Recognizing Members of the Tournament Equilibrium Set is NP-hard

A recurring theme in the mathematical social sciences is how to select the "most desirable" elements given a binary dominance relation on a set of alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions that have been proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive tournament solution concept refining both the Banks set and Dutta's minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard.Comment: 9 pages, 3 figure

Robust Bounds on Choosing from Large Tournaments

Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability.Comment: Appears in the 14th Conference on Web and Internet Economics (WINE), 201

The Complexity of Computing Minimal Unidirectional Covering Sets

Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure

The Banks set and the Uncovered Set in budget allocation problems

We examine how a society chooses to divide a given budget among various regions, projects or individuals. In particular, we characterize the Banks set and the Uncovered Set in such problems. We show that the two sets can be proper subsets of the set of all alternatives, and at times are very pointed in their predictions. This contrasts with well-known "chaos theorems," which suggest that majority voting does not lead to any meaningful predictions when the policy space is multidimensional

Complex Transition to Cooperative Behavior in a Structured Population Model

Cooperation plays an important role in the evolution of species and human societies. The understanding of the emergence and persistence of cooperation in those systems is a fascinating and fundamental question. Many mechanisms were extensively studied and proposed as supporting cooperation. The current work addresses the role of migration for the maintenance of cooperation in structured populations. This problem is investigated in an evolutionary perspective through the prisoner's dilemma game paradigm. It is found that migration and structure play an essential role in the evolution of the cooperative behavior. The possible outcomes of the model are extinction of the entire population, dominance of the cooperative strategy and coexistence between cooperators and defectors. The coexistence phase is obtained in the range of large migration rates. It is also verified the existence of a critical level of structuring beyond that cooperation is always likely. In resume, we conclude that the increase in the number of demes as well as in the migration rate favor the fixation of the cooperative behavior

Generalizations of the General Lotto and Colonel Blotto Games

In this paper, we generalize the General Lotto game (budget constraints satisfied in expectation) and the Colonel Blotto game (budget constraints hold with probability one) to allow for battlefield valuations that are heterogeneous across battlefields and asymmetric across players, and for the players to have asymmetric resource constraints. We completely characterize Nash equilibrium in the generalized version of the General Lotto game and then show how this characterization can be applied to identify equilibria in the Colonel Blotto version of the game. In both games, we find that there exist sets of non-pathological parameter configurations of positive Lebesgue measure with multiple payoff nonequivalent equilibria