Published in Behavioral and Brain Sciences
Volume 26, Number 2: 139-153 (April 2003)
© 2003 Cambridge University Press
Archaeology and cognitive evolution
Cooperation, psychological game theory, and limitations of rationality in social interaction
Andrew M. Colman
School of Psychology, University of Leicester, Leicester LE1 7RH, United Kingdom
amc@le.ac.uk www.le.ac.uk/home/amc
Abstract: Rational choice theory enjoys unprecedented popularity and influence in the behavioral and social sciences, but it generates intractable problems when applied to socially interactive decisions. In individual decisions, instrumental rationality is defined in terms of expected utility maximization. This becomes problematic in interactive decisions, when individuals have only partial control over the outcomes, because expected utility maximization is undefined in the absence of assumptions about how the other participants will behave. Game theory therefore incorporates not only rationality but also common knowledge assumptions, enabling players to anticipate their co-players’ strategies. Under these assumptions, disparate anomalies emerge. Instrumental rationality, conventionally interpreted, fails to explain intuitively obvious features of human interaction, yields predictions starkly at variance with experimental findings, and breaks down completely in certain cases. In particular, focal point selection in pure coordination games is inexplicable, though it is easily achieved in practice; the intuitively compelling payoff-dominance principle lacks rational justification; rationality in social dilemmas is self-defeating; a key solution concept for cooperative coalition games is frequently inapplicable; and rational choice in certain sequential games generates contradictions. In experiments, human players behave more cooperatively and receive higher payoffs than strict rationality would permit. Orthodox conceptions of rationality are evidently internally deficient and inadequate for explaining human interaction. Psychological game theory, based on nonstandard assumptions, is required to solve these problems, and some suggestions along these lines have already been put forward.
Keywords: backward induction, Centipede game, common knowledge, cooperation, epistemic reasoning, game theory, payoff dominance, pure coordination game, rational choice theory, social dilemma
Were such things here as we do speak about?
Or have we eaten on the insane root
That takes the reason prisoner?
Macbeth (I.iii.84)
1. Introduction
Is rational social interaction possible? This may seem a surprising question, given the Apollonian flavor of the contemporary behavioral and social sciences. Rational choice theory (RCT) is the cornerstone of neoclassical economics (Arrow et al., 1996; Elster, 1986; Sugden, 1991b). In political science, RCT began to mushroom after the publication of Social Choice and Individual Values (Arrow, 1963) and transformed the discipline within a few decades (Friedman, 1996; Green & Shapiro, 1994; Ordeshook, 1986). In sociology, Weber’s (1922) analyses of law and economics as models of rationality prepared the ground for the germination of RCT ideas half a century later (Abell, 1991; Coleman & Fararo, 1992; Hollis, 1987; Moser, 1990). Theories of behavioral ecology (Dawkins, 1989; Krebs & Davies, 1987), and particularly the evolution of social behavior (Maynard Smith, 1984), were revolutionized by the introduction of RCT-based game theory in the early 1970s (Maynard Smith & Price, 1973), and even jurisprudence has been influenced by RCT (Raz, 2000).
1.1. Rationality in psychology
In psychology, the picture is admittedly more complex. Since the publication of Freud’s earliest metapsychological writings, and in particular his adumbration of the distinction between two principles of mental functioning, the reality principle and the pleasure principle (Freud, 1911), only the first of which he believed to be functionally rational, psychologists have paid particular attention to irrational aspects of thought and behavior. But psychologists have generally assumed, usually tacitly, that rationality is normal, whereas irrationality is, in some sense, abnormal or pathological.1
1.2. Bounded rationality
This article is not concerned merely with the accuracy of RCT in predicting human behavior. The concept of bounded rationality (Simon, 1957) has been widely accepted and corroborated by experimental evidence. Our bounded rationality obliges us to use rough-and-ready rules of thumb (heuristics) that can lead to predictable errors and judgmental biases, many of which have been investigated empirically (Bell, Raiffa, & Tversky, 1988; Kahneman, Slovic, & Tversky, 1982), but that allow us to solve problems quickly and efficiently (Gigerenzer & Goldstein, 1996; Gigerenzer, Todd, & the ABC Research Group, 1999). For example, a simple rule of win-stay, lose-change can lead to the evolution of mutually beneficial cooperation in a group ofplayers who are ignorant not only of the payoff structure of the game but even of the fact that they are involved with other players in a strategic interaction (Coleman, Colman, & Thomas, 1990).
1.3. Evolutionary game theory
Game-theoretic equilibrium points can thus be arrived at by entirely non-rational evolutionary processes. The basic concepts of game theory can be mapped to the elements of the theory of natural selection as follows. Players correspond to individual organisms, strategies to organisms’ genotypes, and payoffs to the changes in their Darwinian fitness–the numbers of offspring resembling themselves that they transmit to future generations. In evolutionary game theory interpreted biologically, the players do not choose their strategies (genotypes) rationally or even deliberately, but different profiles of strategies lead to different payoffs, and natural selection mimics deliberate choice. Maynard Smith and Price (1973) introduced the concept of theevolutionarily stable strategy(ESS) to handle such games. It is a strategy with the property that if most members of a population adopt it, then no mutant strategy can invade the population by natural selection, and it is therefore the strategy that we should expect to see commonly in nature. An ESS is invariably a Nash equilibrium (see 5.1 below), and therefore a type of game-theoretic solution, but not every Nash equilibrium is an ESS.
Evolutionary game theory deals with social as well as biological evolution. It has been studied intensively since the 1970s, and the theory is well understood (Hofbauer & Sigmund, 1998; Samuelson, 1997). Even purely analytic studies can solve problems and provide useful insights. A simple example with psychological relevance is an evolutionary model of Antisocial Personality Disorder, based on a multi-player Chicken game, that provided an explanation for the low but stable prevalence of this disorder in widely diverse societies (Colman & Wilson, 1997; see also Colman, 1995b). Evolutionary games have also been studied empirically (Maynard Smith, 1984), and above all computationally, by running strategies against one another and transmitting copies of these strategies to future generations according to their accumulating payoffs (Axelrod, 1984, 1997, chaps. 1, 2; Nowak, May, & Sigmund, 1995; Nowak & Sigmund, 1993). Evolutionary game theory deals with non-rational strategic interaction driven by mindless adaptive processes resembling trial and error, and it is therefore not directly relevant to this article. Populations of insects, plants, and even computer programs can evolve to game-theoretic equilibrium points, and cooperation can evolve without rational decision making.This article, however, focuses on whether full rationality can be applied to social interaction.
1.4. Outline of the argument
When a decision involves two or more interactive decision makers, each having only partial control over the outcomes, an individual may have no basis for rational choice without strong assumptions about how the other(s) will act. This complicates the picture and leads to problems, in some cases even to the breakdown of the standard concept of rationality.2 This article brings together a heterogeneous and disparate collection of arguments and evidence suggesting that rationality, conventionally defined, is not characteristic of human social interaction in general.
The remainder of the article is organized as follows. Section 2 outlines the nature of rationality and its formalization. Section 3 focuses more specifically on game theory, and section 4 on game theory’s underlying assumptions. Section 5 argues that selection of focal points and of payoff-dominant equilibria is inexplicable according to game-theoretic rationality, and that a key solution concept for cooperative games is not always applicable. Section 6 is devoted to social dilemmas, in which rationality is self-defeating and human decision makers are paradoxically more successful than ideally rational agents. Section 7 deals with backward induction in sequential games where the standard concept of rationality appears incoherent. Section 8 introduces psychological game theory and outlines some nonstandard contributions designed to overcome these problems. Finally, section 9 draws the threads together and summarizes the conclusions.
2. Nature of rationality
What is rationality? Broadly speaking, it involves thinking and behaving reasonably or logically, and it comes in several guises (Manktelow & Over, 1993). Rational beliefs are those that are internally consistent,3 and rational arguments are those that obey the rules of logic. Rational preferences and decisions require more detailed explanation.
2.1 Rational preferences
Suppose a universe of alternatives includes a subset A of alternatives that are available in a particular decision context. Decision theorists generally assume that an agent’s rational preferences obey the following conditions.
(1) Completeness: For every pair of alternatives ai and aj in A, the agent either prefers ai to aj, or prefers aj to ai, or is indifferent between ai and aj.
(2) Transitivity: Given alternatives ai, aj, and ak in A, an agent who considers ai to be at least as preferable as aj, and aj at least as preferable as ak, considers ai to be at least as preferable as ak.
(3) Context-free ordering: If an agent who considers ai to be at least as preferable as aj in A, then that agent considers ai to be at least as preferable as aj in an enlarged set A' containing all the elements in A plus additional elements from the universe of alternatives.
These three conditions are collectively called the weak ordering principle (McClennen, 1990, chap. 2). We had to begin with a subset A to give meaning to the third condition, which would otherwise be implied by the first.4 Given preferences that satisfy this tripartite principle, a rational decision maker always chooses a maximally preferable alternative (which may not be unique, hence “a” rather than “the”). The formalization of this in expected utility theory will be discussed in 2.3 to 2.6 below. Experimental evidence suggests that human decision makers frequently violate the second and third conditions (Doyle, et al., 1999; Huber, Payne, & Puto, 1982; Slovic & Lichtenstein, 1983; Tversky, 1969), although they tend to modify their intransitive preferences, at least, when their violations are pointed out to them.
2.2. Rational decisions
Rational decisions or choices are those in which agents act according to their preferences, relative to their knowledge and beliefs at the time of acting. This is instrumental rationality (or means-end rationality), and it can be traced back to the Scottish Enlightenment writings of Hume (1739-1740/1978) and Smith (1776/1910). Hume gave the most frequently quoted account of it in his Treatise of Human Nature (2.III.iii):
Reason is, and ought only to be the slave of the passions, and can never pretend to any other office than to serve and obey them. . . . A passion can never, in any sense, be call’d unreasonable, but when founded on a false supposition, or when it chuses means insufficient for the design’d end. (pp. 415-416)
Hume conceived of reason as a faculty affording the means for achieving goals that are not themselves afforded by reason. Russell (1954) summed this up lucidly: “‘Reason’ has a perfectly clear and precise meaning. It signifies the choice of the right means to an end that you wish to achieve. It has nothing whatever to do with the choice of ends” (p. 8).
2.3. Expected utility theory
Formally, decisions that maximize expected utility (EU) are rational decisions. The theory of EU was first presented in axiomatic form by von Neumann and Morgenstern (1947) in an appendix to the second edition of Theory of Games and Economic Behavior. It is based on the weak ordering principle (see 2.1 above), extended to gambles or lotteries among outcomes. It is assumed that a player can express a preference or indifference not only between any pair of outcomes, but also between an outcome and a gamble involving a pair of outcomes, or between a pair of gambles, and that the weak ordering principle applies to these preferences also.
This necessitates a further assumption, called the independence principle (McClennen, 1990, chap. 3). If g1, g2, and g3 are any three gambles, and 0 < p £ 1, then g1 is preferred to g2 if and only if a gamble involving g1 with probability p and g3 with probability 1 – p is preferred to a gamble involving g2 with probability p and g3 with probability 1 – p. From this independence principle, together with the weak ordering principle, it is possible to define a function u(g) that assigns a numerical expected utility to every outcome and gamble, such that the expected utility of a gamble is equal to the sum of the utilities of its components, weighted by their probabilities. It can then be proved that agents who maximize u(g) are acting according to their preferences and are thus manifesting instrumental rationality. Von Neumann-Morgenstern utilities are measured on an interval scale, with an arbitrary zero point and unit of measurement, like temperature measured on a Fahrenheit or Celsius scale, and are therefore unique up to a strictly increasing linear (affine) transformation. This means that two utility scales u and u¢ represent the same preferences if a and b are arbitrary constants, a > 0, and u¢ = au + b. It follows that maximizing u¢ is equivalent to maximizing u. Harless and Camerer (1994) and Starmer (2000) have comprehensively reviewed EU theory and several alternative “non-expected utility” theories and related empirical findings (see also Camerer, 1995; Fishburn, 1988; Frisch & Clemen, 1994; Hey & Orme, 1994; Lea, Tarpy, & Webley, 1987; Machina, 1987, 1991; Sosa & Galloway, 2000; Taylor, 1996).
2.4. Subjective expected utility theory
In Bayesian game theory (initiated by Harsanyi, 1967-1968), expected utilities are based on subjective probabilities rather than objective relative frequencies, and what is maximized is subjective expected utility (SEU). In SEU theory, utilities obey the axioms formulated by Savage (1954)5 or one of the alternative axiom systems that have been proposed. Savage built on the axioms of von Neumann and Morgenstern (1947), who introduced the independence principle, and Ramsey (1931), who showed how to define subjective probabilities in terms of preferences among gambles. Rational decisions are those that maximize EU, whether objective or subjective.
2.5. Utility maximization
Utility maximization has a straightforward interpretation in individual decision making. The choice of alternative ai is rational if the possible alternatives are a1, ..., am, and there are foreseeable outcomes c1, ..., cm, such that a1 leads reliably to c1, ..., and am leads reliably to cm, and no outcome has a higher utility for the decision maker than ci. A choice is thus rational if no alternative yields a preferable outcome.
2.6. Expected utility maximization
According to the theory of revealed preference, popular with economists, a person who is observed to choose alternative ai, and to reject aj is said to have revealed a preference of ai over aj and a higher utility for ai than aj, and choice behavior therefore maximizes expected utility tautologically.6 If chance plays a part, and the choice of ai leads not to a definite outcome ci but to a foreseeable probability distribution over the set of outcomes, then a decision maker who chooses an alternative that maximizes the weighted average expected utility (EU) is acting rationally. But if the decision is interactive and the outcome is determined by two or more decision makers, then the interpretation of instrumental rationality is unclear because, except in special cases, an individual cannot maximize EU in any obvious way. In interactive decisions, EU maximization is undefined without further assumptions.
3. Game theory
The necessary assumptions are provided by game theory, the framework within which interactive decisions are modeled. This is a mathematical theory applicable to any social interaction involving two or more decision makers (players), each with two or more ways of acting (strategies), such that the outcome depends on the strategy choices of all the players, each player having well-defined preferences among the possible outcomes, enabling corresponding von Neumann-Morgenstern utilities (payoffs) to be assigned. The definition is inclusive, embracing as it does a wide range of social interactions.
3.1. Abstraction and idealization
A game is a mathematical abstraction functioning as an idealization of a social interaction. An actual interaction is invariably too complex and ephemeral to be comprehended clearly; thus it is replaced by a deliberately simplified abstraction in which the rules and basic elements (players, strategies, payoffs) are explicitly defined, and from which other properties can be inferred by logical reasoning alone. These inferences apply to the idealized game, not directly to the social interaction that it purports to model, and they are valid, provided that the reasoning is sound, whether or not the game models the original interaction accurately. But if it does not, usually because of faulty judgments about which features to ignore, then its relevance and usefulness are limited. To be both relevant and useful, a game must incorporate the important properties of the interaction and must also generate inferences that are not obvious without its help.
3.2. Normative theory
The primary objective of game theory is to determine what strategies rational players should choose to maximize their payoffs. The theory is therefore primarily normative rather than positive or descriptive. The founding game theorists stated this explicitly (von Neumann, 1928, p. 1; von Neumann & Morgenstern, 1944, pp. 31-33). So did Luce and Raiffa (1957), when they introduced game theory to social scientists:
We feel that it is crucial that the social scientist recognize that game theory is not descriptive, but rather (conditionally) normative. It states neither how people do behave nor how they should behave in an absolute sense, but how they should behave if they wish to achieve certain ends. (p. 63, italics in original)
3.3. Positive theory
If game theory were exclusively normative, then it would have limited relevance to the (empirical) behavioral and social sciences, because a normative theory cannot be tested empirically, and evolutionary game theory (see 1.3 above) would be pointless. Arguably, game theory becomes a positive theory by the addition of a bridging hypothesis of weak rationality, according to which people try to do the best for themselves in any given circumstances. To err is human, and deviations from perfect rationality are inevitable, because of computational limitations or bounded rationality (see 1.2 above), incomplete specification of problems (Berkeley & Humphreys, 1982; Dawes, 2000), or systematic irrationality (Stanovich & West, 2000).7 But none of this is inconsistent with the hypothesis that people try to act rationally. The addition of this hypothesis provides game theory with a secondary objective, to make testable predictions, and this justifies the thriving enterprise of experimental gaming.
The literature of experimental gaming (reviewed by Colman, 1995a; Kagel & Roth, 1995, chaps 1-4; Pruitt & Kimmel, 1977) testifies to the fruitfulness of empirical research within a broadly game-theoretic framework. Some important phenomena, such as the clash between individual and collective rationality (see section 6 below), cannot even be formulated clearly without the conceptual framework of game theory.
4. Standard assumptions
To give meaning to rational choice in games, it is necessary to introduce assumptions, not only about the players’ rationality, but also about their knowledge. The following assumptions are fairly standard8 and are often called common knowledge and rationality (CKR):
CKR1. The specification of the game, including the players’ strategy sets and payoff functions, is common knowledge in the game, together with everything that can be deduced logically from it and from CKR2.
CKR2. The players are rational in the sense of expected utility (EU) theory (see 2.3 to 2.6 above), hence they always choose strategies that maximize their individual expected utilities, relative to their knowledge and beliefs at the time of acting. (By CKR1 this too is common knowledge in the game.)
The concept of common knowledge was introduced by Lewis (1969, pp. 52-68) and formalized by Aumann (1976). A proposition is common knowledge among a set of players if every player knows it to be true, knows that every other player knows it to be true, knows that every other player knows that every other player knows it to be true, and so on. Lewis originally wrote “ad infinitum” rather than “and so on” and commented that “this is a chain of implications, not the steps in anyone’s actual reasoning” (p. 53). In fact, nothing is gained by carrying the knowledge beyond the nth degree when there are n players (Binmore, 1992, pp. 467-472), and in some games players can reason to solutions with fewer degrees (Bicchieri, 1993, chap. 4). Even three or four degrees may seem impossibly demanding, but according to one interpretation, full common knowledge is an everyday phenomenon arising, for example, whenever a public announcement is made so that everyone knows it, knows that others know it, and so on (Milgrom, 1981).9 Common knowledge is crucially different from every player merely knowing a proposition to be true. The celebrated muddy children problem (Fagin et al., 1995, pp. 3-7) exposes this distinction dramatically (for a formal but simple proof, see Colman, 1998, pp. 361-362).
4.1. Implications of the theory
The orthodox belief about the standard assumptions has been summed up by Binmore (1994a):
Game theorists of the strict school believe that their prescriptions for rational play in games can be deduced, in principle, from one-person rationality considerations without the need to invent collective rationality criteria–provided that sufficient information is assumed to be common knowledge. (p. 142)
That is a fair statement of the belief that this article calls into question. The paragraphs that follow provide diverse reasons for doubting its validity. The implications of strict rationality for games is an important and intrinsically interesting problem. Aumann (2000) has argued that “full rationality is not such a bad assumption; it is a sort of idealization, like the ideas of perfect gas or frictionless motion; . . . no less valid than any other scientific idealization” (p. 139). In a survey of the foundations of decision theory, Bacharach and Hurley (1991) wrote: “Von Neumann and Morgenstern (1944) . . . set out to derive a theory of rational play in games from one of rational individual decision-making. Their successors have not deviated from the faith that this can be done” (pp. 3-4). But there are reasons to suspect that this faith may be misplaced.
5. Focal points and payoff dominance
Let us examine the implications of the CKR assumptions in the most trivial convention game that we may call Heads or Tails. Two people independently choose heads or tails, knowing that if they both choose heads or both tails, then each will receive a payoff of five units of utility, otherwise their payoffs will be zero. This is a pure coordination game, because the players’ payoffs are identical in every outcome, and the players are motivated solely to coordinate their strategies. Figure 1 shows the payoff matrix.
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II | |
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Heads |
Tails |
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I |
Heads |
5, 5 |
0, 0 |
|
Tails |
0, 0 |
5, 5 | |
Figure 1. Heads or Tails
Player I chooses between the rows, Player II between the columns, and the numbers in each cell represent the payoffs, the first conventionally being Player I’s and the second Player II’s, though in this game they are always equal. In the games discussed in this article, no harm comes from thinking of the payoffs as US dollars, pounds sterling, euros, or other monetary units. In general, this amounts to assuming that the players are risk-neutral within the range of payoffs in the game, so that utility is a strictly increasing linear function of monetary value, though that is immaterial in this trivial game.
5.1. Nash equilibrium
The players hope to coordinate on either (Heads, Heads) or (Tails, Tails). These are Nash equilibria or equilibrium points. In a two-person game, an equilibrium point is a pair of strategies that are best replies to each other, a best reply being a strategy that maximizes a player’s payoff, given the strategy chosen by the other player.
If a game has a uniquely rational solution, then it must be an equilibrium point. Von Neumann and Morgenstern (1944, pp. 146-148) presented a celebrated Indirect Argument to prove this important result; Luce & Raiffa (1957, pp. 63-65) gave the most frequently cited version of it; and Bacharach (1987, pp. 39-42) proved it from formal axioms. Informally, the players are rational utility-maximizers (by CKR2). Any rational deduction about the game must be common knowledge (by CKR1)–Bacharach named this the transparency of reason. It implies that, if it is uniquely rational for Player I to choose Strategy X and Player II Strategy Y, then X and Y must be best replies to each other, because each player anticipates the other’s strategy and necessarily chooses a best reply to it. Because X and Y are best replies to each other, they constitute an equilibrium point by definition. Therefore, if a game has a uniquely rational solution, then it must be an equilibrium point. Whether or not rational players can reason from the standard assumptions to an equilibrium solution is another matter altogether. When the logic of this problem was examined carefully, it became clear that the CKR assumptions are sometimes more than what is required and sometimes insufficient to allow players to reason to an equilibrium solution (Antonelli & Bicchieri; Bicchieri, 1993; Bicchieri & Antonelli, 1995; Samet, 1996).
5.2. Indeterminacy, refinements, and the core
Nash (1950a, 1951) formalized the equilibrium concept and proved that every finite game has at least one equilibrium point, provided that mixed strategies (probability distributions over the pure strategy sets) are taken into consideration. This does not always help a player to choose a strategy, as the game of Heads or Tails shows. In that game, (Heads, Heads) and (Tails, Tails) are equilibrium points, and there is also a mixed-strategy equilibrium in which each player chooses randomly with a probability of 1/2 assigned to each pure strategy (by tossing a coin, for example). But what should a rational player do? Any model of evolutionary game theory (see 1.3 above) with a stochastic or noise component would converge on one or other of the pure-strategy equilibrium points, but rational choice remains indeterminate. This exposes a fundamental weakness of classical game theory, namely its systematic indeterminacy.10
Various refinements of Nash equilibrium have been proposed to deal with the indeterminacy problem. The most influential is the subgame-perfect equilibrium, proposed by Selten (1965, 1975), but it and other refinements are merely palliative. The Holy Grail is a theory that invariably selects a single equilibrium point, but its status as a solution would rest on a dubious assumption of rational determinacy in games (see 6.6 below).
Numerous solution concepts have been suggested for cooperative games–games in which players are free to negotiate coalitions based on binding and enforceable agreements governing the division of a payoff. Nash (1950b) pioneered an approach involving the reformulation of cooperative games as non-cooperative ones and the search for equilibrium solutions in the reformulated games, but this Nash program ran up against the indeterminacy problem. The most fundamental and influential solution concept for cooperative games is the core (Gillies, 1953). An outcome x of a cooperative game is said to dominate another outcome y if there is a potential coalition that has both the motive and the power to enforce x. The core of a cooperative game is the set of undominated outcomes. The core satisfies individual, coalition, and collective rationality, inasmuch as it includes only divisions of the payoff such that the players receive at least as much as they could guarantee for themselves by acting independently, every proper subset of the players receives at least as much as it could guarantee for itself by acting together, and the totality of players receives at least as much as it could guarantee for itself by acting collectively as a grand coalition, so that nothing is wasted. But there are many games in which no division satisfies all these requirements and the core is therefore empty. For example, if three people try to divide a sum of money among themselves by majority vote, then any proposed division can be outvoted by a coalition with the will and the power to enforce a solution that is better for both of its members.11 Rational social interaction, at least as defined by the core, is simply infeasible in these circumstances. Other solution concepts for cooperative games suffer from similar pathologies.
In the non-cooperative game of Heads or Tails, rational players are forced to choose arbitrarily, with a probability of successful coordination of 1/2 and an expected payoff of 2.5. Can they do better than that?
5.3. Focal points
Of course they can. Going beyond the mathematical properties of the game and delving into its psychology, if both players perceive heads to be more salient than tails, in other words if they both recognize (Heads, Heads) as a focal point, and if both believe this to be common knowledge, then both will unhesitatingly choose heads, and they will coordinate successfully. This was first pointed out by Schelling (1960, chap. 3), who reported the results of informal experiments in which 86 per cent of participants chose heads. This implies a probability of coordination of .86 ´ .86 or approximately 3/4, and hence an expected payoff of approximately 3.7–a big improvement.
According to Schelling (1960), what enables players to focus on heads is the “conventional priority, similar to the convention that dictates A, B, C, though not nearly so strong” (p. 64) of heads over tails. Mehta, Starmer, and Sugden (1994a, 1994b) replicated his finding in England, where 87 per cent of players chose heads. Both studies also included several more difficult pure coordination games, some with infinite strategy sets, in which players frequently coordinated on focal points without difficulty. For example, suppose that you have to meet a stranger at a specified place on a specified day but neither of you has been told the time of the meeting. What time would you choose to optimize your chances of coordinating? Most people focus unhesitatingly on 12 noon (Schelling, 1960, p. 55).
5.4. Hume’s example
The idea of a focal point can be traced back to a discussion by Hume (1739-40/1978, 3.II.iii) of a pure coordination game played by a German, a Frenchman, and a Spaniard, who come across three bottles of wine, namely Rhenish, Burgundy, and port, and “fall a quarrelling about the division of them” (pp. 509-510n). There are 27 ways of assigning three bottles to three people, or six permutations if each person gets exactly one bottle. Hume pointed out that the obvious focal point among these alternatives is to “give every one the product of his own country” (ibid.).12
The focal point of Heads or Tails emerges from its representation within the common language shared by the players (Crawford & Haller, 1990). Considered in the abstract, this game, or the problem of the unspecified meeting time, or Hume’s problem of the three wines, has no focal point. To remove the common language, including the culturally determined labeling of strategies, is to filter out the focal points, reducing the prospects of coordination to chance levels.
5.5. Gilbert’s argument
The salient focal points are obvious in Heads or Tails, the unspecified meeting time, and Hume’s problem of the three wines. Nevertheless, it turns out that their selection cannot be justified rationally. Gilbert (1989b) showed that “if human beings are–happily–guided by salience, it appears that this is not a consequence of their rationality” (p. 61) and that “mere salience is not enough to provide rational agents with a reason for action (though it would obviously be nice, from the point of view of rational agency, if it did)” (p. 69, italics in original).
Gilbert’s proof is easy to follow though hard to swallow. The focal point of Heads or Tails is obviously (Heads, Heads), and to clarify the argument, let us assume that the players have previously agreed on this, so it is common knowledge. Under the CKR2 rationality assumption, Player I will choose heads, given any reason for believing that Player II will choose heads, to ensure a payoff of 5 rather than 0. But in the absence of any reason to expect Player II to choose heads, Player I has no reason to choose it or not to choose it. The fact that (Heads, Heads) is a focal point is not a valid reason for Player I to choose heads, because heads is best only if Player II chooses it also. Because the salience of (Heads, Heads) does not give Player I a reason to choose heads, it cannot give Player I a reason to expect Player II to choose heads. Both players are in exactly the same quandary, lacking any reason for choosing heads in the absence of a reason to expect the co-player to choose it. The argument goes round in circles without providing the players with any rational justification for playing their parts in the focal-point equilibrium, in spite of its salience and intuitive appeal.
This is an excellent example of the fundamental thesis of this article, that the concept of utility maximization cannot be applied straightforwardly to interactive decisions.
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II | |
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H |
L |
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I |
H |
6, 6 |
0, 0 |
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L |
0, 0 |
3, 3 | |
Figure 2. Hi-Lo Matching game
5.6. Payoff dominance
Gilbert’s (1989b) argument applies even to games with structurally inbuilt payoff-dominant (or Pareto-dominant) equilibrium points. These games have focal points that do not depend on any common language, pace Crawford and Haller (1990). Payoff dominance is illustrated most simply in the Hi-Lo Matching game (Figure 2). If both players choose H, each gains six units; if both choose L, each gains three units; otherwise neither gains anything. The two obvious equilibria are HH, with payoffs of (6, 6) and LL, with payoffs of (3, 3). (There is also a mixed-strategy equilibrium in which each player chooses 1/3 H and 2/3 L, with expected payoffs of 2 units each.) Rational players prefer HH to LL, because HH payoff-dominates LL. An equilibrium point payoff-dominates another if it yields a higher payoff to both players. It is obviously a structural focal point.
The payoff-dominance principle is the assumption that, if one equilibrium point payoff-dominates all others in a game, then rational players will play their parts in it.13 Harsanyi and Selten’s (1988) general theory of equilibrium selection is based on it, together with a secondary risk-dominance principle,14 and most game theorists accept its intuitive force (e.g., Bacharach, 1993; Crawford & Haller, 1990; Farrell, 1987, 1988; Gauthier, 1975; Janssen, 2001; Lewis, 1969; Sugden, 1995, 2000). Empirical tests of the payoff-dominance principle have yielded mixed results (e.g., Cooper et al., 1990; van Huyck et al., 1990). But, astonishingly, a straightforward extension of Gilbert’s (1989b) argument reveals that a player has a reason to choose H if and only if there is a reason to expect the co-player to choose H, and there is no such reason, because both players face the identical quandary. The fact that HH is the optimum equilibrium (indeed, the optimum outcome) for both players is not ipso facto a reason for Player I to expect Player II to choose H, because H is not a utility-maximizing choice for Player II in the absence of any reason to expect Player I to choose it, and vice versa (Casajus, 2001; Colman, 1997; Gilbert, 1990; Hollis & Sugden, 1993; Sugden, 1991b). This is a startling failure of game theory. When one first appreciates the force of the argument, one feels like pinching oneself.
A common initial reaction is to try to justify the choice of H in Figure 2 on the grounds that “The best I can get from choosing H is better than the best I can get from choosing L, and the worst is no worse, therefore I should choose H”. To see the fallacy in this naive maximax reasoning, consider the slightly modified game shown in Figure 3. In this version, Strategy L gives Player II a higher payoff whatever Player I chooses, therefore a rational Player II will certainly choose it. By the transparency of reason, Player I will anticipate this and will therefore also choose L, hence the rational solution is unambiguously LL. But the maximax argument (“The best I can get from choosing H is better than the best I can get from choosing L, and the worst is no worse, therefore I should choose H”) would still lead Player I to choose H, and that is manifestly absurd.
|
|
|
II | |
|
|
|
H |
L |
|
I |
H |
6, 6 |
0, 7 |
|
L |
0, 0 |
3, 3 | |
Figure 3. Modified Hi-Lo Matching game
A more sophisticated fallacy is the attempt to justify choosing H in the Hi-Lo Matching game (Figure 2) by assigning subjective probabilities to the co-player’s strategies. The specific probabilities are immaterial, so let us suppose Player I assumes (perhaps by the Principle of Insufficient Reason) that Player II’s strategies are equally probable. If this assumption were valid, then Player I would indeed have a reason (SEU maximization) to choose H, but a simple reductio proof exposes the error. By the transparency of reason, Player I’s intention to choose H would be common knowledge and would induce Player II to choose the best reply, namely H, with certainty, contradicting Player I’s initial assumption.
5.7. Coordination without rationality
Under the CKR knowledge and rationality assumptions, coordination by focal point selection ought to be impossible, yet it occurs quite frequently in everyday social interaction. Even in games with blindingly obvious payoff-dominant focal points, players have no rational justification for choosing the corresponding strategies. Orthodox game-theoretic rationality is powerless to explain these phenomena.
6. Social dilemmas
Social dilemmas are games in which individual and collective interests conflict. The simplest is the familiar two-person Prisoner’s Dilemma game (PDG). The general N-player Prisoner’s Dilemma (NPD), of which the PDG is a special case, was discovered simultaneously and independently by Dawes (1973), Hamburger (1973), and Schelling (1973). Social dilemmas have generated a vast amount of theoretical and empirical research (reviewed by Colman, 1995a, chaps 6, 7, 9; Dawes, 1980, 1988, chap. 9; Foddy et al., 1999; Ledyard, 1995; Nozick, 1993, pp. 50-59; Rapoport, 1989, chaps 12, 14; Schroeder, 1995; van Lange et al., 1992; and van Vugt, 1998; among others).
6.1. Self-defeating strategeies
The peculiarity of the Prisoner’s Dilemma game is that, of the two strategies available to each player, one is uniquely rational, yet each player fares better if both choose the other. To borrow a felicitous epithet from Parfit (1979, 1984, chap. 1), rationality is self-defeating in the PDG, and in social dilemmas in general.
6.2. Prisoner’s Dilemma formalization
The PDG (Figure 4) was discovered in 1950 by Flood and Dresher. What defines it as a PDG are the relative rather than the absolute values of the payoffs, hence the numbers 4, 3, 2, and 1 are used for simplicity, though they are assumed to be utilities.
|
|
|
II | |
|
|
|
C |
D |
|
I |
C |
3, 3 |
1, 4 |
|
D |
4, 1 |
2, 2 | |
Figure 4. Prisoner’s Dilemma game
6.3. Lifelike interpretation
The name Prisoner’s Dilemma comes from an interpretation involving two prisoners, introduced by Tucker in a seminar in the Psychology Department of Stanford University in 1950, the most familiar published version being Luce and Raiffa’s (1957, pp. 94-97). The story is too well known to repeat, and the following alternative interpretation, based on an idea of Hofstadter’s (1983), will help to fix the idea of a game as an abstract structure applicable to a potentially unlimited set of interactions.
Player I is keen to buy a packet of illegal drugs from Player II, and Player II is keen to sell it. They have agreed a price that suits them both, but because of the nature of the trade, it must take place without face-to-face contact. Player I promises to leave an envelope full of money in a dead-letter drop, such as a litter bin in a park. Player II promises to leave an envelope full of drugs in another, distant, dead-letter drop at the same time. Each player faces a choice between cooperating (leaving the promised envelope) or defecting (neglecting to leave it). If both players cooperate and choose C, then the payoffs are good for both (3, 3). If both defect and choose D, the payoffs are worse for each (2, 2). And if one player cooperates while the other defects, then the outcome is worst for the cooperator and best for the defector, hence the payoffs are (1, 4) or (4, 1), depending on who cooperates and who defects.
6.4. Ubiquity of social dilemmas
Many everyday two-person interactions have the strategic structure of the PDG. Rapoport (1962) discovered it in Puccini’s opera Tosca. Lumsden (1973) showed empirically that the Cyprus conflict shared the preference structure of an indefinitely repeated PDG. The PDG is a standard model of bilateral arms races and duopoly competition. Many other two-person interactions involving cooperation and competition, trust and suspicion, threats, promises, and commitments are PDGs.
6.5. Strategic dominance
How should a rational player act in the PDG? There are two main arguments in favor of defecting (choosing D). They apply to the standard one-shot PDG. For indefinitely iterated PDGs, a folk theorem establishes a vast number of equilibrium points, including many leading to joint cooperation (see Binmore, 1992, pp. 373-376, for a clear proof), and evolutionary experiments have reported high levels of cooperation (Axelrod, 1984, 1997, chaps. 1, 2; Kraines & Kraines, 1995; Nowak, May, & Sigmund, 1995; Nowak & Sigmund, 1993). The finitely iterated PDG presents a different problem altogether, to be discussed in 7.1 below.
The most powerful reason for defecting in the one-shot PDG is strategic dominance. The D strategies are strongly dominant for both players inasmuch as each player receives a higher payoff by choosing D than C against either counterstrategy of the co-player. Player I receives a higher payoff by choosing D than C whether Player II chooses C or D, hence D is a strongly dominant strategy for Player I and, by symmetry, the same applies to Player II. It is in the interest of each player to defect whatever the other player might do.
It is generally agreed that a rational agent will never choose a dominated strategy. Dixit and Nalebuff (1991, p. 86) identified the avoidance of dominated strategies as one of the four basic rules of successful strategic thinking, and it has been proved that the only strategies in two-person games that can be rationalized–justified in terms of consistent beliefs about the co-player’s beliefs–are those that survive a process of successively deleting strongly dominated strategies (Bernheim, 1984; Pearce, 1984).
Strategic dominance is a simplified version of the sure-thing principle,15 first propounded by Savage (1951) and incorporated into his decision theory as an axiom, with the comment: “I know of no other extralogical principle governing decisions that finds such ready acceptance” (Savage, 1954, p. 21). The strategic dominance principle, in its strong form, can be deduced from elementary axioms of game theory (Bacharach, 1987), although the (weak) sure-thing principle cannot (McClennen, 1983).
In individual decision making, the sure-thing principle seems intuitively compelling, except in certain pathological though interesting cases based on Simpson’s paradox (Shafir, 1993) or Newcomb’s problem (Campbell & Sowden, 1985), or in situations in which players’ actions are not independent (Jeffrey, 1983, pp. 8-10). But when the strategic dominance principle is applied to the PDG, the conclusion seems paradoxical, because if both players choose dominated C strategies, then each receives a higher payoff than if both choose dominant D strategies. The DD outcome resulting from the choice of dominant strategies is Pareto-inefficient in the sense that another outcome (CC) would be preferred by both players.
6.6. Argument from Nash equilibrium
The second major argument for defection focuses on the fact that DD is the PDG’s only equilibrium point. It is obvious in Figure 4 that D is a best reply to D and that there is no other equilibrium point. From the Indirect Argument (see 5.1 above), if the PDG has a uniquely rational solution, then, because it must be an equilibrium point, it must therefore be this equilibrium point.
It is often considered axiomatic that every game has a uniquely rational solution that could, in principle, be deduced from basic assumptions (Harsanyi, 1962, 1966; Harsanyi & Selten, 1988; Weirich, 1998). When it is expressed formally, this existence postulate is called the principle of rational determinacy. Nash (1950a, 1950b, 1951) assumed it tacitly at first, then in a later article (Nash, 1953) introduced it explicitly as the first of seven axioms: “For each game . . . there is a unique solution” (p. 136). Although it is widely accepted, Bacharach (1987) pointed out that it remains unproven, and this blocks the inference that a game’s unique equilibrium point must necessarily be a uniquely rational solution, because the game may have no uniquely rational solution. Sugden (1991a) presented several reasons for skepticism about the principle of rational determinacy. However, in the PDG, we know from the dominance argument that joint defection must be uniquely rational, and it is therefore paradoxical that irrational players who cooperate end up better off.
6.7. Experimental evidence
Joint defection is uniquely rational in the PDG. Binmore (1994a) devoted a long chapter (chap. 3) to refuting fallacies purporting to justify cooperation. But as a prediction about human behavior, this fails miserably in the light of experimental evidence (reviewed by Colman, 1995a, chap 7; Good, 1991; Grzelak, 1988; Rapoport, 1989, chap. 12; among others). In the largest published PDG experiment (Rapoport & Chammah, 1965), almost 50 per cent of strategy choices were cooperative, and even in experiments using one-shot PDGs, many players cooperate, to their mutual advantage. Game theory fails as a positive theory in the PDG, because human decision makers do not follow its rational prescriptions.
6.8. Three-player Prisoner’s Dilemma
The simplest multi-player Prisoner’s Dilemma, using payoffs of 4, 3, 2, and 1 for convenience once again, is the three-player NPD shown in Table 1. The first row of Table 1 shows that the payoff to each C-chooser is 3 if all three players choose C, and in that case the payoff to each (non-existent) D-chooser is undefined, hence the dash in the last column. In second row, if two players choose C and the remaining player chooses D, then the payoff is 2 to each C-chooser and 4 to the D-chooser, and so on.
Table 1. Three-player Prisoner’s Dilemma
|
Number choosing C |
Number choosing D |
Payoff to each C-chooser |
Payoff to each D-chooser |
|
3 |
0 |
3 |
– |
|
2 |
1 |
2 |
4 |
|
1 |
2 |
1 |
3 |
|
0 |
3 |
– |
2 |
6.9. Defining properties
The defining properties of the NPD are as follows:
(1) Each player chooses between two options that may be labeled C (cooperate) and D (defect).
(2) The D option is strongly dominant for each player: each obtains a higher payoff by choosing D than C no matter how many of the others choose C.
(3) The dominant D strategies intersect in an equilibrium point that is Pareto-inefficient: the dominant D strategies are best replies to one another, but the outcome is better for every player if all choose their dominated C strategies.
The NPD in Table 1 has all three properties, and so does the two-person PDG in Figure 4, which can now be seen as a special case. The NPD is ubiquitous, especially in situations involving conservation of scarce resources and contributions to collective goods. Examples of defection include negotiating a wage settlement above the inflation rate, neglecting to conserve water during a drought or fuel during a fuel shortage, over-fishing, increasing armaments in a multilateral arms race, and bolting for an exit during an escape panic. In each case, individual rationality mandates defection regardless of the choices of the others, but each individual is better off if everyone cooperates.
6.10. More experimental evidence