Straightforwardness of Game Forms with Lotteries as Outcomes

A GAME FORM IS ANY SYSTEM which makes an outcome depend on individual actions of some kind, called strategies. Prime examples are systems of voting. Where voting consists of each person's marking a ballot, a strategy is simply a way of marking one's ballot. In the case of sequential voting on a number of motions, a strategy will be a contingency plan which tells how to vote on each motion as a function of the way the votes on previous motions have gone. By no means all game forms, though, are systems of voting; any system through which people interact exemplifies a game form.' This paper deals with game forms with a special property: that each person, no matter what his preferences are, can choose his strategy without regard to what he expects others to do. Such a game form will be called straightforward. Straightforward game forms of one kind are characterized by Gibbard 2. Call a game form determinate if it makes an outcome depend on individual strategies in a way that involves no element of chance. In the case of a determinate game form, a strategy is dominant for a player with respect to a weak ordering of the alternatives iff no matter what anyone else does, the strategy secures an outcome at least as high in that weak ordering as is any other lottery that that player can secure, with the strategies of everyone else held fixed. A determinate game form is straightforward iff each player, for each weak ordering of the alternatives, has a strategy which is dominant with respect to that weak ordering. The only straightforward determinate game forms, it turns out, are trivial: a straightforward determinate game form either restricts the attainable outcomes in advance to no more than two, or makes one player a dictator among attainable outcomes. The game forms in this paper are of a more general kind. They are systems which make an outcome depend on individual strategies in a way that may involve chance. The systems to be considered have finitely many alternatives, finitely many players, and finitely many strategies for each player. A game form