Coordination in the world of models

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 individuals face the aforemen- tioned coordination problem for the first time. Moreover, it serves as a representa- tion of the possible outcomes they may reach after making their choices. As we have mentioned, two pure strategy equilibria, (left, left) and (right, right), are considered as states where individuals have no intention to deviate, that is, change their strategies. Moreover, if many individuals are involved, these equilibrium points represent alternative conventions. In order to explain the emergence of conventions, or how individuals coordinate, one has to explain how concordant mutual expectations emerge.

For a certain equilibrium (e.g. left, left) to get established and maintained, agents need to ‘know’ what the others will be doing. In other words, every indi- vidual should know that every other has a good reason to play a certain strategy (e.g. left) and that this is common knowledge:

So if a convention, in particular, holds as an item of common knowledge, then to belong to the population in which that convention holds – to be party to it – is to know, in some sense, that it holds. If a regularity R is a convention in population P, then it must be true, and common knowledge in P, that R satisfies the defining conditions for a convention. If it is common knowledge that R satisfies them, then everyone in P has a reason to believe that it is true, and common knowledge in P, that R satisfies them; which is to say that everyone in P must have a reason to believe that R is a convention.

(Lewis 1969: 61)

We have argued that given that the Nash equilibria of the driving game are formally indistinguishable from each other, it is not possible to explain how indi- viduals would rationally succeed in reaching one of them unless they succeed by chance using mixed strategies. Moreover, the reasoning behind the use of a mixed strategy does not allow us to argue that if an equilibrium point is reached it may be maintained if the game is repeated. That is, if individuals have no clue about what to expect from the other player and do not update their expectations with the information they have acquired in previous plays, we cannot explain how they may expect others to play a certain pure equilibrium strategy. Hence, we cannot explain how concordant mutual expectations are established.

The problem of explaining why individuals play a certain equilibrium strategy is based on a deeper problem in game theory: it is commonly argued that Nash equilibrium does not follow from the assumption of rationality of the players, but it is a consequence of the additional assumptions imposed on the players. It is more generally argued that the notion of Nash equilibrium is based on the as- sumption that players are able to anticipate others’ actions (Bernheim 1984).

Aumann argues:

Nash equilibrium does make sense if one starts by assuming that, for some specified reason, each player knows which strategies the other players are using.

(Aumann 1987: 2)

Justifying Nash equilibrium, or explaining how it gets established, has been an important problem for game theorists that has led to the literature known as the refinements literature. Although this problem is usually considered as being dis- tinct from that of equilibrium selection, it is closely related (Harsanyi and Selten 1988; Samuelson 1998).10 Both questions are relevant if we wish to explain the emergence of conventions. In the refinements literature, many suggestions have been made on how to render an equilibrium rational without relying on the as- sumption of common knowledge. The orthodox justifications11 (which are based on static games) fail to explain how and why individuals would play certain equi- librium strategies (Colman and Bacharach 1997; Crawford 1997: 210–211; Kan- dori, Mailath and Rob 1993: 29; Janssen 1998a: 12). For example, Aumann and Brandenburger (1995) and Brandenburger (1992) assume that individuals have coordinated expectations,12 yet, as Janssen (1998a: 9) argues, the justification of the Nash equilibrium in this context requires an explanation of how individuals acquire coordinated expectations. In fact, in order to justify Nash equilibrium in the context of a game one has to make assumptions about agents’ expectations or knowledge about others and every such assumption would be in need of further explanation.

Generally, the problem of justifying the Nash equilibrium and explaining why and how agents would choose a salient strategy are similar problems. In order to justify the Nash equilibrium one has to explain, in a sensible manner, why agent I would expect agent II to choose the Nash strategy, and expect agent II to expect himself (agent I) to expect agent II to play the Nash strategy, and so on. In order to explain why a salient strategy is chosen one has to explain why agent I would expect agent II to choose the salient strategy, and expect agent II to expect himself (agent I) to expect agent II to play the salient strategy, and so on. For example, if two game theorists play a game with a unique Nash equilibrium, they would consider the Nash equilibrium as a salient option because they would mutually expect the other to play his part in the Nash solution, given that they know that their co-player is a game theorist as well.13 This is because ‘the salience of any particular mode of behaviour depends critically upon whether that salience is uni- versally recognised’ (Bernheim 1984: 1010). Yet if two individuals who do not have knowledge of game theory play the same game, we would have no good rea- son to believe that they would play their Nash strategies. Neither the emergence of the Nash outcome nor the selection of a salient equilibrium among multiple equilibria can be explained without explaining how agents come to believe that the others will behave in a particular way. A satisfactory explanation of how a certain Nash equilibrium gets established seems to require a model of learning or of how agents form and update their expectations. Particularly, one has to explain why agents would consider a certain equilibrium as being focal or salient.

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

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