Game theory and conventions: Focal points

Schelling’s (1960) focal point argument and Lewis’s concept of salience has been interpreted by game theorists in different ways. Game theorists have tried to in- tegrate the idea that individuals may be successful in coordination games if a certain strategy stands out as an obvious option into the formal structure of the game. Two well-known standard examples of this approach are the arguments from Pareto dominance and risk dominance (see Appendix IV.4). It has been ar- gued that if one of the Nash equilibria, Pareto, dominates others then one may argue that agents might use this as a coordination device (Harsanyi and Selten 1988).14 That is, Pareto-dominant equilibrium may be considered as a focal-point equilibrium. Yet, in some games Pareto-dominant equilibrium is not always an obvious solution. For example, in the stag hunt game (see Appendix IV.4) agents may perceive the Pareto-dominant equilibrium as a risky option and try to play their parts for the risk-dominant equilibrium (Carlsson and van Damme 1993ab15, cf. Harsanyi and Selten 1988). Nevertheless, unless agents are assumed to know how the other player thinks, neither Pareto dominance nor risk dominance seems to be a compelling argument for equilibrium selection.

Another line of research in the line of justifying or rationalising focal points focuses on the attributes of the different alternatives that are present in the game set-up.17 For example, suppose that you are a participant of a select-a-ball experi- ment. You and your co-player are located in different rooms and you are asked to choose one ball among three balls. Two of the balls are red, one is green, and the red balls are indistinguishable. If you are able to coordinate on the same ball you will get fifty euros, if not you get nothing.18 It is argued that since individuals cannot discriminate between the red balls (principle of insufficient reason) they should choose the green ball (see Bacharach 1991; Janssen 1998a: 15, 2001a,b). This approach tells us that individuals may rationally play their part in a focal- point equilibrium, given that there is a unique strategy which is Pareto optimal and that the individuals are able to cluster the indistinguishable alternatives together (principle of coordination).

There were some attempts to study the importance of labelling and framing in coordination games. Sugden (1995) distinguishes between the strategic structure of the game and the way in which the game is described or labelled for players. The result is that the particular way in which the game is described (or perceived) influences the outcome of a coordination game. Yet he assumes that the labelling procedure is common knowledge among the agents and it remains to be explained how the labelling procedure becomes common knowledge.20 Bacharach and Ber- nasconi (1997), on the other hand, try to formalise the different ways in which strategies may be framed. They generalise Bacharach’s (1991) idea that players’ options are acts under descriptions and they are distinguished by the concepts the players use to specify them. This model permits to conceptualise the pos- sible differences in agents’ perceptions and for this reason it is a step further in understanding how these differences may influence the outcome of a coordination game. Like Janssen (2001b), this model focuses on the attributes of the alternative strategies and how players of the game perceive these attributes. Yet, unfortu- nately, the model is only able to ‘predict’ the outcome of simple coordination games (see Appendix IV.5).

Formal models of focal points commonly focus on whether one may explain the selection of the focal-point equilibrium without assuming common knowledge, or common history. In these models, the modeller predetermines the salient or focal option. For example, in the select-a-ball game a ‘focal’ alternative, green, was em- bedded in the game set-up.21 What remains to be explicated in this context is the conditions under which rational individuals would choose the focal alternative. These conditions are expressed as principles, such as the principles of insufficient reason, coordination, rarity preference, etc. Although formal approaches to focal points justify the intuitive idea that rational individuals would choose the green object (the odd one) in the three ball version of the select-a-ball game, they do not help us solve the equilibrium selection problem presented by games with two Pareto non-comparable Nash equilibria, such as the driving game in Table 8.1.

Consider a version of the select-a-ball game where there are only two balls: one red and one green. In this game, there is no unique strategy that is Pareto op- timal. For this reason, the alternative strategies remain formally indistinguishable from the point of view of the models examined above. Yet remember Schelling’s argument that real individuals are more successful in coordination than the model theoretic agents in games. Then, according to him, connotations of green and red, as well as culture, history and experience of the players might influence the way in which this coordination problem would be solved in the real world. For example, given the existing conventions concerning the colours on warning tags and traffic symbols, if both agents think that the colour ‘red’ is more prominent than ‘green’ and expect the other to do the same they might be able to coordinate by selecting the red ball.22 That is, the existing conventions might influence the way individuals play this game. (A similar argument is developed for Bacharach and Bernasconi’s model in Appendix IV.5.)

In sum, static models (discussed above) study the conditions under which co- ordination may be possible, rather than focusing on the mechanisms of coordina- tion. This is closely related with our distinction between the end-state and process interpretations of the invisible hand (see Chapter 5). Standard game theory, refine- ments and models of focal points introduce the conditions under which a certain equilibrium is plausible, but the emergence of conventions remains unexplained.23 We learn from these models that successful coordination is a plausible outcome of a coordination game if conditions such as ‘common knowledge’, ‘correlated expectations’, or ‘shared frames’ hold. Successful coordination is compatible with rationality only under these conditions. If we believe that individuals have a ten- dency to behave rationally ceteris paribus we should take these results seriously. This framework implies that explaining the emergence of a particular convention requires the introduction of further factors (e.g. existing conventions, history, etc.) because the structure of the game does not necessarily tell us how such an equilib- rium may be reached. Bernheim makes a similar point:

The economist’s predilection for equilibria frequently arises from the belief that some underlying dynamic process (often suppressed in formal models) moves a system to a point from which it moves no further. However, where there are no equilibrating forces, equilibrium in this sense is not a relevant concept. Since each strategic choice is resolved for all time at a specific point during the play of a game, the game itself provides no dynamic for equilibra- tion.

(Bernheim 1984: 1008)

As a model of conventions, the driving game in Table 8.1 does not explain how conventions may emerge, but merely provides a framework for analysing some of the properties of conventions. The explanation of the emergence of convention appears to require that we bring in more ingredients to this model and consider the process of emergence of conventions. The next section discusses whether ‘learn- ing’ may explain the possibility of coordination.

1. Rationality and learning

The driving game presented in Table 8.1 is a one-shot game. It may be considered as a representation of the state of affairs when two (‘clueless’) individuals face the aforementioned coordination problem for the first time. A two-player one-shot game cannot be a good model of the emergence of driving conventions. At most, it describes the relevant coordination problem for two individuals, but not how it is solved. Coordination of two drivers is not enough to create a convention: for a convention to exist there should be many drivers who are coordinating on one of the equilibrium points and who are expecting the others to do the same. Hence, the relevant game-theoretic concept here would be a multi-player repeated game, for example, where n individuals repeatedly meet in pairs and play the one-shot driving game in Table 8.1 (which is called the ‘stage game’ in this context). How- ever, even if we present the game in this form, standard (non-evolutionary) game theory is not very helpful in explaining the emergence of conventions, or in show- ing how one of the two possible equilibrium points is reached. The perfectly ra- tional model players who are able to reason about all possibilities in the repeated game would fail to bring about a convention for they would have no clue about what exactly to expect from others given the structure of the game (also see Bern- heim 1984: 1008–1009). Obviously, random play (e.g. playing a mixed strategy) of all agents would not bring about a convention. Moreover, even if all (or most of) the agents would be able to coordinate on one of the equilibria by chance, this equilibrium would not be stable and would not constitute a convention. That is, unless agents update their expectations or learn to play in a certain way as they re- peatedly play the game, we cannot explain how a certain equilibrium point would be self-supporting.

Given the concepts of salience, precedence and focal points, explanation of individuals’ success in coordination and emergence of conventions necessitates the study of learning in coordination games. The mechanisms of imitation, rein- forcement and best reply dynamics have been employed in various forms to study the consequences of individual learning behaviour in coordination games.26 There is a large number of models that use different assumptions concerning how indi- viduals learn in repeated games. It is not necessary to give a full account of this literature here.27 It will suffice to examine some of the important ideas in order to give a flavour of the models that focus on learning and evolution in the context of a coordination problem.

Standard justifications of the Nash solution concept and solutions to the equi- librium selection problem fail to explicate why rational individuals would play their part in the Nash equilibrium or choose a certain Nash equilibrium among many. Assuming ‘common knowledge’, ‘correlated expectations’ or ‘shared frames’ does not help us explain how real-world agents are able to coordinate, and individual dynamics of coordination have to be examined to explain the emer- gence of conventions as unintended consequences of human action. An important question is whether rational individuals who learn from experience might arrive at a coordination equilibrium. Or whether we could explain the emergence of a con- vention without restricting individuals’ expectations and learning behaviour with a certain form of common knowledge assumption. Goyal and Janssen (1996), who study similar questions, argue that rationality alone does not suffice to explain coordination even if individuals are able to learn.28 The idea behind this argument is the following: in order to ensure coordination in the next period, every agent has to take into account the previous plays of other players. However, since every player knows that the other players are using the information gathered in previous plays to form their expectations for the next period, in order to ensure coordina- tion every player has to know how the others are forming their expectations. The problem is that the outcome of the previous encounters does not restrict the type of hypotheses they might entertain about each other. In other words, as Goyal and Janssen argue, at any point in time one may entertain an infinite number of hypotheses about others, which are consistent with their information. Thus, unless the modeller restricts the number of these hypotheses, rationality does not ensure that players learn how to coordinate.

Crawford and Haller (1990) and Kalai and Lehrer (1993a,b), on the other hand, argue that rational individuals can learn to coordinate. While Crawford and Haller assume that there are optimal rules for learning, Kalai and Lehrer put certain re- strictions on individuals’ prior beliefs. Yet these assumptions (restrictions) imply that rationality alone cannot ensure coordination. More specifically, Goyal and Janssen (1996) argue that even if there may be optimal rules for learning how to coordinate, these rules are not unique. That is, if agents’ learning behaviour is not coordinated at the outset they might not be able to coordinate. Similarly, Kalai and Lehrer’s model indicate that agents’ prior beliefs have to be coordinated to ensure their success in coordination. Both models imply that pre-existent conven- tions are necessary for individuals’ success in coordination. Goyal and Janssen’s argument is consistent with our interpretation of the literature on refinements and focal points: in the model world, coordination is only possible (i.e. individuals might be able to learn to coordinate) if conditions such as common knowledge, shared background or correlated expectations hold. Again, from the perspective of explaining real world coordination problems, this means that the knowledge of pre-existing conventions is necessary to explain how coordination is achieved.

Note that Goyal and Janssen’s argument supports Lewis, Schelling and others who argue that explanation of the emergence of conventions is an empirical mat- ter. Yet one may still argue that models of learning point at certain dynamics in the explanation of successful coordination. Schotter’s model (see Chapter 6) shows that under some conditions simple learning might bring about conventions. Craw- ford and Haller’s model indicates that if we can ensure that individuals are using a certain type of learning rule they may be able to coordinate. Kalai and Lehrer’s model can be interpreted as saying that agents drawn from a population with shared conventions may learn to coordinate and bring about conventions. That is, a learning mechanism may bring about conventions under certain conditions. Of course, this alone does not explain the emergence of any particular convention. These models suggest certain possibilities.

Nevertheless, many argue that rationality is not a good criterion either. It is argued that if we want to understand how real individuals achieve coordination we should consider more realistic models of real individuals (e.g. Marimon 1997: 278–282; Young 1998). A typical model that follows this suggestion has been examined in Chapter 6. Let us return to the driving game in order to recall the results of models with bounded rationality.

2. Bounded rationality and learning

In Chapter 6 we examined Young’s (1998) model of emergence of money that assumes bounded rationality. In brief, his model involves a dynamic known as fictitious play,29 in which each player constructs a simple statistical model of what the other people are doing based on fragmentary information on what they did in the past. The idea is roughly as follows: each player observes the actions that the others have chosen up to a given time t. Then player i computes the observed frequency distribution for his sample size and chooses a best reply to this distribu- tion. The outcome of this process is that after some time individuals coordinate on either (left, left) or (right, right). Both of these equilibrium pairs may be con- sidered as conventions because when individuals reach a state where everyone chooses the same strategy, their best reply to this state of affairs would be to continue playing the same strategy. The model also says that the outcome of this process depends on its initial states, that is, it is non-ergodic. Similar results ap- ply if one of the equilibrium pairs Pareto-dominates the other; for example, when driving on the right yields a higher payoff than driving on the left.30

The above model says that it is possible in this set-up that one of the alterna- tive conventions emerges. Yet real individuals make mistakes and this may be incorporated into this model by introducing small persistent stochastic shocks. These shocks represent the ‘mistakes’ of the players and / or other reasons they may choose an action other than the one indicated by the history of play. In our case we may simply assume that every player chooses his best reply strategy with a high probability (1 – e) and with probability e she chooses another strat- egy (Young 1998). Foster and Young (1990) argue that when there are shocks of this sort the dynamic system spends most of its time in certain (Nash) equilibria than in others. They have called such an equilibrium a stochastically stable equi- librium.31 The introduction of persistent stochastic shocks changes the results of the model.32 Because of the mistakes (or mutations, if you wish), now there is a positive probability that the system might move from one Nash equilibrium to the other. That is, conventions emerge but they do not stay forever. In the long run, the ‘society’ occasionally switches between alternative conventions. When both conventions are equally desirable, the model cannot tell which of the two conven- tions will emerge. However, if one of the conventions is better than the other, then the system spends most of its time in the Pareto-optimal equilibrium (which is also risk dominant). That is, in this model mistakes ensure that the better conven- tion is followed most of the time. In cases where the Pareto-optimal equilibrium is not also the risk-dominant equilibrium (as in the stag-hunt game), risk-dominant equilibrium is the stochastically stable outcome (Foster and Young 1990; Kandori, Mailath and Rob 1993; Young 1993a). Hence, the model solves the equilibrium selection problem and the risk-dominant equilibrium gets selected.

Ellison (1993) points out an important issue concerning these models. It is argued that the model converges to the risk-dominant equilibrium in the long run. But how long is the long run? If we assume for a moment that the assumptions of the model hold for a particular society in the real world, could we expect to observe the emergence of a convention in a reasonable period of time? Ellison examines the nature of convergence and argues that if individuals interact locally (i.e. if individuals mostly interact with their neighbours) then the dynamics intro- duced by the above model may be plausible for large populations. In brief, the final result of these models is that boundedly rational agents who interact locally might bring about conventions.

Yet it seems to be somewhat puzzling that while rational individuals could not learn to coordinate, myopic individuals can. In fact, bounded rationality assump- tion is a way of constraining individual behaviour. When individuals are myopic and base their decisions on fragmentary information one may dispense with the common knowledge assumption in a dynamic setting. Yet even if individuals are not fully rational they need to form expectations about others. Indeed, they are implicitly assumed to expect others to continue doing what they did in the past. Moreover, they are assumed to form their expectations in a certain manner, for example by constructing a simple statistical model of what the other people are doing based on fragmentary information. When considered from the perspective of the real world, these assumptions are in need of further explanation. One has to justify in one way or the other that this is a plausible assumption about individual behaviour. Briefly, while these models dispense with the common knowledge as- sumption they constrain individual behaviour in another way. It should also be noted that there is no guarantee that real individuals would conceive the problem situation as described in the model. They may consider alternative strategies or entertain different hypotheses about other individuals’ future behaviour. Thus, assuming bounded rationality does not help us avoid the questions concerning existing institutions and conventions. Or in other words, assumptions concerning learning behaviour are problematic in a similar manner to assumptions concern- ing common knowledge. In the latter one has to explain how common knowledge (or common priors, correlated expectations) is acquired, in the former one has to explain why individuals form their expectations in that particular manner. Yet a single model cannot explain everything at the same time. Learning models study how concordant mutual expectations may emerge and for this reason they fare better than static models with respect to explicating the mechanisms that may bring about conventions.

One may also ask whether the agents in these models learn anything at all. For example, Fudenberg and Levine (1998: 143) argue that agents in these models have information about the current state and this is the only thing they care about. They respond to this information, yet they do not learn anything about others at all. Fudenberg and Levine suggest that the assumptions concerning the agents’ learn- ing behaviour can be viewed as an approximation to a model where individuals are less perfectly informed. More properly, these models ask what would happen if individuals respond to fragmentary information concerning the history of play. Note that these models define an individual mechanism that may be called a best reply mechanism, or a fictitious play mechanism, and show that their interaction may bring about conventions. The interaction of the individual level mechanisms forms an aggregate level mechanism that may be called the mechanisms of the accumulation of the precedent. Thus, these models explicate how ‘precedence’ may help individuals solve coordination problems. Although precedents may ac- cumulate in different ways in the real world, these models suggest a particular way in which they may relate to individual mechanisms.

Source: Aydinonat N. Emrah (2008), The Invisible Hand in Economics: How Economists Explain Unintended Social Consequences, Routledge; 1st edition.

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