% sensitive (e.g., all sick are identified as sick) and 100% specific (e.g., no healthy are identified as sick); however, theoretically any predictor will possess a minimum error bound known as the Bayes error rate.

QMRIn general, Positive = identified and negative = rejected. Therefore:

True positive = correctly identified

False positive = incorrectly identified

True negative = correctly rejected

False negative = incorrectly rejected

Let us consider a group with P positive instances and N negative instances of some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows

QMRIn statistics, polychoric correlation is a technique for estimating the correlation between two theorised normally distributed continuous latent variables, from two observed ordinal variables. Tetrachoric correlation is a special case of the polychoric correlation applicable when both observed variables are dichotomous. These names derive from the polychoric and tetrachoric series which are used for estimation of these correlations. These series' were mathematical expansions once but not anymore.

QMRFisher's exact test[1][2][3] is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample sizes are small, it is valid for all sample sizes. It is named after its inventor, Ronald Fisher, and is one of a class of exact tests, so called because the significance of the deviation from a null hypothesis (e.g., P-value) can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity, as with many statistical tests.

The test is useful for categorical data that result from classifying objects in two different ways; it is used to examine the significance of the association (contingency) between the two kinds of classification. So in Fisher's original example, one criterion of classification could be whether milk or tea was put in the cup first; the other could be whether Dr Bristol thinks that the milk or tea was put in first. We want to know whether these two classifications are associated – that is, whether Dr Bristol really can tell whether milk or tea was poured in first. Most uses of the Fisher test involve, like this example, a 2 × 2 contingency table. The p-value from the test is computed as if the margins of the table are fixed, i.e. as if, in the tea-tasting example, Dr Bristol knows the number of cups with each treatment (milk or tea first) and will therefore provide guesses with the correct number in each category. As pointed out by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table.

A two by two table is a quadrant

With large samples, a chi-squared test can be used in this situation. However, the significance value it provides is only an approximation, because the sampling distribution of the test statistic that is calculated is only approximately equal to the theoretical chi-squared distribution. The approximation is inadequate when sample sizes are small, or the data are very unequally distributed among the cells of the table, resulting in the cell counts predicted on the null hypothesis (the "expected values") being low. The usual rule of thumb for deciding whether the chi-squared approximation is good enough is that the chi-squared test is not suitable when the expected values in any of the cells of a contingency table are below 5, or below 10 when there is only one degree of freedom (this rule is now known to be overly conservative[4]). In fact, for small, sparse, or unbalanced data, the exact and asymptotic p-values can be quite different and may lead to opposite conclusions concerning the hypothesis of interest.[5][6] In contrast the Fisher exact test is, as its name states, exact as long as the experimental procedure keeps the row and column totals fixed, and it can therefore be used regardless of the sample characteristics. It becomes difficult to calculate with large samples or well-balanced tables, but fortunately these are exactly the conditions where the chi-squared test is appropriate.

For hand calculations, the test is only feasible in the case of a 2 × 2 contingency table. However the principle of the test can be extended to the general case of an m × n table,[7][8] and some statistical packages provide a calculation (sometimes using a Monte Carlo method to obtain an approximation) for the more general case.[9]

QMRFor example, a sample of teenagers might be divided into male and female on the one hand, and those that are and are not currently dieting on the other. We hypothesize, for example, that the proportion of dieting individuals is higher among the women than among the men, and we want to test whether any difference of proportions that we observe is significant. The data might look like this:

Men Women Row total

Dieting 1 9 10

Non-dieting 11 3 14

Column total 12 12 24

The question we ask about these data is: knowing that 10 of these 24 teenagers are dieters, and that 12 of the 24 are female, and assuming the null hypothesis that men and women are equally likely to diet, what is the probability that these 10 dieters would be so unevenly distributed between the women and the men? If we were to choose 10 of the teenagers at random, what is the probability that 9 or more of them would be among the 12 women, and only 1 or fewer from among the 12 men?

Before we proceed with the Fisher test, we first introduce some notation. We represent the cells by the letters a, b, c and d, call the totals across rows and columns marginal totals, and represent the grand total by n. So the table now looks like this:

Men Women Row Total

Dieting a b a + b

Non-dieting c d c + d

Column Total a + c b + d a + b + c + d (=n)

The formula above gives the exact hypergeometric probability of observing this particular arrangement of the data, assuming the given marginal totals, on the null hypothesis that men and women are equally likely to be dieters. To put it another way, if we assume that the probability that a man is a dieter is P, the probability that a woman is a dieter is p, and we assume that both men and women enter our sample independently of whether or not they are dieters, then this hypergeometric formula gives the conditional probability of observing the values a, b, c, d in the four cells, conditionally on the observed marginals (i.e., assuming the row and column totals shown in the margins of the table are given). This remains true even if men enter our sample with different probabilities than women. The requirement is merely that the two classification characteristics—gender, and dieter (or not) -- are not associated.

For example, suppose we knew probabilities P,Q,p,q with P+Q=p+q=1 such that (male dieter, male non-dieter, female dieter, female non-dieter) had respective probabilities (Pp,Pq,Qp,Qq) for each individual encountered under our sampling procedure. Then still, were we to calculate the distribution of cell entries conditional given marginals, we would obtain the above formula in which neither p nor P occurs. Thus, we can calculate the exact probability of any arrangement of the 24 teenagers into the four cells of the table, but Fisher showed that to generate a significance level, we need consider only the cases where the marginal totals are the same as in the observed table, and among those, only the cases where the arrangement is as extreme as the observed arrangement, or more so. (Barnard's test relaxes this constraint on one set of the marginal totals.) In the example, there are 11 such cases. Of these only one is more extreme in the same direction as our data; it looks like this:

QMRWhile Barnard retracted his test in a published paper,[5] most researchers prefer using Barnard's exact test over Fisher's exact test for analyzing 2×2 contingency tables. The only exception is when the true sampling distribution of the table is hypergeometric. Barnard's test can be applied to larger tables, but the computation time increases and the power advantage quickly decreases.[6] It remains unclear which test statistic is preferred when implementing Barnard's test; however, most test statistics yield uniformly more powerful tests than Fisher's exact test.[7]

The two by two table is the quadrant

Barnard's test is used to test the independence of rows and columns in a contingency table. The test assumes each response is independent. Under independence, there are three types of study designs that yield a 2×2 table. To distinguish the different types of designs, suppose a researcher is interested in testing whether a treatment quickly heals an infection. One possible study design would be to sample 100 infected subjects, randomly give them the treatment or the placebo, and see if the infection is still present after a set time. This type of design is common in cross-sectional studies. Another possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and see if the infection is still present after a set time. This type of design is common in case-control studies. The final possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and stop the experiment once a set number of subjects has healed from the infection. This type of design is uncommon, but has the same structure as the lady tasting tea study that lead to R. A. Fisher creating the Fisher's Exact test. The probability of a 2×2 table under the first study design is given by the multinomial distribution; the second study design is given by the product of two independent binomial distributions; the third design is given by the hypergeometric distribution.

The difference between Barnard's exact test and Fisher's exact test is how they handle the nuisance parameter(s) of the common success probability when calculating the p-value. Fisher's test avoids estimating the nuisance parameter(s) by conditioning on the margins, an approximately ancillary statistic. Barnard's test considers all possible values of the nuisance parameter(s) and chooses the value(s) that maximizes the p-value. Both tests have sizes less than or equal to the type I error rate. However, Barnard's test can be more powerful than Fisher's test because it considers more 'as or more extreme' tables by not conditioning on both margins. In fact, one variant of Barnard's test, called Boschloo's test, is uniformly more powerful than Fisher's exact test.[3] A more detailed description

chi squared statistical tests also involve quadrant tables

Game Theory Chapter

Four Forms of Breakout Strategy

Finkelstein et al (Finkelstein et al 2009, p- 20) defines four different forms for the breakout strategy to be applied and the breakout strategy have four different conditions. The breakout strategy can be used in a market that is narrow and its beginning (and as such subordinate) to the established market and their industries. Typically the two forms of conditions (taking by storm, expanding horizon) are to be found in the narrow market. In the broad market then the laggard to leader and shifting shape form is to be found.

Forms of Breakout Strategies

Thereto does Finkelstein et al defines four forms of breakout companies and as such also four forms of breakout strategies. The four different break out forms are organized on emergent markets and established markets. The emergent markets are characterized by that the enterprise that is within them is able to work with a new form of technology, service or market space that is adaptable to the product or service the enterprise sells. The established markets are characterized by that they are mature markets such as there are enterprises that competes on gaining market share. Typically will the industry go into a spiral where the products are seen as commodities and as such commodities are sold primarily based on the price of the product.

Growth Opportunities

For the particular markets the forms of breakouts are characterized as the true originals, revolutionaries, the wave raiders and the big improvements.

The originals usually shapes and develop a new market e.g., by inventing new superior services e.g., they go from a red ocean to a blue ocean. In the same time the revolutionaries in an established market re-invent their enterprise in a way that makes the enterprise able to compete by differentiating the products or services.

The wave riders work with the assumption that they find market space and the market space over time will expand. The big improvers include enterprises that is able to define a business model and value proposition. Likewise is it able to refashion its business model.

QMRGridiron: Fantasy Football was a football-themed collectible card game first published in 1995 by Upper Deck.

Game overview[edit]

Gridiron simulates a football game that takes place in a dystopian "near-future", and features a violent and over-the-top backstory comparing the new style of the game to Roman gladiator combat. Each player uses a 60-card deck, primarily composed of Plays and Actions, which are further divided into Offense and Defense, along with Team cards, which encompass Star Players at various positions, Coaches, Franchises, Traditions and Formations that provide additional benefits. The goal is to be the player with the most points at the end of the game, which consists of a predetermined number of "drives", and opponents take turns playing offense and defense while attempting to score touchdowns and field goals without losing possession of the ball.

The actual gameplay closely follows traditional football, requiring the offensive player to progress 10 yards within 4 downs, and tracks the field position on a special playmat included in the starter decks. Cards feature icons corresponding to four traits (Power, Speed, Skill, and Mental) which are used to determine the success or failure of various plays. Plays which complete for higher yardage require more symbols to succeed, and defensive plays can reduce the yardage gained on a play. Players can punt, score safeties, or go for a 2-point conversion following a touchdown, and certain questionable tactics also include a "Referee Alert", used to try to give the offending player a penalty. The rules include options for both single games and fully organized leagues and seasons.

QMRArcade machines[edit] Dance Dance Revolution is a very popular video game played on a quadrant grid.

A standard Dance Dance Revolution arcade machine consists of two parts, the cabinet and the dance platform. The cabinet has a wide bottom section, which houses large floor speakers and glowing neon lamps. Above this sits a narrower section that contains the monitor, and on top is a lighted marquee graphic, with two small speakers and flashing lights on either side. Below the monitor are two sets of buttons (one for each player), each consisting of two triangular selection buttons and a center rectangular button, used mainly to confirm a selection or start the game.

The dance stage, divided into 9 sections, 4 of them in the cardinal directions contain pressure sensors for the detection of steps.

The dance stage is a raised metal platform divided into two sides. Each side houses a set of four acrylic glass pads[9] arranged and pointing in the orthogonal directions (left, up, down and right), separated by metal squares. Each pad sits atop four pressure activated switches, one at each edge of each pad, and a software-controlled cold cathode lamp illuminating the translucent pad. A metal safety bar in the shape of an upside-down "U" is mounted to the dance stage behind each player. Some players make use of this safety bar to help maintain proper balance, and to relieve weight from the legs so that arrows can be pressed with greater speed and accuracy.

Some DDR cabinets are equipped with Sony PlayStation memory card slots, allowing the player to insert a compatible memory card before starting a game and save their high scores to the card. Additionally, the equivalent home versions of DDR allow players to create and save custom step patterns (edits) to their memory card — the player can then play those steps on the arcade machine if the same song exists on that machine. This feature is supported in 2ndMix through Extreme. SuperNova didn't support memory card slots. However, it introduced Konami's internet based link system e-Amusement to the series, which can save stats and unlocks for individual players (but cannot store edits). This functionality however, could only be used in Japan. During the North American release of Dance Dance Revolution SuperNOVA 2, an e-Amuse capable machine was made available at a Brunswick Zone Arcade in Naperville, Illinois. Both it and another machine located in the Konami offices of El Segundo, California are currently the only e-Amuse capable machines in the United States.

The Solo arcade cabinet is smaller and contains only one dance pad, modified to include six arrow panels instead of four (the additional panels are "upper-left" and "upper-right"). These pads generally don't come with a safety bar, but include the option for one to be installed at a later date. The Solo pad also lacks some of the metal plating that the standard pad has, which can make stepping difficult for players who are used to playing on standard machines. An upgrade was available for Solo machines called the "Deluxe pad", which was closer to the standard cabinet's pad. Additionally Solo machines only incorporate two sensors, located horizontally in the center of the arrow, instead of four sensors (one on each edge).

Educational use[edit]

Systems for multiple simultaneous players were introduced into UK schools in 2007 by Cyber coach. Such systems can connect up to 64 dance pads at any one time without wires.[6] The systems have been incorporated into school lessons as well as for extra curricular use. The major advantage of these systems is that they appeal to those not normally inclined towards exercise.[7] In this way, these systems are seen as a valuable tool in combating childhood obesity.

A dance pad has been also used in experimental systems for navigating through virtual environments, such as those reproducing historical buildings. [8]

Other games[edit]

Some games that can be played with dance pads do not involve pressing the arrow buttons on the pad to keep with the rhythm of a song:

Many games developed for the NES Power Pad

Breakthrough Gaming Presents: Axel

The minigames of Dance Dance Revolution: Mario Mix

Kraft Rocking the Boat

Kraft Soccer Striker

Kraft Virtual Dojo

Exult - using Dance Pad Walking controller mod

DDR-A-Mole, a game similar to Whac-A-Mole [9]

Various mini-games in Dance Praise 2: The ReMix (DanceTris, a Tetris simulator, is included).

The 2014 rogue-like rhythm game Crypt of the NecroDancer

QMRA cross-in-square or crossed-dome plan was the dominant architectural form of middle- and late-period Byzantine churches, featuring a square centre with an internal structure shaped like a cross, topped by a dome.

The first cross-in-square churches were probably built in the late 8th century, and the form has remained in use throughout the Orthodox world until the present day. In the West, Donato Bramante's first design (1506) for St. Peter's Basilica was a centrally planned cross-in-square under a dome and four subsidiary domes.

In German, such a church is a Kreuzkuppelkirche, or ‘cross-dome church’. In French, it is an église à croix inscrite, ‘church with an inscribed cross’

A cross-in-square church is centered around a quadratic naos (the ‘square’) which is divided by four columns or piers into nine bays (divisions of space). The inner five divisions form the shape of a quincunx (the ‘cross’).[1] The central bay is usually larger than the other eight, and is crowned by a dome which rests on the columns. The four rectangular bays that directly adjoin this central bay are usually covered by barrel vaults; these are the arms of the "cross" which is inscribed within the "square" of the naos. The four remaining bays in the corner are usually groin-vaulted. The spatial hierarchy of the three types of bay, from the largest central bay to the smallest corner bays, is mirrored in the elevation of the building; the domed central bay is taller than the cross arms, which are in turn taller than the corner bays.[2]

To the west of the naos stands the narthex, or entrance hall, usually formed by the addition of three bays to the westernmost bays of the naos. To the east stands the bema, or sanctuary, often separated from the naos by templon or, in later churches, by an iconostasis. The sanctuary is usually formed by three additional bays adjoining the easternmost bays of the naos, each of which terminates in an apse crowned by a conch (half-dome). The central apse is larger than those to the north and south. The term bema is sometimes reserved for the central area, while the northern section is known as the prothesis and the southern as the diakonikon.[3]

Although evidence for Byzantine domestic architecture is scant, it appears that the core unit of the cross-in-square church (nine bays divided by four columns) was also employed for the construction of halls within residential structures.[4]

QMRThe flag of the Solomon Islands features a quincunx of stars. A quincunx is a cross made of five dots

QMRRepresentation of games[edit]

See also: List of games in game theory

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".)[3] A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Normal form[edit]

Player 2

chooses Left Player 2

chooses Right

Player 1

chooses Up 4, 3 –1, –1

Player 1

chooses Down 0, 0 3, 4

Normal form or payoff matrix of a 2-player, 2-strategy game

Main article: Normal-form game

The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.[5]

QMRThe prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely "rational" individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it, "prisoner's dilemma" (Poundstone, 1992), presenting it as follows:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

If A and B each betray the other, each of them serves 2 years in prison

If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa)

If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)

It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get, and that their decision will not affect their reputation in the future. Because betraying a partner offers a greater reward than cooperating with him, all purely rational self-interested prisoners would betray the other, and so the only possible outcome for two purely rational prisoners is for them to betray each other.[1] The interesting part of this result is that pursuing individual reward logically leads both of the prisoners to betray, when they would get a better reward if they both kept silent. In reality, humans display a systematic bias towards cooperative behavior in this and similar games, much more so than predicted by simple models of "rational" self-interested action.[2][3][4][5] A model based on a different kind of rationality, where people forecast how the game would be played if they formed coalitions and then they maximize their forecasts, has been shown to make better predictions of the rate of cooperation in this and similar games given only the payoffs of the game.[6]

An extended "iterated" version of the game also exists, where the classic game is played repeatedly between the same prisoners, and consequently, both prisoners continuously have an opportunity to penalize the other for previous decisions. If the number of times the game will be played is known to the players, then (by backward induction) two classically rational players will betray each other repeatedly, for the same reasons as the single shot variant. In an infinite or unknown length game there is no fixed optimum strategy, and Prisoner's Dilemma tournaments have been held to compete and test algorithms.

The prisoner's dilemma game can be used as a model for many real world situations involving cooperative behaviour. In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games: for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it merely difficult or expensive, not necessarily impossible, to coordinate their activities to achieve cooperation.

QMRTHe prisoner’s dilemma and Nash equilibriums are quadrant matrices, and game theory’s foundation is these quadrant matrices. The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely "rational" individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it, "prisoner's dilemma" (Poundstone, 1992), presenting it as follows:

If A and B each betray the other, each of them serves 2 years in prison

If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa)

If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)

Strategy for the prisoners' dilemma[edit]

Both cannot communicate, they are separated in two individual rooms. The normal game is shown below:

Prisoner B stays silent (cooperates) Prisoner B betrays (defects)

Prisoner A stays silent (cooperates) Each serves 1 year Prisoner A: 3 years

Prisoner B: goes free

Prisoner A betrays (defects) Prisoner A: goes free

Prisoner B: 3 years Each serves 2 years

Here, regardless of what the other decides, each prisoner gets a higher reward by betraying the other ("defecting"). The reasoning involves an argument by dilemma: B will either cooperate or defect. If B cooperates, A should defect, because going free is better than serving 1 year. If B defects, A should also defect, because serving 2 years is better than serving 3. So either way, A should defect. Parallel reasoning will show that B should defect.

In traditional game theory, some very restrictive assumptions on prisoner behaviour are made. It is assumed that both understand the nature of the game, and that despite being members of the same gang, they have no loyalty to each other and will have no opportunity for retribution or reward outside the game. Most importantly, a very narrow interpretation of "rationality" is applied in defining the decision-making strategies of the prisoners. Given these conditions and the payoffs above, prisoner A will betray prisoner B. The game is symmetric, so Prisoner B should act the same way. Since both "rationally" decide to defect, each receives a lower reward than if both were to stay quiet. Traditional game theory results in both players being worse off than if each chose to lessen the sentence of his accomplice at the cost of spending more time in jail himself.

Generalized form[edit]

The structure of the traditional Prisoners’ Dilemma can be generalized from its original prisoner setting. Suppose that the two players are represented by the colors, red and blue, and that each player chooses to either "Cooperate" or "Defect".

If both players cooperate, they both receive the reward R for cooperating. If both players defect, they both receive the punishment payoff P. If Blue defects while Red cooperates, then Blue receives the temptation payoff T, while Red receives the "sucker's" payoff, S. Similarly, if Blue cooperates while Red defects, then Blue receives the sucker's payoff S, while Red receives the temptation payoff T.

This can be expressed in normal form:

Canonical PD payoff matrix

Cooperate Defect

Cooperate R, R S, T

Defect T, S P, P

and to be a prisoner's dilemma game in the strong sense, the following condition must hold for the payoffs:

T > R > P > S

The payoff relationship R > P implies that mutual cooperation is superior to mutual defection, while the payoff relationships T > R and P > S imply that defection is the dominant strategy for both agents. That is, mutual defection is the only strong Nash equilibrium in the game (i.e. the only outcome from which each player could only do worse by unilaterally changing strategy). The dilemma then is that mutual cooperation yields a better outcome than mutual defection but it is not the rational outcome because from a self-interested perspective, the choice to cooperate, at the individual level, is irrational.

Special case: Donation game[edit]

The "donation game"[7] is a form of prisoner's dilemma in which cooperation corresponds to offering the other player a benefit b at a personal cost c with b > c. Defection means offering nothing. The payoff matrix is thus

Cooperate Defect

Cooperate b-c, b-c -c, b

Defect b, -c 0, 0

Note that 2R>T+S (i.e. 2(b-c)>b-c) which qualifies the donation game to be an iterated game (see next section).

The donation game may be applied to markets. Suppose X grows oranges, Y grows apples. The marginal utility of an apple to the orange-grower X is b, which is higher than the marginal utility (c) of an orange, since X has a surplus of oranges and no apples. Similarly, for apple-grower Y, the marginal utility of an orange is b while the marginal utility of an apple is c. If X and Y contract to exchange an apple and an orange, and each fulfills their end of the deal, then each receive a payoff of b-c. If one "defects" and does not deliver as promised, the defector will receive a payoff of b, while the cooperator will lose c. If both defect, then neither one gains or loses anything.

The iterated prisoners' dilemma[edit]

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If two players play prisoners' dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoners' dilemma.

In addition to the general form above, the iterative version also requires that 2R > T + S, to prevent alternating cooperation and defection giving a greater reward than mutual cooperation.

The iterated prisoners' dilemma game is fundamental to some theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled by a multi-player, iterated, version of the game. It has, consequently, fascinated many scholars over the years. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoners' dilemma has also been referred to as the "Peace-War game".[8]

If the game is played exactly N times and both players know this, then it is always game theoretically optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof is inductive: one might as well defect on the last turn, since the opponent will not have a chance to later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on. The same applies if the game length is unknown but has a known upper limit.

Unlike the standard prisoners' dilemma, in the iterated prisoners' dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. Within standard economic theory, though, this is the only correct answer. The superrational strategy in the iterated prisoners' dilemma with fixed N is to cooperate against a superrational opponent, and in the limit of large N, experimental results on strategies agree with the superrational version, not the game-theoretic rational one.

For cooperation to emerge between game theoretic rational players, the total number of rounds N must be random, or at least unknown to the players. In this case 'always defect' may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.

Strategy for the iterated prisoners' dilemma[edit]

Interest in the iterated prisoners' dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). In it he reports on a tournament he organized of the N step prisoners' dilemma (with N fixed) in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

Axelrod discovered that when these encounters were repeated over a long period of time with many players, each with different strategies, greedy strategies tended to do very poorly in the long run while more altruistic strategies did better, as judged purely by self-interest. He used this to show a possible mechanism for the evolution of altruistic behaviour from mechanisms that are initially purely selfish, by natural selection.

The winning deterministic strategy was tit for tat, which Anatol Rapoport developed and entered into the tournament. It was the simplest of any program entered, containing only four lines of BASIC, and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, the player does what his or her opponent did on the previous move. Depending on the situation, a slightly better strategy can be "tit for tat with forgiveness." When the opponent defects, on the next move, the player sometimes cooperates anyway, with a small probability (around 1–5%). This allows for occasional recovery from getting trapped in a cycle of defections. The exact probability depends on the line-up of opponents.

By analysing the top-scoring strategies, Axelrod stated several conditions necessary for a strategy to be successful.

Nice

The most important condition is that the strategy must be "nice", that is, it will not defect before its opponent does (this is sometimes referred to as an "optimistic" algorithm). Almost all of the top-scoring strategies were nice; therefore, a purely selfish strategy will not "cheat" on its opponent, for purely self-interested reasons first.

Retaliating

However, Axelrod contended, the successful strategy must not be a blind optimist. It must sometimes retaliate. An example of a non-retaliating strategy is Always Cooperate. This is a very bad choice, as "nasty" strategies will ruthlessly exploit such players.

Forgiving

Successful strategies must also be forgiving. Though players will retaliate, they will once again fall back to cooperating if the opponent does not continue to defect. This stops long runs of revenge and counter-revenge, maximizing points.

Non-envious

The last quality is being non-envious, that is not striving to score more than the opponent.

The optimal (points-maximizing) strategy for the one-time PD game is simply defection; as explained above, this is true whatever the composition of opponents may be. However, in the iterated-PD game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations. For example, consider a population where everyone defects every time, except for a single individual following the tit for tat strategy. That individual is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy for that individual is to defect every time. In a population with a certain percentage of always-defectors and the rest being tit for tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.

In the strategy called Pavlov, win-stay, lose-switch, If the last round outcome was P,P, a Pavlov player switches strategy the next turn, which means P,P would be considered as a failure to cooperate.[citation needed] For a certain range of parameters[specify], Pavlov beats all other strategies by giving preferential treatment to co-players which resemble Pavlov.

Deriving the optimal strategy is generally done in two ways:

Bayesian Nash Equilibrium: If the statistical distribution of opposing strategies can be determined (e.g. 50% tit for tat, 50% always cooperate) an optimal counter-strategy can be derived analytically.[9]

Monte Carlo simulations of populations have been made, where individuals with low scores die off, and those with high scores reproduce (a genetic algorithm for finding an optimal strategy). The mix of algorithms in the final population generally depends on the mix in the initial population. The introduction of mutation (random variation during reproduction) lessens the dependency on the initial population; empirical experiments with such systems tend to produce tit for tat players (see for instance Chess 1988), but no analytic proof exists that this will always occur.

Although tit for tat is considered to be the most robust basic strategy, a team from Southampton University in England (led by Professor Nicholas Jennings and consisting of Rajdeep Dash, Sarvapali Ramchurn, Alex Rogers, Perukrishnen Vytelingum) introduced a new strategy at the 20th-anniversary iterated prisoners' dilemma competition, which proved to be more successful than tit for tat. This strategy relied on collusion between programs to achieve the highest number of points for a single program. The university submitted 60 programs to the competition, which were designed to recognize each other through a series of five to ten moves at the start.[10] Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector. If the program realized that it was playing a non-Southampton player, it would continuously defect in an attempt to minimize the score of the competing program. As a result,[11] this strategy ended up taking the top three positions in the competition, as well as a number of positions towards the bottom.

This strategy takes advantage of the fact that multiple entries were allowed in this particular competition and that the performance of a team was measured by that of the highest-scoring player (meaning that the use of self-sacrificing players was a form of minmaxing). In a competition where one has control of only a single player, tit for tat is certainly a better strategy. Because of this new rule, this competition also has little theoretical significance when analysing single agent strategies as compared to Axelrod's seminal tournament. However, it provided the framework for analysing how to achieve cooperative strategies in multi-agent frameworks, especially in the presence of noise. In fact, long before this new-rules tournament was played, Richard Dawkins in his book The Selfish Gene pointed out the possibility of such strategies winning if multiple entries were allowed, but he remarked that most probably Axelrod would not have allowed them if they had been submitted. It also relies on circumventing rules about the prisoners' dilemma in that there is no communication allowed between the two players, which the Southampton programs arguably did with their opening "ten move dance" to recognize one another; this only reinforces just how valuable communication can be in shifting the balance of the game.

QMRRisk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game.1 When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction (i.e. is less risky). This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

The payoff matrix in Figure 1 provides a simple two-player, two-strategy example of a game with two pure Nash equilibria. The strategy pair (Hunt, Hunt) is payoff dominant since payoffs are higher for both players compared to the other pure NE, (Gather, Gather). On the other hand, (Gather, Gather) risk dominates (Hunt, Hunt) since if uncertainty exists about the other player's action, gathering will provide a higher expected payoff. The game in Figure 1 is a well-known game-theoretic dilemma called stag hunt. The rationale behind it is that communal action (hunting) yields a higher return if all players combine their skills, but if it is unknown whether the other player helps in hunting, gathering might turn out to be the better individual strategy for food provision, since it does not depend on coordinating with the other player. In addition, gathering alone is preferred to gathering in competition with others. Like the Prisoner's dilemma, it provides a reason why collective action might fail in the absence of credible commitments.

QMRFriend or Foe?[edit]

Friend or Foe? is a game show that aired from 2002 to 2005 on the Game Show Network in the USA. It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, Friend or Foe has a rewards model between prisoner's dilemma and the game of Chicken.

The rewards matrix is

Cooperate Defect

Cooperate 1, 1 0, 2

Defect 2, 0 0, 0

This payoff matrix has also been used on the British television programmes Trust Me, Shafted, The Bank Job and Golden Balls, and on the American shows Bachelor Pad and Take It All. Game data from the Golden Balls series has been analyzed by a team of economists, who found that cooperation was "surprisingly high" for amounts of money that would seem consequential in the real world, but were comparatively low in the context of the game.[32]

QMRIterated snowdrift[edit]

Researchers from the University of Lausanne and the University of Edinburgh have suggested that the "Iterated Snowdrift Game" may more closely reflect real-world social situations. Although this model is actually a chicken game, it will be described here. In this model, the risk of being exploited through defection is lower, and individuals always gain from taking the cooperative choice. The snowdrift game imagines two drivers who are stuck on opposite sides of a snowdrift, each of whom is given the option of shoveling snow to clear a path, or remaining in their car. A player's highest payoff comes from leaving the opponent to clear all the snow by themselves, but the opponent is still nominally rewarded for their work.

This may better reflect real world scenarios, the researchers giving the example of two scientists collaborating on a report, both of whom would benefit if the other worked harder. "But when your collaborator doesn’t do any work, it’s probably better for you to do all the work yourself. You’ll still end up with a completed project."[33]

Example Snowdrift Payouts (A, B)

B cooperates B defects

A cooperates 200, 200 100, 300

A defects 300, 100 0, 0

Example PD Payouts (A, B)

B cooperates B defects

A cooperates 200, 200 -100, 300

A defects 300, -100

0, 0

See

QMRIn game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.[1] If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium. The reality of the Nash equilibrium of a game can be tested using experimental economics method.

Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision while Will's decision remains unchanged, and Will is making the best decision he can, taking into account Amy's decision while Amy's decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long the other party's decision remains unchanged.

QMRIn game theory, the stag hunt is a game that describes a conflict between safety and social cooperation. Other names for it or its variants include "assurance game", "coordination game", and "trust dilemma". Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag. This is taken to be an important analogy for social cooperation.

The stag hunt differs from the Prisoner's Dilemma in that there are two pure strategy Nash equilibria: when both players cooperate and both players defect. In the Prisoner's Dilemma, in contrast, despite the fact that both players cooperating is Pareto efficient, the only Nash equilibrium is when both players choose to defect.

An example of the payoff matrix for the stag hunt is pictured in Figure 2.

Stag Hare

Stag a, a c, b

Hare b, c d, d

Fig. 1: Generic symmetric stag hunt

Stag Hare

Stag 2, 2 0, 1

Hare 1, 0 1, 1

Fig. 2: Stag hunt example

Formal definition[edit]

Formally, a stag hunt is a game with two pure strategy Nash equilibria—one that is risk dominant and another that is payoff dominant. The payoff matrix in Figure 1 illustrates a generic stag hunt, where a>b\ge d>c. Often, games with a similar structure but without a risk dominant Nash equilibrium are called assurance game. For instance if a=2, b=1, c=0, and d=1. While (Hare, Hare) remains a Nash equilibrium, it is no longer risk dominant. Nonetheless many would call this game a stag hunt.

Reaction-correspondence-stag-hunt.jpg

In addition to the pure strategy Nash equilibria there is one mixed strategy Nash equilibrium. This equilibrium depends on the payoffs, but the risk dominance condition places a bound on the mixed strategy Nash equilibrium. No payoffs (that satisfy the above conditions including risk dominance) can generate a mixed strategy equilibrium where Stag is played with a probability higher than one half. The best response correspondences are pictured here.

The stag hunt and social cooperation[edit]

"Nature and Appearance of Deer" taken from "Livre du Roy Modus", created in the 14th century

Although most authors focus on the prisoner's dilemma as the game that best represents the problem of social cooperation, some authors believe that the stag hunt represents an equally (or more) interesting context in which to study cooperation and its problems (for an overview see Skyrms 2004).

There is a substantial relationship between the stag hunt and the prisoner's dilemma. In biology many circumstances that have been described as prisoner's dilemma might also be interpreted as a stag hunt, depending on how fitness is calculated.

Cooperate Defect

Cooperate 2, 2 0, 3

Defect 3, 0 1, 1

Fig. 3: Prisoner's dilemma example

It is also the case that some human interactions that seem like prisoner's dilemmas may in fact be stag hunts. For example, suppose we have a prisoner's dilemma as pictured in Figure 3. The payoff matrix would need adjusting if players who defect against cooperators might be punished for their defection. For instance, if the expected punishment is −2, then the imposition of this punishment turns the above prisoner's dilemma into the stag hunt given at the introduction.

QMRNormal-form game

From Wikipedia, the free encyclopedia

(Redirected from Payoff matrix)

In game theory, normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.

In static games of complete, perfect information, a normal-form representation of a game is a specification of players' strategy spaces and payoff functions. A strategy space for a player is the set of all strategies available to that player, whereas a strategy is a complete plan of action for every stage of the game, regardless of whether that stage actually arises in play. A payoff function for a player is a mapping from the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility—often cardinal in the normal-form representation) of a player, i.e. the payoff function of a player takes as its input a strategy profile (that is a specification of strategies for every player) and yields a representation of payoff as its output.

An example[edit]

A normal-form game

Player 1 \ Player 2 Player 2 chooses left Player 2 chooses right

Player 1 chooses top 4, 3 −1, −1

Player 1 chooses bottom 0, 0 3, 4

The matrix to the right is a normal-form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).

Other representations[edit]

Often, symmetric games (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.

Both players

Stag Hare

Stag 3, 3 0, 2

Hare 2, 0 2, 2

Just row

Stag Hare

Stag 3 0

Hare 2 2

Uses

Dominated strategies[edit]

The Prisoner's Dilemma

Cooperate Defect

Cooperate −1, −1 −5, 0

Defect 0, −5 −2, −2

The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma (to the right), we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for good. However, if they both defect, they will both be locked up for a shorter time. One can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect).

Sequential games in normal form[edit]

Both extensive and normal form illustration of a sequential form game with subgame imperfect and perfect Nash equilibriium marked with red and blue respectively.

A sequential game

Left, Left Left, Right Right, Left Right, Right

Top 4, 3 4, 3 −1, −1 −1, −1

Bottom 0, 0 3, 4 0, 0 3, 4

These matrices only represent games in which moves are simultaneous (or, more generally, information is imperfect). The above matrix does not represent the game in which player 1 moves first, observed by player 2, and then player 2 moves, because it does not specify each of player 2's strategies in this case. In order to represent this sequential game we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left and Right. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:

Left if player 1 plays Top and Left otherwise

Left if player 1 plays Top and Right otherwise

Right if player 1 plays Top and Left otherwise

Right if player 1 plays Top and Right otherwise

On the right is the normal-form representation of this game.

QMRIn game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. Coordination games are a formalization of the idea of a coordination problem, which is widespread in the social sciences, including economics, meaning situations in which all parties can realize mutual gains, but only by making mutually consistent decisions. A common application is the choice of technological standards.

For a classic example of a coordination game, consider the 2-player, 2-strategy game, with the payoff matrix shown on the right (Fig. 1).

Left Right

Up A, a C, c

Down B, b D, d

Fig. 1: 2-player coordination game

If this game is a coordination game, then the following inequalities in payoffs hold for player 1 (rows): A > B, D > C, and for player 2 (columns): a > c, d > b. In this game the strategy profiles {Left, Up} and {Right, Down} are pure Nash equilibria, marked in gray. This setup can be extended for more than two strategies (strategies are usually sorted so that the Nash equilibria are in the diagonal from top left to bottom right), as well as for a game with more than two players.

Examples[edit]

A typical case for a coordination game is choosing the sides of the road upon which to drive, a social standard which can save lives if it is widely adhered to. In a simplified example, assume that two drivers meet on a narrow dirt road.Both have to swerve in order to avoid a head-on collision. If both execute the same swerving maneuver they will manage to pass each other, but if they choose differing maneuvers they will collide. In the payoff matrix in Fig. 2, successful passing is represented by a payoff of 10, and a collision by a payoff of 0.

In this case there are two pure Nash equilibria: either both swerve to the left, or both swerve to the right. In this example, it doesn't matter which side both players pick, as long as they both pick the same. Both solutions are Pareto efficient. This is not true for all coordination games, as the pure coordination game in Fig. 3 shows. Pure (or common interest) coordination is the game where the players both prefer the same Nash equilibrium outcome, here both players prefer partying over both staying at home to watch TV. The {Party, Party} outcome Pareto dominates the {Home, Home} outcome, just as both Pareto dominate the other two outcomes, {Party, Home} and {Home, Party}.

Left Right

Left 10, 10 0, 0

Right 0, 0 10, 10

Fig. 2: Choosing sides

Party Home

Party 10, 10 0, 0

Home 0, 0 5, 5

Fig. 3: Pure coordination game

Party Home

Party 10, 5 0, 0

Home 0, 0 5, 10

Fig. 4: Battle of the sexes

Stag Hare

Stag 10, 10 0, 8

Hare 8, 0 7, 7

Fig. 5: Stag hunt

QMRIn game theory, battle of the sexes (BoS) is a two-player coordination game. Imagine a couple that agreed to meet this evening, but cannot recall if they will be attending the opera or a football match (and the fact that they forgot is common knowledge). The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go?

The payoff matrix labeled "Battle of the Sexes (1)" is an example of Battle of the Sexes, where the wife chooses a row and the husband chooses a column. In each cell, the first number represents the payoff to the wife and the second number represents the payoff to the husband.

This representation does not account for the additional harm that might come from not only going to different locations, but going to the wrong one as well (e.g. he goes to the opera while she goes to the football game, satisfying neither). To account for this, the game is sometimes represented as in "Battle of the Sexes (2)".

Some authors refer to the game as Bach or Stravinsky and designate the players simply as Player 1 and Player 2, rather than assigning gender.[1]

It is made up of quadrant grids

Burning money[edit]

Opera Football

Opera 4,1 0,0

Football 0,0 1,4

Unburned

Opera Football

Opera 2,1 -2,0

Football -2,0 -1,4

Burned

Interesting strategic changes can take place in this game if one allows one player the option of "burning money" – that is, allowing that player to destroy some of her utility. Consider the version of Battle of the Sexes pictured here (called Unburned). Before making the decision the row player can, in view of the column player, choose to set fire to 2 points making the game Burned pictured to the right. This results in a game with four strategies for each player. The row player can choose to burn or not burn the money and also choose to play Opera or Football. The column player observes whether or not the row player burns and then chooses either to play Opera or Football.

If one iteratively deletes weakly dominated strategies then one arrives at a unique solution where the row player does not burn the money and plays Opera and where the column player plays Opera. The odd thing about this result is that by simply having the opportunity to burn money (but not actually using it), the row player is able to secure her favored equilibrium. The reasoning that results in this conclusion is known as forward induction and is somewhat controversial. For a detailed explanation, see [1] p8 Section 4.5. In brief, by choosing not to burn money, the player is indicating she expects an outcome that is better than any of the outcomes available in the "burned" version, and this conveys information to the other party about which branch she will take.

Decisions are said to be ambiguous if there are no objective probabilities given and it is difficult or impossible to assign subjective probabilities to events. Kelsey and le Roux (2015) [2] report an experimental test of the influence of ambiguity on behaviour in a Battle of Sexes game which has an added safe strategy, R, available for Player 2 (see Table). The paper studies the behaviour of subjects in the presence of ambiguity and attempts to determine whether subjects playing the Battle of Sexes game prefer to choose an ambiguity safe option.

The value of x, which is the safe option available to Player 2, varies in the range 60-260. For some values of x, the safe strategy (option R) is dominated by a mixed strategy of L and M, and thus would not be played in a Nash equilibrium. For some higher values of x the game is dominance solvable. The effect of ambiguity-aversion is to make R (the ambiguity-safe option) attractive for Player 2. R is never chosen in Nash equilibrium for the parameter values considered. However it may be chosen when there is ambiguity. Moreover for some values of x, the games are dominance solvable and R is not part of the equilibrium strategy. For a detailed explanation, see [3].

It was found that R is chosen quite frequently by subjects. While the Row Player randomises 50:50 between her strategies, the Column Player shows a marked preference for avoiding ambiguity and choosing his ambiguity-safe strategy. Thus, the results provide evidence that ambiguity influences behaviour in the games.

QMRAn evolutionarily stable strategy (ESS) is a strategy which, if adopted by a population in a given environment, cannot be invaded by any alternative strategy that is initially rare. It is relevant in game theory, behavioural ecology, and evolutionary psychology. An ESS is an equilibrium refinement of the Nash equilibrium. It is a Nash equilibrium that is "evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear.

First published as a specific term in the 1972 book by John Maynard Smith,[1] the ESS is widely used in behavioural ecology and economics, and has been used in anthropology, evolutionary psychology, philosophy, and political science.

QMRExamples of differences between Nash Equilibria and ESSes[edit]

Cooperate Defect

Cooperate 3, 3 1, 4

Defect 4, 1 2, 2

Prisoner's Dilemma

A B

A 2, 2 1, 2

B 2, 1 2, 2

Harm thy neighbor

In most simple games, the ESSes and Nash equilibria coincide perfectly. For instance, in the Prisoner's Dilemma there is only one Nash equilibrium, and its strategy (Defect) is also an ESS.

Some games may have Nash equilibria that are not ESSes. For example, in Harm thy neighbor both (A, A) and (B, B) are Nash equilibria, since players cannot do better by switching away from either. However, only B is an ESS (and a strong Nash). A is not an ESS, so B can neutrally invade a population of A strategists and predominate, because B scores higher against B than A does against B. This dynamic is captured by Maynard Smith's second condition, since E(A, A) = E(B, A), but it is not the case that E(A,B) > E(B,B).

C D

C 2, 2 1, 2

D 2, 1 0, 0

Harm everyone

Swerve Stay

Swerve 0,0 -1,+1

Stay +1,-1 -20,-20

Chicken

Nash equilibria with equally scoring alternatives can be ESSes. For example, in the game Harm everyone, C is an ESS because it satisfies Maynard Smith's second condition. D strategists may temporarily invade a population of C strategists by scoring equally well against C, but they pay a price when they begin to play against each other; C scores better against D than does D. So here although E(C, C) = E(D, C), it is also the case that E(C,D) > E(D,D). As a result, C is an ESS.

Even if a game has pure strategy Nash equilibria, it might be that none of those pure strategies are ESS. Consider the Game of chicken. There are two pure strategy Nash equilibria in this game (Swerve, Stay) and (Stay, Swerve). However, in the absence of an uncorrelated asymmetry, neither Swerve nor Stay are ESSes. There is a third Nash equilibrium, a mixed strategy which is an ESS for this game (see Hawk-dove game and Best response for explanation).

This last example points to an important difference between Nash equilibria and ESS. Nash equilibria are defined on strategy sets (a specification of a strategy for each player), while ESS are defined in terms of strategies themselves. The equilibria defined by ESS must always be symmetric, and thus have fewer equilibrium points. both are made up of quadrants

QMRNice Guys Finish First (BBC Horizon television series) is a 1986 documentary by Richard Dawkins which discusses selfishness and cooperation, arguing that evolution often favors co-operative behaviour, and focusing especially on the tit for tat strategy of the prisoner's dilemma game. The film is approximately 45 minutes long and was produced by Jeremy Taylor.

The twelfth chapter in Dawkins' book The Selfish Gene (added in the second edition, 1989) is also named Nice Guys Finish First and explores similar material.

Overview[edit]

In the opening scene, Richard Dawkins responds very precisely to what he views as a misrepresentation of his first book, The Selfish Gene. In particular, the response of the right wing for using it as justification for social Darwinism and laissez-faire economics (free-market capitalism). Dawkins has examined this issue throughout his career and focused much of his documentary The Genius of Charles Darwin on this very issue.

The concept of reciprocal altruism is a central theme of this documentary. Dawkins also examines the tragedy of the commons, and the dilemma that it presents. He uses the large area of common land Port Meadow in Oxford, England, which has been hurt by overgrazing as an example of the tragedy of the commons. Fourteen academics as well as experts in game theory submitted their own computer programs to compete in a tournament to see who would win in the prisoner's dilemma. The winner was tit for tat, a program based on "equal retaliation", and Dawkins illustrates the four conditions of tit for tat.

Unless provoked, the agent will always cooperate.

If provoked, the agent will retaliate.

The agent is quick to forgive.

The agent must have a good chance of competing against the opponent more than once.

In a second trial, this time of over sixty applicants, tit for tat won again.

QMRKopelman, Weber, & Messick (2002), in a review of the experimental research on cooperation in commons dilemmas, identify nine classes of independent variables that influence cooperation in commons dilemmas: social motives, gender, payoff structure, uncertainty, power and status, group size, communication, causes, and frames. They organize these classes and distinguish between psychological individual differences (stable personality traits) and situational factors (the environment). Situational factors include both the task (social and decision structure) and the perception of the task.[42]

Empirical findings support the theoretical argument that the cultural group is a critical factor that needs to be studied in the context of situational variables.[43] Rather than behaving in line with economic incentives, people are likely to approach the decision to cooperate with an appropriateness framework.[44] An expanded, four factor model of the Logic of Appropriateness,[45][46] suggests that the cooperation is better explained by the question: "What does a person like me (identity) do (rules) in a situation like this (recognition) given this culture (group)?"

QMRIn game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain (or loss) of utility is exactly balanced by the losses (or gains) of the utility of the other participant(s). If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally (see marginal utility).

In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality,[1] or with Nash equilibrium.

The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game).[2]

Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.

Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.

The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent’s payoff at a favorable cost to himself rather to prefer more than less. The punishing-the-opponent standard can be used in both zero-sum games (i.e. warfare game, Chess) and non-zero-sum games (i.e. pooling selection games).

For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.

Example[edit]

A zero-sum game

A B C

1 30, -30 -10, 10 20, -20

2 10, -10 20, -20 -20, 20

A game's payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right or above.

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red's reasoning and choice of action 1, and goes for action B, so as to win 10 points? Or if Red in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

Émile Borel and John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, while Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.

Solving[edit]

The Nash equilibrium for a two-player, zero-sum game can be found by solving a linear programming problem. Suppose a zero-sum game has a payoff matrix M where element M_{i,j} is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (see ref. [2], page 740) by solving the following linear program to find a vector u:

Minimize:

\sum _{i}u_{i}

Subject to the constraints:

u ≥ 0

Mu ≥ 1.

The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the Mu vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.

If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.

The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it's positive), then solving the resulting game.

If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.[citation needed]

Universal Solution[edit]

If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or incredible after the play is started, such as Poker, there is no NE strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it’s interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.[

Non-zero-sum[edit]

Economics[edit]

Many economic situations are not zero-sum, since valuable goods and services can be created, destroyed, or badly allocated in a number of ways, and any of these will create a net gain or loss of utility to numerous stakeholders. Specifically, all trade is by definition positive sum, because when two parties agree to an exchange each party must consider the goods it is receiving to be more valuable than the goods it is delivering. In fact, all economic exchanges must benefit both parties to the point that each party can overcome its transaction costs, or the transaction would simply not take place.[citation needed]

There is some semantic confusion in addressing exchanges under coercion. If we assume that "Trade X", in which Adam trades Good A to Brian for Good B, does not benefit Adam sufficiently, Adam will ignore Trade X (and trade his Good A for something else in a different positive-sum transaction, or keep it). However, if Brian uses force to ensure that Adam will exchange Good A for Good B, then this says nothing about the original Trade X. Trade X was not, and still is not, positive-sum (in fact, this non-occurring transaction may be zero-sum, if Brian's net gain of utility coincidentally offsets Adam's net loss of utility). What has in fact happened is that a new trade has been proposed, "Trade Y", where Adam exchanges Good A for two things: Good B and escaping the punishment imposed by Brian for refusing the trade. Trade Y is positive-sum, because if Adam wanted to refuse the trade, he theoretically has that option (although it is likely now a much worse option), but he has determined that his position is better served in at least temporarily putting up with the coercion. Under coercion, the coerced party is still doing the best they can under their unfortunate circumstances, and any exchanges they make are positive-sum.[citation needed]

There is additional confusion under asymmetric information. Although many economic theories assume perfect information, economic participants with imperfect or even no information can always avoid making trades that they feel are not in their best interest. Considering transaction costs, then, no zero-sum exchange would ever take place, although asymmetric information can reduce the number of positive-sum exchanges, as occurs in "The Market for Lemons".[citation needed]

Psychology[edit]

The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing our personal interests can enhance our collective well-being, but in others personal interest results in mutually destructive behavior.

Complexity[edit]

It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.

Extensions[edit]

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1)th player representing the global profit or loss.[5]

Misunderstandings[edit]

Zero-sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games.[1]

Zero-sum mentality[edit]

In community psychology "Zero-sum mentality" is a way of thinking that assumes all games are zero-sum: that for every winner there is a loser; for every gain there is a loss.[citation needed]

QMRIn game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrixes - matrix A describing the payoffs player 1 and matrix B describing the payoffs of player 2.[1]

Player 1 is often called the "row player" and player 2 the "column player". If player 1 has m possible actions and player 2 n possible actions, then each of the two matrixes has m rows by n columns. when the row player selects the i-th action and the column player selects the j-th action, the payoff to the row player is A[i,j] and the payoff to the column player is B[i,j].

The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector x of length m such that: \sum_{i=1}^m x_i = 1. Similarly, a mixed strategy for the column player is a non-negative vector y of length m such that: \sum_{j=1}^m y_j = 1. When the players play mixed strategies with vectors x and y, the expected payoff of the row player is: x^T A y and of the column player: x^T B y.

Nash equilibrium in bimatrix games[edit]

Every bimatrix game has a Nash equilibrium in (possibly) mixed strategies. Finding such a Nash equilibrium is a special case of the Linear complementarity problem and can be done in finite time by the Lemke–Howson algorithm.[1]

There is a reduction from the problem of finding a Nash equilibrium in a bimatrix game to the problem of finding a competitive equilibrium in an economy with Leontief utilities.[2]

In game theory, a simultaneous game is a game where each player chooses his action without knowledge of the actions chosen by other players. Normal form representations are usually used for simultaneous games.

Real Life Example[edit]

Rock-Paper-Scissors, a widely played hand game, is a real life example of a simultaneous game. Both make a decision at the same time, randomly, without prior knowledge of the opponent's decision. There are two players in this game and each of them has 3 different strategies to make decision; the combination of strategy proﬁles forms a 3×3 table. We will display Player 1’s strategies as rows and Player 2’s strategies as columns. In the table, the numbers in red represent the payoff to Player 1, the numbers in blue represent the payoff to Player 2. Hence, the pay off for a 2 player game in Rock-Paper-Scissors will look like this:

Rockpaperscissorspayoff.png

In game theory terms, Prisoner dilemma is an example of simultaneous game.

QMRIn animals[edit]

Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in long term partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, guppies inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors by tit for tat strategy.[citation needed]

Vampire bats are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior:[23]

C/C: "Reward: I get blood on my unlucky nights, which saves me from starving. I have to give blood on my lucky nights, which doesn't cost me too much."

D/C: "Temptation: You save my life on my poor night. But then I get the added benefit of not having to pay the slight cost of feeding you on my good night."

C/D: "Sucker's Payoff: I pay the cost of saving your life on my good night. But on my bad night you don't feed me and I run a real risk of starving to death."

D/D: "Punishment: I don't have to pay the slight costs of feeding you on my good nights. But I run a real risk of starving on my poor nights."

QMRNon-credible threat

From Wikipedia, the free encyclopedia

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (August 2009)

Illustration that shows the difference between a SPNE and another NE. The blue equilibrium is not subgame perfect because player two makes a non-credible threat at 2(2) to be unkind (U).

A non-credible threat is a term used in game theory and economics to describe a threat in a sequential game that a rational player would actually not carry out, because it would not be in his best interest to do so.

For a simple example, suppose person A walks up, carrying a bomb, to another person B. A tells B he will set off the bomb, killing them both, unless B gives him all his money. If A is rational and non-suicidal he stands nothing to gain from setting off the bomb, so his threat cannot be considered credible. On the other hand, a person in the situation of B might give A his money, fearing that A is not rational, or might even be suicidal.

A non-credible threat is made on the hope that it will be believed, and therefore the threatening undesirable action will not need to be carried out. For a threat to be credible within an equilibrium, whenever a node is reached where a threat should be fulfilled, it will be fulfilled. Those Nash equilibria that rely on non-credible threats can be eliminated through backward induction, the remaining equilibria are called Subgame perfect Nash equilibria.

Quadrant Model of Reality

Monday, February 22, 2016

Quadrant Model of Reality Book 14 Philosophy and Game Theory

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