Positivity and convexity in incomplete cooperative games

TL;DR

This paper systematically studies incomplete cooperative games, focusing on positivity and convexity, providing characterizations, explicit formulas, and new perspectives on extensions and related theories.

Contribution

It generalizes positivity results to broader classes, characterizes extendability, and links incomplete games to cooperative interval games.

Findings

Characterization of non-extendability to positive games.

Explicit formulas for special incomplete game classes.

Full description of symmetric convex extensions.

Abstract

Incomplete cooperative games generalise the classical model of cooperative games by omitting the values of some of the coalitions. This allows to incorporate uncertainty into the model and study the underlying games as well as possible payoff distribution based only on the partial information. In this paper we perform a systematic study of incomplete games, focusing on two important classes of cooperative games: positive and convex games. Regarding positivity, we generalise previous results for a special class of minimal incomplete games to general setting. We characterise non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. The results are then used to obtain explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with considering the case of non-negative minimal incomplete games. Then we survey existing results in the related theory of set functions, namely providing context to the problem of completing partial functions. We provide a characterisation of extendability and a full description of the set of symmetric convex extensions. The set serves as an approximation of the set of convex extensions. Finally, we outline an entirely new perspective on a connection between incomplete cooperative games and cooperative interval games.