Showing papers by "Pradeep Dubey published in 1979"

Mathematical Properties of the Banzhaf Power Index

Abstract: The Banzhaf index of power in a voting situation depends on the number of ways in which each voter can effect a “swing” in the outcome. It is comparable—but not actually equivalent—to the better-known Shapley-Shubik index, which depends on the number of alignments or “orders of support” in which each voter is pivotal. This paper investigates some properties of the Banzhaf index, the main topics being its derivation from axioms and its behavior in weighted-voting models when the number of small voters tends to infinity. These matters have previously been studied from the Shapley-Shubik viewpoint, but the present work reveals some striking differences between the two indices. The paper also attempts to promote better communication and less duplication of mathematical effort by identifying and describing several other theories, formally equivalent to Banzhaf’s, that are found in fields ranging from sociology to electrical engineering. An extensive bibliography is provided.

Price-Quantity Strategic Market Games

Abstract: THIS PAPER IS yet another in a rapidly growing series (e.g. [2]-[11]) on strategic approaches to economic equilibrium. Our aim here is to make precise the remark implicitly due to Bertrand [1 or 10, Ch. 4, 5] that if the agents in an economy use price-setting strategies then the strategic Nash equilibria will, in fact, be Walrasian; and this without any assumption of a "large number of small agents." However, in our models, not only prices but also quantities are set by the agents. Thus it might actually be more appropriate to call them "Bertrand-Cournot" types of models. We begin with a standard Walras exchange economy with a finite number of traders and commodities. This is recast as a game in strategic form in essentially two different ways. There is a trading-post for each commodity to which traders send contingent statements about how much they wish to buy and sell, and at what prices. In Model 1, the trading point is determined by the intersection of the aggregate supply and demand curves. In Model 2, trade takes place so as to meet as many contingent statements as possible. Each buyer whose orders are filled pays the price he quoted, using a fiat money which can be borrowed costlessly and limitlessly. But after trade is over there is a settlement of accounts and a penalty is levied on those who are bankrupt. No attempt is made to model the penalty in any detail. It is simply described in the form of a disutility. But this in turn may be imagined to stem from confiscation of assets (see Remark 5) or the necessity of procuring highly-priced loans, etc. Call a noncooperative equilibrium "active" if it turns out that no trader is isolated, i.e., trapped as the sole buyer or seller at some trading-post. Then our results may be described as follows. In Model 1, the active2 N.E. of the game coincide with the C.E. of the market; furthermore there is a subset of tight, active N.E. which also coincide with the C.E., and each N.E. in this subset is strong

Value Theory without Efficiency

Abstract: A semivalue is a symmetric positive linear operator on a space of games, which leaves the additive games fixed. Such an operator satisfies all of the axioms defining the Shapley value, with the possible exception of the efficiency axiom. The class of semivalues is completely characterized for the space of finite-player games, and for the space pNA of nonatomic games.(This abstract was borrowed from another version of this item.)