Steven J. Brams

The 3/2's Rule in Presidential Campaigning

The purpose of this article is to assess the effect of the winner-take-all feature of the Electoral College on the allocation of resources by candidates to the states in a presidential campaign. Conceptualizing the campaign as a two-person zero-sum infinite game, it is found that the main effect of this feature is to induce candidates to allocate campaign resources roughly in proportion to the 3/2s power of the electoral votes of each state, which creates a peculiar bias that makes voters living in the largest states as much as three times as attractive campaign targets as voters living in the smallest states. Empirically, it is shown that the 3/2s rule explains quite well the time allocations of presidential and vice-presidential candidates in the 1960, 1964, 1968, and 1972 campaigns; for presidential campaigns in 1976 and 1980, optimal allocations are indicated for all fifty states and the District of Columbia. A comparison with optimal allocations under a system of direct popular-vote election of the president reveals that such a system would be less susceptible to manipulative strategies than the Electoral College as well as being compatible with the egalitarian principle of “one man, one vote.”

Power and size: A new paradox

An analysis of several different indices of voting power reveals that the voting power of a member of a weighted voting body may increase, rather than decrease, when new members are added to the original body. Real instances of this phenomenon, called the paradox of new members, are shown to have occurred when new states were added to the U.S. Electoral College and new countries to the European Community Council of Ministers. Conditions for the existence of the paradox, and probabilities of its occurrence in small and moderate-size voting bodies, are given. Efficient algorithms for the calculation of the voting power indices, based on generating functions, are also outlined.

Superpower games : applying game theory to superpower conflict

Routledge Performance Practitioners is a series of introductory guides to the key theatre-makers of the 20th century. Each volume explains the background to and the work of one of the major influences on 20th- and 21st-century performance. These compact, well-illustrated and clearly written books unravel the contribution of modern theatres most charismatic innovators, through: personal biography; explanation of key writings; description of significant productions; and reproduction of practical exercises.

Nonmyopic Equilibria in 2×2 Games

In this paper we shall define a new concept of equilibrium in elementary normal-form, or matrix, games, based on the idea that players can look ahead and anticipate where a process might end up if they are allowed to make an indefinite number of sequential moves and countermoves from any outcome in a game. We call this concept a nonmyopic equilibrium and shall show how it differs from the more myopic equilibrium concepts of Nash and Stackelberg, which consider only unilateral deviations of players, or single best responses by players to another player’s strategy choice. We shall also point out that solution concepts in cooperative game theory are encumbered by a similar myopia. Because the concept of a nonmyopic equilibrium is a rather subtle one, we shall, after reviewing previous research in section 2, restrict our analysis to 2 x 2 ordinal games, in which each player can rank the four possible outcomes from best to worst. After defining this concept in section 3, and comparing it with other equiIibrium concepts in section 4, we shall show in section 5 that it exists in 37 of the 78 distinct 2 x 2 games (47 percent), generally coinciding with pure-strategy Nash equilibria in these games. In two of the 37 games, however, the nonmyopic equilibria do not

Going from Theory to Practice: The Mixed Success of Approval Voting

Approval voting (AV) is a voting system in which voters can vote for, or approve of, as many candidates as they like in multicandidate elections. In 1987 and 1988, four scientific and engineering societies, collectively comprising several hundred thousand members, used AV for the first time. Since then, about half a dozen other societies have adopted AV. Usually its adoption was seriously debated, but other times pragmatic or political considerations proved decisive in its selection. While AV has an ancient pedigree, its recent history is the focus of this paper. Ballot data from some of the societies that adopted AV are used to compare theoretical results with experience, including the nature of voting under AV and the kinds of candidates that are elected. Although the use of AV is generally considered to have been successful in the societies—living up to the rhetoric of its proponents—AV has been a controversial reform. AV is not currently used in any public elections, despite efforts to institute it, so its success should be judged as mixed. The chief reason for its nonadoption in public elections, and by some societies, seems to be a lack of key “insider” support.

Transaction Flows in the International System

When we speak of an international system , we start with the presumption that there is something habitual and regular about the behavior of the nations that constitute it. Unfortunately, the concept of an international system has had a singularly hollow ring in the works of many scholars who have employed the term. It is frequently compared to an incredibly complicated watch or thermostat, or alternatively it is defined so abstractly that it would appear to have no specific empirical referents—and therefore practically everything in one way or another would qualify as a “system.” The abstract and shadowy significance of the concept in international relations studies has retarded its usefulness for exploring the regularities that underlie the interactions of nations. More than ever before, however, the actions of nations have multiple reverberations on each other and can be ascribed meaning only within the context of the relations of many nations with each other. Because the configuration of inter-nation relations has become increasingly complex, it has become more and more difficult to trace out these relations and determine what structure, if any, there is in the “system.” We shall see later that any definition of a system is arbitrary to the extent that its inclusion and exclusion rules are arbitrary. If we can specify the simplifying assumptions which create this arbitrariness, however, then the problematic cases included or excluded in a system or component subsystems can usually be identified and explained. This approach seems preferable to positing systems criteria that are either ambiguous or non-operational, enriching the vocabulary but not the analysis.

Applied Game Theory

This paper is concerned with the formalization of the notion of power in the context of two person negotiations. The notion of power is defined as follows: the amount of power of A over B is related to the level of achievement of Bs objectives in the interaction A-B. As for the origin of power, it is taken as related to the strategies available to both parties in their interaction huch as rewards, punishments, etc.). The development of these ideas relies mainly on the concepts of game theory but specific factors drawn from the psychological and psychosociologicalliterature, such as the notion of representation, arc explicitely introduced in the model. Besides the formalization, the paper includes a detailed discussion of two examples enhancing the role of representation in power analysis. An axiomatic treatment of the model is briefly reported in the appendix.

Efficient Fair Division Help the Worst off or Avoid Envy

Two or more players rank a set of indivisible items from best to worst. An efficient allocation of items is characterized, which may satisfy such properties as maximin, Borda maximin, and envy-avoidance. Whereas the two maximin properties are in conflict with envy-avoidance, there is always an efficient allocation that does not ensure envy, but it may not be maximin or Borda maximin. Computer calculations show that maximin allocations lead to envy quite often, but Borda maximin allocations do so only rarely. Implications of the theoretical findings for real-world fair-division problems are discussed.

Competitive Fair Division

Several indivisible goods are to be divided among two or more players, whose bids for the goods determine their prices. An equitable assignment of the goods at competitive prices is given by a fair‐division procedure, called the Gap Procedure, that ensures (1) nonnegative prices that never exceed the bid of the player receiving the good; (2) Pareto optimality, though coupled with possible envy; (3) monotonicity, such that higher bids never hurt in obtaining a good; (4) sincere bids that preclude negative utility; and (5) prices that are partially independent of the amounts bid (as in a Vickrey auction). A variety of applications are discussed.

Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity

Abstract. This paper focuses on the fair division of a set of indivisible items between two people when both have the same linear preference order on the items but may have different preferences over subsets of items. Surprisingly, divisions that are envy-free, Pareto-optimal, and ensure that the less well-off person does as well as possible (i.e., are equitable) can often be achieved. Preferences between subsets are assumed to satisfy axioms of qualitative probability without implying the existence of additive utilities, which is treated as a special case. Algorithms that render fair division practicable are proposed, and their vulnerability to strategic manipulation is investigated.