Abstract: Publication date: May 2019Source: Mathematical Social Sciences, Volume 99Author(s): G. Bergantiños, A. Navarro-Ramos We consider minimum cost spanning tree problems with multiple sources. We propose a cost allocation rule based on a painting procedure. Agents paint the edges on the paths connecting them to the sources. We prove that the painting rule coincides with the folk rule.

Abstract: Publication date: May 2019Source: Mathematical Social Sciences, Volume 99Author(s): Laura Kasper, Hans Peters, Dries Vermeulen We identify the maximal voting correspondence which is Condorcet Consistent and satisfies two participation conditions, namely the Top Property and the Bottom Property — thereby extending a result in Pérez (2001). The former participation condition says that if an alternative is in the chosen set at a profile of rankings and a ranking is added with that alternative on top, then it remains to be a member of the chosen set. The latter says that if an alternative is not in the chosen set at a profile of rankings and a ranking is added with that alternative at bottom, then the alternative is again not in the chosen set. In particular, voting functions (single-valued voting correspondences) with these three properties select from this maximal correspondence, and we demonstrate several ways in which this can or cannot be done.

Abstract: Publication date: May 2019Source: Mathematical Social Sciences, Volume 99Author(s): Susumu Kawanaka, Naoyuki Kamiyama In this paper, we consider the problem of testing substitutability of weak preferences. For this problem, Aziz, Brill, and Harrenstein proposed an O(ℓ3u2+ℓ2u2s2)-time algorithm, where u is the size of the ground set, ℓ is the number of acceptable sets, and s is the maximum size of an equivalent class. In this paper, we propose an O(ℓ3u+ℓ2u2s)-time algorithm for this problem. Our algorithm is based on a generalization of the characterization of substitutability of strict preferences given by Croitoru and Mehlhorn.

Abstract: Publication date: Available online 19 March 2019Source: Mathematical Social SciencesAuthor(s): Hans Gersbach, Hans Haller We develop a model that combines competitive exchange of private commodities across endogenously formed groups with public good provision and global collective decisions. There is a tension between local and global collective decisions that can cause non-existence of competitive equilibria with endogenous household formation and public choice. In particular, we show that group formation and collective decisions on public goods may destabilize each other, even if there exist favorable conditions for matching on the one hand, and for global collective decisions on the other hand. We establish sufficient conditions for the existence of competitive equilibria with endogenous household formation and public choice and illustrate a variety of phenomena when households take local collective decisions and have a voice in global collective decisions.

Abstract: Publication date: Available online 19 March 2019Source: Mathematical Social SciencesAuthor(s): Umberto Grandi, Daniel Hughes, Francesca Rossi, Arkadii Slinko The Gibbard-Satterthwaite theorem states that for any non-dictatorial voting system there will exist an election where a voter, called a manipulator, can change the election outcome in their favour by voting strategically. When a given preference profile admits several manipulators, voting becomes a game played by these voters, who have to reason strategically about each other’s actions. To complicate the game even further, some voters, called countermanipulators, may try to counteract potential actions of manipulators. Previously, voting manipulation games have been studied mostly for the Plurality rule. We extend this to k-Approval voting rules. However, unlike previous studies, we assume that voters are boundedly rational and do not think beyond manipulating or countermanipulating. We classify all 2-by-2 games that can be encountered by these strategic voters, and investigate the complexity of arbitrary voting manipulation games, identifying conditions on strategy sets that guarantee the existence of a Nash equilibrium in pure strategies.

Abstract: Publication date: Available online 18 March 2019Source: Mathematical Social SciencesAuthor(s): Francesco De Sinopoli, Giovanna Iannantuoni, Elena Manzoni, Carlos Pimienta We introduce a model with strategic voting in a parliamentary election with proportional representation and uncertainty about the voter’s preferences. In any equilibrium of the model, most of the voters only vote for those parties whose positions are extreme. In the resulting parliament, a consensus government forms and the policy maximizing the sum of utilities of the members of the government is implemented.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Joseph M. Abdou, Hans Keiding In the present work, we consider a basic model of political structure, given through its agents or forces and the viable configurations of agents as collective bodies of decision making. When the set of all agents is not viable, a compromise must be searched for. We model a political structure as a simplicial complex where a viable configuration is a simplex. A represented compromise is a viable configuration obtained by the withdrawal of some agents in favor of other agents acting as representatives. A delegated compromise is a more elaborated version of a compromise obtained by iteration of the process of delegation. Existence of such compromises depends on the discrete topology of the simplicial complex. In the paper, we study represented and delegated compromises in their dependence on the combinatorial structure of the viable configurations, and in particular we show that existence of a delegated compromise is equivalent to strong contractibility of the simplicial complex.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Ata Atay, Marina Núñez We analyze the extent to which two known results of the relationship between the core and the stable sets for two-sided assignment games can be extended to three-sided assignment games. We find that the dominant diagonal property is necessary for the core to be a stable set and, likewise, sufficient when each sector of the three-sided market has two agents. Unlike the two-sided case, the union of the extended cores of all the μ-compatible subgames with respect to an optimal matching μ may not be a von Neumann–Morgenstern stable set.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Elnaz Bajoori, Dries Vermeulen In second-price auctions with interdependent values, bidders do not necessarily have dominant strategies. Moreover, such auctions may have many equilibria. In order to rule out the less intuitive equilibria, we define the notion of distributional strictly perfect equilibrium (DSPE) for Bayesian games with infinite type and action spaces. This equilibrium is robust against arbitrary small perturbations of strategies. We apply DSPE to a class of symmetric second-price auctions with interdependent values. We show that the efficient equilibrium defined by Milgrom (1981) is a DSPE, while a class of less intuitive, inefficient, equilibria introduced by Birulin (2003) is not.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Noah Giansiracusa, Cameron Ricciardi We use the United States Supreme Court as an illuminative context in which to discuss three different spatial voting preference models: an instance of the widely used single-peaked preferences, and two models that are more novel in which vote outcomes have a strength in addition to a location. We introduce each model from a formal axiomatic perspective, briefly discuss practical motivation for each in terms of judicial behavior, prove mathematical relationships among the voting coalitions compatible with each model, and then study the two-dimensional setting by presenting computational tools for working with the models and by exploring these with judicial voting data from the Supreme Court.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Chantal Marlats This paper explores the robustness of predictions made in long but finitely repeated games. The robustness approach used in this paper is related to the idea that a modeler may not have absolute faith in his model: The payoff matrix may not remain the same at all dates and may vary temporarily from time to time with an arbitrarily small probability. Therefore, he may require not rejecting an outcome if it is an equilibrium in some game arbitrarily close to the original one. It is shown that the set of feasible and rational payoffs is the (essentially) unique robust equilibrium payoff set when the horizon is sufficiently large. Consequently, cooperation can arise as an equilibrium behavior in a game arbitrarily close to the standard prisoner’s dilemma if the horizon is finite but sufficiently long.

Abstract: Publication date: March 2019Source: Mathematical Social Sciences, Volume 98Author(s): Pablo Amorós A possibly partial jury has to choose the winner of a competition. A deserving winner exists and her identity is common knowledge among the jurors, but it is not known by the planner. Jurors may be biased in favor (friend) or against (enemy) some contestants. We study the conditions based on the configuration of the jury such that it is possible to implement the deserving winner in Nash equilibrium when we restrict ourselves to mechanisms that satisfy two conditions: (1) each juror only has to announce the contestant she thinks should win the competition, and (2) announcing the deserving winner is always an equilibrium. We refer to this notion as natural implementation. We show that knowledge of jurors’ friends or jurors’ enemies may help to naturally implement the deserving winner. In particular, our results suggest that knowledge of jurors’ friends may be more useful to the planner than knowledge of jurors’ enemies in terms of facilitating natural implementation.

Abstract: Publication date: Available online 28 February 2019Source: Mathematical Social SciencesAuthor(s): Beth Bjorkman, Sean Gravelle, Jonathan K. Hodge The notion of separability is important in a variety of fields, including economics, political science, computer science, and operations research. Separability formalizes the idea that a decision-maker’s preferred ordering of outcomes on some dimensions within a multidimensional alternative space may depend on the values of other dimensions. The character admissibility deals with the construction of preferences that are separable on specific sets of dimensions. In this paper, we develop a graph-theoretic approach to the character admissibilty problem, using Hamiltonian paths to generate preference orderings. We apply this method specifically to hypercube graphs, defining the class of cubic preferences. We then explore how the algebraic structure of the group of symmetries of the hypercube impacts the separability structures exhibited by cubic preferences. We prove that the characters of cubic preferences satisfy set theoretic properties distinct from those produced by previous methods, and we define two functions to construct cubic preferences. Our methods have potential applications to a variety of multiple-criteria decision-making problems, including multiple-question referendum elections.

Abstract: Publication date: Available online 17 October 2018Source: Mathematical Social SciencesAuthor(s): Umut Mert Dur Many school districts in the U.S. assign students to schools via the Boston mechanism. The Boston mechanism is not strategy-proof, and it is easy to manipulate. We slightly modify the Boston mechanism and show that the modified version outperforms the Boston mechanism in terms of strategy-proofness. In particular, the Boston mechanism is manipulable whenever the modified version is, but the modified version is not necessarily manipulable whenever the Boston mechanism is. We define a weaker form of consistency and characterize the modified Boston mechanism by this weaker form and a new axiom called respect of priority of the top-ranking students.