关键词: TU-博弈, 边际性, 超图结构合作博弈, Position值

Abstract: Cooperative game theory examines how individuals cooperate with each other to achieve a common goal. In cooperative games, players cooperate with each other by integrating their efforts, resources, and skills to maximize the collective good. The transferable utility cooperative game, the TU-game, refers to a situation where any collection of players can be rewarded for cooperating to form a viable coalition, and it focuses on how the benefits (costs) of a grand coalition are shared (apportioned) among players. It is usually assumed that all players can communicate with each other, and the Shapley value is a well-known allocation rule based on this assumption. However, in the real world, cooperation between players is limited by subjective and objective factors such as differences in players’ abilities, cultural diversity, time and resources constraints. To describe this phenomenon, MYERSON (1977) introduced a class of TU-games with cooperation structures given by communication graphs on the player set, in which it is assumed that only connected players can fully cooperate, and the Shapley value of the so-called graph-restricted games induced by the graph cooperation structure is defined as a new allocation rule, i.e., the Myerson value. Subsequently, MEESSEN (1988) proposes another important allocation rule for graph-restricted games, namely the position value. This allocation rule is based on the graph structure, emphasizing the role of links in the graph. It treats the links connecting players as players themselves and allocates the Shapley value to these links, and then distributing the Shapley value obtained by each link equally to the two players connecting that link.
The Shapley value focuses on the player’s marginal contribution to each coalition. A player’s marginal contribution is the difference in value before and after that player joins each coalition. The Shapley value assigned to each player is exactly the average of all of the player’s marginal contributions. In 1985, YOUNG first introduced the concept of marginality, which states that the same player with the same marginal contribution should receive equal payments in two different games. Since then, marginality and its derived forms have gradually been applied to the characterizations of values in TU-games.
In graph-restricted games, academics introduce PL-marginality based on the axiom of marginality, which is a variant of marginality that takes into account the influence of players and their connected links. Specifically, it considers the possibility that a player joins a coalition and then communicates with that coalition. In this paper, we consider the position value for hypergraph games from the same perspective and generalize PL-marginality from graph-restricted games to hypergraph games by introducing PH-marginality. In reality, the hypergraph structure is more general and thus can model more complex communication relationships between players. Secondly, by weakening the property of partial balanced conference contributions in the hypergraph games, it applies only to necessary players. By means of efficiency, partial balanced conference contributions for necessary players, and PH-marginality, this paper provides a new characterization of the position value for hypergraph games.
Finally, applications of the position value for hypergraph games are considered. In this paper, we describe the problem of optimizing the allocation of network resources among devices in a local area network (LAN) as a hypergraph game model, and tailor network optimization solutions for different application scenarios based on the device’s position value.
In the future, a natural question is whether the inscription method of marginality can be generalized to the characterization of other allocation rules in hypergraph games.

Key words: TU-game, marginality, hypergraph games, the position value