In this paper we propose a Bayesian game to allocate resources. In this game, there are $c$ units of resources to be allocated to $n$ players. Agent $i$ has a demand of $V_i$ units of resources and takes action $X_i$ according to a strategy function $s_i$, \ie $X_i=s_i(V_i)$. Payoffs are setup such that player $i$ is contented with no more than $V_i$ units of resources. We assume that resources are granted to the players on a smallest-request-first and all-or-nothing basis. For this game with two players, we analyze the equilibrium strategy functions mathematically within the family of alternating identity-and-flat (AIF) functions. We show that Nash equilibrium profiles consist of two identity functions, two AIF functions with a common switch point, or two AIF functions with one and three switch points, respectively. For an $n$-player game with a large $n$ and a large $c_n$ of order $O(n)$, we present a mean-field first order approximation and a second-order Gaussian approximation for its equilibrium strategy function. The first-order analysis obtains an equilibrium AIF function with one switch point. In Gaussian analysis of large games, we propose a construction algorithm. This construction algorithm begins in searching within the family of AIF functions. If a gradient conflict condition occurs, the game enters a chattering regime, in which players play a continuous, strictly increasing strategy function that is not an identity nor a flat function. Conceptually one can view the chattering regime as if players alternate between a slope-one strategy and a flat strategy infinitely fast in order to sustain a high payoff. We prove that the construction algorithm always obtains a Nash equilibrium and terminates in a finite number of steps. We present several numerical examples for the two player game as well as the Gaussian model.

翻译：本文提出了一种贝叶斯博弈模型用于资源分配。在该博弈中，共有 $c$ 个单位的资源需分配给 $n$ 个玩家。玩家 $i$ 具有 $V_i$ 单位的资源需求，并根据策略函数 $s_i$ 采取行动 $X_i$，即 $X_i=s_i(V_i)$。收益设置使得玩家 $i$ 在获得不超过 $V_i$ 单位的资源时感到满意。我们假设资源按照最小请求优先且全有或全无的原则分配给玩家。对于双玩家博弈，我们讨论了交替恒等-平坦（AIF）函数族内的均衡策略函数数学形式。我们证明纳什均衡轮廓分别由两个恒等函数、具有公共开关点的两个AIF函数、或分别具有一个与三个开关点的两个AIF函数组成。对于具有大量 $n$ 和阶数为 $O(n)$ 的大 $c_n$ 的 $n$ 人博弈，我们提出了其均衡策略函数的平均场一阶近似和高斯二阶近似。一阶分析得到了具有一个开关点的均衡AIF函数。在大博弈的高斯分析中，我们提出了一种构造算法。该构造算法首先在AIF函数族内进行搜索。若出现梯度冲突条件，博弈进入颤动状态，此时玩家采用连续严格递增的非恒等非平坦策略函数。从概念上看，颤动状态可理解为玩家以无限快速交替执行斜率为1的策略与平坦策略以维持高收益。我们证明该构造算法始终能获得纳什均衡并在有限步内终止。我们给出了双玩家博弈及高斯模型的若干数值算例。