IV - General Model of Repeated Games

Summary

THE MODEL

In this chapter we introduce formally the general model of repeated games.

We start with a non-cooperative game G and define a new game Г_∞, a play of which is an infinite sequence of plays of G.

In fact, it appears in many applications that current moves not only influence the current payoff but also the future play, hence some state variable of the model. This is the reason why stochastic games appear in a natural way.

Moreover we have to describe the information available to the players. There may be some differences between their initial knowledge of the characteristics: initial state, preferences, even transition law. This is taken into account in the framework of games with incomplete information.

Finally it is necessary for a full description of the game to specify what additional information is transmitted to the players after each stage of the play. It is easy to see that assuming the knowledge of the other players' strategies is unrealistic. A more plausible weaker assumption may be that only the actual moves are observed. An even weaker assumption is that only the individual player's payoff is known to him. Even more generally, we may consider a model without full monitoring of previous moves, or of the outcomes, or even of the players' own payoffs. This leads to the notion of signals that may depend in a random way on the actual moves and state.

To integrate all such effects it is sufficient to define a state- and move-dependent lottery that selects at every stage the signals for the players, their payoffs, and the next state of nature. If one wants in addition to incorporate the effect of information lags, this transition may also depend on the past events. In fact, we will see that this quite huge construction can be reduced to a simple and convenient form (cf. Proposition IV.2.3, p. 183).

It follows from the above presentation that this model is an adequate description of a stationary multistage game in the sense that its formulation is time-shift invariant (which may require adding new states and payoffs, if necessary) and needs only some counter to let the stage of the game be known to the players.