Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firms Location Problem
Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firms Location Problem
复制标题
3企业选址问题混合策略纳什均衡的存在性及计算
DOI：
--
复制
发表时间：
1982
期刊：
影响因子：
--
通讯作者：
A. Shaked
中科院分区：
文献类型：
--
作者：
A. Shaked
闭区间上企业的选址问题是由Hotling[3]引入的，后来由Eaton&Lipsey[2]研究。公司以固定价格销售同质产品，分布在区间内的客户每人从距离他们最近的公司购买一件产品，公司的目标是最大化客户数量。众所周知，当客户沿区间均匀分布且企业数量不是3家时，选址问题存在纯策略纳什均衡(见[2])。三家公司的情况是特殊的，因为不存在纯粹的策略纳什均衡，因为在任何情况下，极端的一家公司(不位于另外两家公司之间)可以通过靠近或远离另两家公司来增加收入。混合策略的存在并不明显，因为当两个或多个公司位于同一点时，收益函数是不连续的。在最近的一篇论文中，Dasgupta和Maskin[1]研究了一类博弈，这些博弈都源于经济问题，而且它们的收益函数都有一些间断。它们证明了在所研究的博弈中，支付函数的不连续性并不妨碍混合策略纳什均衡的存在。特别是，前面描述的这类选址问题具有这种“弱”间断性，因此具有混合策略纳什均衡。尽管他们的证明是建设性的，提供了一种将这些均衡接近为有限博弈均衡的极限的方法，但所涉及的方程很快就变得无法管理。在本文中，我们计算了3家公司选址问题的混合策略纳什均衡。假设客户沿[0，1]均匀分布，则混合策略是[0，1]上的概率度量。问题的对称性表明，解将是双重对称的，因为所有公司都选择相同的混合策略，并且该策略围绕-(对称位置被给予相同的概率)对称。Dasgupta和Maskin将他们的分析局限于双对称解，并证明了在任何双对称均衡中的策略都是无原子的。鉴于博弈的对称性，在达斯古普塔和马斯金的分析之后，我们将在这里将自己限制在这样的双重对称
Location problems of firms on a closed interval were introduced by Hotelling [3] and later investigated by Eaton & Lipsey [2]. Firms sell a homogeneous product at a fixed price, customers distributed along the interval buy one unit each from the firm nearest to them and firms aim to maximize the number of their customers. It is well known that where the customers are evenly distributed along the interval and the number of firms is other than 3 the location problem has a pure strategy Nash equilibrium (see [2]). The case of 3 firms is peculiar in that no pure strategy Nash equilibrium exists, for in any situation one of the extreme firms (not located between the other two) can increase its revenue by either moving towards or away from the other two firms. The existence of mixed strategies is not obvious because the payoff functions are discontinuous whenever two or more firms are located at the same point. In a recent paper Dasgupta and Maskin [1] investigate a class of games, all derived from economic problems and all with some discontinuities in their payoff functions. They demonstrate that the discontinuities in the payoff functions do not prevent the existence of mixed strategies Nash equilibria in the games investigated. In particular, location problems of the type described earlier have such "weak" discontinuities and hence possess a mixed strategy Nash equilibrium. Although their proof is constructive, offering a method of approaching these equilibria as a limit of equilibria of finite games, the equations involved very quickly become unmanageable. In this note we compute a mixed strategy Nash equilibrium for the 3 firms location problem. Let the customers be evenly distributed along [0, 1] then a mixed strategy is a probability measure over [0, 1]. The symmetry of the problem suggests that the solution will be doubly symmetric in that all firms choose the same mixed strategy and that this strategy is symmetric around - (symmetric locations are given the same probabilities). Dasgupta and Maskin confine their analysis to doubly symmetric solutions and prove that the strategies in any doubly symmetric equilibrium are atomless. Given the symmetric nature of the game and following the analysis of Dasgupta and Maskin we shall confine ourselves here to such doubly sym