The paradox of multiple elections

Assume that voters must choose between voting yes (Y) and voting no (N) on three propositions on a referendum. If the winning combination is NYY on the first, second, and third propositions, respectively, the paradox of multiple elections is that NYY can receive the fewest votes of the 2³ = 8 combinations. Several variants of this paradox are illustrated, and necessary and sufficient conditions for its occurrence, related to the “incoherence” of support, are given. The paradox is shown, via an isomorphism, to be a generalization of the well-known paradox of voting. One real-life example of the paradox involving voting on propositions in California, in which not a single voter voted on the winning side of all the propositions, is given. Several empirical examples of variants of the paradox that manifested themselves in federal elections – one of which led to divided government – and legislative votes in the US House of Representatives, are also analyzed. Possible normative implications of the paradox, such as allowing voters to vote directly for combinations using approval voting or the Borda count, are discussed. Received: 31 July 1996 / Accepted: 1 October 1996     

The undercut procedure: an algorithm for the envy-free division of indivisible items

We propose a procedure for dividing a set of indivisible items between two players. We assume that each player’s preference over subsets of items is consistent with a strict ranking of the items, and that neither player has information about the other’s preferences. Our procedure ensures an envy-free division—each player receives a subset of items that it values more than the other player’s complementary subset—given that an envy-free division of “contested items,” which the players would choose at the same time, is possible. We show that the possibility of one player’s undercutting the other’s proposal, and implementing the reduced subset for himself or herself, makes the proposer “reasonable,” and generally leads to an envy-free division, even when the players rank items exactly the same. Although the undercut procedure is manipulable and its envy-free allocation may be Pareto-inferior, each player’s maximin strategy is to be truthful. Applications of the procedure are discussed briefly.     

A decision support system for the graph model of conflicts

A comprehensive decision support system called GMCA (Graph Model for Conflict Analysis) implementing the multi-player graph model for analyzing conflicts is developed. GMCA contains algorithms for the rapid computation of a wide range of solution concepts, thereby enabling decision makers to take account of the diversity of human behavior. Using an engineering case study, the key features of GMCA are illustrated.     

A case-based distance method for screening in multiple-criteria decision aid

Screening is a process of multiple-criteria decision aid (MCDA) in which a large set of alternatives is reduced to a smaller set that most likely contains the best choice. We propose screening using a distance model calibrated on the basis of the decision-maker's own judgement. Viewing MCDA as preference aggregation based on consequence data, we define consequence and preference expressions (values and weights) and describe how they are aggregated. Then we define screening and explain some of its properties. Using an appropriate definition of distance, our case-based distance method screens a set of alternatives using criterion weights and a distance threshold obtained by quadratic optimization using the decision-maker's selection of alternatives from a test set. This case-based method can elicit the decision maker's preferences more expeditiously and accurately than direct inquiry. An application in water supply planning is used to demonstrate the procedure.     

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.     

Backward Induction Is Not Robust: The Parity Problem and the Uncertainty Problem

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed.     

Non-strict ordinal 2 × 2 games: A comprehensive computer-assisted analysis of the 726 possibilities

Game theory has provided many tools for the study of social conflict. The 2 × 2 game has been found to be a particularly useful model. This paper describes the enumeration and analysis of all 726 distinct 2 × 2 games. A computer is used to generate the complete set, and a wide variety of maximin, equilibrium and stability calculations is performed for each player for every outcome in every game. The resulting data set is of great value for both the modeling and analysis of social conflict.     

Theory and implementation of coalitional analysis in cooperative decision making

Stability definitions for describing human behavior under conflict when coalitions may form are generalized within the Graph Model for Conflict Resolution and algebraic formulations of these definitions are provided to allow computer implementation. The more general definitions of coalitional stabilities relax the assumption of transitive graphs capturing movements under the control of decision makers, either independently or cooperatively, and allow the convenient expansion to the case of coalitions of the four basic individual stabilities consisting of Nash stability, general metarationality, symmetric metarationality, and sequential stability. To permit the various coalitional stabilities to be efficiently calculated and conveniently encoded within a decision support system, algebraic expressions for the coalitional stabilities are provided in this research. Furthermore, a range of the theorems establish the mathematical credibility of employing the innovative algebraic approach to conflict resolution when coalitions are present. Finally, a conflict over the proposed exportation of bulk water from Lake Gisborne within the Canadian Province of Newfoundland and Labrador is modelled and analyzed to illustrate the practical application of the different coalitional stabilities and the strategic insights they provide.     

The use of costless inspection in enforcement

The process whereby a reluctant decisionmaker may be induced to behave in accordance with a rule, regulation, or law, is examined in a general context. It is assumed that an enforcement agency can make use of a costless inspection process, which would provide a free, but possibly inaccurate, assessment of the subject's behavior. The agency also has some power of punishment. When should evidence be collected by the agency, and how should it be used? How can an enforcement system be structured so as to maximize the level of compliance? How does compliance depend on the subject's private gain for violation, the level of punishment, the risk of errors (of both kinds) in the inspection report, and other parameters? These questions and others are addressed using non-cooperative game models. Answers give new insights into real-world enforcement problems in areas as diverse as environmental regulation and arms control.