One for all, all for one---von Neumann, Wald, Rawls, and Pareto

Applications of the maximin criterion extend beyond economics to statistics, computer science, politics, and operations research. However, the maximin criterion---be it von Neumann's, Wald's, or Rawls'---draws fierce criticism due to its extremely pessimistic stance. I propose a novel concept, dubbed the optimin criterion, which is based on (Pareto) optimizing the worst-case payoffs of tacit agreements. The optimin criterion generalizes and unifies results in various fields: It not only coincides with (i) Wald's statistical decision-making criterion when Nature is antagonistic, (ii) the core in cooperative games when the core is nonempty, though it exists even if the core is empty, but it also generalizes (iii) Nash equilibrium in $n$-person constant-sum games, (iv) stable matchings in matching models, and (v) competitive equilibrium in the Arrow-Debreu economy. Moreover, every Nash equilibrium satisfies the optimin criterion in an auxiliary game

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06

Shapley Meets Shapley

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page

Why and How Identity Should Influence Utility

This paper provides an argument for the advantage of a preference for identity-consistent behaviour from an evolutionary point of view. Within a stylised model of social interaction, we show that the development of cooperative social norms is greatly facilitated if the agents of the society possess a preference for identity consistent behaviour. As cooperative norms have a positive impact on aggregate outcomes, we conclude that such preferences are evolutionarily advantageous. Furthermore, we discuss how such a preference can be integrated in the modelling of utility in order to account for the distinctive cooperative trait in human behaviour and show how this squares with the evidence

Matching games: the least core and the nucleolus

A matching game is a cooperative game defined by a graph $G=(V,E)$. The player set is $V$ and the value of a coalition $S \subseteq V$ is defined as the size of a maximum matching in the subgraph induced by $S$. We show that the nucleolus of such games can be computed efficiently. The result is based on an alternative characterization of the least core which may be of independent interest. The general case of weighted matching games remains unsolved. \u

Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games

Cooperative games provide a framework for fair and stable profit allocation in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are such solution concepts that characterize stability of cooperation. In this paper, we study the algorithmic issues on the least-core and nucleolus of threshold cardinality matching games (TCMG). A TCMG is defined on a graph $G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$ meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the problems of computing least-core value, finding and verifying least-core payoff are all polynomial time solvable. We also provide a general characterization of the least core for a large class of TCMG. Next, based on Gallai-Edmonds Decomposition in matching theory, we give a concise formulation of the nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When the threshold $T$ is relevant to the input size, we prove that the nucleolus can be obtained in polynomial time in bipartite graphs and graphs with a perfect matching