On the core of m$m$‐attribute games
研究一类特殊的可转移效用合作博弈，其中每个玩家拥有多个可加属性，联盟价值仅取决于属性向量。给出了核心非空的条件，并建立了与凸性的联系。
We study a special class of cooperative games with transferable utility (TU), called <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> ‐ attribute games . Every player in an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> ‐attribute game is endowed with a vector of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> ‐attribute games is that their characteristic function is defined by a continuous attribute function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>π</mml:mi> <mml:annotation encoding="">$\pi$</mml:annotation> </mml:semantics> </mml:math> —the value of a coalition depends only on evaluation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>π</mml:mi> <mml:annotation encoding="">$\pi$</mml:annotation> </mml:semantics> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>π</mml:mi> <mml:annotation encoding="">$\pi$</mml:annotation> </mml:semantics> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> ‐attribute game embedded in attribute function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>π</mml:mi> <mml:annotation encoding="">$\pi$</mml:annotation> </mml:semantics> </mml:math> has a nonempty core, and a set of necessary and sufficient conditions that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>π</mml:mi> <mml:annotation encoding="">$\pi$</mml:annotation> </mml:semantics> </mml:math> should satisfy for the embedded game to be convex. We also develop several sufficient conditions for nonemptiness of the core of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation> </mml:semantics> </mml:math> ‐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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics definitionURL="" encoding=""> <mml:mi>m</mml:mi> <mml:annotation encoding="">$m$</mml:annotation>