Monitoring cooperative equilibria in a stochastic differential game

This paper deals with a class of equilibria which are based on the use of memory strategies in the context of continuous-time stochastic differential games. In order to get interpretable results, we will focus the study on a stochastic differential game model of the exploitation of one species of fish by two competing fisheries. We explore the possibility of defining a so-called cooperative equilibrium, which will implement a fishing agreement. In order to obtain that equilibrium, one defines a monitoring variable and an associated retaliation scheme. Depending on the value of the monitoring variable, which provides some evidence of a deviation from the agreement, the probability increases that the mode of a game will change from a cooperative to a punitive one. Both the monitoring variable and the parameters of this jump process are design elements of the cooperative equilibrium. A cooperative equilibrium designed in this way is a solution concept for a switching diffusion game. We solve that game using the sufficient conditions for a feedback equilibrium which are given by a set of coupled HJB equations. A numerical analysis, approximating the solution of the HJB equations through an associated Markov game, enables us to show that there exist cooperative equilibria which dominate the classical feedback Nash equilibrium of the original diffusion game model.This research was supported by FNRS-Switzerland, NSERC-Canada, FCAR-Quebec.     

The equilibria of a multiple objective game

For a multiple objective game, we introduce its cooperative, non-cooperative, hybrid and quasi-hybrid solution concepts and prove their existence. JEL #: C70, C71, C72The author is indebted to Truman Bewley, Pradeep Dubey, Zhimin Huang, Tatsuro Ichiishi, David Pearce, Herbert E. Scarf, Lloyd Shapley, Martin Shubik, Lin Zhou and the Editor and an anonymous referee ofThe International Journal of Game Theory for their valuable dicussions and suggestions. The support of the 1990–91 Anderson Prize Fellowship from the Cowles Foundation for Research in Economics at Yale University is also appreciated.     

On stationary equilibria of a single-controller stochastic game

We consider a two-person, general-sum, rational-data, undiscounted stochastic game in which one player (player II) controls the transition probabilities. We show that the set of stationary equilibrium points is the union of a finite number of sets such that, every element of each of these sets can be constructed from a finite number of extreme equilibrium strategies for player I and from a finite number of pseudo-extreme equilibrium strategies for player II. These extreme and pseudo-extreme strategies can themselves be constructed by finite (but inefficient) algorithms. Analogous results can also be established in the more straightforward case of discounted single-controller games.     

Complicated equilibria for game problems

We suggest two new notions of game equilibrium for arbitrary conflict problems (antagonistic, noncooperative, cooperative, static, and dynamic ones), which permit one to find a unique strongest equilibrium in some cases in which it is impossible to find such a solution with the use of all earlier known notions of equilibrium.     

Useful equilibria for game theory

We suggest two new notions of equilibria for arbitrary game problems, both static and dynamic ones, whose definitions contain no artificial norms for the behavior of the players. In examples of static and differential games, we demonstrate the possibilities of these equilibria and a technique for finding them.     

Stability of equilibria in evolution game models

Translated from Sistemnoe Programmirovanie i Modeli Issledovaniya Operatsii, pp. 174–182, 1993.     

Strong equilibria of a repeated game with randomized strategies

A version of Aumann's (1976) model of a repeated game with randomized strategies is studied. The pure strategy set of each player is assumed to be a compact metric spacnd complexities due to the information structures are explicitly handled. It is shown that one can extend Aumann's argument to this setup and still prove the Aumann Proposition on equivalence of the β-core of a one-shot game with correlated strategies and the strong equilibrium utility allocations of the associated repeated game with randomized strategies. To this extended version of the Aumann Proposition, the author's theorem for nonemptiness of the β-core with correlated strategies is applicable, so this version is non-vacuous.     

Finding all pure strategy Nash equilibria in a planar location game

In this paper, we deal with a planar location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. If firms set the equilibrium prices in the second stage, the game is reduced to a location game for which pure strategy Nash equilibria are studied assuming that the marginal delivered cost is proportional to the distance between the customer and the facility from which it is served. We present characterizations of local and global Nash equilibria. Then an algorithm is shown in order to find all possible Nash equilibrium pairs of locations. The minimization of the social cost leads to a Nash equilibrium. An example shows that there may exist multiple Nash equilibria which are not minimizers of the social cost.     

Pure equilibria in a simple dynamic model of strategic market game

We present a discrete model of two-person constant-sum dynamic strategic market game. We show that for every value of discount factor the game with discounted rewards possesses a pure stationary strategy equilibrium. Optimal strategies have some useful properties, such as Lipschitz property and symmetry. We also show value of the game to be nondecreasing both in state and discount factor. Further, for some values of discount factor, exact form of optimal strategies is found. For β less than , there is an equilibrium such that players make large bids. For β close to 1, there is an equilibrium with small bids. Similar result is obtained for the long run average reward game.     

Determining all Nash equilibria in a (bi-linear) inspection game

This paper addresses a “game” between an inspection agency and multiple inspectees that are subject to random inspections by that agency. We provide explicit (easily computable) expressions for all possible Nash equilibria and verify that none is left out. In particular, our results characterize situations when there exists a unique Nash equilibrium. We also explore special features of the Nash equilibria and the solution of the problem the inspection agency faces in a non-strategic environment.     

Nash equilibria in the two-player kidney exchange game

Kidney exchange programs have been set in several countries within national, regional or hospital frameworks, to increase the possibility of kidney patients being transplanted. For the case of hospital programs, it has been claimed that hospitals would benefit if they collaborated with each other, sharing their internal pools and allowing transplants involving patients of different hospitals. This claim led to the study of multi-hospital exchange markets. We propose a novel direction in this setting by modeling the exchange market as an integer programming game. The analysis of the strategic behavior of the entities participating in the kidney exchange game allowed us to prove that the most rational game outcome maximizes the social welfare and that it can be computed in polynomial time.     

On the set of proper equilibria of a bimatrix game

In this paper it is proved that the set of proper equilibria of a bimatrix game is the finite union of polytopes. To that purpose we split up the strategy space of each player into a finite number of equivalence classes and consider for a given >0 the set of all -proper pairs within the cartesian product of two equivalence classes. If this set is non-empty, its closure is a polytope. By considering this polytope as goes to zero, we obtain a (Myerson) set of proper equilibria. A Myerson set appears to be a polytope.     

A theorem on the number of Nash equilibria in a bimatrix game

We show that ify is an odd integer between 1 and 2ⁿ ? 1, there is ann × n bimatrix game with exactlyy Nash equilibria (NE). We conjecture that this 2ⁿ ? 1 is a tight upper bound on the number of NEs in a “nondegenerate”n × n game.     

Monotone and bounded interval equilibria in a coordination game with information aggregation

We analyze how private learning in a class of games with common stochastic payoffs affects the form of equilibria, and how properties such as player welfare and the extent of strategic miscoordination relate across monotone and non-monotone equilibria. Researchers typically focus on monotone equilibria. We provide conditions under which non-monotone equilibria also exist, where players attempt to coordinate to obtain the stochastic payoff whenever signals are in a bounded interval. In bounded interval equilibria (BIE), an endogenous fear of miscoordination discourages players from coordinating to obtain the stochastic payoff when their signals suggest coordination is most beneficial. In contrast to monotone equilibria, expected payoffs from successful coordination in BIE are lower than the ex-ante expected payoff from ignoring signals and always trying to coordinate to obtain the stochastic payoff. We show that BIE only exist when, absent private information, the game would be a coordination game.     

Sustainability of pension systems with voluntary participation

Motivated by declining support for mandatory participation in pension arrangements, we explore whether the intergenerational risk-sharing benefits that these arrangements offer suffice to ensure their survival when participation becomes voluntary. Funded systems with asset buffers are particularly interesting since these buffers make contributions more sensitive to financial returns. Equilibria are characterised by thresholds on the young’s willingness to contribute. Standard values for our parameters yield two such equilibria; only the one with the higher threshold is consistent with the initial young being prepared to start the system. An advancement relative to the related literature is that the equilibria feature a non-zero probability of collapse. Finally, we explore the social welfare maximising values for the pension parameters for various levels of uncertainty and risk aversion.     

Pure-strategy Nash equilibria in an advertising game with interference

Two manufacturers produce substitutable goods for a homogeneous market. The advertising efforts of the two manufacturers determine the demand for the goods and interfere negatively with each other. The demand of each good is a piecewise linear function of the product goodwill, and the latter is a linear function of advertising efforts. In a game with two competing profit-maximizing manufacturers who have access to a set of several advertising media, the pure-strategy Nash equilibria are characterized and their existence is shown.     

Voter turnout as a participation game: An experimental investigation

Financial support by the Netherlands' Organization for Scientific Research (NWO) is gratefully acknowledged. We would like to thank Jeff Banks, Eric van Damme, Werner Güth, Claudia Keser, Theo Offerman, Mark Olson, J?rgen Wit, and two anonymous referees for useful comments, and Otto Perdeck for computational assistance.     

Definition and properties of cooperative equilibria in a two-player game of infinite duration

A two-player multistage game, with an infinite number of stages is considered. The concepts of overtaking and weakly overtaking payoff sequences are introduced. The class of strategies considered consists of memory strategies, which are based on the past history of the control and the initial state from where the game has been played. Weak equilibria are defined in this class of strategies. It is then shown how such equilibria can be constructed by composing into a trigger strategy a nominal cooperative control sequence and two threat strategies representing the announced retaliation by each player in the case where the other player does not play according to the nominal control. When the threats consists of a feedback equilibrium pair, the resulting cooperative equilibrium is perfect. Another result shows that, if each player can use a most effective threat based on a saddle-point feedback strategy, then any weak equilibrium in the class of memory strategies is in some sense related to this particular kind of equilibrium in the class of trigger strategies.Dedicated to G. LeitmannThis research was supported by SSHRC Grant No. 410-81-0722 and FCAC Grant No. EQ-428 to the first author. This research has also been made possible by a financial support from the University of Puerto Rico.