济贫问题的随机分配模型

Stochastic Assigning Model of Aiding the Poor Game

对美国Columbia大学HerbertRobbins教授提出的济贫问题进行了研究 ,该问题要求计算每个人获得的钱数的方差 .先将此问题转换为非齐次马尔可夫链模型 ,推导出一步转移概率矩阵 ,依此证明了当钱数趋于无穷大时 ,方差趋于零 ;给出了计算K =3和N =4时问题的精确解 ;最后给出了N ,K为更大数值的Monte Carlo模拟解 。

The problem of Aiding the Poor Game put forward by Prof.Herbert Robbins at Columbia University of the USA was researched.In this problem,the variance of the money provided for each individual needs to be calculated.In this research,the problem was transformed into a non-homogeneous Markov chain model,and the one-step transition probability matrix was deduced,thus proving that the variance approaches zero when the money approaches infinite.Moreover,the accuracy solution of the problem,when K=3 and N=4,was given.The Monte-Carlo simulation solution,when N and K are both comparatively great,was finally given and the coherence of the simulation solution and accurate solution was verified.