15 - Stability of equilibria

Summary

At a Nash equilibrium a balance is struck between the players’ strategies: each player chooses a strategy that is optimal for her, given the strategies chosen by the other players. The equilibrium concept does not deal with the question of how such a balance is created, or what will happen if it is upset. In other words, the equilibrium is a static concept and does not address the question of what dynamics (if any) could possibly cause the players to choose, or to gradually approach, the equilibrium strategies.

A large number of dynamic processes are conceivable in which, over time, each player repeatedly updates her choice, and in so doing studies the moves of the other players and the payoffs she is getting. Of course, different processes will correspond to different assumptions concerning the players’ degree of sophistication, the information available to them, the memory resources and computational capability at their disposal, and so on. Accordingly, an important branch of modern game theory is called Learning in Games and this topic is one of the frontiers of game-theoretic research.

We will now proceed to describe two key types of updating processes.

Updating processes

Eductive processes

In eductive processes, the game is played one single time. Before it starts, each player mulls over various possible strategy profiles that both she and the other players may adopt, and in an iterative process progressively narrows down the possibilities that appear to her as reasonable.

In preceding chapters, we have already encountered two such types of eductive processes:

• iterative elimination of strongly dominated strategies;

• iterative elimination of weakly dominated strategies.

We have also seen that in certain games (such as Divvying up the Jackpot) the eductive process of iterative elimination of weakly dominated strategies can lead the players to exclude certain Nash equilibria.