# On the core of m$m$‐attribute games

Ulaş Özen, Marco Slikker and Greys Sošić

Production and Operations Management, 2022, vol. 31, issue 4, 1770-1787

Abstract: We study a special class of cooperative games with transferable utility (TU), called m$m$‐attribute games. Every player in an m$m$‐attribute game is endowed with a vector of m$m$ attributes that can be combined in an additive fashion; that is, if players form a coalition, the attribute vector of this coalition is obtained by adding the attributes of its members. Another fundamental feature of m$m$‐attribute games is that their characteristic function is defined by a continuous attribute function π$\pi$—the value of a coalition depends only on evaluation of π$\pi$ on the attribute vector possessed by the coalition, and not on the identity of coalition members. This class of games encompasses many well‐known examples, such as queueing games and economic lot‐sizing games. We believe that by studying attribute function π$\pi$ and its properties, instead of specific examples of games, we are able to develop a common platform for studying different situations and obtain more general results with wider applicability. In this paper, we first show the relationship between nonemptiness of the core and identification of attribute prices that can be used to calculate core allocations. We then derive necessary and sufficient conditions under which every m$m$‐attribute game embedded in attribute function π$\pi$ has a nonempty core, and a set of necessary and sufficient conditions that π$\pi$ should satisfy for the embedded game to be convex. We also develop several sufficient conditions for nonemptiness of the core of m$m$‐attribute games, which are easier to check, and show how to find a core allocation when these conditions hold. Finally, we establish natural connections between TU games and m$m$‐attribute games.

Date: 2022
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https://doi.org/10.1111/poms.13643

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