# Common Knowledge Rationality Games

# Common Knowledge Rationality Games

When two or more agents knowingly interact in the sense that each knows how the outcomes for them depend not just on their own actions (strategies) but also on the actions of the others, they are playing a game. Each is rational when acting instrumentally to maximize his or her subjectively expected utility associated with the outcomes, and the game is a *common knowledge of rationality* (CKR) game when, in addition, (1) each knows that each is rational, and (2) each knows that each knows that each is rational, and so on in an infinite chain of recursively built knowledge.

The purpose of CKR is to place each agent, so to speak, in the mind of others with the result that no one will behave in a manner that surprises. In particular, CKR licenses the iterative deletion of dominated strategies. Strategies are dominated when they yield a worse outcome in every eventuality than some other available strategy (for example, the cooperative strategy in a prisoners’ dilemma game).

Thus in a two-person game, when *A* knows that *B* is rational, he or she knows that *B* will not play a strategy that is dominated (so any such strategy can be effectively deleted). *B*, knowing this, also knows that *A* will not select a dominated strategy in terms of the payoffs that remain once *B* ’s dominated strategies have been removed, and any such strategy of *A* can now also be ignored and so on. The strategies that remain are now referred to as *rationalizable* (Pearce 1984; Bernheim 1984); and in some, but far from all, games the result is a single strategy for each player (i.e., a unique prediction for rational agents will do).

It was sometimes thought that CKR delivered something potentially stronger: the Nash equilibrium solution concept, which identifies rational action with strategies that are best responses to each other. It is now typically accepted that in general the Nash equilibrium solution concept has to be motivated not only by CKR but also an assumption of common priors whereby rational agents hold a common view as to how a game will be played rationally. If there is such a unique, albeit possibly probabilistic, way in which each rational agent will play the game, then it will be apparent that, with CKR, rational actions must be best replies to each other (otherwise at least one agent would not be acting rationally).

CKR is sometimes modified so that agents only engage in some fixed level of reasoning of this sort. Thus first-order CKR refers to the case where *A* knows that *B* is rational and vice versa. Second-order CKR has in addition that *A* knows that *B* knows that *A* is rational and so on. Given the brain’s limited processing capacity, this is often more appealing than full-blown CKR; and in experiments, it seems that most people rarely engage in more than second-order CKR (see Camerer 2003).

**SEE ALSO** *Collective Action Games; Dynamic Games; Evolutionary Games; Game Theory; Nash Equilibrium; Noncooperative Games; Prisoner’s Dilemma (Economics); Screening and Signaling Theory Games; Strategic Games*

## BIBLIOGRAPHY

Bernheim, B. Douglas. 1984. Rationalizable Strategic Behavior. *Econometrica* 52: 1007–1028.

Camerer, Colin. 2003. *Behavioral Game Theory: Experiments in Strategic Interaction*. Princeton, NJ: Princeton University Press.

Pearce, David G. 1984. Rationalizable Strategic Behavior and the Problem of Perfection. *Econometrica* 52: 1029–1050.

*Shaun P. Hargreaves Heap*

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