What about incomplete information?

This is a very common and interesting situation in reality, but, very difficult to make optimal choices if you do not know the adversary or the environment.

Many games invented by humans have this property. This is what makes interesting these otherwise boring games: Poker in complete information? Please!... Chess tries to be as in complete and perfect information as possible: you even know the ELO score of the other player.

If we don’t know some exact characteristic of the other player, for example, but we have some idea about its distribution, we can use game theory and [Savage, 1954] tells us that we can always form some probability distribution in our mind (subjective probability).

I don’t know if my competitor has low of high cost, for example, but I know/think that the chances are $50-50$

This is a risky situation, but we can work with it, thanks to [Harsanyi, 1967-68] (“Nobel” prize in economics 1994, together with Nash and Selten).

Harsanyi introduces a first fictitious player, “Nature”, who plays before the real players and choses the type of my competitor, but I cannot observe the choice of the player Nature: An incomplete information game becomes an imperfect information one.

We know how to solve these games!

The believes of players about the game and other players very become important in such a situation, and players’ choices may also reveal information about their characteristics. For example, a low cost firm will in general fix a lower price than a high cost firm, but it can also try to copy a low cost one to influence the believes of the competitor (signaling);

So the equilibria strongly depends on believes and learning of players during the game

Hence a new solution concepts: Bayesian Nash equilibrium (for static games) and Bayesian perfect equilibrium (for dynamic games), in which we determine the optimal strategies and the corresponding rational believes in the same time:

The strategies must be optimal given the believes, and the believes must be rational given the strategies.

The equilibria correspond to fixed points in the joint strategies and believes space.

This is a very smart, but demanding equilibrium concept: If I don’t know the (subjective!) beliefs of other players, how can I anticipate their behavior? we also end up with a profusion of equilibria: too many degrees of freedom when we include the believes in the equilibrium definition, and many fixed points (commonly an infinity of them) become possible. In many cases, it is not easy to really conclude anything useful from these results. This considerable multiplicity also called for even more demanding equilibrium concepts (like the Sequential equilibrium concept of [Kreps, 1982], for example), in order to try to eliminate some equilibria. For more information, see, for example, [Vega-Redondo, 2003] or [Myerson, 1991].