Nicholas R. Miller

Game Theory and Political Theory

Formal political theory seeks to develop formal, mathematical models of political and economic processes. This book attempts to integrate the last twenty years of development in this field. Professor Ordeshook uses the modern developments in the theory of games (decision making with multiple, interactive decision makers) as the basis for the synthesis. Topics covered include models of elections and of committee processes, the demand and supply of public goods, and surveys of game theory and social-choice theory. Game Theory and Political Theory is designed as a textbook for graduate courses in formal political theory and political economy.

Pluralism and Social Choice

Pluralist political theory identifies certain patterns of political preferences as promoting the “stability” of democratic political systems and others as threatening to such stability. Social choice theory likewise identifies certain patterns of political preferences as leading to “stability” in social choice under majority rule and related collective decision rules, and other patterns as leading to “unstable” social choice. But the preference patterns identified by pluralist theory as promoting stability are essentially those identified by social choice theory as entailing instability. Thus the notions of stability and the implicit normative criteria associated with the two theories are very close to being logically incompatible. This incompatibility suggests that the social choice ideal of collective rationality may not be one that we should endorse. Indeed, the generic instability of the pluralist political process and its consequent collective irrationality may contribute to the stability of pluralist political systems.

The uncovered set in spatial voting games

In a majority rule voting game, the uncovered set is the set of alternatives each of which can defeat every other alternative in the space either directly or indirectly at one remove. Research has suggested that outcomes under most reasonable agenda processes (both sincere and sophisticated) will be confined to the uncovered set. Most research on the uncovered set has been done in the context of voting games with a finite number of alternatives and relatively little is known about the properties of the uncovered set in spatial voting games.We examine the geometry of the uncovered set in spatial voting games and the geometry of two important subsets of the uncovered set, the Banks set and the Schattschneider set. In particular, we find both general upper and lower limits on the size of the uncovered set, and we give the exact bounds of the uncovered set for situations with three voters. For situations with three voters, we show that the Banks set is identical to the uncovered set.

Centripetal forces in spatial voting: On the size of the Yolk

The yolk, the smallest circle which intersects all median lines, has been shown to be an important tool in understanding the nature of majority voting in a spatial voting context. The center of the yolk is a natural ‘center’ of the set of voter ideal points. The radius of the yolk can be used to provide bounds on the size of the feasible set of outcomes of sophisticated voting under standard amendment procedure, and on the limits of agenda manipulation and cycling when voting is sincere. We show that under many plausible conditions the yolk can be expected to be small. Thus, majority rule processes in spatial voting games will be far better behaved than has commonly been supposed, and the possible outcomes of agenda manipulations will be generally constrained. This result was first conjectured by Tullock (1967).

Election Inversions under Proportional Representation

It has been shown by Peter Kurrild-Klitgaard, using several empirical examples under the Danish electoral system, that proportional representation (PR) can produce ‘election inversions’ such that a coalition of parties collectively supported by a majority of voters fails to win a majority of parliamentary seats. However, Kurrild-Klitgaards examples result from imperfections in the Danish PR system introduced to serve goals other than proportionality. In this article, Kurrild-Klitgaards analysis is carried a step further by showing that election inversions can occur even under the purest type of PR – namely, one with (i) a single national constituency, (ii) no explicit seat threshold, and (iii) a highly proportional electoral formula. Inversions result from the unavoidable ‘whole number problem’. Recent election data from Israel and the Netherlands is examined and examples of inversions under their relatively pure PR systems are found. Inversions are also found after recalculating seat allocations without a threshold, and on the basis of the most proportional electoral formulas and when the analysis is restricted to seat-winning parties. Kurrild-Klitgaards Danish data is then re-examined in the same fashion, as is the most recent apportionment of seats in the United States House of Representatives, and more examples of inversions are found.

WHAT TO DO ABOUT ELECTION INVERSIONS UNDER PROPORTIONAL REPRESENTATION

An election inversion occurs when a party, or coalition of parties, wins more seats in the legislature than another party, or coalition of parties, despite the fact that the other party, or coalition of parties, won a larger number of votes cast in an election. It is well known that such inversions can and do occur under the kind of ‘majoritarian’ electoral systems commonly used in English-speaking countries. It has recently been observed that such inversions not only can and do occur also under party-list proportional representation (PR) systems, but also that they are unavoidable due to the ‘whole number problem’. This finding raises the question of whether and how the problem of election inversions under PR systems might be solved or mitigated. We consider a number of options and point out their relative advantages and disadvantages.

A Priori Voting Power and the US Electoral College

This chapter uses the Banzhaf power measure to calculate the a priori voting power of individual voters under the existing Electoral College system for electing the President of the United States, as well as under variants of this system in which electoral votes are either apportioned among the states in a different manner or cast by the states in a different manner. While the present winner-take-all manner of casting state electoral gives a substantial advantage to voters in the largest states, this advantage is diluted by the small-state advantage in apportionment. Moreover, most of the alternative Electoral College plans that have been proposed to remedy this large-state advantage give an equally substantial voting power advantage to voters in small states. Direct popular election of the President uniquely maximizes and equalizes individual voting power.

The structure of the Banks set

We consider a new solution set for majority voting tournaments recently proposed by Banks (1985), and we examine its internal structure. In particular, we demonstrate that, in the absence of a Condorcet winner, there is always a cycle including precisely the points in the Banks set. We introduce the concept of “external stability” in order to facilitate analysis.

Cycle avoiding trajectories, strategic agendas, and the duality of memory and foresight: An informal exposition*

This paper considers the notion of cycle avoiding trajectories in majority voting tournaments and shows that they underlie and guide several apparently disparate voting processes. The set of alternatives that are maximal with respect to such trajectories constitutes a new solution set of considerable significance. It may be dubbed the Banks set, in recognition of the important paper by Banks (1985) that first made use of this set. The purpose of this paper is to informally demonstrate that the Banks set is a solution set of broad relevance for understanding group decision making in both cooperative and non-cooperative settings and under both sincere and sophisticated voting. In addition, we show how sincere and sophisticated voting processes can be viewed as mirror images of one another — embodying respectively, “dmemory” and “foresight.” We also show how to develop the idea of a “sophisticated agenda,” one in which the choice of what alternatives to propose is itself a matter of strategic calculation.

Election Inversions by the U.S. Electoral College

An election inversion occurs when the candidate (or party) that wins the most votes from an electorate fails to win the most electoral votes (or parliamentary seats) and therefore loses the election. Public commentary commonly uses terms such as “reversal of winners,” “wrong winner,” “divided verdict,” and “misfire” to describe this phenomenon; the academic social choice literature adds such terms as “repre- sentative inconsistency,” “compound majority paradox,” “referendum paradox,” and “majority deficit.” Election inversions can occur under any two-tier electoral system, including the U.S. Electoral College. As is well known, the Electoral College actually produced a “wrong winner” in the 2000 Presidential election, and it has done so twice before.