Tuesday, July 5, 2011

An annoying mistake people make using game theory

There's a lot of confusion between actions and strategies, at least in the minds (and presentations and papers, sadly) of some analytical modelers.

In a game each agent $i$ has a set of actions $\mathcal{A}_i$. For example, in the prisoners' dilemma, the actions are {Defect,Cooperate}; in the matching pennies game they are {H,T}.

A strategy for player $i$, $\sigma_i$, can be a simple action, in the case of pure strategies. For example, the strategy for the prisoners' dilemma is to Defect always, a pure strategy. So in this very particular case, the observed action, say $A_i \in \mathcal{A}_i$ coincides with the strategy $\sigma_i$.

A strategy can be a distribution $\sigma_i(A_i) \in \Delta(\mathcal{A}_i)$ over actions $A_i \in \mathcal{A}_i$, which is the case with mixed strategies. The balanced matching pennies game has a unique Nash equilibrium where both players play H with 1/2 probability and T with 1/2 probability.

And this is where a lot of modelers get confused.

I've heard (and read, sadly) modelers say "we never see mixed strategies, so we are going to look at equilibria with pure strategies only." (Usually even this statement is wrong. What they are looking at are "equilibria" in which players are forced to play pure strategies, which is different. These are usually not equilibria at all: typically they have competitive best responses in mixed strategies that dominate the "equilibrium" one.)

Of course you don't see mixed strategies. You never see any strategy; all you can see are actions. What you see in pure strategies is an action that happens to coincide with the strategy. In the matching pennies game, any play is executed by drawing from the distribution an action; that is what you see, say H. There's still an underlying $\sigma_i(H) = \sigma_i(T)=1/2$, but it is not visible; it must be inferred from the structure of the game's payoffs.

And, of course, a strategy can be a distribution $\sigma_i(A_i|\mathcal{I}_i)$ over $\mathcal{A}_i$ that is a function of information set of player $i$ at the time of play, $\mathcal{I}_i$, which makes things even more complicated. And more error-prone.

Some times during my first game theory course I thought all the formalism was a bit pedantic. Then I met people who didn't learn game theory properly, and realized that the formalism is there for a reason.

It removes the confusion.