Convex Interval Games

Convex interval games are introduced and characterizations are given. Some economic situations leading to convex interval games are discussed. The Weber set and the Shapley value are defined for a suitable class of interval games and their relations with the interval core for convex interval games are established. The notion of population monotonic interval allocation scheme (pmias) in the interval setting is introduced and it is proved that each element of the Weber set of a convex interval game is extendable to such a pmias. A square operator is introduced which allows us to obtain interval solutions starting from the corresponding classical cooperative game theory solutions. It turns out that on the class of convex interval games the square Weber set coincides with the interval core

Open-ended Learning in Symmetric Zero-sum Games

Zero-sum games such as chess and poker are, abstractly, functions that evaluate pairs of agents, for example labeling them `winner' and `loser'. If the game is approximately transitive, then self-play generates sequences of agents of increasing strength. However, nontransitive games, such as rock-paper-scissors, can exhibit strategic cycles, and there is no longer a clear objective -- we want agents to increase in strength, but against whom is unclear. In this paper, we introduce a geometric framework for formulating agent objectives in zero-sum games, in order to construct adaptive sequences of objectives that yield open-ended learning. The framework allows us to reason about population performance in nontransitive games, and enables the development of a new algorithm (rectified Nash response, PSRO_rN) that uses game-theoretic niching to construct diverse populations of effective agents, producing a stronger set of agents than existing algorithms. We apply PSRO_rN to two highly nontransitive resource allocation games and find that PSRO_rN consistently outperforms the existing alternatives.Comment: ICML 2019, final versio

Monotonicity Problems of Interval Solutions and the Dutta-Ray Solution for Convex Interval Games

This paper examines several monotonicity properties of value-type interval solutions on the class of convex interval games and focuses on the Dutta-Ray (DR) solution for such games. Well known properties for the classical DR solution are extended to the interval setting. In particular, it is proved that the interval DR solution of a convex interval game belongs to the interval core of that game and Lorenz dominates each other interval core element. Consistency properties of the interval DR solution in the sense of Davis-Maschler and of Hart-Mas-Colell are verified. An axiomatic characterization of the interval DR solution on the class of convex interval games with the help of bilateral Hart-Mas-Colell consistency and the constrained egalitarianism for two-person interval games is given.cooperative interval games;convex games;the constrained egalitarian solution;the equal division core;consistency

Convex Fuzzy Games and Participation Monotonic Allocation Schemes

AMS classifications: 90D12; 03E72Convex games;Core;Decisionmaking;Fuzzy coalitions;Fuzzy games;Monotonic allocation schemes;Weber set

Convex fuzzy games and participation monotonic allocation schemes

90D12;03E72cooperative games

Monotonicity of Social Optima With Respect to Participation Constraints.

In this paper we consider solutions which select from the core. For games with side payments with at least four players, it is well-known that no core-selection satifies monotonicity for all coalitions; for the particular class of core-selections found by maximizing a social welfare function over the core, we investigate whether such solutions are monotone for a given coalition. It is shown that if this is the case then the solution actually maximizes aggregate coalition payoff on the core. Furthermore, the social welfare function to be maximized exhibits larger marginal social welfare with respect to the payoff of any member of the coalition. The results may be used to show that there are no monotonic core selection rules of this type in the context of games without side payments.coalitional games; monotonicity; core; social welfare

Tree-connected Peer Group Situations and Peer Group Games

A class of cooperative games is introduced which arises from situations in which a set of agents is hierarchically structured and where potential individual economic abilities interfere with the behavioristic rules induced by the organization structure.These games form a cone generated by a specific class of unanimity games, namely those based on coalitions called peer groups. Different economic situations like auctions, communication situations, sequencing situations and flow situations are related to peer group games.For peer group games classical solution concepts have nice properties.auctions;cooperative games;peer groups