Payoffs Of Players Research Articles

Breakdown of Cooperation in Iterative Real-Time Trust Dilemmas

We study a class of trust dilemmas with symmetric players that evolve in real-time. In these games, as long as all the n players continue to cooperate, the payoff function increases exponentially over time. Simultaneously, however, the temptation to defect also increases at the same rate. The first player to defect receives the present value of the payoff function whereas each of the other n-1 players only receives a fraction d of that player's payoff. We report the results of an iterated trust dilemma that varies the values of n and d in a between-subjects design. In all cases, trust-based cooperation breaks down in the population over iterations of the stage game with the rate of breakdown sharply increasing in n and slowly decreasing in d.

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.

Shirking and Squandering in Sharing Games

In a sharing game the players' choices yield a revenue, each player's choice carries a cost, and a player's payoff is a portion of the revenue minus the player's cost. Such games are appealingly simple devices for partially aligning individual incentives with organizational goals, but their equilibria may be inefficient, i.e., at an equilibrium the surplus (revenue minus the sum of the costs) may not be maximal. Sharing games as a general class have not been well studied. We start a general theory of sharing games by going beyond the common economic setting, where strategy sets are continua and cost and revenue functions are smooth. We include games in which some of a player's strategies are equally costly, and revenue changes when he switches from one of them to another. We consider several large classes of reward functions, including nondecreasing residual (NDR) functions, in which residual (revenue minus rewards) does not drop when revenue increases. That class includes budget-balancing functions, where residual is always zero. To focus the discussion, we examine a ``Folk Claim", which asserts that at every inefficient equilibrium shirking, in some sense, occurs. We show that in NDR games a complementarity condition indeed insures that no one squanders at equilibrium (spends more than at an efficient profile). But when we drop complementarity, the situation changes sharply, and there are games with compelling equilibria, at which some players squander. The shirking/squandering distinction is particularly important in tracing the effect of technical improvement on the surplus shortfall at a sharing game's equilibrium. The paper also obtains conditions for existence of (pure-strategy) equilibria and finds, in particular, that every finite game in which rewards are linearly related has an equilibrium.

The effect of quantum noise on the restricted quantum game

It has recently been established that quantum strategies have great advantage over classical ones in quantum games. However, quantum states are easily affected by the quantum noise resulting in decoherence. In this paper, we investigate the effect of quantum noise on the restricted quantum game in which one player is restricted in classical strategic space, another in quantum strategic space and only the quantum player is affected by the quantum noise. Our results show that in the maximally entangled state, no Nash equilibria exist in the range of 0<p⩽0.422 (p is the quantum noise parameter), while two special Nash equilibria appear in the range of 0.422<p<1. The advantage that the quantum player diminished only in the limit of maximum quantum noise. Increasing the amount of quantum noise leads to the increase of the classical player's payoff and the reduction of the quantum player's payoff, but is helpful in forming two Nash equilibria.

Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems.

A differential game of joint implementation of environmental projects

This paper proposes a two-player, finite-horizon differential game model to analyze joint implementation in environmental projects, one of the flexible mechanisms considered in the Kyoto Protocol. Our results show that allowing for foreign investments could in some cases increase the welfares of both parties involved in the project. Further, imposing an environmental target constraint does not necessarily deteriorate the payoffs of both players.

TECHNICAL NOTE: NONTRANSFERABLE INDIVIDUAL PAYOFFS IN COOPERATIVE STOCHASTIC DIFFERENTIAL GAMES

This note develops a mechanism for the derivation of individual player's payoff functions in cooperative stochastic differential games with nontransferable payoffs. An illustrative example is provided.

The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Comment

HARRIS (1985) HAS SHOWN that subgame-perfect equilibria exist in deterministic continuous games with perfect information.' A recent influential paper by Harris, Reny, and Robson (1995) shows that public randomization ensures the existence of subgameperfect equilibria in continuous games with almost perfect information. The authors exhibit an example of a game with almost perfect information in which no subgameperfect equilibrium exists without public randomization. In addition, their Proposition 36 argues that public randomization is not required in games with perfect information. Contrary to this proposition, we give here an example of a continuous game with perfect information in which no subgame-perfect equilibrium exists if public randomization is not allowed. The result of Harris (1985) implies that our example must be a game in which Nature is an active player. Intrapersonal games for consumers with changing preferences are usually games of this type, as in, for example, stochastic versions of Peleg and Yaari (1973) and Goldman (1980). The example has five stages. In stage 1, player 1 chooses al E [0, 1]. In stage 2, player 2 chooses a2 E [0, 1]. In stage 3, Nature chooses x by randomizing uniformly over the interval [-2 + a, + a2, 2 - a, - a2]. After this, players 3 and 4 move sequentially. The subgame following a history (a,, a2, x) and the associated payoffs for all four players are shown in Figure 1. In the following, let a and ,B denote the probabilities with which players 3 and 4 choose U and u, respectively. Consider first the subgame defined by a, = a2 = 1. Nature's move is degenerate in this subgame: x = 0. The set of subgame-perfect equilibrium paths in the resulting subgame is characterized by three segments of mixing probabilities: a = 0 and /3 E [0, .5]; a E [0, 1] and ,B = .5; and a = 1 and f3 E [.5, 1]. The set of equilibrium expected payoffs for players 1 and 2 implied by these three segments is given by

Individually rational pure strategies in large games

In “randomly selected” large finite normal form games, pure strategy “strongly individually rational” outcomes (all payoffs exceeding pure strategy minimax payoffs) exist with high probability. As a corollary, pure strategy Nash equilibria also exist with high probability in certain infinitely repeated games. Specifically, the pure strategy Folk Theorem, which can be true in a vacuous sense, has nontrivial substance with probability approaching one as pure strategy sets increase in cardinality, provided strategy set growth rates are not too different. To prove this, and noting a possible “bounded rationality” interpretation, we begin with a study of “approximate equilibria” in finite normal form games. Given a positive integer l, these are pure strategy outcomes where each player's payoff is among the l largest available given strategy choices of opponents. When pure strategy sets are large, such outcomes are highly probable and usually have payoffs dominating pure minimax payoffs.

Some Asymptotic Results in Discounted Repeated Games of One-Sided Incomplete Information

The paper analyzes the Nash equilibria of two-person discounted repeated games with one-sided incomplete information and known own payoffs. If the informed player is arbitrarily patient, relative to the uninformed player, then the characterization for the informed player's payoffs is essentially the same as that in the undiscounted case. This implies that even small amounts of incomplete information can lead to a discontinuous change in the equilibrium payoff set. For the case of equal discount factors, however, and under an assumption that strictly individually rational payoffs exist, a result akin to the Folk Theorem holds when a complete information game is perturbed by a small amount of incomplete information.

Computation of Stackelberg Trajectories in a Class of Two-Person Linear Differential Games with Terminal Players' Payoffs and Polygonal Constrainings for Controls

The report suggests a numerical method for constructing Stackelberg trajectories in a linear two-person n-dimensional differential game with terminal payoffs of players and polygonal constrainings for controls. Formalization of players' strategies in the game is based on formalization and the results of positional antagonistic differential games theory, developed by N. N. Krasovskii and his scientific school. The game is reduced to a game on the plane and the problem is transformed to solving a non-standard optimal control problem. For the approximation of the trajectories sets for this problem a set of computational geometry algorithms in plane is used, including convex hull construction, union and intersection of polygons and a Minkowski sum for polygons.

Unity is Strength: An Experimental Study of Decentralized and Collective Bargaining

We present an experiment where two players bargain with a third player. They can bargain either separately or form a joint venture to bargain collectively. Our theoretical benchmark solution predicts decentralized bargaining, as only one player has an interest in forming a joint venture. However, we observe a significant amount of collective bargaining. Collective bargaining, when compared with decentralized bargaining, has no significant effect on the payoffs of the players in the joint venture but reduces the payoff of the third player.

Stable and efficient bargaining networks

Abstract. We examine a bargaining game among players connected by a network, where successively potential partners negotiate over terms of a project. For any network structure and ordering of selection of bargaining partners, there is a unique subgame perfect equilibrium outcome which coincides with the standard Rubinstein split. Based on this, we derive expected payoffs for players as a function of the network. We the characterize the pairwise stable and efficient bargaining networks. These two sets do not always coincide and in many situations pairwise stable networks are over-connected.

Axiomatizations of Permission Values for Games with a Hierarchical Permission Structure using Split Neutrality

Recently, cooperative game theory has been applied to various economic allocation problems in which players are not fully anonymous but belong to some relational structure. One of the most developed models in this respect are communications situations or (symmetric) network situations in which players can only cooperate if there are sufficient communication links in the communication network. Another class of applications considers situations in which the players are hierarchically ordered, i.e. they are part of a structure of asymmetric relations. Examples are auctions, airport games, sequencing situations, the water distribution problem and hierarchically structured firms. This paper is about games with permission structure being a general game theoretic model to study situations with asymmetric relations between the players. We provide new axiomatic characterizations of the Shapley permission values and the first characterizations of the Banzhaf permission values using split properties which say something about the payoffs of players if we split certain players in two.

A rough set approach to estimating the game value and the Shapley value from data

A value of a game v is a function which to each coalition S assigns the value v(S) of this coalition, meaning the expected pay-off for players in that coalition. A classical approach of von Neumann...

The Excluded Player's Payoff in Coalition Formation

In this study we investigated how the excluded player's payoff in coalition formation affects coalition behavior. Results of two experiments showed that a decrease in excluded player's payoff decreased the number of proposed small coalitions. This effect was moderated by social value orientations. Prosocials decreased their preference for being members of a small coalition when the excluded player's payoff decreased. Proselfs preferred being members of small coalitions regardless of the excluded player's payoff. These results provided a different perspective than previous research that generally focused on the members of a coalition, and assumed that coalition is foremost a game in which players wants to maximize their own outcome. Results are discussed and related to research on social exclusion, the do-no-harm principle, and social value orientations.

The Stability of Hedonic Coalition Structures

We consider the partitioning of a society into coalitions in purely hedonic settings, i.e., where each player's payoff is completely determined by the identity of other members of her coalition. We first discuss how hedonic and nonhedonic settings differ and some sufficient conditions for the existence of core stable coalition partitions in hedonic settings. We then focus on a weaker stability condition: individual stability, where no player can benefit from moving to another coalition while not hurting the members of that new coalition. We show that if coalitions can be ordered according to some characteristic over which players have single-peaked preferences, or where players have symmetric and additively separable preferences, then there exists an individually stable coalition partition. Examples show that without these conditions, individually stable coalition partitions may not exist. We also discuss some other stability concepts, and the incompatibility of stability with other normative properties. Journal of Economic Literature Classification Numbers: C71, A14, D20.

Nash Equilibrium and Subgame Perfection in Observable Queues

A subgame perfection refinement of Nash equilibrium is suggested for games of the following type: each of an infinite number of identical players selects an action using his private information on the system's state; any symmetric strategy results in a discrete Markov chain over such states; the player's payoff is a function of the state, the selected action, and the common strategy selected by the other players. The distinction between equilibria which are subgame perfect and those which are not, is made apparent due to the possibility that some states are transient. We illustrate the concept by considering several queueing models in which the number of customers in the system constitutes the state of the system.

Optimal Imperfect Commitments in a Negotiation with a Deadline

One major finding of the bargaining literature is that the more irrevocable a commitment is, the better for the committed player. One notable exception is Fershtman and Seidmann (1993) bilateral negotiation where the commitment not only does not improve the committed player's payoff but also it results in inefficient delays. In this paper we relax two assumptions of the bargaining literature related to the uncertainty and irrevocability of commitments. We allow players not only to possess imperfect commitments but also to choose the degree of imperfection of their commitment. When these assumptions are incorporated in Fershtman and Seidmann (1993)'s model, we obtain that, on the one hand, the delays disappear and an immediate agreement is reached. On the other hand, the commitment becomes effective, that is, it improves the payoff of the committed player.

Large Nonanonymous Repeated Games

E. J. Green (1980, J. Econ. Theory 22, 155–182) and H. Sabourian (1990, J. Econ. Theory 51, 92–110) studied repeated games where a player's payoff depends on his actions and an anonymous aggregate outcome, and show that long-run players behave myopically in any equilibrium of such games. In this paper we extend these results to games where the aggregate outcome is not necessarily an anonymous function of players' actions, and where players' strategies may depend nonanonymously on signals of other players' behavior. Our argument also provides a conceptually simpler proof of Green and Sabourian's results, showing how their analysis is driven by general bounds on the number of pivotal players in a game.