Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

Banzhaf Measures for Games with Several Levels of Approval in the Input and Output

An axiomatic characterization of ‘a Banzhaf score’ notion is provided for a class of games called (j,k) simple games with a numeric measure associated to the output set, i.e., games with n players, j ordered qualitative alternatives in the input level and k possible ordered quantitative alternatives in the output. Three Banzhaf measures are also introduced which can be used to determine a player's ‘a priori’ value in such a game. We illustrate by means of several real world examples how to compute these measures. Research partially supported by Grant BFM 2003-01314 of the Science and Technology Spanish Ministry and the European Regional Development Fund.     

Value theory without symmetry

We investigate quasi-values of finite games – solution concepts that satisfy the axioms of Shapley (1953) with the possible exception of symmetry.  Following Owen (1972), we define “random arrival', or path, values: players are assumed to “enter' the game randomly, according to independently distributed arrival times, between 0 and 1; the payoff of a player is his expected marginal contribution to the set of players that have arrived before him.  The main result of the paper characterizes quasi-values, symmetric with respect to some coalition structure with infinite elements (types), as random path values, with identically distributed random arrival times for all players of the same type.  General quasi-values are shown to be the random order values (as in Weber (1988) for a finite universe of players).  Pseudo-values (non-symmetric generalization of semivalues) are also characterized, under different assumptions of symmetry. Received: April 1998/Revised version: February 2000     

Multiplicative Correlations

A multivariate distribution is said to have multiplicative correlation if the correlation matrix R = (r _ij ) is written as r _ij = δ _i δ _j or r _ij = −δ _i δ _j (i ≠ j) for a parameter vector . We first determine feasible values for and show that variables with such a correlation matrix can always be decomposed into a common “signal” variable plus individual “noise” variables. It is also shown that a special case of this correlation matrix implies a sum constraint among variables and vice versa. Such properties illustrate why many multivariate distributions have such a correlation structure. Furthermore, several invariance properties lead to simple relations among several multivariate distributions.     

The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game “Shoot Later, Shoot First!”

We consider the following “silent duel” of m players with a possible economic interpretation. Each player has one “bullet”, which she can shoot at any time during the time interval [0,1]. The probability that the i-th player hits the “target” at moment t is given by an increasing accuracy function f _i (t). The winner is the player who hits the target first. Under natural assumptions on the functions f _i (t) we prove the existence and uniqueness of a Nash equilibrium point in this game, and we provide an explicit construction of this equilibrium. This construction allows us to obtain exact solutions for many specific examples. Some of them are presented.This work was partly supported by RBRF grants 03-01-00479.     

A value for games withn players andr alternatives

We study value theory for a class of games called games withn players andr alternatives. In these games, each of then players must choose one and only one of ther alternatives. A linear, efficient value is obtained using three characterizations, two of which are axiomatic. This value yields an a priori evaluation for each player relative to each alternative.     

An alternative characterization of the weighted Banzhaf value

We provide a new characterization of the weighted Banzhaf value derived from some postulates in a recent paper by Radzik, Nowak and Driessen [7]. Our approach owes much to the work by Lehrer [4] on the classical Banzhaf value based on the idea of amalgamation of pairs of players and an induction construction of the value. Compared with the approach in [7] we consider two new postulates: a weighted version of Lehrer’s “2-efficiency axiom” [4] and a generalized “null player out” property studied in terms of symmetric games by Derks and Haller [2]. Received: December 1997/final version: October 1999     

On consistent solutions for strategic games

Nash equilibria for strategic games were characterized by Peleg and Tijs (1996) as those solutions satisfying the properties of consistency, converse consistency and one-person rationality.  There are other solutions, like the ɛ-Nash equilibria, which enjoy nice properties and appear to be interesting substitutes for Nash equilibria when their existence cannot be guaranteed. They can be characterized using an appropriate substitute of one-person rationality. More generally, we introduce the class of “personalized” Nash equilibria and we prove that it contains all of the solutions characterized by consistency and converse consistency. Received January 1996/Final version December 1996     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

On equilibrium in pure strategies in games with many players

We demonstrate that, if there are sufficiently many players, any Bayesian equilibrium of an incomplete information game can be “ε-purified” . That is, close to any Bayesian equilibrium there is an approximate Bayesian equilibrium in pure strategies. Our main contribution is obtaining this result for games with a countable set of pure strategies. In order to do so we derive a mathematical result, in the spirit of the Shapley–Folkman Theorem, permitting countable strategy sets. Our main assumption is a “large game property,” dictating that the actions of relatively small subsets of players cannot have large affects on the payoffs of other players. E. Cartwright and M. Wooders are indebted to Phillip Reny, Frank Page and two anonymous referees for helpful comments.     

Interactive epistemology II: Probability

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or “player.” One is interested not only in each person's knowledge and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space Ω of states of the world, and for each player i, partitions ℐ_i of Ω and probability distributions π_i(·; ω) on Ω for each state ω of the world. The atom of ℐ_i containing a given state ω represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω; the probability distributions π_i(·; ω) represents i's beliefs at the state ω. The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent.  In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space Ω, the partitions ℐ_i, and the probability distributions π_i(·; ω) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.     

Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

On a metric generalization of ramsey’s theorem

An increasing sequence of realsx=〈x _i :i<ω〉 is simple if all “gaps”x _i +1−x _i are different. Two simple sequencesx andy are distance similar ifx _i +1−x _i <x _j +1−x _j if and only ify _i +1−y _i <y _j +1−y _j for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequencey distance similar tox all of whose pairs are of the same color. We also consider many related problems and generalizations. Partially supported by OTKA-4269. Partially supported by NSF grant STC-91-19999. Partially supported by OTKA-T-020914, NSF grant CCR-9424398 and PSC-CUNY Research Award 663472.     

Values of games with a continuum of players

A definition of the “Shapley value” of games with a continuum of players and a formula for this value are given for a certain class of games, regarding them as limits of games with a finite number of players. The research described in this paper was partially supported by the U.S. Office of Naval Research, Logistics and Mathematical Statistics Branch, under Contract Number N 62558-3586.     

Zeros of linear recurrence sequences

Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with i≠j, α _i /α _j is not a root of unity. Define . We introduce a partition of into subsets (1≤i≤m), which induces a decomposition of f into , so that, for 1≤i≤m, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have

Independent mistakes in large games

Economic models usually assume that agents play precise best responses to others' actions. It is sometimes argued that this is a good approximation when there are many agents in the game, because if their mistakes are independent, aggregate uncertainty is small. We study a class of games in which players' payoffs depend solely on their individual actions and on the aggregate of all players' actions. We investigate whether their equilibria are affected by mistakes when the number of players becomes large. Indeed, in generic games with continuous payoff functions, independent mistakes wash out in the limit. This may not be the case if payoffs are discontinuous. As a counter-example we present the n players Nash bargaining game, as well as a large class of “free-rider games.” Received: November 1997/Final version: December 1999