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Abstract:
Ordinally single-peaked preferences are distinguished from cardinally single-peaked preferences, in which all players have a similar perception of distances in some one-dimensional ordering. While ordinal single-peakedness can lead to disconnected coalitions that have a "hole" in the ordering, cardinal single-peakedness precludes this possibility, based on two models of coalition formation:

- Fallback (FB): Players seek coalition partners by descending lower and lower in their preference rankings until a majority coalition forms.

- Build-Up (BU): Similar to FB, except that when nonmajority subcoalitions form, they fuse into composite players, whose positions are defined cardinally and who are treated as single players in the convergence process.

FB better reflects the unconstrained, or nonmyopic, possibilities of coalition formation, whereas BU - because all subcoalition members must be included in any majority coalition that forms - restricts combinatorial possibilities and tends to produce less compact majority coalitions. Applications of the models to legislatures, parliamentary coalitions, and military alliances are discussed.

Coalition formation, dynamic analysis, single-peakedness, legislatures

Abstract:
Players are assumed to rank each other as coalition partners.

Two processes of coalition formation are defined and illustrated:

- Fallback (FB): Players seek coalition partners by descending lower and lower in their preference rankings until some majority coalition, all of whose members consider each other mutually acceptable, forms.

- Build-up (BU): Same descent as FB, except only majorities whose members rank each other highest form coalitions.

BU coalitions are stable in the sense that no member would prefer to be in another coalition, whereas FB coalitions, whose members need not rank each other highest, may not be stable. BU coalitions are bimodally distributed in a random society, with peaks around simple majority and unanimity the distributions of majorities in the US Supreme Count and in the US House of Representatives follow this pattern. The dynamics of real-life coalition-formation processes are illustrated by two Supreme Court cases.

Coalition dynamics, Fallback bargaining, Manipulability, Legislatures, US Supreme Court

Abstract:
Democracy resolves conflicts in difficult games like Prisoners' Dilemma and Chicken by stabilizing their cooperative outcomes. It does so by transforming these games into games in which voters are presented with a choice between a cooperative outcome and a Pareto-inferior noncooperative outcome. In the transformed game, it is always rational for voters to vote for the cooperative outcome, because cooperation is a weakly dominant strategy independent of the decision rule and the number of voters who choose it. Such games are illustrated by 2-person and n-person public-goods games, in which it is optimal to be a free rider, and a biblical story from the book of Exodus.

Democracy, voting, game theory, public goods, cooperation, Bible

Abstract:
Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each others' preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is nonmonotonicty - choosing earlier may hurt rather than help a party. We introduce a mechanism that combines sequential choices with a structured form of trading that results in sincere choices for two parties. Although there are difficulties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to making consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.

Proportional Representation, apportionment, divisor methods, Sincere and Sophisticated Choices, Envy Free Allocations, Sports Drafts

Abstract:
Three models are presented in which two players agree to share power in a particular ratio, but either player may subsequently "fire" at the other, as in a duel, to try to eliminate it. The players have positive probabilities of eliminating each other by firing. If neither is successful, the agreement stays in place; if one is successful, that player obtains all the power; if each eliminates the other, both players get nothing. In Model I, the game is played once, and in Model II it is repeated, with discounting of future payoffs. Although there are conditions under which each player would prefer not to shoot, satisfying these conditions for one player precludes satisfying them for the other, so at least one player will always have an incentive to shoot. In anticipation, its rival would prefer to shoot, too, so there will be a race to preempt. In Model III, a damage factor caused by shooting, whether successful or not, is introduced into Model II. This mitigates the incentive to shoot but does not eliminate it entirely. The application of the models to conflicts, especially civil wars, is discussed.

power sharing, duel, repeated game, civil war

Abstract:
Power sharing is modeled as a duel over some prize. Each of two players may either share the prize in some ratio or fire at the other player - either in sequence or simultaneously - and eliminate it with a specified probability. If one player eliminates the other without being eliminated itself, it captures the entire prize, but the prize is damaged over time when there is shooting. Simultaneous shooting, which is more damaging than sequential shooting, tends to induce the players to share the prize and expand their opportunities for sharing it. It was effectively implemented by the superpowers with the doctrine of launch on warning during the Cold War, and it was strengthened by the development of second-strike capability. Deterring terrorism has proved a different matter, because terrorists are difficult to detect and present few targets that can be damaged.

Power sharing, duel, game, deterrence, terrorism

Abstract:
Assume that players strictly rank each other as coalition partners. We propose a procedure whereby they 'fall back' on their preferences, yielding internally compatible, or coherent, majority coalition(s), which we call fallback coalitions. If there is more than one fallback coalition, the players common to them, or kingmakers, determine which fallback coalition will form. The players(s) who are the first to be acceptable to all other members of a fallback coalition are the leader(s) of that coalition. The effects of different preference assumptions - particularly, different kinds of single-peakedness - and of player weights on the number of coherent coalitions, their connectedness, and which players become kingmakers and leaders are investigated. The fallback procedure may be used (i) empirically to identify kingmakers and leaders or (ii) normatively to select them. We illustrate the model by applying it to coalition formation on the U.S. Supreme Court, 2005-2008.

Coalition, Fallback Procedure, Kingmaker, Leader, Cardinally Single-Peaked, Ordinally Single-Peaked

Abstract:
We study the problem of dissolving an equal-entitlement partnership when the objective is to minimize maximum regret. We initially focus on the family of linear-pricing mechanisms and derive regret-optimizing strategies. We also demonstrate that there exist linear-pricing mechanisms satisfying ex-post efficiency. Next, we analyze a binary-search mechanism which is ex-post individually rational. We discuss connections with the standard Bayesian-Nash framework for both linear and binary-search mechanisms. On a more general level, we show that if entitlements are unequal, ex-post efficiency and ex-post individual rationality impose significant restrictions on permissible mechanisms. In particular, they rule out both linear and binary-search mechanisms.

Partnership dissolution, fair division, minimax regret, ex-post efficiency, ex-post rationality, Bayesian-Nash equilibrium

Abstract:
We propose a procedure for dividing indivisible items between two players in which each player ranks the items from best to worst. It ensures that each player receives a subset of items that it values more than the other player's complementary subset, given that such an envy-free division 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, each player's maximin strategy is to be truthful. Applications of the undercut procedure are briefly discussed.

Fair division, indivisible items, envy-freeness, manipulability, maximin strategy

Abstract:
Cutting a cake -- or any heterogeneous, divisible good -- fairly has received much attention in recent years, but fair division of a pie into wedge-shaped sectors has received far less. Whereas cake-cutting is applicable to land division, pie-cutting is more applicable to the division of an island's shoreline into connected lots, or a daily cycle into on-call periods. Unlike cake division, we show that there may be no envy-free and efficient division of a pie among n players using n radial cuts (the minimum number), nor may there be an envy-free allocation that is equitable (each player receives the same value in its measure), though an envy-free and equitable allocation is always possible. Thus, pie-cutting is harder than cake-cutting - not a piece of cake.

Fair division, cake-cutting, pie-cutting, envy-freeness, efficiency, equitability, divisible good

Abstract:
Abstract will be provided by author.

Abstract:
Abstract will be provided by author.

Abstract:
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.