Power, Prices, and Incomes in Voting Systems

For a lobbyist bribing a legislature, equilibrium prices for the legislators are shown to exist for any voting game without veto players. The resulting expected incomes of the legislators provide a new measure of their relative power

The Market Value of a Game

A correspondence is observed between a class of n-person cooperative games and production functions with fixed, discrete factor inputs. This correspondence motivates a simple way of valuing the players (or factors): the players, or factor representatives, set prices on themselves in the face of a market of buyers. A noncooperative price-setting game results for which equilibrium prices always exist. Interpreted as a cooperative game it always has a core, which reduces to the core of the original game if the latter is nonempty. This concept was originally applied to the problem of determining the relative value of the players in a voting game when a market exists for their votes

Social Justice and Individual Choice

Consider a divisible resource or cost that is to be fairly distributed among a group, and suppose that various members of the group have different opinions about what a fair distribution might be. We exhibit a class of mechanisms that aggregate individual opinions into a group opinion, and which have the property that no one can manipulate the size of his own share. There is a unique such mechanism that satisfies a variant of Arrow's conditions for social choice functions. We illustrate its application to distributing dues among the member countries of IIASA. Other potential applications include distributing shares in formerly state-owned enterprises, and in allocating voting power among different states or regions in a federal system of government

Lobbying and Campaigning with Applications to the Measure of Power

Two models are developed of the relative power of voters in a voting body, both based on the idea of lobbyists or interest groups creating a "market" for votes. In the first model, a single lobbyist attempts to put together a winning coalition at least cost, and the voters play an n-person game among themselves to determine who will be part of this coalition. Equilibrium prices for the voters result (unless there is a "veto player"); in fact, a novel and particularly strong equilibrium concept for the solution of this class of games obtains. The expected payoffs to the voters constitute one way of measuring their relative power, and the results are compared with traditional measures like the Shapley-Shubik index and the Banzhaf index. In the second model, two lobbyists are imagined to be in competition for votes. A 2-person game results, and the expected payoffs to the voters (if an equilibrium exists) can be said to constitute a measure of power in a competitive environment. Explicit solutions to the model are obtained when the opposing lobbyists are unevenly matched in terms of resources, and the results are applied to such diverse problems as Presidential campaigning for the U.S. Electoral College and the setting of legislators' salaries

Optimal Group Decisions

Consider a group of individuals who must rank a set of decisions or choices. If the members of the group disagree, how should their opinions be reconciled into a group ordering? Historically we may discern two ways of answering this question. The "relativist" approach, which is the dominant one in the modern social choice literature, holds that differences of opinion arise largely from differences in preferences or values. Hence the objective should be to strike a fair compromise between differences of opinion. The "rationalist" approach, which was an article of faith among the eighteenth-century founders of voting theory, holds that differences of opinion arise from misperceptions about the true merit of different decisions. For them the goal was to find the ordering that is most likely to be "correct" or "true". These two positions are not incompatible. Under suitable conditions, in fact, they yield the same method, which was first proposed in a rudimentary form by Condorcet. We show that it can be characterized by a slight weakening of the independence of irrelevant alternative condition

On Permutations and Permutation Polytopes

Permutation polytopes arise in a class of problems in which the objective is to find an optimal complete ordering of some given alternatives, subject to a linear objective criterion. In this paper an easy characterization is given of neighbors on permutation polytopes. Using this characterization it is shown that the graph of any such polytope is Hamiltonian, and that the diameter is two. The methods used are combinatorial in nature

Cost Allocation and Demand Revelation in Public Enterprises

A classic problem in both the public economics and game theory literature is how to allocate the joint costs of a public enterprise equitably among the customers. Traditional normative solutions, like Ramsey pricing or the Shapley value, have the distinct disadvantage of requiring full information about demand, which in practice may not be known. This paper describes a simple noncooperative bidding mechanism that discovers the efficient set to serve and simultaneously allocates costs. Consumers bid to be served and the game regulator offers to serve that' coalition that maximizes net surplus. It is shown that a Nash equilibrium -- indeed a strong Nash equilibrium -- for this noncooperative game always exists, no matter what the cost function, and the resulting set of consumers served is economically efficient. The resulting allocations constitute a normative solution concept for cooperative games that is apparently new and generalizes the core in a natural way. The principal application is to determine prices and output levels for a regulated public enterprise with incomplete information about demand

Valuation of Games and Other Discrete Production Processes by Competitive Bidding

The values of the players in an n-person cooperative game are analyzed by considering a simple auction model in which outside "entrepreneurs" bid to acquire control of the players. This bidding procedure always has a Nash equilibrium in pure strategies, thus yielding a concept of "market values" for the players. This class of values is easy to characterize and contains the core of the game. The model applies to various valuation problems (such as estate auctions, the setting of wage structures for laborers, or the valuation of divisions of a corporation) in which indivisible factors are present and there may be increasing returns to scale