Misplaced Pages

#792207

124-469: For ease of exposition, the simple example illustrated above can be considered, where the proposer has two options: a fair split, or an unfair split. The argument given in this section can be extended to the more general case where the proposer can choose from many different splits. A Nash equilibrium is a set of strategies (one for the proposer and one for the responder in this case), where no individual party can improve their reward by changing strategy. If

248-584: A game theory context stable equilibria now usually refer to Mertens stable equilibria. If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are: Examples of game theory problems in which these conditions are not met: In his Ph.D. dissertation, John Nash proposed two interpretations of his equilibrium concept, with

372-641: A strategy  – an action plan based on what has happened so far in the game – and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. In

496-576: A Nash equilibrium is a best response to the other players' strategies in that equilibrium. Formally, let S i {\displaystyle S_{i}} be the set of all possible strategies for player i {\displaystyle i} , where i = 1 , … , N {\displaystyle i=1,\ldots ,N} . Let s ∗ = ( s i ∗ , s − i ∗ ) {\displaystyle s^{*}=(s_{i}^{*},s_{-i}^{*})} be

620-485: A brain imaging experiment by Sanfey et al., stingy offers (relative to fair and hyperfair offers) differentially activated several brain areas, especially the anterior insular cortex , a region associated with visceral disgust . If Player 1 in the ultimatum game anticipates this response to a stingy offer, they may be more generous. An increase in rational decisions in the game has been found among experienced Buddhist meditators . fMRI data show that meditators recruit

744-428: A chimpanzee has been measured at a general range of 282–500 cm . The human brain, in contrast, is about three times larger, with a reported average volume of about 1330 cm . Chimpanzees reach puberty between the age of eight and ten years. A chimpanzee's testicles are unusually large for its body size, with a combined weight of about 4 oz (110 g) compared to a gorilla's 1 oz (28 g) or

868-399: A choice of 3 strategies and where each strategy is a route from A to D (one of ABD , ABCD , or ACD ). The "payoff" of each strategy is the travel time of each route. In the graph on the right, a car travelling via ABD experiences travel time of 1 + x 100 + 2 {\displaystyle 1+{\frac {x}{100}}+2} , where x {\displaystyle x}

992-455: A community, the position of an individual and the influence the individual has on others dictates a definite social hierarchy . Chimpanzees live in a leaner hierarchy wherein more than one individual may be dominant enough to dominate other members of lower rank. Typically, a dominant male is referred to as the alpha male . The alpha male is the highest-ranking male that controls the group and maintains order during disputes. In chimpanzee society,

1116-424: A coordination game is the stag hunt . Two players may choose to hunt a stag or a rabbit, the stag providing more meat (4 utility units, 2 for each player) than the rabbit (1 utility unit). The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, the stag hunter will totally fail, for a payoff of 0, whereas the rabbit hunter will succeed, for

1240-541: A failure to inhibit a desire to punish the first player for making an unfair offer. Morewedge, Krishnamurti, and Ariely (2014) found that intoxicated participants were more likely to reject unfair offers than sober participants. As intoxication tends to exacerbate decision makers' prepotent response, this result provides support for the self-control account, rather than the altruistic punishment account. Other research from social cognitive neuroscience supports this finding. However, several competing models suggest ways to bring

1364-433: A finite set of actions. The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point

SECTION 10

#1732773315793

1488-411: A function of the strategies. The strategy profile s ∗ {\displaystyle s^{*}} is a Nash equilibrium if A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak : a player might be indifferent among several strategies given the other players' choices. It is unique and called a strict Nash equilibrium if the inequality

1612-404: A future state or event. In October 1960, Jane Goodall observed the use of tools among chimpanzees . Recent research indicates that chimpanzees' use of stone tools dates back at least 4,300 years (about 2,300 BC). One example of chimpanzee tool usage behavior includes the use of a large stick as a tool to dig into termite mounds, and the subsequent use of a small stick altered into a tool that

1736-408: A game in which Carol and Dan are also players, (A, B, C, D) is a Nash equilibrium if A is Alice's best response to (B, C, D), B is Bob's best response to (A, C, D), and so forth. Nash showed that there is a Nash equilibrium, possibly in mixed strategies , for every finite game. Game theorists use Nash equilibrium to analyze the outcome of the strategic interaction of several decision makers . In

1860-582: A gift to Frederick Henry, Prince of Orange in 1640, and were followed by a few of its brethren over the next several years. Scientists described these first chimpanzees as " pygmies ", and noted the animals' distinct similarities to humans. The next two decades, a number of the creatures were imported into Europe, mainly acquired by various zoological gardens as entertainment for visitors. Charles Darwin 's theory of natural selection (published in 1859) spurred scientific interest in chimpanzees, as in much of life science , leading eventually to numerous studies of

1984-919: A group. Community female acceptance is necessary for alpha male status; females must ensure that their group visits places that supply them with enough food. A group of dominant females will sometimes oust an alpha male which is not to their preference and back another male, in whom they see potential for leading the group as a successful alpha male. The mating system within each community is polygynandrous , with each male and female possibly having multiple sexual partners. Chimpanzees make tools and use them to acquire foods and for social displays; they have sophisticated hunting strategies requiring cooperation, influence and rank; they are status conscious, manipulative and capable of deception; they can learn to use symbols and understand aspects of human language including some relational syntax , concepts of number and numerical sequence; and they are capable of spontaneous planning for

2108-426: A human's 1.5 ounces (43 g). This relatively great size is generally attributed to sperm competition due to the polygynandrous nature of chimpanzee mating behaviour . In the wild, chimpanzees live to their 30s, while some captured chimps have reached an age of 70 years and older . Chimpanzees are known for possessing great amount of muscle strength, especially in their arms. However, compared to humans

2232-504: A language of "the Angolans" (apparently from a Bantu language ; reportedly modern Vili (Civili) , a Zone H Bantu language, has the comparable ci-mpenzi ). The spelling chimpanzee is found in a 1758 supplement to Chamber's Cyclopædia . The colloquialism "chimp" was most likely coined some time in the late 1870s. The chimpanzee was named Simia troglodytes by Johann Friedrich Blumenbach in 1776. The species name troglodytes

2356-562: A low offer. It could also be the case that the second player, by having the power to reject the offer, uses such power as leverage against the first player, thus motivating them to be fair. The classical explanation of the ultimatum game as a well-formed experiment approximating general behaviour often leads to a conclusion that the rational behavior in assumption is accurate to a degree, but must encompass additional vectors of decision making. Behavioral economic and psychological accounts suggest that second players who reject offers less than 50% of

2480-620: A manner that is similar to that of human nonverbal communication, using vocalizations, hand gestures, and facial expressions. There is some evidence that they can recreate human speech. Research into the chimpanzee brain has revealed that when chimpanzees communicate, an area in the brain is activated which is in the same position as the language center called Broca's area in human brains. Adult common chimpanzees, particularly males, can be very aggressive. They are highly territorial and are known to kill others of their species. Chimpanzees also engage in targeted hunting of smaller primates, such as

2604-439: A mechanical apparatus. Some studies have found significant differences between cultures in the offers most likely to be accepted and most likely to maximize the proposer's income. In one study of 15 small-scale societies, proposers in gift-giving cultures were more likely to make high offers and responders were more likely to reject high offers despite anonymity, while low offers were expected and accepted in other societies, which

SECTION 20

#1732773315793

2728-415: A payoff of 1. The game has two equilibria, (stag, stag) and (rabbit, rabbit), because a player's optimal strategy depends on their expectation on what the other player will do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they think the other will hunt the rabbit, they too will hunt the rabbit. This game is used as an analogy for social cooperation, since much of

2852-448: A pure strategy for each player or might be a probability distribution over strategies for each player. Nash equilibria need not exist if the set of choices is infinite and non-compact. For example: However, a Nash equilibrium exists if the set of choices is compact with each player's payoff continuous in the strategies of all the players. Rosen extended Nash's existence theorem in several ways. He considers an n-player game, in which

2976-399: A refinement that eliminates equilibria which depend on non-credible threats . Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeated , or what happens if a game is played in the absence of complete information . However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is

3100-538: A separate group at all, they were referred to as "pygmy" or "gracile chimpanzees". Together with humans , gorillas , and orangutans they are part of the family Hominidae (the great apes, or hominids ). Native to sub-Saharan Africa , chimpanzees and bonobos are currently both found in the Congo jungle , while only the chimpanzee is also found further north in West Africa. Both species are listed as endangered on

3224-426: A set of strategies such that each player's strategy is optimal given the choices of the others. A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing

3348-478: A small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold: If these cases are both met, then a player with the small change in their mixed strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for

3472-722: A sort of mattress, which is supported by strong branches for a foundation, and then lined with softer leaves and twigs; the minimum diameter is 5 metres (16 ft) and may be located at a height of 3 to 45 metres (10 to 150 ft). Both day and night nests are built, and may be located in groups. A study in 2014 found that the muhimbi tree is favoured for nest building by chimpanzees in Uganda due to its physical properties, such as bending strength, inter-node distance, and leaf surface area. Studies have shown chimpanzees engage in apparently altruistic behaviour within groups. Some researchers have suggested that chimpanzees are indifferent to

3596-496: A strategic interaction, the outcome for each decision-maker depends on the decisions of the others as well as their own. The simple insight underlying Nash's idea is that one cannot predict the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do taking into account what the player expects the others to do. Nash equilibrium requires that one's choices be consistent: no players wish to undo their decision given what

3720-525: A strategy profile, a set consisting of one strategy for each player, where s − i ∗ {\displaystyle s_{-i}^{*}} denotes the N − 1 {\displaystyle N-1} strategies of all the players except i {\displaystyle i} . Let u i ( s i , s − i ∗ ) {\displaystyle u_{i}(s_{i},s_{-i}^{*})} be player i's payoff as

3844-501: Is a Cartesian product of convex sets S 1 ,..., S n , such that the strategy of player i must be in S i . This represents the case that the actions of each player i are constrained independently of other players' actions. If the following conditions hold: Then a Nash equilibrium exists. The proof uses the Kakutani fixed-point theorem . Rosen also proves that, under certain technical conditions which include strict concavity,

Ultimatum game - Misplaced Pages Continue

3968-412: Is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto efficient . As a result of these requirements, strong Nash

4092-432: Is a non-negative real number. Nash's existing proofs assume a finite strategy set, but the concept of Nash equilibrium does not require it. A game can have a pure-strategy or a mixed-strategy Nash equilibrium. In the latter, not every player always plays the same strategy. Instead, there is a probability distribution over different strategies. Suppose that in the Nash equilibrium, each player asks themselves: "Knowing

4216-463: Is a reference to the Troglodytae (literally "cave-goers"), an African people described by Greco-Roman geographers . Blumenbach first used it in his De generis humani varietate nativa liber ("On the natural varieties of the human genus") in 1776, Linnaeus 1758 had already used Homo troglodytes for a hypothetical mixture of human and orangutan . The bonobo, in the past also referred to as

4340-404: Is also often modelled using a continuous strategy set. Suppose the proposer chooses a share S of a pie to offer the receiver, where S can be any real number between 0 and 1, inclusive. If the receiver accepts the offer, the proposer's payoff is (1-S) and the receiver's is S . If the receiver rejects the offer, both players get zero. The unique subgame perfect equilibrium is ( S =0, Accept). It

4464-407: Is an easy numerical way to identify Nash equilibria on a payoff matrix. It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the payoff pair of the cell, is the maximum of

4588-408: Is an n-tuple such that each player's mixed strategy maximizes [their] payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for

4712-575: Is choosing to get nothing rather than something, that individual must not be acting solely to maximize their economic gain, unless one incorporates economic applications of social, psychological, and methodological factors (such as the observer effect ). Several attempts have been made to explain this behavior. Some suggest that individuals are maximizing their expected utility , but money does not translate directly into expected utility. Perhaps individuals get some psychological benefit from engaging in punishment or receive some psychological harm from accepting

4836-475: Is divided into four subspecies , while P. paniscus is undivided. Based on genome sequencing , these two extant Pan species diverged around one million years ago. The most obvious differences are that chimpanzees are somewhat larger, more aggressive and male-dominated, while the bonobos are more gracile, peaceful, and female-dominated. Their hair is typically black or brown. Males and females differ in size and appearance. Both chimpanzees and bonobos are some of

4960-521: Is important from a sociological perspective, because it illustrates the human unwillingness to accept injustice . The tendency to refuse small offers may also be seen as relevant to the concept of honour . The extent to which people are willing to tolerate different distributions of the reward from " cooperative " ventures results in inequality that is, measurably, exponential across the strata of management within large corporations. See also: Inequity aversion within companies . An early description of

5084-486: Is named after American mathematician John Forbes Nash Jr . The same idea was used in a particular application in 1838 by Antoine Augustin Cournot in his theory of oligopoly . In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A Cournot equilibrium occurs when each firm's output maximizes its profits given

Ultimatum game - Misplaced Pages Continue

5208-437: Is part of the subfamily Homininae , to which humans also belong. The lineages of chimpanzees and humans separated in a process of speciation between roughly five to twelve million years ago, making them humanity's closest living relative. Research by Mary-Claire King in 1973 found 99% identical DNA between human beings and chimpanzees. For some time, research modified that finding to about 94% commonality, with some of

5332-414: Is seen as more important than any economic reward. Others have proposed the social status of the responder may be part of the payoff. Another way of integrating the conclusion with utility maximization is some form of inequity aversion model (preference for fairness). Even in anonymous one-shot settings, the economic-theory suggested outcome of minimum money transfer and acceptance is rejected by over 80% of

5456-443: Is strict so one strategy is the unique best response: The strategy set S i {\displaystyle S_{i}} can be different for different players, and its elements can be a variety of mathematical objects. Most simply, a player might choose between two strategies, e.g. S i = { Yes , No } . {\displaystyle S_{i}=\{{\text{Yes}},{\text{No}}\}.} Or,

5580-400: Is the maximum of the row. If these conditions are met, the cell represents a Nash equilibrium. Check all columns this way to find all NE cells. An N×N matrix may have between 0 and N×N pure-strategy Nash equilibria. The concept of stability , useful in the analysis of many kinds of equilibria, can also be applied to Nash equilibria. A Nash equilibrium for a mixed-strategy game is stable if

5704-408: Is the number of cars traveling on edge AB . Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to their travel time. For

5828-469: Is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium. A refined Nash equilibrium known as coalition-proof Nash equilibrium (CPNE) occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by iterated strict dominance and on

5952-537: Is used to "fish" the termites out of the mound. Chimpanzees are also known to use smaller stones as hammers and a large one as an anvil in order to break open nuts. In the 1970s, reports of chimpanzees using rocks or sticks as weapons were anecdotal and controversial. However, a 2007 study claimed to reveal the use of spears, which common chimpanzees in Senegal sharpen with their teeth and use to stab and pry Senegal bushbabies out of small holes in trees. Prior to

6076-411: Is weak because the receiver's payoff is 0 whether they accept or reject. No share with S > 0 is subgame perfect, because the proposer would deviate to S' = S - ϵ {\displaystyle \epsilon } for some small number ϵ {\displaystyle \epsilon } and the receiver's best response would still be to accept. The weak equilibrium is an artifact of

6200-731: Is worth noting that the instructions offered to proposers in this study explicitly state, "if the responder's goal is to earn as much money as possible from the experiment, they should accept any offer that gives them positive earnings, no matter how low," thus framing the game in purely monetary terms. Generous offers in the ultimatum game (offers exceeding the minimum acceptable offer) are commonly made. Zak, Stanton & Ahmadi (2007) showed that two factors can explain generous offers: empathy and perspective taking. They varied empathy by infusing participants with intranasal oxytocin or placebo (blinded). They affected perspective-taking by asking participants to make choices as both player 1 and player 2 in

6324-585: The IUCN Red List of Threatened Species , and in 2017 the Convention on Migratory Species selected the chimpanzee for special protection. The chimpanzee ( P. troglodytes ), which lives north of the Congo River , and the bonobo ( P. paniscus ), which lives south of it, were once considered to be the same species, but since 1928 they have been recognized as distinct. In addition, P. troglodytes

SECTION 50

#1732773315793

6448-453: The Nash equilibrium is the most commonly-used solution concept for non-cooperative games . A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of Cournot , who in 1838 applied it to his model of competition in an oligopoly . If each player has chosen

6572-551: The Pareto frontier is a CPNE. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core . Nash proved that if mixed strategies (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be

6696-407: The expectation of the player who did the change, if the other player's mixed strategy is still (50%,50%)), then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%). Stability is crucial in practical applications of Nash equilibria, since the mixed strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in

6820-585: The red colobus and bush babies . Males who acquire the meat may share it with females to have sex or for grooming. In February 2013, a study found that chimpanzees solve puzzles for entertainment. Chimpanzees, as well as other apes, had also been purported to have been known to ancient writers, but mainly as myths and legends on the edge of European and Near Eastern societal consciousness. Apes are mentioned variously by Aristotle . The English word ape translates Hebrew קוף (qof) in English translations of

6944-497: The subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. The coordination game is a classic two-player, two- strategy game, as shown in the example payoff matrix to the right. There are two pure-strategy equilibria, (A,A) with payoff 4 for each player and (B,B) with payoff 2 for each. The combination (B,B) is a Nash equilibrium because if either player unilaterally changes their strategy from B to A, their payoff will fall from 2 to 1. A famous example of

7068-470: The trust game , and net splits tend to be more equitable. The "reverse ultimatum game" gives more power to the responder by giving the proposer the right to offer as many divisions of the endowment as they like. Now the game only ends when the responder accepts an offer or abandons the game, and therefore the proposer tends to receive slightly less than half of the initial endowment. Incomplete information ultimatum games: Some authors have studied variants of

7192-465: The "competitive ultimatum game" there are many proposers and the responder can accept at most one of their offers: With more than three (naïve) proposers the responder is usually offered almost the entire endowment (which would be the Nash Equilibrium assuming no collusion among proposers). In the "ultimatum game with tipping", a tip is allowed from responder back to proposer, a feature of

7316-543: The "pygmy chimpanzee", was given the species name of paniscus by Ernst Schwarz (1929), a Greek-style diminutive of the theonym Pan used by Cicero . There are two species of the genus Pan , both previously called simply chimpanzees: [REDACTED] Unknown [REDACTED] [REDACTED] Unknown [REDACTED] gibbons (family Hylobatidae) orangutans (genus Pongo ) gorillas (genus Gorilla ) humans (genus  Homo ) chimpanzees (genus  Pan ) The genus Pan

7440-578: The 'dominant male' sometimes is not the largest or strongest male but rather the most manipulative and political male that can influence the goings on within a group. Male chimpanzees typically attain dominance by cultivating allies who will support that individual during future ambitions for power. The alpha male regularly displays by puffing his normally slim coat up to increase view size and charge to seem as threatening and as powerful as possible; this behaviour serves to intimidate other members and thereby maintain power and authority, and it may be fundamental to

7564-598: The 1960s these tests were repeated and chimpanzees were found to have twice the strength of a human when it came to pulling weights. The reason for the higher strength seen in chimpanzees compared to humans are thought to come from longer skeletal muscle fibers that can generate twice the work output over a wider range of motion compared to skeletal muscle fibers in humans. It is suspected that human observers can influence chimpanzee behaviour. For this reason researchers sometimes prefer camera traps and remote microphones rather than human observers. Anatomical differences between

SECTION 60

#1732773315793

7688-557: The Bible ( 1 Kings 10:22), but the word may refer to a monkey rather than an ape proper. The diary of Portuguese explorer Duarte Pacheco Pereira (1506), preserved in the Portuguese National Archive ( Torre do Tombo ), is probably the first written document to acknowledge that chimpanzees built their own rudimentary tools. The first of these early transcontinental chimpanzees came from Angola and were presented as

7812-402: The absolute amount of the offer is low. The concept here is that if the amount to be split were 10 million dollars, a 9:1 split would probably be accepted rather than rejecting a 1 million-dollar offer. Essentially, this explanation says that the absolute amount of the endowment is not significant enough to produce strategically optimal behaviour. However, many experiments have been performed where

7936-675: The adoption of technical standards , and also the occurrence of bank runs and currency crises (see coordination game ). Other applications include traffic flow (see Wardrop's principle ), how to organize auctions (see auction theory ), the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations (see tragedy of the commons ), natural resource management, analysing strategies in marketing, penalty kicks in football (see matching pennies ), robot navigation in crowds, energy systems, transportation systems, evacuation problems and wireless communications. Nash equilibrium

8060-417: The alpha male's holding on to his status. Lower-ranking chimpanzees will show respect by submissively gesturing in body language or reaching out their hands while grunting. Female chimpanzees will show deference to the alpha male by presenting their hindquarters. Female chimpanzees also have a hierarchy, which is influenced by the position of a female individual within a group. In some chimpanzee communities,

8184-457: The amount at stake do so for one of two reasons. An altruistic punishment account suggests that rejections occur out of altruism: people reject unfair offers to teach the first player a lesson and thereby reduce the likelihood that the player will make an unfair offer in the future. Thus, rejections are made to benefit the second player in the future, or other people in the future. By contrast, a self-control account suggests that rejections constitute

8308-498: The amount of strength reported in media and popular science is greatly exaggerated with numbers of four to eight times the muscle strength of a human. These numbers stem from two studies in 1923 and 1926 by a biologist named John Bauman. These studies were refuted in 1943 and an adult male chimpanzee was found to pull about the same weight as an adult man. Corrected for their smaller body sizes, chimpanzees were found to be stronger than humans but not anywhere near four to eight times. In

8432-903: The amount offered was substantial: studies by Cameron and Hoffman et al. have found that higher stakes cause offers to approach closer to an even split, even in a US$ 100 game played in Indonesia , where average per-capita income is much lower than in the United States . Rejections are reportedly independent of the stakes at this level, with US$ 30 offers being turned down in Indonesia, as in the United States, even though this equates to two weeks' wages in Indonesia. However, 2011 research with stakes of up to 40 weeks' wages in India showed that "as stakes increase, rejection rates approach zero". It

8556-544: The animals in the wild and captivity. The observers of chimpanzees at the time were mainly interested in behaviour as it related to that of humans. This was less strictly and disinterestedly scientific than it might sound, with much attention being focused on whether or not the animals had traits that could be considered 'good'; the intelligence of chimpanzees was often significantly exaggerated, as immortalized in Hugo Rheinhold 's Affe mit Schädel (see image, left). By

8680-402: The authors suggested were related to the ways that giving and receiving were connected to social status in each group. Proposers and responders from WEIRD (Western, educated, industrialized, rich, democratic) societies are most likely to settle on equal splits. Some studies have found significant effects of framing on game outcomes. Outcomes have been found to change based on characterizing

8804-432: The benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation. Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with

8928-430: The bonobo. The average captive chimpanzee sleeps 9 hours and 42 minutes per day. Contrary to what the scientific name ( Pan troglodytes ) may suggest, chimpanzees do not typically spend their time in caves, but there have been reports of some of them seeking refuge in caves because of the heat during daytime. Chimpanzees live in large multi-male and multi-female social groups , which are called communities. Within

9052-399: The chimp's feet have broader soles and shorter toes. The bonobo has proportionately longer upper limbs and walks upright more often than does the common chimpanzee. Both species can walk upright on two legs when carrying objects with their hands and arms. The chimpanzee is tailless; its coat is dark; its face, fingers, palms of the hands, and soles of the feet are hairless. The exposed skin of

9176-566: The chimpanzees, where she primarily studied the members of the Kasakela chimpanzee community . Her discovery that chimpanzees made and used tools was groundbreaking, as humans were previously believed to be the only species to do so. The most progressive early studies on chimpanzees were spearheaded primarily by Wolfgang Köhler and Robert Yerkes , both of whom were renowned psychologists. Both men and their colleagues established laboratory studies of chimpanzees focused specifically on learning about

9300-404: The column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium. We can apply this rule to a 3×3 matrix: Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash equilibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A), 40 is the maximum of the first column and 25 is the maximum of

9424-433: The common chimp's long arms span one and a half times the body's height. The bonobo is slightly shorter and thinner than the common chimpanzee, but has longer limbs. In trees, both species climb with their long, powerful arms; on the ground, chimpanzees usually knuckle-walk , or walk on all fours, clenching their fists and supporting themselves on the knuckles. Chimpanzees are better suited for walking than orangutans, because

9548-667: The common chimpanzee and the bonobo are slight. Both are omnivorous adapted to a mainly frugivorous diet . Yet sexual and social behaviours are markedly different. The common chimpanzee has a troop culture based on beta males led by an alpha male , and highly complex social relationships. The bonobo, on the other hand, has egalitarian , nonviolent , matriarchal , sexually receptive behaviour . Bonobos frequently have sex, sometimes to help prevent and resolve conflicts. Different groups of chimpanzees also have different cultural behaviour with preferences for types of tools. The common chimpanzee tends to display greater aggression than does

9672-459: The cultural preferences of the players within the optimized utility function of the players in such a way as to preserve the utility maximizing agent as a feature of microeconomics . For example, researchers have found that Mongolian proposers tend to offer even splits despite knowing that very unequal splits are almost always accepted. Similar results from other small-scale societies players have led some researchers to conclude that " reputation "

9796-594: The difference occurring in noncoding DNA , but more recent knowledge puts the difference in DNA between humans, chimpanzees and bonobos at just about 1%–1.2% again. The chimpanzee fossil record has long been absent and thought to have been due to the preservation bias in relation to their environment. However, in 2005, chimpanzee fossils were discovered and described by Sally McBrearty and colleagues. Existing chimpanzee populations in West and Central Africa are separate from

9920-459: The discovery of tool use by chimpanzees, humans were believed to be the only species to make and use tools; however, several other tool-using species are now known. Nest-building, sometimes considered to be a form of tool use, is seen when chimpanzees construct arboreal night nests by lacing together branches from one or more trees to build a safe, comfortable place to sleep; infants learn this process by watching their mothers. The nest provides

10044-411: The end of the 19th century, chimpanzees remained very much a mystery to humans, with very little factual scientific information available. In the 20th century, a new age of scientific research into chimpanzee behaviour began. Before 1960, almost nothing was known about chimpanzee behaviour in their natural habitats. In July of that year, Jane Goodall set out to Tanzania 's Gombe forest to live among

10168-594: The equilibrium is unique. Nash's result refers to the special case in which each S i is a simplex (representing all possible mixtures of pure strategies), and the payoff functions of all players are bilinear functions of the strategies. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal . Nash equilibrium may also have non-rational consequences in sequential games because players may "threaten" each other with threats they would not actually carry out. For such games

10292-451: The face, hands, and feet varies from pink to very dark in both species, but is generally lighter in younger individuals and darkens with maturity. A University of Chicago Medical Centre study has found significant genetic differences between chimpanzee populations. A bony shelf over the eyes gives the forehead a receding appearance, and the nose is flat. Although the jaws protrude, a chimp's lips are thrust out only when it pouts. The brain of

10416-537: The following payoff matrix: In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%, 100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where

10540-426: The game begins at the green square, it is in player 1's interest to move to the purple square and it is in player 2's interest to move to the blue square. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria: (0,0), (1,1), (2,2), and (3,3). There

10664-443: The game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium. Finally in the eighties, building with great depth on such ideas Mertens-stable equilibria were introduced as a solution concept . Mertens stable equilibria satisfy both forward induction and backward induction . In

10788-535: The graph on the right, if, for example, 100 cars are travelling from A to D , then equilibrium will occur when 25 drivers travel via ABD , 50 via ABCD , and 25 via ACD . Every driver now has a total travel time of 3.75 (to see this, a total of 75 cars take the AB edge, and likewise, 75 cars take the CD edge). Notice that this distribution is not, actually, socially optimal. If the 100 cars agreed that 50 travel via ABD and

10912-547: The infant. According to a literature summary by James W. Harrod, evidence for chimpanzee emotivity includes display of mourning ; "incipient romantic love "; "rain dances" ; appreciation of natural beauty (such as a sunset over a lake); curiosity and respect towards other wildlife (such as the python , which is neither a threat nor a food source to chimpanzees); altruism toward other species (such as feeding turtles); and animism , or "pretend play", when chimpanzees cradle and groom rocks or sticks. Chimpanzees communicate in

11036-592: The intellectual abilities of chimpanzees, particularly problem-solving . This typically involved basic, practical tests on laboratory chimpanzees, which required a fairly high intellectual capacity (such as how to solve the problem of acquiring an out-of-reach banana). Notably, Yerkes also made extensive observations of chimpanzees in the wild which added tremendously to the scientific understanding of chimpanzees and their behaviour. Yerkes studied chimpanzees until World War II , while Köhler concluded five years of study and published his famous Mentality of Apes in 1925 (which

11160-738: The major human fossil sites in East Africa; however, chimpanzee fossils have been reported from Kenya , indicating that both humans and members of the Pan clade were present in the East African Rift Valley during the Middle Pleistocene . The chimpanzee's arms are longer than its legs. The male common chimp stands up to 1.2 m (3.9 ft) high. Male adult wild chimps weigh between 40 and 60 kg with females weighing between 27 and 50 kg. When extended,

11284-718: The most social great apes, with social bonds occurring throughout large communities. Fruit is the most important component of a chimpanzee's diet; but they will also eat vegetation, bark, honey, insects and even other chimpanzees or monkeys. They can live over 30 years in both the wild and captivity. Chimpanzees and bonobos are equally humanity's closest living relatives. They use a variety of sophisticated tools and construct elaborate sleeping nests each night from branches and foliage. Their learning abilities have been extensively studied. There may even be distinctive cultures within populations. Field studies of Pan troglodytes were pioneered by primatologist Jane Goodall . The genus name Pan

11408-424: The objective of showing how equilibrium points can be connected with observable phenomenon. Pan (genus) Pan troglodytes Pan paniscus The genus Pan consists of two extant species: the chimpanzee and the bonobo . Taxonomically, these two ape species are collectively termed panins . The two species were formerly collectively called "chimpanzees" or "chimps"; if bonobos were recognized as

11532-453: The other 50 through ACD , then travel time for any single car would actually be 3.5, which is less than 3.75. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess's paradox . This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win

11656-457: The others are deciding. The concept has been used to analyze hostile situations such as wars and arms races (see prisoner's dilemma ), and also how conflict may be mitigated by repeated interaction (see tit-for-tat ). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes ), and whether they will take risks to achieve a cooperative outcome (see stag hunt ). It has been used to study

11780-476: The output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally. The modern concept of Nash equilibrium is instead defined in terms of mixed strategies , where players choose a probability distribution over possible pure strategies (which might put 100% of

11904-404: The player is indifferent between switching and not), then the equilibrium is classified as a weak or non-strict Nash equilibrium . The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough. Strong Nash equilibrium allows for deviations by every conceivable coalition. Formally, a strong Nash equilibrium

12028-474: The player who changed. In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes their probabilities (which would neither benefit or damage

12152-547: The players. An explanation which was originally quite popular was the "learning" model, in which it was hypothesized that proposers' offers would decay towards the sub game perfect Nash equilibrium (almost zero) as they mastered the strategy of the game; this decay tends to be seen in other iterated games. However, this explanation ( bounded rationality ) is less commonly offered now, in light of subsequent empirical evidence. It has been hypothesized (e.g. by James Surowiecki ) that very unequal allocations are rejected only because

12276-420: The posterior insular cortex (associated with interoception ) during unfair offers and show reduced activity in the anterior insular cortex compared to controls. People whose serotonin levels have been artificially lowered will reject unfair offers more often than players with normal serotonin levels. People who have ventromedial frontal cortex lesions were found to be more likely to reject unfair offers. This

12400-420: The probabilities for each player are (50%, 50%). An application of Nash equilibria is in determining the expected flow of traffic in a network. Consider the graph on the right. If we assume that there are x {\displaystyle x} "cars" traveling from A to D , what is the expected distribution of traffic in the network? This situation can be modeled as a " game ", where every traveler has

12524-426: The probability on one pure strategy; such pure strategies are a subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior , but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with

12648-435: The proposer always makes an unfair offer, the responder will do best by always accepting the offer, and the proposer will maximize their reward. Although it always benefits the responder to accept even unfair offers, the responder can adopt a strategy that rejects unfair splits often enough to induce the proposer to always make a fair offer. Any change in strategy by the proposer will lower their reward. Any change in strategy by

12772-458: The proposer's role as giving versus splitting versus taking, or characterizing the game as a windfall game versus a routine transaction game. The highly mixed results, along with similar results in the dictator game , have been taken as both evidence for and against the Homo economicus assumptions of rational, utility-maximizing, individual decisions. Since an individual who rejects a positive offer

12896-411: The proposer's strategy set would be all integers between 0 and 100, inclusive for their choice of offer, S . This would have two subgame perfect equilibria: (Proposer: S =0, Accepter: Accept), which is a weak equilibrium because the acceptor would be indifferent between their two possible strategies; and the strong (Proposer: S =1, Accepter: Accept if S >=1 and Reject if S =0). The ultimatum game

13020-431: The responder to accept the offer. So, the second set of Nash equilibria above is not subgame perfect: the responder can choose a better strategy for one of the subgames. The simplest version of the ultimatum game has two possible strategies for the proposer, Fair and Unfair. A more realistic version would allow for many possible offers. For example, the item being shared might be a dollar bill, worth 100 cents, in which case

13144-555: The responder will result in the same reward or less. Thus, there are two sets of Nash equilibria for this game: However, only the first set of Nash equilibria satisfies a more restrictive equilibrium concept , subgame perfection . The game can be viewed as having two subgames: the subgame where the proposer makes a fair offer, and the subgame where the proposer makes an unfair offer. A perfect-subgame equilibrium occurs when there are Nash Equilibria in every subgame, that players have no incentive to deviate from. In both subgames, it benefits

13268-469: The same purpose. Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many solution concepts ('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not ' credible '. In 1965 Reinhard Selten proposed subgame perfect equilibrium as

13392-402: The second row. For (A,B), 25 is the maximum of the second column and 40 is the maximum of the first row; the same applies for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns. This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the pair

13516-407: The smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give up two points to the other. This game has a unique pure-strategy Nash equilibrium: both players choosing 0 (highlighted in light red). Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if

13640-399: The strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" For instance if a player prefers "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in

13764-416: The strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?" If every player's answer is "Yes", then the equilibrium is classified as a strict Nash equilibrium . If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e.

13888-465: The strategy of each player i is a vector s i in the Euclidean space R . Denote m := m 1 +...+ m n ; so a strategy-tuple is a vector in R . Part of the definition of a game is a subset S of R such that the strategy-tuple must be in S . This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when S

14012-572: The strategy set might be a finite set of conditional strategies responding to other players, e.g. S i = { Yes | p = Low , No | p = High } . {\displaystyle S_{i}=\{{\text{Yes}}|p={\text{Low}},{\text{No}}|p={\text{High}}\}.} Or, it might be an infinite set, a continuum or unbounded, e.g. S i = { Price } {\displaystyle S_{i}=\{{\text{Price}}\}} such that Price {\displaystyle {\text{Price}}}

14136-560: The strategy space being continuous. The first experimental analysis of the ultimatum game was by Werner Güth , Rolf Schmittberger, and Bernd Schwarze: Their experiments were widely imitated in a variety of settings. When carried out between members of a shared social group (e.g., a village, a tribe, a nation, humanity) people offer "fair" (i.e., 50:50) splits, and offers of less than 30% are often rejected. One limited study of monozygotic and dizygotic twins claims that genetic variation can have an effect on reactions to unfair offers, though

14260-579: The study failed to employ actual controls for environmental differences. It has also been found that delaying the responder's decision leads to people accepting "unfair" offers more often. Common chimpanzees behaved similarly to humans by proposing fair offers in one version of the ultimatum game involving direct interaction between the chimpanzees. However, another study also published in November 2012 showed that both kinds of chimpanzees ( common chimpanzees and bonobos ) did not reject unfair offers, using

14384-537: The tendency of some wrens to forage in dark crevices. The International Commission on Zoological Nomenclature adopted Pan as the only official name of the genus in 1895, though the "cave-dweller" connection was able to be included, albeit at the species level ( Pan troglodytes – the common chimpanzee) for one of the two species of Pan . The first use of the name "chimpanze" is recorded in The London Magazine in 1738, glossed as meaning "mockman" in

14508-542: The traditional economic principle that consumers are rational and utility-maximising. This started a variety of research into the psychology of humans. Since the ultimatum game's development, it has become a popular economic experiment , and was said to be "quickly catching up with the Prisoner's Dilemma as a prime showpiece of apparently irrational behavior" in a paper by Martin Nowak , Karen M. Page, and Karl Sigmund . In

14632-489: The ultimatum game in which either the proposer or the responder has private information about the size of the pie to be divided. These experiments connect the ultimatum game to principal-agent problems studied in contract theory . The pirate game illustrates a variant with more than two participants with voting power, as illustrated in Ian Stewart 's "A Puzzle for Pirates". Nash equilibrium In game theory ,

14756-413: The ultimatum game is by Nobel laureate John Harsanyi in 1961, who footnotes Thomas Schelling's 1960 book, The Strategy of Conflict on its solution by dominance methods. Harsanyi says, Josh Clark attributes modern interest in the game to Ariel Rubinstein, but the best-known article is the 1982 experimental analysis of Güth, Schmittberger, and Schwarze. Results from testing the ultimatum game challenged

14880-413: The ultimatum game, with later random assignment to one of these. Oxytocin increased generous offers by 80% relative to placebo. Oxytocin did not affect the minimum acceptance threshold or offers in the dictator game (meant to measure altruism). This indicates that emotions drive generosity. Rejections in the ultimatum game have been shown to be caused by adverse physiologic reactions to stingy offers. In

15004-519: The welfare of unrelated group members, but a more recent study of wild chimpanzees found that both male and female adults would adopt orphaned young of their group. Also, different groups sometimes share food, form coalitions, and cooperate in hunting and border patrolling. Sometimes, chimpanzees have adopted young that come from unrelated groups. And in some rare cases, even male chimpanzees have been shown to take care of abandoned infant chimpanzees of an unrelated group, though in most cases they would kill

15128-457: The young females may inherit high status from a high-ranking mother. Dominant females will also ally to dominate lower-ranking females: whereas males mainly seek dominant status for its associated mating privileges and sometimes violent domination of subordinates, females seek dominant status to acquire resources such as food, as high-ranking females often have first access to them. Both genders acquire dominant status to improve social standing within

15252-458: Was first introduced by Lorenz Oken in 1816. While Oken did not give a rationale for his choice, it is generally thought to have been inspired by the name of the Greek god Pan . An alternative Theranthropus was suggested by Brookes 1828 and Chimpansee by Voigt 1831. Troglodytes was not available, as it had been given as the name of a genus of wren in 1809, for "cave-dweller", reflecting

15376-485: Was suggested to be due to the abstractness and delay of the reward, rather than an increased emotional response to the unfairness of the offer. Other authors have used evolutionary game theory to explain behavior in the ultimatum game. Simple evolutionary models, e.g. the replicator dynamics , cannot account for the evolution of fair proposals or for rejections. These authors have attempted to provide increasingly complex models to explain fair behavior. The ultimatum game

#792207