Information about Moves in Extensive Games. I,
YALE UNIV NEW HAVEN CT COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
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In this paper, we explore the relation between information patterns and Nash Equilibria in extensive games. By information we mean what players know about each others moves. Also we confine ourselves throughout to pure strategies. Our main result is that in games in which the level of information is intrinsically low, the Nash outcomes are invariant of the information. The extensive game model is of fundamental importance and captures the interplay between information and decision-making. Yet we find that its definition, as set forth by Kuhn 1953, is insufficient from certain points-of-view. It is unable to incorporate games with a continuum of players. Also it often makes for an unnaturally complex representation. For instance, a game in which n players more simultaneously can be described in the Kuhn framework. But first we would have to order the players artificially and then have them move in sequence with suitably enlarged information sets. If we try to carry this out when n is not finite but a continuum, the difficulty of the procedure becomes clear. Therefore, we are led to develop a variant model which has the feature that several players can move simultaneously at any position in the game. Games of the type in Kuhns paper are, of course, included as a special case of our set-up. In Section 2 we develop our model and illustrate it with examples. In Section 3 we show that among all possible information patterns in a game there is a unique minimal one. This is done under the assumption of perfect recall. Otherwise the conclusion is false. In the rest of the paper, we focus on the effect on Nash Equilibria that is caused solely by changes in the information pattern of an extensive game.
- Numerical Mathematics