GAME THEORY

Game theory is the analysis of choices made by individuals, institutions, or governments, which are termed players; the results of one player's choice depend on the choices made by the others. Anticipations by players about how others may respond or may anticipate their actions thus influence choices of actions. An important attempt to use game theory involved the formation of nuclear deterrence strategy by the United States during the Cold War (1945–1990). However, game theory has many more general implications that go beyond those involving intentional choice.

Despite the fact that game theory matured only toward the end of the twentieth century, it has become a central tool in some of the behavioral sciences and doubtless will extend its influence into all disciplines that attempt to explain the behavior of living organisms. Indeed, game theory provides a language that transcends and potentially unites the various disciplines that deal with human behavior. Moreover, it provides an experimental methodology that allows for the rigorous construction and testing of strategic interaction because it forces an experimenter to be explicit in defining the actions available to the subjects, the payoffs, and the distribution of information among the subjects.

An Illuminating Example

A fox is chasing a rabbit through a wooded area. Foxes are faster than rabbits, and so if the rabbit runs in a straight line, it will be caught and eaten. The rabbit therefore periodically veers left or right, gaining ground on the fox. If the rabbit changes course too rapidly, its average forward movement will be so slow that it will be caught, but if it changes course too slowly, the fox will be so close that a small misstep by the rabbit will lead to its immediate demise. Therefore, the rabbit must choose the average rate of veering to optimize its probability of escaping.

In game theory it is said that the rabbit has actions: R_t = "Veer Right after t seconds" and L_t = "Veer Left after t seconds." The rabbit also wants to randomize its choice of Veer Right and Veer Left, because if the fox discovers a pattern in the rabbit's movement, it may be able to anticipate the rabbit's next move, thereby gaining ground on it. The proper mix of Veer Left and Veer right is doubtless 50 percent Left and 50 percent Right, for the fox potentially could exploit an imbalance in either direction.

However, suppose that there is an open field some distance to the east of the wood and that foxes run much faster than rabbits do in an open field. Then the fox might run constantly a little to the west of the rabbit, forcing the rabbit to turn east more often than it turns west. The rabbit in turn may risk being caught by veering west more frequently than it would otherwise, trying to keep away from the open field. It can be seen that both the rabbit and the fox choose actions to maximize the probability of winning, with each anticipating the effect of its actions on the other. This is the type of situation studied in game theory.

How important is game theory? It is central to understanding life in all its varied forms. This may sound excessive, but one must step back from this interaction between a rabbit and a fox to ask more basic questions. For example, why are rabbits bilaterally symmetrical about the axis along which their movement is most rapid and energy-efficient (left leg and right leg symmetrically placed and equally strong, left eye and right eye symmetrically placed and of equal size and discriminating capacity, and single external body parts such as the nose and tail arrayed along the axis of movement)? The answer is that if rabbits had strength biased to the right, it would be easier for them to jump left than jump right, and that would give an advantage to their natural predators, the foxes. Foxes are bilaterally symmetrical for similar reasons. Game theory thus explains important facts about life that otherwise appear arbitrary and incomprehensible.

This simple game theoretic argument explains a major fact about the organization of life. Animals that run to escape predators or capture prey have body forms that are bilaterally symmetrical about a vertical axis along the direction of their most rapid motion. This applies to most animals and fish but not to plants, which do not run and are radially symmetrical, or to squid, octopuses, and other sea creatures whose primary motion is up and down.

To avoid the conclusion that game theory deals only with conflict, one can consider an example that is called the Cooperation Game. A group of ten hunters in a village spread out in the jungle every day to look for large game. They hunt individually, climbing tall trees and waiting quietly and attentively for long hours until the prey appears. At the end of the day the hunters share the day's kill equally. Of course, each hunter could spend the day sleeping in a tree. Suppose that by working each hunter adds an average of 3,000 calories to the total kill, of which his share is 300, but expends 500 calories of energy hunting as opposed to sleeping. A selfish hunter thus will sleep rather than hunt, saving 200 calories but costing the other group members 2,700 calories. This is a game in which there are n players and each player (i.e., each hunter) has two actions: Work or Shirk. If m hunters Work, the Shirkers' payoff is 3,000m/n each, whereas the Workers' payoff is 3,000m/n -500.

A best response of a player in a game can be defined as a strategy (in this case an action) that maximizes that player's payoff in light of the strategies of the other players. It is easy to see that a self-interested player's best response in this game is to Shirk no matter what the other players do. A Nash equilibrium of a game is defined as a choice of strategies made by the players such that each is a best response to the other players' choices. It is clear that in the Cooperation Game there is only one Nash equilibrium, in which everyone shirks (m = 0) and no one eats.

Suppose another rule is added to the game. If a hunter is caught shirking, he is punished by being prohibited from hunting and sharing the kill for two days. Further, suppose the probability of being caught shirking is 0.50. To see that having everyone hunt is now a Nash equilibrium, one must decide whether a single hunter in a group of ten could do better by shirking and risking getting caught. The hunter saves 200 calories by shirking, but half the time he is caught and then loses two days' payoff, which is 5,400 calories. Thus, that hunter loses an average of 2,500 calories a day by shirking, and so his best response to hunt with the others. The conclusion is that with this new punishing mechanism full cooperation by each hunter becomes a Nash equilibrium.

History and Analytics of Game Theory

Game theory presupposes rational choice theory because it assumes that players have rational preferences in regard to the game's outcomes. It also presupposes rational decision theory because choice under conditions of uncertainty is the rule in most game situations. Because rational choice theory and decision theory were codified only in the late twentieth century, it is not surprising that game theory is still an incomplete and rather underdeveloped science. Before about 1950 games were assumed to be zero-sum; that means that what one player loses, the other player wins. The rabbit-fox game described earlier is zero-sum, but the hunter game is not because with the proper strategies all the hunters gain by cooperating.

With the zero-sum assumption cooperation never leas to a gain, and this would undercut some of the major contributions of game theory to the understanding of cooperation in biology and economics. Moreover, the three mathematicians who developed game theory—Ernst Zermelo (1871–1953), Stefan Banach (1892–1945), and John von Neumann (1903–1957)—assumed that each player will choose a strategy that minimizes the maximum gain for an opponent. This so-called minimax analysis cannot be extended to more general strategic contexts.

Modern game theory was born in 1950 after the publication of a paper by the young Princeton mathematician John F. Nash, Jr. (b. 1928; winner of a Nobel Prize in economics in 1994), who introduced the novel idea of a game equilibrium as a set of mutual best responses. The central term in modern game theory, the Nash equilibrium, acknowledges his work. Several conceptual problems had to be cleared up before game theory could attain a central position in the behavioral sciences. In 1965 Reinhard Selten (b. 1930; winner of a Nobel Prize in economics in 1994) developed the concept of equilibrium refinement, which showed why certain Nash equilibria are likely to be of empirical relevance and others are not. In 1967 and 1968 John Harsanyi (1920–2000; winner of a Nobel Prize in economics in 1994) showed how to apply game theory when the players have incomplete knowledge of the other players and the payoffs.

Until the 1980s it was believed by many people that game theory could be applied only to highly intelligent, so-called rational players because an analysis of the best responses is intellectually demanding. However, in 1972 the biologist John Maynard Smith (1920–2004) applied game theoretic notions to explaining animal conflict, a process that culminated in his publication of Evolution and the Theory of Games (1982). The innovation here is the idea that evolution can provide an alternative to high-level mental reasoning. For instance, rabbits veer optimally when chased by foxes not because each rabbit logically compares and empirically tests the alternatives but because running behavior is encoded in a rabbit's genes and those genes which render the rabbit most capable of eluding the fox are favored by natural selection in successive generations of rabbits. Inefficient genes simply become fox food.

The Ultimatum Game and Altruistic Preferences

An example of such research is the ultimatum game, in which under conditions of complete anonymity two players separately are shown a sum of money, say, $10. One of the players, called the proposer, is instructed to offer any number of dollars from $1 to $10 to the second player, who is called the responder. The proposer can make only one offer, and the game is never repeated with the same players facing each other. The responder can accept or reject this offer. If the responder accepts the offer, the money is shared accordingly. If the responder rejects the offer, both players receive nothing.

If the responder cares only about her own payoff in the game (it is said that she is self-regarding in this case) and the proposer knows or supposes this, the proposer will make the responder the minimum offer of $1, the responder will accept, and the game will be over. However, when the game actually is played, the self-regarding outcome almost never is attained or even approximated. In fact, as many replications of this experiment have documented, under varying conditions and with varying amounts of money, proposers routinely offer responders very substantial amounts (50 percent of the total generally is the modal offer) and responders frequently reject offers below 30 percent (Camerer 2003).

Are these results culturally dependent? Do they have a strong genetic component, or do all "successful" cultures transmit similar values of reciprocity to individuals? Alvin Roth (Roth, Prasnikar, Okuno-Fujiwara, and Zamir 1991) conducted ultimatum games in four different countries (the United States, Yugoslavia, Japan, and Israel) and found that although the level of offers differed slightly in different countries, the probability of an offer being rejected did not. This indicates that both proposers and responders have the same notion of what is considered fair in that society and that proposers adjust their offers to reflect that common notion. The differences in the levels of offers across countries were relatively small.

This ultimatum game result, along with that of many other similar games, suggests that many human subjects are strong reciprocators. Strong reciprocators come to strategic interactions with a propensity to cooperate (altruistic cooperation), respond to cooperative behavior by maintaining or increasing their level of cooperation, and responds to noncooperative behavior by punishing the "offenders" even at a cost to themselves and even when they cannot reasonably expect future personal gains to flow from the imposition of such punishment (this is called altruistic punishment).

Behavior in the ultimatum game thus conforms to the strong reciprocity model: Fair behavior in the ultimatum game among college students is a fifty-fifty split. Responders reject offers under 40 percent as a form of altruistic punishment of a norm-violating proposer. Proposers offer 50 percent because they are altruistic cooperators or 40 percent because they fear rejection. To support this interpretation it can be noted that if the offers in an ultimatum game are generated by a computer rather than by the proposer and if the respondents know this, low offers very rarely are rejected (Blount 1995). Moreover, in a variant of the game in which a responder's rejection leads to the responder getting nothing but allows the proposer to keep the share she suggested for herself, responders infrequently reject offers and proposers make considerably smaller offers.

The strong reciprocator is not a representative of one of the types of human nature found in traditional political philosophy. A strong reciprocator thus is neither the selfless altruist of utopian theory in the tradition of Jean-Jacques Rousseau (1712–1778) or that of Karl Marx (1818–1883) nor the selfish hedonist found in traditional economics and described by the economist Adam Smith (1723–1790) in The Wealth of Nations (1776). Such a person is a conditional cooperator whose penchant for reciprocity can be elicited in circumstances in which pure selfishness would dictate a different action. Indeed, the strong reciprocator is more akin to the empathetic individual found in Adam Smith's other important work, The Theory of the Moral Sentiments (1759) except that Smith there emphasizes the sweet side of human nature, playing down the willingness to punish transgressions that is uncovered routinely in behavioral games.

Social Dilemmas

Another important behavioral game that sheds light on human nature and increases people's understanding of human social interaction is the social dilemma. A social dilemma is a group interaction in which all the players benefit if they all cooperate but each individual has an incentive to shirk and benefit from the cooperation of others.

An experimental representation of a social dilemma is the so-called public goods game. A typical public goods game consists of a number of rounds, say, ten. In each round each subject is grouped with several other subjects, say, three others. Each subject is given a certain amount of money, say, $20. Each subject, unseen by the others, then places a fraction of his or her money in a common account and puts the remainder in his or her private account. The experimenter then tells the subjects how much was contributed to the common account adds to the money in the common account enough so that, when divided among the four players, the private account of each subject can be increased by a fraction, say, 40 percent, of the players' original contribution to in the common account. Thus, if a subject contributes his or her whole $20 to the common account, the experimenter adds an additional $12, so each of the four group members will receive ($20 + $12)/4 = $8 at the end of the round. In effect, by putting the whole endowment into the common account, a player loses $12 and the other three group members gain in total $24 (= $8 × 3).

A self-regarding player will contribute nothing to the common account. However, only a fraction of subjects conform to the self-regarding model. The subjects begin by contributing on average about half of their endowments to the public account. The level of contributions decays over the course of the ten rounds until in the final rounds most players behave in a self-regarding manner. This is exactly what is predicted by the strong reciprocity model. Because they are altruistic contributors, strong reciprocators start out by contributing to the common pool, but in response to the norm violation on the part of the self-regarding types they begin to refrain from contributing.

How can it be known that the decay of cooperation in the public goods game is due to cooperators punishing free riders by refusing to contribute? Subjects often report this behavior retrospectively. More compelling, however, is the fact that when subjects are given a more constructive way of punishing defectors, they use it in a manner that helps sustain cooperation. For instance, Ernst Fehr and Simon Gächter (2000) set up an experimental situation in which the possibility of punishment for personal gain was removed completely. They used six- and ten-round public goods games with groups of four and with costly punishment allowed at the end of each round, employing three different methods of assigning members to groups.

They found that when costly punishment is permitted, cooperation does not deteriorate; indeed, if the same players stay together for the whole session, despite strict anonymity cooperation increases almost to full cooperation even on the final round. In effect, even though the groups had some selfish players, there was a sufficiently large fraction of strong reciprocators to ensure that it was not in the interest of the selfish to act selfishly.

The Epistemological Foundations of Game Theory

One can characterize the choice situation facing an agent in terms of its level of complexity. The least complex situation occurs when an agent must choose from a set of fixed alternatives. Analytically complete axiomatic models of choice in this situation are well developed and empirically successful. Of intermediate complexity is a situation in which an agent must choose from a set of alternatives, each of which is a probability distribution over determinate outcomes. Analytically complete axiomatic models of choice in this situation are also well developed and empirically successful, although some important anomalies in human behavior have been noted in regard to decision theory. The most complex situation is the one described by game theory: An agent's choices affect not only that agent but other agents as well, the other agents also are engaged in making choices that affect themselves and others, and all agents take into account the strategic nature of their interactions. One of the most widely known attempts to illustrate such a game theoretic situation is the Prisoner's Dilemma.

It would be gratifying to have a fully successful analytical model of strategic interaction applicable to the highly complex level, but despite the efforts of theoreticians since the second half of the twentieth century, none exists. Ignoring the Prisoner's Dilemma for now, one can consider three simple games that dramatize the problems in developing such a theory, which then can be used to outline some important contributions to the epistemological underpinnings of game theory.

EVEN-ODD GAME. The first is the simple Even-Odd game. This game has two players, each of whom can show either one finger (One) or two fingers (Two). The two players show their fingers simultaneously, with player 1 winning if his choice matches that of the other player (i.e., if One-One or Two-Two occurs) and player 2 winning if her choice does not match it (i.e., if One-Two or Two-One occurs). Figure 1 shows the normal form of this game (the normal form specifies the moves that each player can make and the payoffs for each player as a function of the moves of both players).

This game obviously has no Nash equilibria in the "pure" strategies: One and Two. However it does have a unique Nash equilibrium in which each player plays One with probability 1/2 and plays Two with probability 1/2. Doubtless many people remember this solution from schoolyard days, when they learned to "mix up" their choices so that an opponent could not discover their next move. The problem is that this game is played just once (it is a one-shot game). Hence, if a player's opponent randomizes as suggested by the Nash equilibrium, it does not matter what the first player does: The expected payoff is zero whether the first player chooses One, Two, or a probability distribution over One and Two. However, the same is true for the opponent. Therefore, there is no reason for either player to randomize, yet that is the solution suggested by game theory.

An important step toward dealing with this problem is to note that each player chooses a best response not to the actual strategy of the other players but to his or her own conjecture about what the other players will do. Robert Aumann and Adam Brandenburger (1995) prove the following theorem for a two-player game. Suppose ?₁ is player 1's conjecture concerning player 2's strategy and ?₂ is player 2's conjecture concerning player 1's strategy. If both players know each other's conjectures and each knows that the other is rational (i.e., chooses a best response to his or her conjecture), (?₂,?₁) is a Nash equilibrium.

BATTLE OF THE SEXES. This is a fine solution for Odd or Even, which has only one Nash equilibrium. However, one must consider another famous game, the Battle of the Sexes, which is depicted in Figure 2. In this game Rowena and Colin love each other and get one point by being together. However, Rowena loves the ballet and Colin loves gambling. Each gets a point for attending his or her favorite event. Thus, if both go to the opera, Rowena gets 2 and Colin gets 1, whereas if they both go gambling, Colin gets 2 and Rowena gets 1. Moreover, when they are not together, it is assume that they are so unhappy that each gets zero. It is easy to find two Nash equilibria: Both go gambling, and both go to the opera. It turns out that there is also a third Nash equilibrium in which each party goes to his or her favorite place with probability 2/3 and to the other's favorite place with probability 1/3. This is called a mixed strategy equilibrium.

To see that One gambling with probability 2/3 and Two gambling with probability 1/3 is a Nash equilibrium, one should note that the expected payoff to One from gambling equals 2 × 1/3 + 0 × 2/3 = 2/3, whereas the expected payoff to One from ballet equals 0 × 1/3 + 1 × 2/3 = 2/3. Because these probabilities are equal, One can do no better than his probability 2/3 gambling, probability 1/3 ballet strategy, and a similar argument holds for Two.

In the case of Battle of the Sexes it is unreasonable to posit that each player knows the other's conjecture because there is no way of explaining how this mutual knowledge would have come about. Indeed, it is not even plausible to suppose that the players have conjectures concerning what the other will do unless there is more to the social situation than has been explained. Moreover, the players still have no incentive to play according to their partners' conjectures (Binmore 1988).

The problem becomes even more implausible when there are more than two players. In this case Aumann and Brandenburger (1995) show that if all players assign the same probability distribution to player types, it is known mutually that all players are rational (i.e., choose best responses), and the players' conjectures are commonly known, these conjectures form a Nash equilibrium. One says that a fact is commonly known if all players know the fact, all know that the others know it, all know that all know that the others know it, and so on (Lewis 1969).

CENTIPEDE GAME. There are simple games in which the very notion of rationality and the adequacy of the concept of the Nash equilibrium are brought into question. Consider, for instance, the Centipede Game. The players, Mutt and Jeff, start out with $2 each, and they alternate rounds. On the first round Mutt can defect (D) by stealing $2 from Jeff, and the game is over. Otherwise Mutt cooperates (C) by not stealing and receives an additional $1. Then Jeff can defect (D) and steal $2 from Mutt, and the game is over, or he can cooperate (C) and receive an additional $1. This continues until one player or the other defects or until each player has $100. The game tree is illustrated in Figure 3.

This game has only one Nash equilibrium outcome, in which Mutt defects on the first round. To see this, let round k be the first round in which either player defects in a Nash equilibrium. If k > 1, the other player's best response is to defect on round k - 1. Of course, common sense indicates that this is not the way real players would act in this situation, and empirical evidence corroborates this (McKelvey and Palfrey 1992). People in this game will cooperate up to round 90 and beyond before considering defecting.

It would be difficult to fault players for not being rational in this case because they do much better playing the way they do rather than the way dictated by the Nash equilibrium concept. The concept of rationality is problematized for the following reason: If Jeff believes Mutt is rational, Jeff will defect in round 2. This is why Mutt defects in round 1. But suppose Mutt cooperates in round 1. Then Jeff will recognize that his assumption concerning Mutt must be false. Jeff probably will say to himself, "I don't know what strategy Mutt is using, but since he cooperated once, perhaps if I cooperate now, Mutt will cooperate a second time." Thus, Jeff will tend to cooperate in round 2. Now Mutt, who is very smart, can foresee what Jeff will be thinking and hence will cooperate even if he is rational. One can conclude that agents who use best responses will not play the Nash equilibrium in this game. It is easy to see the problem by referring to the analysis of Aumann and Brandenburger (1995): The two players do not know each other's conjectures.

Evolutionary Game Theory

To this point the focus has been on so-called classical game theory, which depicts the strategic interaction among a number of rational agents. The interaction is socially disembodied, with the agents having neither history nor substance outside this particular interaction. All socially relevant aspects of the actors must be captured by their beliefs and conjectures, which are totally disembodied and unmotivated. A similar degree of social minimality has given rise to powerful models of decision making when strategic interaction is absent, as in rational choice theory and decision theory. However, this does not extend to game theory, in which a more socially embedded approach is needed to derive plausible results.

The most promising alternative foundation for strategic interaction is known as evolutionary game theory (Maynard Smith 1982, Samuelson 1997, Gintis 2000). The central actors in evolutionary game theory are not players but strategies. Suppose a group of agents periodically plays a certain classical game G. One assumes a large population of agents, each of whom adopts a particular strategy in playing G. One does not assume that the strategies represented in the population are in any way optimal, although one does assume that there is enough random variation and mutation across time that all pure strategies are represented.

In each period agents from the population are assigned randomly to play G. Their scores are tallied, and the change in the population over time is governed by an evolutionary dynamic in the sense that agents whose strategies are very successful tend to be copied by agents whose strategies are less successful. Thus, the population ecology of strategies moves over time in accordance with the notion of survival of the fittest. This is called a replicator dynamic (Hofbauer and Sigmund 1998, Gintis 2000).

The fundamental theorem of evolutionary game theory is that every equilibrium point of an evolutionary dynamic is a Nash equilibrium. This provides a justification for the concept of the Nash equilibrium without the need for the epistemological assumptions of classical game theory. Moreover, evolutionary game theory shows that many Nash equilibria of classical game theory are not evolutionarily stable and thus cannot explain observable social behavior.

A case in point is the Centipede Game described earlier in this entry. The author of this entry has created a computer program to simulate the evolution of behavior in the Centipede Game (this is called an agent-based simulation). The author created a population of 200 agents, each supplied with a strategy sk of the following form: "cooperate until round k, then defect." Initially, these strategies are assigned randomly to the agents, and they play 300,000 rounds, with a mutation rate of 0.001 (a mutant assumes a random strategy sk where 1 ? k ? 101). The results are shown in Figure 4.

It can be seen that cooperation quickly increases until after only a few rounds the average payoff is more than 95. Then cooperation erodes, as might be expected, until the average payoff dips below 80. At that point a pair of agents who choose strategies near k = 100 do very well, and those strategies grow at the expense of the strategies that involve defection on rounds near k = 80. Cooperation shoots back up to nearly perfect. The cycle repeats for 300,000 rounds and shows no signs of changing its basic character.

Even though the only Nash equilibrium of the stage game uses strategy s₁, it can be seen that the evolutionary dynamic never remotely approaches this equilibrium. This is the case because the Nash equilibrium involves such poor payoffs that even a small number of mutant players can invade a population of all-defectors, and the system quickly ramps up to almost full cooperation (changing the mutation rate does not alter this result). Thus, evolutionary game theory shows that the behavior observed when people play the Centipede Game is easy to model in a dynamic framework.

Game Theory and Ethics

Game theory has been applied to ethical theory by John Harsanyi (1992). Harsanyi (1920–2000; winner of a Nobel Prize in economics in 1994) develops a theory of justice very close to that of the philosopher John Rawls (1921–2002) and shows that it can be derived from basic game theoretic reasoning. Other important contributions to the game theoretic analysis of ethics include those of Brian Skyrms (1996) and Ken Binmore (1998).

Perhaps the first indication that game theory would be important to ethical theory was the famous tit-for-tat computer competition run by Robert Axelrod (Axelrod and Hamilton 1981). Axelrod asked what a successful strategy in the repeated Prisoner's Dilemma might look like. In that game the dominant strategy is to defect. However, if the game is repeated several times, players may be able to use the threat of defecting in the future to induce their partners to cooperate in the present.

Axelrod recruited fourteen game theorists from economics, mathematics, and the behavioral and computer sciences to submit computerized strategies for playing 200 rounds of the Prisoner's Dilemma. Those strategies were paired with each other in a round robin tournament with the result that the absolutely simplest strategy won. This strategy was tit-for-tat, supplied by Anatol Rapoport, a mathematician at the University of Toronto. Tit-for-tat cooperates on the first move and then does whatever its partner did on the previous move. Tit-for-tat is thus a simple reciprocity enforcer, cooperating when its partner cooperates and defecting when its partner defects.

After publishing these results (Axelrod and Hamilton 1981) Axelrod decided to stage a second tournament. More than sixty researchers from six countries submitted new programs, many of which were aimed explicitly at defeating tit-for-tat. Nevertheless, tit-fortat again won handily.

This result relates to ethical theory because it shows the success of a strategy that is nice (never defect first), punishing (always retaliate against a defector), and forgiving (always revert to cooperating if your partner cooperates). These responses, of course, represent three important ethical principles. A fourth common ethical principle—always turn the other cheek—certainly would not fare well in this encounter, as it would be beaten by any program that could detect "wimps" (those who do not punish) and defect consistently in playing against them.

It is clear that the ethical principles behind the strong reciprocity associated with social dilemmas represent a higher development of tit-for-tat. Whereas tit-for-tat applies only to dyadic relationships, strong reciprocity applies to n-player social dilemmas.

HERBERT GINTIS

SEE ALSO Artificial Morality;Choice Behavior;Decision Theory;Rational Choice theory.

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