A well-known issue in game theory is that of multiple Nash equilibria in coordination games.

For example, consider the following game. Players 1 and 2 both choose either "high" or "low". If both choose "high", they both get 10 utility; if both choose "low", they both get 5 utility; if they choose different actions, then they get no utility. There are 3 Nash equilibria: one where both always play "high", one where both always play "low", and one where each plays "high" with 1/3 probability and "low" with 2/3 probability. While (high, high) is obviously the best Nash equilibrium, unilaterally switching to high when the current equilibrium is (low, low) is probably unwise.

You could imagine TDT arguments based on symmetry between the different players to argue that "high" is the only rational choice, but it is easy to imagine asymmetric variants of this problem where the symmetry arguments are much less clear.

The presence of suboptimal Nash equilibria looks like a serious problem for reconciling individual and collective rationality, even when players have common payoffs. If everyone is going to play "low", then playing "low" is individually rational (at least according to CDT); however, the collective of agents fails to act rationally due to a coordination failure.

What I hoped to do before coming up with the idea in this post is to somehow reduce collective rationality to individual rationality. Can we prescribe individual rationality conditions to each player, such that if each player follows these conditions, then a near-optimal outcome for everyone is attained?

Somewhat surprisingly, the answer is "kind of." It certainly seems like solving coordination problems requires global optimization instead of just local optimization, but under appropriate conditions, local optimization is actually sufficient.

Problem setup

We will consider the setting of normal-form common-payoff games. Let $n$ be the number of players. Let $A$ be some finite set of actions; each player $i$ will select the action $A_{i}$ . Let $U : A^{n} \to R$ be the shared utility function. Each player receives the same utility $U (a_{1}, \dots, a_{n})$ .

I will now assume that the players play some number of practice rounds before the "real" round. These practice rounds are very similar to those in the fictitious play algorithm. Ontologically, what are these practice rounds? One interpretation is that they are actual rounds

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Group Rationality

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