Abstract
Do conventions need to be common knowledge in order to work? David Lewis builds this requirement into his definition of a convention. This paper explores the extent to which his approach finds support in the game theory literature. The knowledge formalism developed by Robert Aumann and others militates against Lewis’s approach, because it shows that it is almost impossible for something to become common knowledge in a large society. On the other hand, Ariel Rubinstein’s Email Game suggests that coordinated action is no less hard for rational players without a common knowledge requirement. But an unnecessary simplifying assumption in the Email Game turns out to be doing all the work, and the current paper concludes that common knowledge is better excluded from a definition of the conventions that we use to regulate our daily lives.
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Notes
All equilibria in two-person, zero-sum games are interchangeable and payoff-equivalent.
John Maynard Smith (1982) defines an evolutionarily stable strategy as a best reply to itself that is a better reply to any alternative best reply than the alternative best reply is to itself, but biologists don’t seem to worry much about the small print involving alternative best replies.
When outcomes other than just winning or losing can arise, it is necessary to interpret the payoffs as Von Neumann and Morgenstern (1944) utilities.
Lewis says that a pure strategy is strictly dominated if it is never a best reply to any strategy combination available to the other players. With this weak definition, it is false that a strictly dominated strategy is never used with positive probability in equilibrium.
When (K4) is assumed, (K0) and (K3) are redundant.
If the true state x lies in a truism T that implies KE, we first show that Alice knows that E has occurred. But if x is in T, then x is in KE, whether or not T is a truism. We next show that if Alice knows that E has occurred, then a truism T has occurred that implies E. Take T = KE. The event T is a truism, because (K3) says that T implies KT. The truism T must have occurred, because to say that Alice knows that E has occurred means that the true state x lies in KE = T.
Wherever something is asserted to be known in the standard theory, say instead that it is believed with probability at least p.
We identify a player’s informational state with the numbers of messages the player thinks it possible may have been sent. Thus {0,1} is the state in which Alice thinks either 0 or 1 messages have been sent. If Alice plays the default action DOVE in this state, it is optimal for Bob to play DOVE at {1,2}. On finding himself in this informational state, Bob believes it more likely that the number of messages is 1 rather than 2, because the second message can only go astray if the first message is received. Can it then be optimal for him to play HAWK? The most favorable case is when each of the two alternatives is equally likely, and Alice is planning to play DOVE in the informational state {2,3}. Bob might as well then be playing against someone playing each strategy in the ordinary Stag Hunt Game with equal probability, so his optimal reply is hawk, which he knows corresponds to DOVE at {1,2}. Similarly, Bob’s play of DOVE in the informational state {1,2} implies that Alice plays DOVE at {2,3}. And so on.
Mathematicians say that there is a discontinuity at infinity. That is to say, when we take the limit as N approaches infinity, we don’t get the same result as when we set N equal to infinity. (Monderer and Samet (1989) argue that we should be taking the limit as p approaches one of the common p-belief operator.)
One can restrict the number of Nash equilibria by imposing costs of sending and receiving messages, but this doesn’t affect the basic result.
The exact Nash equilibria don’t all approximate the Nash bargaining solution N because our computer implementation didn’t allow the players to vary their demands continuously.
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Binmore, K. Do Conventions Need to Be Common Knowledge?. Topoi 27, 17–27 (2008). https://doi.org/10.1007/s11245-008-9033-4
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DOI: https://doi.org/10.1007/s11245-008-9033-4