Mark A. Satterthwaite

Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions

Consider a committee which must select one alternative from a set of three or more alternatives. Committee members each cast a ballot which the voting procedure counts. The voting procedure is strategy-proof if it always induces every committee member to cast a ballot revealing his preference. I prove three theorems. First, every strategy-proof voting procedure is dictatorial. Second, this paper’s strategy-proofness condition for voting procedures corresponds to Arrow’s rationality, independence of irrelevant alternatives, nonnegative response, and citizens’ sovereignty conditions for social welfare functions. Third, Arrow’s general possibility theorem is proven in a new manner. 1. INTR~OUOTI~N

Efficient Mechanisms for Bilateral Trading

We consider bargaining problems between one buyer and one seller for a single object. The seller’s valuation and the buyer’s valuation for the object are assumed to be independent random variables, and each individual’s valuation is unknown to the other. We characterize the set of allocation mechanisms that are Bayesian incentive compatible and individually rational, and show the general impossibility of ex post efficient mechanisms without outside subsidies. For a wide class of problems we show how to compute mechanisms that maximize expected total gains from trade, and mechanisms that can maximize a broker’s expected profit. Journal of Economic Literature Classification Number: 026.

The equivalence of strong positive association and strategy-proofness

Consider a group that must select one alternative from a set of three or more alternatives. Each member casts a ballot that the voting procedure counts. For a given alternative X, let two ballot profiles C and D have the property that if a member ranks alternative x above alternative y within C, then he also ranks x above that y within D. Strong positive association requires that if the voting procedure selects x when the profile is C, then it must also select x when the profile is D. We prove that strong positive association is equivalent to strategy-proofness. It therefore follows that no voting procedure exists that satisfies strong positive association, nondictatorship, and citizens’ sovereignty. Define a group to be a set N whose ) N 1 elements are the group’s members. They select one element from the set of alternatives, S, by each casting a ballot and then using a voting procedure to count the ballots. A ballot Bi is a strict ordering of the elements within S, e.g., Bi = (xyz) where S = (x, y, z> and x is ranked highest, y second highest, and z lowest. Indifference is not allowed. A voting procedure is a single-valued function v(B, ,..., BJ that evaluates the profile of ballots and selects one element of S as the group’s chosen alternative. Each member i E N has preferences Pi over the set of alternatives S, Preferences, like a ballot, are a complete, asymmetric, and transitive ordering of S. A member’s preferences Pi describe what he truly desires. For example, Pi = (xyz) denotes that individual i most prefers that the group’s choice be x, next prefers that it be y, and least prefers that it be z. An alternative notation for the preference ordering Pi = (xyz) is xPiy, xPiz, and yPiz, where xPi y means individual i prefers x toy. Similarly an alternative notation for the ballot Bi = (xyz) is xBiy, etc. Beyond completeness, asymmetry, and transitivity we place no restrictions, such as single-peakedness, on either

Bilateral trade with the sealed bid k-double auction: Existence and efficiency

For k in the unit interval, the k-double auction determines the terms of trade when a buyer and a seller negotiate transfer of an item. The buyer submits a bid b and the seller submits an offer s. Trade occurs if b exceeds s, at price kb + (1 − k) s. We model trade as a Bayesian game in which each trader privately knows his reservation value, but only has beliefs about the other traders value. Existence of a multiplicity of equilibria is proven for a class of traders beliefs. For generic beliefs, however, these equilibria are shown to be ex ante inefficient.

Competition and Market Power in Option Demand Markets

An increasing number of markets, particularly health insurance, are characterized by option demand. In option demand markets, intermediaries sell networks of upstream suppliers to downstream consumers; in many cases, consumers must attach a value to the network of suppliers before their needs are realized. Building upon a Logit model of demand, we introduce a method for modeling such markets. Ideally, the intermediaries use their expertise and bargaining power to negotiate better terms than could consumers acting on their own. However, the realization of these savings could be thwarted if the upstream suppliers have significant market power, or are allowed to merge and thereby attain market power. We estimate the model using data on inpatient hospital services in San Diego, California and use our results to simulate the effects of hospital mergers. We find that hospital markets are localized in the sense that mergers among suburban hospitals, or mergers among urban hospitals with considerable service overlap, would lead to significant price increases. This is an important finding. The 1990s witnessed a wave of hospital mergers, relatively few of which were challenged by the Federal Trade Commission or the Department of Justice. At trial after trial, the defendant hospitals prevailed; in most of these cases the judge(s) relied upon patient flow data to conclude that the relevant market is large, and therefore the merger would not be anticompetitive. Our results cast serious doubt upon this conclusion.

The Rate of Convergence to Efficiency in the Buyer's Bid Double Auction as the Market Becomes Large

A trader who privately knows his preferences may misrepresent them in order to influence the market price. This strategic behaviour may prevent realization of all gains from trade. In this paper, trade in a simple market with an explicit rule for price formation is modelled as a Bayesian game. We show that the difference between a traders bid and his reservation value is maximally O(1/m) where m is the number of traders on each side of the market. Competitive pressure as m increases thus quickly overcomes the inefficiency private information causes and forces the market towards an efficient allocation.

Social welfare functions when preferences are convex, strictly monotonic, and continuous

The paper shows that if the class of admissible preference orderings is restricted in a manner appropriate for economic and political models, then Arrows impossibility theorem for social welfare functions continues to be valid. Specifically if the space of alternatives is R+n, n ≥ 3, where each dimension represents a different public good and if each persons preferences are restricted to be convex, continuous, and strictly monotonic, then no social welfare function exists that satisfies unanimity, independence of irrelevant alternatives, and nondictatorship.

The optimality of a simple market mechanism

Strategic behavior in a finite market can cause inefficiency in the allocation, and market mechanisms differ in how successfully they limit this inefficiency. A method for ranking algorithms in computer science is adapted here to rank market mechanisms according to how quickly inefficiency diminishes as the size of the market increases. It is shown that trade at a single market-clearing price in the k-double auction is worst-case asymptotic optimal among all plausible mechanisms: evaluating mechanisms in their least favorable trading environments for each possible size of the market, the k-double auction is shown to force the worst-case inefficiency to zero at the fastest possible rate.

Computable Markov-Perfect Industry Dynamics: Existence, Purification, and Multiplicity

We provide a general model of dynamic competition in an oligopolistic industry with investment, entry, and exit. To ensure that there exists a computationally tractable Markov perfect equilibrium, we introduce firm heterogeneity in the form of randomly drawn, privately known scrap values and setup costs into the model. Our game of incomplete information always has an equilibrium in cutoff entry/exit strategies. In contrast, the existence of an equilibrium in the Ericson & Pakes (1995) model of industry dynamics requires admissibility of mixed entry/exit strategies, contrary to the assertion in their paper, that existing algorithms cannot cope with. In addition, we provide a condition on the models primitives that ensures that the equilibrium is in pure investment strategies. Building on this basic existence result, we first show that a symmetric equilibrium exists under appropriate assumptions on the models primitives. Second, we show that, as the distribution of the random scrap values/setup costs becomes degenerate, equilibria in cutoff entry/exit strategies converge to equilibria in mixed entry/exit strategies of the game of complete information. Finally, we provide the first example of multiple symmetric equilibria in this literature.

Individual decisions and group decisions : The fundamental differences

Abstract The paper contains a unified development of Arrows impossibility theorem for rational group decisions, Gibbard and Satterthwaites impossibility theorem for strategy-proof group decisions, and the close reciprocal relationship that exists between these two theorems.