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

[A decision scheme makes the probabilities of alternatives depend on individual strong orderings of them. It is strategy-proof if it logically precludes anyone's advantageously misrepresenting his preferences. It is unilateral if only one individual can affect the outcome, and duple if it restricts the final outcome to a fixed pair of alternatives. Any strategy-proof decision scheme, it is shown, is a probability mixture of schemes each of which is unilateral or duple. If it guarantees Pareto optimal outcomes, it is a probability mixture of dictatorial schemes. If it guarantees ex ante Pareto optimal lotteries, it is dictatorial.]

It has been conjectured that no system of can preclude strategic voting-the securing by a voter of an outcome he prefers through misrepresentation of his preferences. In this paper, for all significant systems of in which chance plays no role, the conjecture is verified. To prove the conjecture, a more general theorem in game theory is proved: a gameform is a game without utilities attached to outcomes; only a trivial game form, it is shown, can guarantee that whatever the utilities of the players may be, each player will have a dominant pure strategy. I SHALL PROVE in this paper that any non-dictatorial scheme with at least three possible outcomes is subject to individual manipulation. By a voting I mean any scheme which makes a community's choice depend entirely on individuals' professed preferences among the alternatives. An individual manipulates the scheme if, by misrepresenting his preferences, he secures an outcome he prefers to the outcome-the choice the community would make if he expressed his true preferences. The result on schemes follows from a theorem I shall prove which covers schemes of a more general kind. Let a gameform be any scheme which makes an outcome depend on individual actions of some specified sort, which I shall call strategies. A scheme, then, is a game form in which a strategy is a profession of preferences, but many game forms are not schemes. Call a strategy dominant for someone if, whatever anyone else does, it achieves his goals at least as well as would any alternative strategy. Only trivial game forms, I shall show, ensure that each individual, no matter what his preferences are, will have available a dominant strategy. Hence in particular, no non-trivial scheme guarantees that honest expression of preferences is a dominant strategy. These results are spelled out and proved in Section 3. The theorems in this paper should come as no surprise. It is well-known that many schemes in common use are subject to individual manipulation. Consider a rank-order scheme: each voter reports his preferences among the alternatives by ranking them on a ballot; first place on a ballot gives an alternative four votes, second place three, third place two, and fourth place one. The alternative with the greatest total number of votes wins. Here is a case in which an individual can manipulate the scheme. There are three voters and four alternatives; voter a ranks the alternatives in order xyzw on his ballot; voter b in order wxyz; and voter c's true preference ordering is wxyz. If c votes honestly, then, the winner is his second choice, x, with ten points. If c pretends that x is his last choice by giving his preference ordering as wyzx, then x gets only eight points, and c's first choice, w, wins with nine points. Thus c does best to misrepresent his