Solutions for cooperative games with r alternatives

We consider games with n players and r alternatives. In these games each player must choose one and only one alternative, reaching an ordered partition of the set of players. An extension of the Shapley value to this framework is studied. Received: 1 November 1997 / Accepted: 24 January 1999     

An axiomatization of the core of cooperative games without side payments

Let M be a set of m players, m≧3, and let Γ be the set of all (finite) games (without side payments) that have a non-empty core. When M is finite, the following four (independent) axioms fully characterize the core on Γ: (i) non-emptiness, (ii) individual rationality, (iii) the reduced game property, and (iv) the converse reduced game property. If M is infinite, then the converse reduced game property is redundant.     

Subdifferentiability and lipschitz continuity in experimental design problems

The calculus of concave functions is a widely accepted tool for optimum experimental design problems. However, as a function of the support points and the weights the design problem fails to be concave. In this paper we make use of generalized gradients in the sense of Rockafellar (1980) and Clarke (1983). A chain rule is presented for the subdifferential of the composition of an information function with the moment matrix mapping. Lipschitz continuity of the global design function is proved and conditions for strict differentiability are given.     

How to share when context matters: The Möbius value as a generalized solution for cooperative games

All quasivalues rest on a set of three basic axioms (efficiency, null player, and additivity), which are augmented with positivity for random order values, and with positivity and partnership for weighted values. We introduce the concept of Möbius value associated with a sharing system and show that this value is characterized by the above three axioms. We then establish that (i) a Möbius value is a random order value if and only if the sharing system is stochastically rationalizable and (ii) a Möbius value is a weighted value if and only if the sharing system satisfies the Luce choice axiom.     

Learning games

This paper presents a model of learning about a game. Players initially have little knowledge about the game. Through playing the same game repeatedly, each player not only learns which action to choose but also constructs a personal view of the game. The model is studied using a hybrid payoff matrix of the prisoner's dilemma and coordination games. Results of computer simulations show that (1) when all the players are slow at learning the game, they have only a partial understanding of the game, but might enjoy higher payoffs than in cases with full or no understanding of the game; (2) when one player is quick in learning the game, that player obtains a higher payoff than the others. However, all can receive lower payoffs than in the case in which all players are slow learners.     

Repeated games in continuous time as extensive form games

Extensive form games modeling continuous-time decisions are plagued with problems of nonexistence and nonuniqueness of outcomes. As a “second-best”, the literature has imposed extraneous restrictions on the strategy sets, raising questions on the nature and interpretation of the resulting formal object, and on which restrictions are appropriate. We provide a “first-best” framework, formalizing continuous-time repeated games as extensive form games incorporating natural conditions from the onset. Every strategy profile induces a unique outcome, without any restrictions on the strategy sets. Further, the unrestricted strategy sets are equivalent to a specific class of strategies in a more naïvely specified continuous-time game.     

Proper strong-Fibonacci games

We define proper strong-Fibonacci (PSF) games as the subset of proper homogeneous weighted majority games which admit a Fibonacci representation. This is a homogeneous, type-preserving representation whose ordered sequence of type weights and winning quota is the initial string of Fibonacci numbers of the one-step delayed Fibonacci sequence. We show that for a PSF game, the Fibonacci representation coincides with the natural representation of the game. A characterization of PSF games is given in terms of their profile. This opens the way up to a straightforward formula which gives the number \(\varPsi (t)\) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of \(\varPsi (t)\) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio \(\varPsi (t+2)/\varPsi (t)\) converges toward the golden ratio \({\varPhi }\).     

On Markov games

In the paper it is demonstrated, how a dynamic programming approach may be useful for the analysis of Markov games or stochastic games. Markov games with finitely many stages are dealt with extensively. The existence of optimal Markov strategies is proven for finite stage Markov games using a shortcut of a proof by D erman for the analogous result for Markov decision processes. For Markov games with a countably infinite number of stages some results are summarized. Here again the results and the methods of proof have much in common with results and proofs for Markov decision processes. Actually the theory of Markov games is a generalisation. The paper contains short introductions into the theories of matrix games and tree games.     

Restricted houseswapping games

Restricted houseswapping games (RHGs) are a generalization of ‘one-sided matching games’, in which we specify a class II* of ‘allowable’ simple trading cycles. The cores of such games may be empty. Given II*, all possible closed RHGs have non-empty cores of II* is ‘strongly balanced’. Examples include the one-sided matching markets (^{Shapley and Scarf. Journal of Mathematical Economics 1974. 1. 23–37}. ^{Tijs et al., OR Spektrum 1984, 6, 119–123}; ^{Quinzii, International Journal of Game Theory 1984, 13, 41–60}) and the two-sided matching markets (^{Gale and Shapley. American Mathematical Monthly 1962. 69, 9–16}; ^{Shapley and Shubik, International Journal of Game Theory 1972, 1, 111–130}: and ^{Demange and Gale Econometrica 1985, 53, 873–888}).We then consider the subclass of RHGs in which there is no transferable resource. In this case, a weaker condition on II*, called ‘weak balancedness’, is sufficient to guarantee core non-emptiness. In addition, if II* is not weakly balanced, then there exists a preference profile such that the strict core of the resultant game is empty.Several other examples are given of II* that are (a) strongly balanced: (b) weakly balanced but not strongly balanced: and (c) not even weakly balanced.Finally, we discuss the issues of equilibrium definition, existence, and core-equilibrium allocation equivalence in RHGs.     

Global Thinking Script Locally

<正>近日翻看旧杂志,在《Times》里看到一篇讲美国情景喜剧在中国落地的文章,题目叫《Global Thinking Script Locally(全球化视野,本地化落地)》,突然觉得这句话很适合用在今天中国的广告行业。     

Locally stable price mechanisms

This paper develops necessary conditions for a price adjustment mechanism to achieve local stability at regular competitive equilibria. Two principal questions are: how closely must a locally stable mechanism be tailored to particular excess demand functions, and can any such mechanism be interpreted as a market adjustment process. In response to the first question, a variant of the (local) Newton method, termed the ‘orthogonal Newton method’ is shown to require, in a dimensional sense, the minimal information about excess demand functions. The second question is answered in the negative by proving the non-existence of any locally stable mechanism with the property that the price of any given commodity is not changed when its own market is in equilibrium. These and other results are obtained by using convergent price paths to generate a homotopy between the adjustment dictated by the mechanism and the actual direction of the equilibrium.     

Ambiguity in election games

We construct a model in which the ambiguity of candidates allows them to increase the number of voters to whom they appeal. We focus our analysis on two points that are central to obtain ambiguity in equilibrium: restrictions on the beliefs that candidates can induce in voters, and intensity of voters' preferences. The first is necessary for a pure strategy equilibrium to exist, while the second is necessary for ambiguity in equilibrium when there exists a Condorcet winner in the set of pure alternatives (e.g. the spatial model of electoral competition), and when candidates' only objective is to win the election. In this last case, an ambiguous candidate may offer voters with different preferences the hope that their most preferred alternative will be implemented. We also show that if there are sufficiently many candidates or parties, ambiguity will not be possible in equilibrium, but a larger set of possible policies increases the chance that at least one candidate will choose to be ambiguous in equilibrium. We would like to thank Alberto Alesina, Antonio Cabrales, Steve Coate, Olivier Compte, Tim Feddersen, Itzhak Gilboa, Joe Harrington, Michel Le Breton, Alessandro Lizzeri, George Mailath, Steve Matthews, Steve Morris, Ignacio Ortuno, Tom Palfrey, Larry Samuelson, Murat Sertel, Fernando Vega, Eyal Winter and an anonymous referee for helpful comments. The first author acknowledges financial support from DGICYT-PB 95-0983. This work was done while the first author was visiting the Center in Political Economy at Washington University, and visiting the Center for Basic Research in the Social Sciences at Harvard University. Their hospitality is gratefully acknowledged. The support of the second author's research by the National Science Foundation is also gratefully acknowledged.     

电子商务之网游

截止2006年,中国网民的数量已经达到13700万人。虽然网民总数多,增长快,但是互联网普及程度还很低。未来的发展空氓粱很大。而网游的力量是不可忽视的,所以将电子商务和网游结合起来是企业拓展盈利的一个新的方向。文中从中国网民现状。中国网游现状,中国电子商务现状及电子商务与网游的结合进行了讨论。     

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.     

Survival in Cournot games

In a model that encompasses a general equilibrium framework, we consider a monopolist (a producer) with subjective beliefs that endogenously hedges against fluctuations in input prices in a complete market. We allow for entries and Cournot competition in this economy, and we study how erroneous beliefs affect long-run survival for those firms. We introduce a notion of entropy of beliefs, and we use it to characterize the class of beliefs for which the monopolist eventually disappears almost surely. When disappearance occurs, the whole market power switches to the entrant making the most accurate predictions in our sense. The class of beliefs for which survival occurs is much broader than that of perfectly competitive settings.     

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).     

Network games with incomplete information

We consider a network game with strategic complementarities where the individual reward or the strength of interactions is only partially known by the agents. Players receive different correlated signals and they make inferences about other players’ information. We demonstrate that there exists a unique Bayesian-Nash equilibrium. We characterize the equilibrium by disentangling the information effects from the network effects and show that the equilibrium effort of each agent is a weighted combinations of different Katz–Bonacich centralities.     

Recurrent crises in global games

Global games have unique equilibria in which aggregate behavior changes sharply when an underlying random fundamental crosses some threshold. This property relies on the existence of dominance regions: all players have a highest and lowest action that, for some fundamentals, is strictly dominant. But if the fundamental follows a random walk, it eventually spends nearly all of its time in these regions: crises gradually disappear. We obtain recurring crises by adding a single large player who lacks dominance regions. We also show that in order to obtain recurring crises, one must either relax dominance regions or restrict to fundamentals that continually return to or cross over a fixed region.     

Elections and strategic positioning games

We formalize the interplay between expected voting behavior and strategic positioning behavior of candidates as a common agency problem in which the candidates (i.e., the principals) compete for voters (i.e., agents) via the issues they choose and the positions they take. A political situation is defined as a feasible combination of candidate positions and expected political payoffs to the candidates. Taking this approach, we are led naturally to a particular formalization of the candidates' positioning game, called a political situation game. Within the context of this game, we define the notion of farsighted stability (introduced in an abstract setting by Chwe 1994) and apply Chwe's result to obtain existence of farsightedly stable outcomes. We compute the farsightedly stable sets for several examples of political situations games, with outcomes that conform to real-world observations.Received: 18 May 2001, Accepted: 22 January 2002, JEL Classification: C7, D7, D8Myrna H. Wooders: http://www.warwick.ac.uk/fac/soc/Economics/wooders/An earlier version of this paper was completed while the first author was visiting the Department of Economics, University of Exeter. The first author gratefully acknowledges Exeter's support and hospitality. Similarly, the second author gratefully acknowledges the support and hospitality of the Centre for Operations Research and Econometrics (CORE) of the University of Louvain-la-Neuve, Belgium and of the University of Cergy-Pontoise, France. Both authors are indebted to Amrita Dhillon, John Duggan and Gilat Levy for helpful comments about references. Both authors thank an anonymous referee for helpful comments.     

Estimation of finite sequential games

I propose a new estimation method for finite sequential games that is efficient, computationally attractive, and applicable to a fairly general class of finite sequential games that is beyond the scope of existing studies. The major challenge is the computation of high-dimensional truncated integration whose domain is complicated by strategic interaction. This complication resolves when unobserved off-the-equilibrium-path strategies are controlled for. Separately evaluating the likelihood contribution of each subgame-perfect equilibrium that generates the observed outcome allows the use of the GHK simulator, a widely used importance-sampling probit simulator. Monte Carlo experiments demonstrate the performance and robustness of the proposed method.