体育赛事对局结果预测

甲乙双方展开系列赛,每局必分胜负,那么,若要分出输赢,平均需要比赛多少局？通过实例对这一问题进行研究：将胜负概率的乘积作为一个整体,由此可获得一个一般性公式,并以斯诺克比赛为例,介绍其应用.     

Phenomena in Inverse Stackelberg Games,Part 1: Static Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

Phenomena in Inverse Stackelberg Games,Part 2: Dynamic Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

Measuring skill in games: several approaches discussed

An aspect of casino games that in general leads to discussions among both participants and spectators, is the relative extent to which a player can positively influence his results by making appropriate strategic choices. This question is closely related to the issue of how to distinguish between games of skill and games of chance. This is an issue that is interesting from a juridical point of view too, since in many countries the legitimacy of exploiting a specific game depends on the category to which it belongs. This paper summarizes recent developments concerning the measurement of skill in games. It points out the elements in the definitions that need closer attention, it illustrates the analysis with examples and it discusses further possibilities.     

A limit theorem for equilibria under ambiguous belief correspondences

Previous literature shows that, in many different models, limits of equilibria of perturbed games are equilibria of the unperturbed game when the sequence of perturbed games converges to the unperturbed one in an appropriate sense. The question of whether such a limit property extends to the equilibrium notions in ambiguous games is not yet as clear as it seems; in fact, previous literature shows that the extension fails in simple examples.     

Strategy-indifferent games of best choice

The game of best choice (or “secretary problem”) is a model for making an irrevocable decision among a fixed number of candidate choices that are presented sequentially in random order, one at a time. Because the classically optimal solution is known to reject an initial sequence of candidates, a paradox emerges from the fact that candidates have an incentive to position themselves immediately after this cutoff which challenges the assumption that candidates arrive in uniformly random order.One way to resolve this is to consider games for which every (reasonable) strategy results in the same probability of success. In this work, we classify these “strategy-indifferent” games of best choice. It turns out that the probability of winning such a game is essentially the reciprocal of the expected number of left-to-right maxima in the full collection of candidate rank orderings. We present some examples of these games based on avoiding permutation patterns of size 3, which involves computing the distribution of left-to-right maxima in each of these pattern classes.     

Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems

This paper is about games where the agents face constraints in the combined strategy space (unlike in standard games where the action sets are defined separately for each player) and about computational methods for solutions to such games. The motivation examples for such games include electricity generation problems with transmission capacity constraints, environmental management to control pollution and internet switching to comply to buffers of bounded capacity. In each such problem a regulator may aim at compliance to standards or quotas through taxes or charges. The relevant solution concept for these games has been known under several names like generalised Nash equilibrium, coupled constraint equilibrium and more. Existing numerical methods converging to such an equilibrium will be explained. Application examples of use of NIRA, which is a suite of Matlab routines that implement one of the methods, will be provided.      

A model of a 2-player stopping game with priority and asynchronous observation

Various models of 2-player stopping games have been considered which assume that players simultaneously observe a sequence of objects. Nash equilibria for such games can be found by first solving the optimal stopping problems arising when one player remains and then defining by recursion the normal form of the game played at each stage when both players are still searching (a 2 × 2 matrix game). The model considered here assumes that Player 1 always observes an object before Player 2. If Player 1 accepts the object, then Player 2 does not see that object. If Player 1 rejects an object, then Player 2 observes it and may choose to accept or reject it. It is shown that such a game can be solved using recursion by solving appropriately defined subgames, which are played at each moment when both players are still searching. In these subgames Player 1 chooses a threshold, such that an object is accepted iff its value is above this threshold. The strategy of Player 2 in this subgame is a stopping rule to be used when Player 1 accepts this object, together with a threshold to be used when Player 1 rejects the object. Whenever the payoff of Player 1 does not depend on the value of the object taken by Player 2, such a game can be treated as two optimisation problems. Two examples are given to illustrate these approaches.     

Independence results in formal topology

We analyze a family of games by using formal topology as a tool. In order to win any game in the family one has to find a sequence of moves leading to one of the final states for that game. Thus, two results are relevant to the topic: to find terminating strategies and/or to show that every strategy is terminating. We will show that the language of formal topology can be useful to represent in a topological framework both of the problems, and in particular that the property of termination of all the strategies for a game is equivalent to the discreteness of a suitable formal space. Finally, we will provide some examples of games which are terminating according to any strategy, that is, such that the associated formal spaces are discrete, but the first order formulas expressing such a discreteness cannot be proved in Peano Arithmetic.     

Useful New Equilibria for Differential Games with Side Interests of the Participants

For conflict static and dynamic problems (described by differential equations) considered either on a game set common for all participants or on partly intersecting game sets, we propose new notions of conflict equilibria which are efficient for seeking the solutions of coalition-free and cooperative games and for specifying the hierarchy of all known equilibria. Several examples are used to show that, without the proposed new notions of equilibrium, an actually fair sharing may be impossible in cooperative games.     

On repeated games without a recursive structure: Existence of limV n

We consider repeated zero-sum games with symmetric incomplete information where at each stage the common signal is either non-revealing or completely revealing. We prove that the sequence of values ofn-stage games converges by approximating the repeated game by a sequence of games in continuous time.     

Unawareness—A gentle introduction to both the literature and the special issue

This article provides a brief survey of the literature on unawareness and introduces the contributions to the special issue on unawareness in Mathematical Social Sciences. First, we provide a brief overview both about epistemic models of unawareness and models of extensive-form games with unawareness. Instead of introducing the approaches in full detail, we illustrate the main differences and similarities with the help of examples. Finally, we discuss the contributions to the special issue on unawareness.     

Cooperative games with stochastic payoffs

This paper introduces a new class of cooperative games arising from cooperative decision making problems in a stochastic environment. Various examples of decision making problems that fall within this new class of games are provided. For a class of games with stochastic payoffs where the preferences are of a specific type, a balancedness concept is introduced. A variant of Farkas' lemma is used to prove that the core of a game within this class is non-empty if and only if the game is balanced. Further, other types of preferences are discussed. In particular, the effects the preferences have on the core of these games are considered.     

The nucleolus and kernel of veto-rich transferable utility games

The process of computing the nucleolus of arbitrary transferable utility games is notoriously hard. A number of papers have appeared in which the nucleolus is computed by an algorithm in which either one or a huge number of huge linear programs have to be solved. We show that on the class of veto-rich games, the nucleolus is the unique kernel element. Veto-rich games are games in which one of the players is needed by coalitions in order to obtain a non-zero payoff. We then provide a fast algorithm which does not use linear programming techniques to compute the nucleolus of these games. Furthermore, we provide a few examples of economic situations which belong to the class of veto-rich games and which are treated in the literature.     

博弈的统计演化分析

首先定义了具有确定分布的随机环境中的基于适应度的平均选择算子,然后证明了主体战略集上的任一概率密度在平均选择算子的迭代过程中收敛于平均适应度函数的最大集上的某一分布,然后就多主体的博弈问题定义了平均选择算子,并以此为基础证明了平均选择算子的不动点就是博弈的纳什均衡.     

Reductions of interval noncooperative games

Noncooperative games of a finite number of persons with interval-valued payoff functions are considered. The concept of an equilibrium situation is introduced. A reduction of such games to deterministic noncooperative games is proposed. Properties of the reduced games are discussed. Interval antagonistic and bimatrix games are examined, and illustrative examples are considered.     

A note on isomorphism and strategic equivalence of cooperative games

In this note, we will give several examples to illustrate that two essential games which are isomorphic are not necessarily S-equivalent when the cores of both games are “small” or empty. In other words, we show that whether two isomorphic games are S-equivalent can not be justified in terms of the “size” of the core.     

Remarks on sensitive equilibria in stochastic games with additive reward and transition structure

A class of stochastic games with additive reward and transition structure is studied. For zero-sum games under some ergodicity assumptions 1-equilibria are shown to exist. They correspond to so-called sensitive optimal policies in dynamic programming. For a class of nonzero-sum stochastic games with nonatomic transitions nonrandomized Nash equilibrium points with respect to the average payoff criterion are also obtained. Included examples show that the results of this paper can not be extented to more general payoff or transition structure.     

The golden number and Fibonacci sequences in the design of voting structures

Some distinguished types of voters, as vetoes, passers or nulls, as well as some others, play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power. In this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters. With this purpose in mind we enumerate these special classes of games and find out that its number always follows a Fibonacci sequence with smooth polynomial variations. As a consequence we find several families of games with the same asymptotic exponential behavior except for a multiplicative factor which is the golden number or its square. From a more general point of view, our studies are related with the design of voting structures with a predetermined importance ranking.     

从北京奥运会看概率在体育中的体现和应用

概率论是从数量上研究随机现象统计规律性的一门数学学科,在科技、管理、经济等领域具有着重要作用.同样,在充满变数的奥运会里面,也有着广泛的应用和体现.从北京奥运会里面的一些问题加以总结,以扩大对于概率论在体育中应用的了解,增加学习概率论的兴趣.