Deterministic multi-player Dynkin games

A multi-player Dynkin game is a sequential game in which at every stage one of the players is chosen, and that player can decide whether to continue the game or to stop it, in which case all players receive some terminal payoff.We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than 1.We prove that a subgame-perfect ε-equilibrium in Markovian strategies exists. If the game is not degenerate this ε-equilibrium is actually in pure strategies.     

Learning a population distribution

This paper introduces a dynamic Bayesian game with an unknown population distribution. Players do not know the true population distribution and assess it based on their private observations using Bayes׳ rule. First, we show the existence and characterization of an equilibrium in which each player׳s strategy is a function not only of the player׳s type but also of experience. Second, we show that each player׳s initial belief about the population distribution converges almost surely to a “correct” belief.     

Comparing means of several treatments with a control when one observation exists per treatment

In an one-way analysis of variance with standard assumptions suppose that only one observation exists per treatment. In addition, assume that one of the treatments is a control group. Because of insufficient observations, the variance of the populations cannot be estimated and hence the usual methods for comparing treatments with the control group fail. In this paper, we present a method to compare treatments with a control when one observation exits per treatment. An algorithm is given to estimate the critical values of the test. The power of the test is investigated by a Monte Carlo simulation; numerical studies show that when there is a treatment whose mean is close to the control group, the power of the test is satisfactory.     

A note on the evaluation of information in zero-sum repeated games

Two players play a zero-sum repeated game with incomplete information. Before the game starts one player receives a private signal that depends on the realized state of nature. The rules that govern the choice of the signal are determined by the information structure of the game. Different information structures induce different values. The value-of-information function of a game associates every information structure with the value it induces. We characterize those functions that are value-of-information functions for some zero-sum repeated game with incomplete information.     

On the finite sample behavior of testing whether new is better than used of a specified age

Hollander, Park and Proschan (1986) proposed a test of new is better than used of a specified age. It is based on large sample normality of the test statistic. There is, however, no study in the literature on its actual size for small and moderate sample sizes. To shed some lights on this, the results of a Monte Carlo simulation study as well as two real data examples are reported and these indicate that the test can have a quite liberal size, especially for small to moderate sample sizes. In order to improve on this weakness, a modified test is proposed and studied. It is noticed that this modified test seems to over-correct the original test to an extent that it becomes unduly conservative sometimes. Hence we propose another modification that combines the original test and the modified test turns out to have its size quite close to the nominal level and is therefore preferable to both the original and modified tests.     

Potential games with incomplete preferences

This paper studies potential games allowing the possibility that players have incomplete preferences and empty best-response sets. We define four notions of potential games, ordinal, generalized ordinal, best-response, and generalized best-response potential games, and characterize them using cycle conditions. We study Nash equilibria of potential games and show that the set of Nash equilibria remains the same when every player’s preferences are replaced with the smallest generalized (best-response) potential relation or a completion of it. Similar results are established about strict Nash equilibria of ordinal and best-response potential games. Lastly, we examine the relations among the four notions of potential games as well as pseudo-potential games.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

$${\phi_p}$$-optimal designs for a linear log contrast model for experiments with mixtures

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S ^k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these -optimal designs behave. Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001.     

例谈探究式教育游戏设计

探究式教育游戏是以游戏为载体,以网络为平台,将知识以游戏任务的形式融入主题游戏并通过交互策略来引导学生探究思维的学习软件。本文以"乐中学"游戏为例阐述了探究式教育游戏的设计流程、基本结构以及应注意的问题。     

Group size and collective action in a binary contribution game

We consider how group size affects the private provision of a public good with non-refundable binary contributions. A fixed amount of the good is provided if and only if the number of contributors reaches an exogenous threshold. The threshold, the group size, and the identical, non-refundable cost of contributing to the public good are common knowledge. Our focus is on the case in which the threshold is larger than one, so that teamwork is required to produce the public good. We show that both expected payoffs and the probability that the public good is obtained in the best symmetric equilibrium are decreasing in group size. We also characterize the limit outcome when group size converges to infinity and provide precise conditions under which the expected number of contributors is decreasing or increasing in group size for sufficiently large groups.     

Coordinating under incomplete information

We show that, in a minimum effort game with incomplete information where player types are independently drawn, there is a largest and smallest Bayesian equilibrium, leading to the set of equilibrium payoffs (as evaluated at the interim stage) having a lattice structure. Furthermore, the range of equilibrium payoffs converges to those of the deterministic complete information version of the game, in the limit as the incomplete information vanishes. This entails that such incomplete information alone cannot explain the equilibrium selection suggested by experimental evidence. We thank Atila Abdulkadiroglu, Hans Carlsson, Ani Guerdjikova and an anonymous referee for helpful comments. Part of this work was done while Asheim was visiting Cornell University, which hospitality is gratefully acknowledged.     

A note on strategies for bandit problems with infinitely many arms

A bandit problem consisting of a sequence of n choices (n) from a number of infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a common distribution F on the interval [0,1] and F is continuous with F(0)=0 and F(1)=1. The goal is to investigate the asymptotic expected failure rates of k-failure strategies, and obtain a lower bound for the expected failure proportion over all strategies presented in Berry et al. (1997). We show that the asymptotic expected failure rates of k-failure strategies when 0<b1 and a lower bound can be evaluated if the limit of the ratio F(1)–F(t) versus (1–t)^b exists as t1^– for some b>0.     

Time consistent side payments in a dynamic game of downstream pollution

This note determines a rule to share a surplus gained when two countries or regions agree to coordinate their policies to reduce downstream pollution. An intertemporal decomposition scheme for the total side payment is proposed. This scheme has the following individual rationality property: in each subgame that starts along the cooperative trajectory, one country is guaranteed to receive a higher payoff in the cooperative solution than in the disagreement solution. For this country another notion of individual rationality obtains: this country will at any instant of time during the play of the game receive a higher payoff in the cooperative solution than in the disagreement solution.     

论游戏与儿童创造力的培养——让儿童回归“田野游戏”

城市化进程、功利化教育理念、科技的发展,让传统的游戏方式逐渐简化成室内智能游戏的特殊模式,违背了儿童的成长和创新性培养的要求。基于这一现象,为了提高儿童创造力,笔者提出了要让儿童回归真实、自然、实践的"田野游戏"。在游戏中,促进儿童生存技能和智力的双重训练,从而培养儿童的创造力和创新精神。     

A method for generating uniformly scattered points on the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-norm unit sphere and its applications

Tashiro (Ann Inst Stat Math 29:295–300, 1977) studied methods for generating unform points on the surface of the regular unit sphere. The L _p -norm unit sphere is a generalization of the regular unit sphere. In this paper we propose a method associated with an algorithm for generating uniformly scattered points on the L _p -norm unit sphere and discuss its applications in statistical simulation, representative points of a wide class of multivariate probability distributions and optimization problems. Some examples are illustrated for these applications. This research was supported by The University of Hong Kong Research Grant and a University of New Haven 2005 and 2006 Summer Faculty Fellowships.     

Export restraints in a model of trade with capital accumulation

This paper examines the impact of voluntary export restraints (VERs) in an international duopoly modeled as a differential game. With a Ramsey capital accumulation dynamics, the game admits multiple steady states, and a VER cannot be ‘voluntarily’ employed by the foreign firm in case of Cournot behavior in demand substitutes. Hence, the dynamic framework confirms the results of the VERs literature with static interaction in output levels. In the case of price behavior, the adoption of an export restraint may increase the profits of both firms if products are substitutes and the steady state is ‘market-driven’. However, contrary to the acquired wisdom based upon the static approach, the dynamic analysis also admits an equilibrium outcome, identified by the Ramsey golden rule, where the incentive to adopt a VER is ruled out, irrespective of whether firms are quantity- or price-setters.     

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

大学生人际关系的思考与对策——女生传话惹祸的心理教育案例

大学校园里经常发生因传话导致矛盾,产生纠纷,这是教育工作者面临的一个难题。本文通过此案例,展示班主任如何运用心理健康教育知识,引导学生自我思考,自我教育,自我成长。     

Generic finiteness of equilibrium payoffs for bimatrix games

It is shown that in any affine space of payoff matrices the equilibrium payoffs of bimatrix games are generically finite.     

Rationalizability in continuous games

Define a continuous game to be one in which every player's strategy set is a Polish space, and the payoff function of each player is bounded and continuous. We prove that in this class of games the process of sequentially eliminating “never-best-reply” strategies terminates before or at the first uncountable ordinal, and this bound is tight. Also, we examine the connection between this process and common belief of rationality in the universal type space of Mertens and Zamir (1985).