arXiv 1603.048932016-03math.OCcs.GT

具有用户群的次模块效用系统Nash均衡的性能界

Performance Bounds for Nash Equilibria in Submodular Utility Systems with User Groups

Yajing Liu, Edwin K. P. Chong, Ali Pezeshki

备注：: This paper was accepted by Journal of Control and Decision

摘要：

在本文中，我们考虑Vetta所考虑的公用事业系统的变化，其中用户被分组在一起。我们的目标是建立用户之间的分组和合作如何影响性能界限。我们考虑两种类型的分组。第一种类型是\cite{Zhang2014}，其中每个用户属于一组与之有社会联系的用户。对于这种类型的效用系统，每个用户的策略最大化其社会群体效用函数，从而产生了\emph{social-aware Nash equilibrium}的概念。我们证明了这个社会效用系统屈服于Vetta对非合作系统的边界结果，从而建立了可证明的社会意识纳什均衡的性能保证。对于第二种类型的分组，将用户集划分为$l$不相交的组，其中组内的用户合作以最大化其组效用函数，从而产生\emph{group Nash equilibrium}的概念。在这种情况下，每个组可以被看作是一个新的用户与向量值的行动，和1/2界的性能组纳什均衡如下从Vetta的结果。但是，正如我们所示，可以建立更严格的边界，包括曲率。通过定义组曲率$c_{k_i}$与$k_i$用户组$i$，我们表明，如果社会效用函数是非减和submodular，那么任何组纳什均衡达到至少$1/(1+\max_{1\leq i\leq l}c_{k_i})$的最优社会效用，这是比没有分组的情况下更紧。作为一种特殊情况，如果每个用户都有相同的行动空间，那么我们有任何组纳什均衡达到至少$1/(1+c_{k^*})$的最优社会效用，其中$k^*$是最少的用户数之间的$l$组。最后，我们提出了一个数据库辅助频谱访问的实用程序系统的例子来说明我们的结果。

英文摘要：

In this paper, we consider variations of the utility system considered by Vetta, in which users are grouped together. Our aim is to establish how grouping and cooperation among users affect performance bounds. We consider two types of grouping. The first type is from \cite{Zhang2014}, where each user belongs to a group of users having social ties with it. For this type of utility system, each user's strategy maximizes its social group utility function, giving rise to the notion of \emph{social-aware Nash equilibrium}. We prove that this social utility system yields to the bounding results of Vetta for non-cooperative system, thus establishing provable performance guarantees for the social-aware Nash equilibrium. For the second type of grouping, the set of users is partitioned into $l$ disjoint groups, where the users within a group cooperate to maximize their group utility function, giving rise to the notion of \emph{group Nash equilibrium}. In this case, each group can be viewed as a new user with vector-valued actions, and a 1/2 bound for the performance of group Nash equilibrium follows from the result of Vetta. But as we show tighter bounds involving curvature can be established. By defining the group curvature $c_{k_i}$ associated with group $i$ with $k_i$ users, we show that if the social utility function is nondecreasing and submodular, then any group Nash equilibrium achieves at least $1/(1+\max_{1\leq i\leq l}c_{k_i})$ of the optimal social utility, which is tighter than that for the case without grouping. As a special case, if each user has the same action space, then we have that any group Nash equilibrium achieves at least $1/(1+c_{k^*})$ of the optimal social utility, where $k^*$ is the least number of users among the $l$ groups. Finally, we present an example of a utility system for database assisted spectrum access to illustrate our results.

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摘要：

英文摘要：

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