The Power of Game Theory

Coordination and Correlation

David Hugh-Jones has a very nice post on his substack about the explanatory power of simple 2x2 games. He discusses the prisoner’s dilemma, the trust game, the battle of the sexes, matching pennies, and the stag hunt. These are all indeed very enlightening games that account for a large variety of social interactions and socioeconomic phenomena. Let me in this post modestly complete the list with two additions: coordination games and the so-called “hawk-dove game.”

Let’s start with coordination games, the prototype is illustrated in Figure 1 below. Coordination games are arguably the simplest games one can conceive of. In those, players’ interests are fully aligned, i.e., their preferences are identical. It follows that the only thing that matters for the players is to coordinate by playing the same strategy. If you assume – as this is generally the case – that the type of game played is itself common knowledge among the players, then they commonly know that they only have to make sure that everybody plays the same way. That sounds pretty simple. After all, if players can communicate before choosing their strategy, the solution is fairly trivial. The impression of triviality disappears however once one realizes that communication itself is a kind of coordination game where players have to agree on the meaning of utterances that they use to send information.

This is the basic insight of David Lewis’s famous study of the role of conventions in coordination games.[1] Lewis’s starting point is a puzzle posed by Quine: if agreements on social practices are possible through the use of language, how is the agreement on language established at first? Lewis argues that this apparent paradox is solved by showing how any agreement, including on language, can be established without the need for prior communication. Lewis’s account largely borrows from Thomas Schelling’s concept of “focal point”[2] by demonstrating our coordination without prior communication is permitted by the “salience” of some strategies and outcomes. The basic idea is that in specific strategic contexts, people share expectations and inferential modes of reasoning, such that from the observation of a given event (say, a ball enters into a basket), everyone will believe that (i) this event has been mutually observed and (ii) everyone else infers another event (say, this basket counts for 2 points). Lewis in particular locates the origins of the shared expectations and inferential modes of reasoning in the precedent, i.e., the fact that past events and behavior have been related in a certain way. Shared expectations ground the existence of conventions, that is, commonly known patterns of behavior that arise in a specific strategic context where alternative patterns could also have existed.

Even if we may discuss the details of Lewis’s account, the basic idea, anticipated by Schelling, remains insightful and applies to a large range of socioeconomic phenomena, from how we greet each other to the way we measure time and, of course, how we can understand each other. The analysis can be extended to other classes of coordination games, such as the “Hi-Lo” game where equilibria can be Pareto-ranked. An interesting aspect is that in this game, the fact that some outcome X is preferred by everyone to another outcome Y is not sufficient to make it rational for individuals to play the strategy that leads to X instead of the one that leads to Y. Shared expectations are again required. This becomes particularly relevant when you put the game into a dynamic setting where a large population of players randomly and repeatedly interact. Once the population is caught in the inferior equilibrium (say, an inferior technology), it is very difficult to steer it toward the superior equilibrium as this requires a massive coordination effort that is difficult to implement over a large scale.

The Hawk-Dove game appeals to similar ideas but models more conflictual interactions – see Figure 2 below. It is in a “mixed-motive” game as the prisoner’s dilemma except for the fact that players don’t have a dominant strategy. The two equilibria in pure strategy, where one player “attacks” and the other “concedes” are asymmetric in terms of preferences – as in the battle of the sexes – the mixed-strategy equilibrium is highly socially inefficient because players will engage in a costly fight a significant amount of time. This game is the right model to study conflicts over resources, which explains the seemingly surprising fact that it has attracted as many biologists as economists.[3]

An interesting question that can be asked is how players settled on one of the equilibria, thus avoiding costly conflicts. Economists have in particular studied this question to understand the evolution and social function of property rights.[4] Again, using a dynamic setting makes the problem even more interesting. Suppose you have a large population of players randomly interacting by pair and updating their strategy choice based on a simple heuristic, for instance by using the strategy that brought the best expected result in the previous iteration. It can be shown that the only evolutionary stable equilibrium in this game corresponds to the mixed-strategy equilibrium where a fraction p of players plays “Concede” and a fraction 1-p plays “Attacks.” This is not very good because attackers will met with probability (1-p)² which generates a social loss. Now, suppose that players can discriminate against each other through the use of a particular credible signal or the public observation of some personal characteristic. More specifically, suppose that in each specific interaction, one of the two players can non-ambiguously be characterized as the “incumbent” and the other the “challenger.” This gives rise to alternative strategies such as “Attack if incumbent, Concede if challenger.” In this setting, such a “correlated strategy” is an evolutionary stable equilibrium: if everyone in the population plays it, there is no way that someone playing an alternative strategy can do better. What you have here is an evolutionary account of the emergence of correlating devices that coordinate individuals’ expectations and behavior to some exogenous signal. Property rights and, as the late Herbert Gintis thoroughly argued, social norms more generally can be analyzed as correlation devices in this sense.[5]                   

Note that that correlation can lead to socially adverse effects. For instance, suppose that the trait that discriminates the members of the population is permanent. For instance, we can divide the population between the “reds” and the “greens” and assume that interactions never take place between players of the same color. Now, at least initially, the reds and the greens will converge toward the same evolutionary stable equilibrium as in the initial game above (a fraction p of reds and greens will play Concede, a fraction 1-p of reds and greens will play “Attack”). But suppose that for no reason other than randomness, a slightly higher fraction of greens than reds play “Concede.” Reds who play “Attack” will do slightly better than those who play “Concede.” More and more reds will settle for “Attack.” In the meantime, greens who play “Concede” will do slightly better than those who play “Attacks” and the gap will increase as the fraction of attacking red players rises. Thus, more and more greens will play “Concede.” The evolutionary stable equilibrium is the state where all reds play “Attack” and all greens play “Concede”. This is a highly unbalanced equilibrium and, presumably, an unfair one because it has no other explanation than randomness. Indeed, it corresponds to an evolutionary version of the phenomenon known as statistical discrimination.

An interesting question is how to account for the origins of correlation devices. In my short story about reds and greens, players’ colors emerge spontaneously as a correlation device by statistical necessity. But suppose that players can be differentiated by other characteristics. What accounts for the fact that one type of characteristic will be used as a correlation device but not the other? The same applies to the distinction between incumbents and challengers. You can basically tell stories here. You can argue that the emergence of correlation devices is purely random, even if eventually biased by naturalistic factors that make the emergence of some devices more likely than others. This is the story told by Brian Skyrms among others.[6] Or you can argue that the “choice” of a correlation device is itself the product of shared expectations and inferential modes of reasoning that are constitutive of some cultures and forms of life,[7] similar to the way Schelling and Lewis explain coordination. These two stories are not necessarily mutually exclusive. The ultimate cause of correlation is probably to search in our evolutionary history. But the best proximate explanation is in our particular forms of life. What we take for granted – the signals that help us coordinate our expectations and behavior – is constitutive of who we are as persons and collectives.      

---

[1] David K. Lewis, Convention: A Philosophical Study (John Wiley and Sons, 1969 [2002]).

[2] Thomas C. Schelling, The Strategy of Conflict (Harvard University Press, 1960 [1981]).

[3] The game is largely discussed by the biologist John Maynard Smith in his classic Evolution and the Theory of Games (Cambridge University Press, 1982).

[4] Robert Sugden, The Economics of Rights, Cooperation and Welfare, 2nd ed. (Palgrave Macmillan, 1986 [2005]).

[5] Herbert Gintis, The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences (Princeton University Press, 2009).

[6] Brian Skyrms, Evolution of the Social Contract (Cambridge University Press, 1996). Brian Skyrms, The Stag Hunt and the Evolution of Social Structure (Cambridge University Press, 2004).

[7] This is what I’m defending in Cyril Hédoin, “A Framework for Community-Based Salience: Common Knowledge, Common Understanding and Community-Membership,” Economics and Philosophy 30, no. 3 (November 2014): 365–95.