Abraham Neyman

• Hebrew University of Jerusalem

This paper characterizes those preferences over bounded infinite utility streams that satisfy the time value of money principle and an additivity property, and the subset of these preferences that, in addition, are either impatient or patient. Based on this characterization, the paper introduces a concept of optimization that is robust to a small i...

The Big Match is a multistage two-player game. In each stage, player 1 hides one or two pebbles in his hand, and his opponent has to guess that number. Player 1 loses a point if player 2 is correct; otherwise, he wins a point. As soon as player 1 hides one pebble, the players cannot change their choices in any future stage. The undiscounted Big Mat...

An absorbing game is a two-person zero-sum repeated game. Some of the entries are “absorbing” in the sense that, following the play of an absorbing entry, with positive probability all future payoffs are equal to that entry's payoff. The outcome of the game is the long-run average payoff. We prove that a two-person zero-sum absorbing game, with eit...

The value is a solution concept for n ‐person strategic games, developed by Nash, Shapley, and Harsanyi. The value of a game is an a priori evaluation of the economic worth of the position of each player, reflecting the players' strategic possibilities, including their ability to make threats against one another. Applications of the value in econom...

In this paper we offer a new approach to modeling strategies of bounded complexity, the so-called factor-based strategies. In our model, the strategy of a player in the multi-stage game does not directly map the set of histories H to the set of her actions. Instead, the player's perception of H is represented by a factor φ : H → X; where X ref...

The Big Match is a multi-stage two-player game. In each stage Player 1 hides one or two pebbles in his hand, and his opponent has to guess that number; Player 1 loses a point if Player 2 is correct, and otherwise he wins a point. As soon as Player 1 hides one pebble, the players cannot change their choices in any future stage. Blackwell and Ferguso...

A game of threats on a finite set of players, N, is a function d that assigns a real number to any coalition, S⊆N, such that d(S)=−d(N∖S). A game of threats is not necessarily a coalitional game as it may fail to satisfy the condition d(∅)=0. We show that analogs of the classic Shapley axioms for coalitional games determine a unique value for games...

We study continuous-time stochastic games, with a focus on the existence of their equilibria that are insensitive to a small imprecision in the specification of players' evaluations of streams of payoffs. We show that the stationary, namely, time-independent, discounting game has a stationary equilibrium and that the discounting game and the more g...

Correlation of players' actions may evolve in the common course of play of a repeated game with perfect monitoring (\online correlation"), and we study the concealment of such correlation from a boundedly rational player. We show that \strong" players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate the online...

We introduce asymptotic analysis of stochastic games with short-stage duration. The play of stage k, k≥0, of a stochastic game Γ δ with stage duration δ is interpreted as the play in time kδ≤t<(k+1)δ and, therefore, the average payoff of the n-stage play per unit of time is the sum of the payoffs in the first n stages divided by nδ, and the λ-disco...

The variation of a martingale $p_0^k=p_0,...,p_k$ of probabilities on a finite (or countable) set $X$ is denoted $V(p_0^k)$ and defined by $V(p_0^k)=E(\sum_{t=1}^k|p_t-p_{t-1}|_1)$. It is shown that $V(p_0^k)\leq \sqrt{2kH(p_0)}$, where $H(p)$ is the entropy function $H(p)=-\sum_xp(x)\log p(x)$ and $\log$ stands for the natural logarithm. Therefore...

Fix a zero-sum repeated game Γ with incomplete information on both sides. It is known that the value of the infinitely repeated game Γ∞ need not exist (Aumann and Maschler 1995). It is proved that any number between the minmax and the maxmin of Γ∞ is the value of a long finitely repeated game Γ n where players’ information about the uncertain numbe...

Every simple monotonic game in bv ' NA is a weighted majority game. Every game v∈bv ' NA has a representation v=u+∑ i=1 ∞ f i ∘μ i where u∈pNA,μ i ∈NA 1 and f i is a sequence of bv ' functions with ∑ j=1 ∞ ∥f i ∥<∞. Moreover, the representation is unique if we require f i to be singular and that for every i≠j,μ i ≠μ j .

Let G=<I,J,g> be a two-person zero-sum game. We examine the two-person zero-sum repeated game G(k,m) in which players 1 and 2 place down finite state automata with k,m states respectively and the payoff is the average per-stage payoff when the two automata face off. We are interested in the cases in which player 1 is "smart" in the sense that k is...

We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertai...

The paper initiates the study of long term interactions where players' bounded rationality varies over time. Time dependent bounded rationality, for player i, is reflected in part in the number [psi]i(t) of distinct strategies available to him in the first t-stages. We examine how the growth rate of [psi]i(t) affects equilibrium outcomes of repeate...

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players' actions and at the beginning of each stage only Player 1 is informed about the state. The results apply to the case of standard signaling where players' stage actions are observable, a...

Robert Aumann has played an essential and indispensable role in shaping game theory and much of economic theory. He promotes a unified view of the very wide domain of rational behaviour, a domain that encompasses areas of many apparently disparate disciplines, like economics, political science, biology, psychology, mathematics, philosophy, computer...

We study the space-and-time automaton-complexity of the CYCLE-LENGTH problem. The input is a periodic stream of bits whose cycle length is bounded by a known number n. The output, a number between 1 and n, is the exact cycle length. We also study a related problem, CYCLE-DIVISOR. In the latter problem the output is a large number that divides the c...

We prove that games with absorbing states with compact action sets have a value.

Consider a repeated two-person game. The question is how much smarter should a player be to effectively predict the moves of the other player. The answer depends on the formal definition of effective prediction, the number of actions each player has in the stage game, as well as on the measure of smartness. Effective prediction means that, no matte...

We examine incentive-compatible mechanisms for fair financing and efficient selection of a public budget (or public good). A mechanism selects the level of the public budget and imposes taxes on individuals. Individuals’ preferences are quasilinear. Fairness is expressed as weak monotonicity (called scale monotonicity) of the tax imposed on an indi...

We study a repeated game with asymmetric information about a dynamic state of nature. In the course of the game, the better-informed player can communicate some or all of his information to the other. Our model covers costly and&sol;or bounded communication. We characterize the set of equilibrium payoffs and contrast these with the communication eq...

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First...

A (pure) strategy in a repeated game is a mapping from histories, or, more generally, signals, to actions. We view the implementation of such a strategy as a computational procedure and attempt to capture in a formal model the following intuition: as the game proceeds, the amount of information (history) to be taken into account becomes large and t...

We introduce a model of communication with dynamic state of nature. We rely on entropy as a measure of information, characterize the set of expected empirical distributions that are achievable. We present applications to games with and without common interests. Classification JEL : C61, C73, D82.

We prove here the existence of a value (of norm 1) on the spaces ′N A and even ′A N, the closure in the variation distance of the linear space spanned by all games f∘μ, where μ is a non-atomic, non-negative finitely additive measure of mass 1 and f a real-valued function on [0,1] which satisfies a much weaker continuity at zero and one.

In honor of L. S. Shapley's eightieth birthday The asymptotic value, introduced by Kannai in 1966, is an asymptotic approach to the notion of the Shapley value for games with infinitely many players. A vector measure game is a game v where the worth v(S) of a coalition S is a function f of μ(S) where μ is a vector measure. Special classes of vector...

In a repeated game with perfect monitoring, correlation among a group of players may evolve in the common course of play (online correlation). Such a correlation may be concealed from a boundedly rational player. The feasibility of such “online concealed correlation” is quantified by the individually rational payoff of the boundedly rational player...

We study a repeated game in which one player, the prophet, acquires more information than another player, the follower, about the play that is going to be played. We characterize the optimal amount of information that can be transmitted online by the prophet to the follower, and provide applications to repeated games played by finite automata, and...

Markov chains1 and Markov decision processes (MDPs) are special cases of stochastic games. Markov chains describe the dynamics of the states of a stochastic game where each player has a single action in each state. Similarly, the dynamics of the states of a stochastic game form a Markov chain whenever the players’ strategies are stationary. Markov...

Let (X,∥∥) be a Banach space and T:X→X a nonexpansive map. This chapter studies asymptotic properties of the orbits of nonexpansive maps defined on a normed space, and relates these properties to properties of the value of two-person zero-sum games and to properties of the minmax of n-person stochastic games. In Section 2 we state the characterizat...

This chapter studies the theory of value of games with infinitely many players.Games with infinitely many players are models of interactions with many players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single...

The existence of the value for stochastic games with finitely many states and actions, as well as for a class of stochastic games with infinitely many states and actions, is proved in [J.-F. Mertens and A. Neyman, Int. J. Game Theory 10, No. 2, 53–66 (1981; Zbl 0486.90096)]. Here we use essentially the same tools to derive the existence of the minm...

We study two-person repeated games in which a player with a restricted set of strategies plays against an unrestricted player. An exogenously given bound on the complexity of strategies, which is measured by the size of the smallest automata that implement them, gives rise to a restriction on strategies available to a player. We examine the asympt...

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numeri...

We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer's strategies while the other player is unrestricted. We will show that if the bound η(n), a function of the number of repetitions n, satisfies the condition η(n)/n → γ (n → ∞), then the maxmin valu...

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space,\({\mathcal{F}}_0 \subset {\mathcal{F}}_1 \subset ... \subset {\mathcal{F}}_n \) an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly\({\mathcal{F}}_n \)-measurable and stronglyP-inte...

We introduce the entropy-based measure of uncertainty for mixed strategies of repeated games—strategic entropy. We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of player 1's strategies while player 2 is unrestricted, as the bound grows to infinity. We apply the...

An exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners' dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be approximated by an equilibrium of a finitely repeated version of that game. The depa...

Every two person repeated game of symmetric incomplete information, in which the signals sent at each stage to both players are identical and generated by a state and moves dependent probability distribution on a given finite alphabet, has an equilibrium payoff.

In honor of R. J. Aumann's 65th birthday The paper studies the implications of bounding the complexity of the strategies players may select, on the set of equilibrium payoffs in repeated games. The complexity of a strategy is measured by the size of the minimal automation that can implement it. A finite automation has a finite number of states and...

Any correlated equilibrium of a strategic game with bounded payoffs and convex strategy sets which has a smooth concave potential, is a mixture of pure strategy profiles which maximize the potential. If moreover, the strategy sets are compact and the potential is strictly concave, then the game has a unique correlated equilibrium.

Four axioms are placed on a correspondence from smooth, non-atomic economies to their allocations. We show that the axiomscategorically determinethe (coincident) competitive-core-value correspondence. Thus any solution is equivalent to the above three if, and only if, it satisfies the axioms. In this sense our result is tantamount to an “equivalenc...

Every two person game of incomplete information in which the information to both player is identical and deterministic has an equilibrium.

This chapter studies the implications of bounding the complexity of players’ strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton has a finite number of states and an initial state. It prescribes the action to be taken as a function of the curre...

The scope of this paper is to present a generalization (due to Autnann and Shapley, 1974) of the Shapley value to the case of a game with a continuum of players. The Shapley value is one of the basic solution concepts of cooperative game theory. It can be viewed as a sort of average or expected outcome, or even as an a priori evaluation of the play...

It is a striking fact that different solutions (such as Walrasian, core and value allocations) become equivalent in perfectly competitive economies (see, e.g., [1], [2], [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [19], [20]). We attempt to understand this phenomenon by making explicit certain crucial properties that are common across t...

It has been remarked that in rational interactions more information to one player, while all others' information remains the same, may reduce his payoff in equilibrium. This classical observation relies on comparing equilibria of two different games. It is argued that this analysis is not tenably performed by comparing equilibria of two different g...

We present some characterizations for the class of non-atomic weighted majority games which are defined on a measurable space (I,C). The characterizations are done within the class of all monotonic simple games which are upper semicontinuous on C and continuous at I with respect to the N A-topology on C. We also use the results on simple games to o...

The paper presents a characterization of continuous cooperative games (set functions) which are monotonic functions of countably additive non-atomic measures. The characterization is done through a natural desirablity relation defined on the set of coalitions of players. A coalition S is at least as desirable as a coalition T (with respect to a giv...

Consider non-atomic vector measure games; i.e., games v of the form v = ƒ ∘(μ1,…,μn, where (μ1,…,μn) is a vector of non-atomic non-negative measures and ƒ is a real-valued function defined on the range of (μ1,…,μn). Games of this form arise, for example, from production models and from finite-type markets. We show that the value of such a game need...

The asymptotic value of a game v with a continuum set of players, I, is defined whenever all the sequences of the Shapley values of finite games that “approximate” v have the same limit. A weighted majority game is a game of the form f ∘ μ where μ is a positive measure and fx = 1 if x ≥ q and fx = 0 otherwise, and q is a real number, 0 <q < μI. In...

Composed in honour of the sixty-fifth birthday of Lloyd Shapley, this volume makes accessible the large body of work that has grown out of Shapley's seminal 1953 paper. Each of the twenty essays concerns some aspect of the Shapley value. Three of the chapters are reprints of the 'ancestral' papers: Chapter 2 is Shapley's original 1953 paper definin...

Composed in honour of the sixty-fifth birthday of Lloyd Shapley, this volume makes accessible the large body of work that has grown out of Shapley's seminal 1953 paper. Each of the twenty essays concerns some aspect of the Shapley value. Three of the chapters are reprints of the 'ancestral' papers: Chapter 2 is Shapley's original 1953 paper definin...

It is shown that the Shapley value of any given game v is characterized by applying the value axioms—efficiency, symmetry, the null player axiom, and either additivity or strong positivity—to the additive group generated by the game ν itself and its subgames.

There is a value (of norm one) on the closed space of games that is generated by all games of bounded variationf o μ, whereμ is a vector of non-atomic probability measures andf is continuous at 0=μ(ø) and atμ(I).

A game-theoretic analysis using the Harsanyi-Shapley nontransferable utility value indicates that the choice of public goods in a democracy is not affected by who has voting rights. This is corroborated by an independent economic argument based on the implicit price of a vote.

The ‘folk theorem’ formalizes the theme that ‘repetition leads to cooperation’. We present an example showing that, even with perfect monitoring, the set of Nash equilibria of the discounted games does not have to converge to the feasible, individually rational set, i.e., this version of the ‘folk theorem’ can break down.

The class of continuous semivalues is completely characterized for various spaces of nonatomic games.

Cooperation in the finitely repeated prisoner's dilemma is justified, without departure from strict utility maximization or complete information, but under the assumption that there are bounds (possibly very large) to the complexity of the strategies that the players may use.

For fixed 1≦p<∞ theL p-semi-norms onR n are identified with positive linear functionals on the closed linear subspace ofC(R n ) spanned by the functions |<ξ, ·>| p , ξ∈R n . For every positive linear functional σ, on that space, the function Φσ:R n →R given by Φσ is anL p-semi-norm and the mapping σ→Φσ is 1-1 and onto. The closed linear span of...

The problem of allocating the production cost of a finite bundle of divisible consumption goods (or services) by means of per unit costs or prices is a basic problem in economics. Recently an axiomatic approach has been proposed by L. J. Billera and D. C. Heath [ibid. 7, 32-39 (1982; Zbl 0509.90009)] and L. J. Mirman and Y. Tauman [ibid. 7, 40-56(1...

Hilbert's metric on a cone K is a measure of distance between the rays of K. Hilbert's metric has many applications, but they all depend on the equivalence between closeness of two rays in the Hilbert metric and closeness of the two unit vectors along these rays (in the usual sense). A necessary and sufficient condition on K for this equivalence to...

It is shown that when resources are privately owned, the institution of voting is irrelevant to the choice of non-exclusive public goods: the total bundle of such goods produced by Society is the same whether or not minority coalitions are permitted to produce them. This is in sharp contrast to the cases of redistribution and of exclusive public go...

The problem of allocating the production cost of a finite bundle of infinitely divisible consumption goods by means of prices is a basic problem in economics. This paper extends the recent axiomatic approach in which one considers a class of cost problems and studies the maps from the class of cost problems to prices by means of the properties thes...

Let $\pi$ be a finite set, $\lambda$ a probability measure on $\pi, 0 < x < 1$ and $a \in \pi$. Let $P(a, x)$ denote the probability that in a random order of $\pi, a$ is the first element (in the order) for which the $\lambda$-accumulated sum exceeds $x$. The main result of the paper is that for every $\varepsilon > 0$ there exist constants $\delt...

Undiscounted nonterminating stochastic games in which the state and action spaces are finite have a value.

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→x f(T n x/n)=limn→x‖T n x/n ‖=α, where α≡inf y∈c ‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichT n x/n converges weakly for allx (infz∈f g(T n x/n-z)→0,...

Stochastic Games have a value.

The asymptotic value of a game v with a continuum of players is defined whenever all the sequences of Shapley values of finite games that “approximate” v have the same limit. In this paper we prove that if v is defined by vS = fμS, where μ is a nonatomic probability measure and f is a function of bounded variation on [0, 1] that is continuous at 0...

The following conditions on a zonoidZ, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoidZ determines them-range of any non-atomic vector measure with rangeZ, where them-range of a vector measure μ is the set ofm-tuples (μ(S 1), …, μ(S m), wher...

It is well known that the set of all zonoids (integrals of line segments) in Rn (n>2) is a closed and nowhere dense subset in the space of all compact, convex and centrally symmetric subsets of Rn. We generalize this result to sets which are the integral ofk-dimensional convex sets, k<n.

The partition value is a new approach to the value concept. It links together the asymptotic and the axiomatic approach. Using this approach we prove the existence of a continuous value on each of the following spaces: bv'NA, A, A * bv'NA, A * bv'NA * bv'NA and the space W spanned by those spaces and ASYMP

A semivalue is a symmetric positive linear operator on a space of games, which leaves the additive games fixed. Such an operator satisfies all of the axioms defining the Shapley value, with the possible exception of the efficiency axiom. The class of semivalues is completely characterized for the space of finite-player games, and for the space pNA...

It is proved that every continuous value is diagonal, which in particular implies that every value on a closed reproducing space is diagonal. We deduce also that there are noncontinuous values.

The question of whether or not the diagonal property is a consequence of the axioms defining the axiomatic value is answered in the negative by means of a counter-example a space and an axiomatic value on it are introduced not possessing the diagonal property.

In the existing literature on bounded rationality in repeated games, sets of feasible strategies are assumed to be independent of time (i.e. stage). In this paper we consider a time-dependent description of strategy sets, growing strategy sets. A growing strategy set is characterized by the way the set of strategies available to a player at each st...

We study a class of games that models costly information transmis- sion in long term interactions. The signaling is restricted to the play of the game and thus have payoff consequences. The auxiliary game: A sequence of actions Xt is chosen by nature, and announced to a player, the prophet, but not to his teammate, the follower. A repeated game the...