Academia.eduAcademia.edu

Outline

Title
Abstract
References
All Topics
Economics
Game Theory

Sequential Correlated Equilibria in Stopping Games

Profile image of Yuval HellerYuval Heller

2009, Social Science Research Network

https://doi.org/10.1287/OPRE.1110.1010
visibility

…

link

1 file

Abstract

In many situations, such as trade in stock exchanges, agents have many opportunities to act within a short interval of time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium.

Related papers

Randomized stopping games and Markov market games
Elzbieta Ferenstein

Mathematical Methods of Operations Research, 2007

We study nonzero-sum stopping games with randomized stopping strategies. The existence of Nash equilibrium and ε-equilibrium strategies are discussed under various assumptions on players random payoffs and utility functions dependent on the observed discrete time Markov process. Then we will present a model of a market game in which randomized stopping times are involved. The model is a mixture of a stochastic game and stopping game. Keywords Stopping games • Stochastic games • Nash equilibrium • Markov chain Research supported by grant PBZ-KBN-016/P03/99.

View PDFchevron_right
A Class of Solvable Stopping Games
luis alvarez

Applied Mathematics and Optimization - APPL MATH OPT, 2008

We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.

View PDFchevron_right
The Generalized Stackelberg Equilibrium of the Two-Person Stopping Game
Krzysztof Szajowski

Journal of the Operations Research Society of China

In modeling the bilateral selection of states of the process, Dynkin (Dokl Akad Nauk USSR 185:241–288, 1969) proposed a two-person game in which players use stopping moments as strategies. The purpose of this work is to present a model of the game in which the players have different information about the process itself, as well as various laws to stop the process and accept its state. The game model uses the stochastic process apparatus, in particular, the ability to create different filters for the same process. The sets of stopping moments based on different filters are not identical, which allows us to model different sets of strategies for players. We show that the follower, by observing the behavior of a rational leader, can recover information that is lost due to the lack of complete observation of the state of the process. In the competition of two opponents for the maximum of the i.i.d. sequence, one of whom has access to full information and the other only knows their relat...

View PDFchevron_right
Stopping games in continuous time
Rida Laraki

2003

We study two-player zero-sum stopping games in continuous time and infinite horizon. We prove that the value in randomized stopping times exists as soon as the payoff processes are right-continuous. In particular, as opposed to existing literature, we do not assume any conditions on the relations between the payoff processes. We also show that both players have simple ε-optimal randomized stopping times; namely, randomized stopping times which are small perturbations of non-randomized stopping times.

View PDFchevron_right
Convergence of the prices of games in problems of optimal stopping of Markovian processes
Vigirdas Mackevičius

Lithuanian Mathematical Transactions, 1974

View PDFchevron_right
On infinite horizon multi-person stopping games with priorities
Elzbieta Ferenstein

Banach Center Publications, 2006

We study nonzero-sum multi-person multiple stopping games with players' priorities. The existence of Nash equilibrium is proved. Examples of multi stopping of Markov chains are considered. The game may also be presented as a special case of a stochastic game which leads to many variations of it, in which stopping is a part of players' strategies.

View PDFchevron_right
Zero-sum stochastic games with stopping and control
Mrinal K. Ghosh

Operations Research Letters, 2007

We study a zero-sum stochastic game where each player uses both control and stopping times. Under certain conditions we establish the existence of a saddle point equilibrium, and show that the value function of the game is the unique solution of certain dynamic programming inequalities with bilateral constraints.

View PDFchevron_right
Incentive compatibility in sender-receiver stopping games
Dries Vermeulen

2020

We introduce a model of sender-receiver stopping games, where the state of the world follows an iid--process throughout the game. At each period, the sender observes the current state, and sends a message to the receiver, suggesting either to stop or to continue. The receiver, only seeing the message but not the state, decides either to stop the game, or to continue which takes the game to the next period. The payoff to each player is a function of the state when the receiver quits, with higher states leading to better payoffs. The horizon of the game can be finite or infinite. We prove existence and uniqueness of responsive (i.e. non-babbling) Perfect Bayesian Equilibrium (PBE) under mild conditions on the game primitives in the case where the players are sufficiently patient. The responsive PBE has a remarkably simple structure, which builds on the identification of an easy-to-implement and compute class of threshold strategies for the sender. With the help of these threshold stra...

View PDFchevron_right
Optimal Stopping with Delayed Information
Bernt Oksendal

Stochastics and Dynamics, 2005

We study a general optimal stopping problem for a strong Markov process in the case when there is a time lag δ > 0 from the time τ when the decision to stop is taken (a stopping time) to the time τ + δ when the system actually stops. Equivalently, we impose the constraint that the admissible times for stopping are stopping (Markov) times with respect to the delayed flow of information. It is shown that such a problem can be reduced to a classical optimal stopping problem by a simple transformation. The results are applied: (i) to find the optimal time to sell an asset (ii) to solve an optimal resource extraction problem, in both cases under delayed information flow.

View PDFchevron_right
A Problem of Sequential Entry and Exit Decisions Combined with Discretionary Stopping
Mihail Zervos

SIAM Journal on Control and Optimization, 2003

We consider a stochastic control problem that has emerged in the economics literature as an investment model under uncertainty. This problem combines features of both stochastic impulse control and optimal stopping. The aim is to discover the form of the optimal strategy. It turns out that this has a priori rather unexpected features. The results that we establish are of an explicit nature. We also construct an example whose value function does not possess C 1 regularity.

View PDFchevron_right

References (66)

  1. Assaf, D., E. Samuel-Cahn. 1998a. Optimal cooperative stopping rules for maximization of the product of the expected stopped values. Stat. Probab. Lett. 38(1) 89-99.
  2. Assaf, D., E. Samuel-Cahn. 1998b. Optimal multivariate stopping rules. J. Appl. Probab. 35(3) 693-706.
  3. Aumann, R. J. 1974. Subjectivity and correlation in randomized strategies. J. Math. Econom. 1(1) 67-96.
  4. Aumann, R. J. 1997. Rationality and bounded rationality. Games Econom. Behav. 21(1) 2-14.
  5. Aumann, R. J., M. Maschler. 1995. Repeated Games with Incomplete Information. MIT Press, Cambridge, MA.
  6. Ben-Porath, E. 1998. Communication without mediation: Expending the set of equilibrium outcomes by "cheap" pre-play procedures. J. Econom. Theory 80(1) 108-122.
  7. Bernstein, F., A. Federgruen. 2004. A general equilibrium model for indus- tries with price and service competition. Oper. Res. 52(6) 868-886.
  8. Bouakiz, M., M. J. Sobel. 1992. Inventory control with an exponential utility criterion. Oper. Res. 40(3) 603-608.
  9. Bulow, J., P. Klemperer. 2001. The generalized war of attrition. Amer. Econom. Rev. 89(1) 175-189.
  10. Chatterjee, K., L. Samuelson. 1988. Bargaining under two-sided incom- plete information: The unrestricted offers case. Oper. Res. 36(4) 605-618.
  11. Christie-David, R., M. Chaudhry, W. Khan. 2002. News releases, mar- ket integration, and market leadership. J. Financial Res. XXV(2) 223-245.
  12. Dhillon, A., J. F. Mertens. 1996. Perfect correlated equilibria. J. Econom. Theory 68(2) 279-302.
  13. Dynkin, E. B. 1969. Game variant of a problem on optimal stopping. Soviet Math.-Doklady 10 270-274.
  14. Ferguson, T. S. 2005. Selection by committee. A. S. Nowak, K. Szajowski, eds. Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Annals of the Interna- tional Society of Dynamic Games, Vol. 7, Part III. Birkhäuser, Boston, 203-209.
  15. Operations Research 60(1), pp. 209-224, © 2012 INFORMS
  16. Forges, F. 1986. An approach to communication equilibria. Econometrica 54(6) 1375-1385.
  17. Fudenberg, D., J. Tirole. 1985. Preemption and rent equalization in the adoption of new technology. Rev. Econom. Stud. LII(3) 383-401.
  18. Fudenberg, D., J. Tirole. 1986. A theory of exit in duopoly. Econometrica 54(4) 943-960.
  19. Glickman, H. 2004. Cooperative stopping rules in multivariate problems. Sequential Anal. 23(3) 427-449.
  20. Heller, Y. 2010a. Minority-proof cheap-talk protocol. Games Econom. Behav. 69(2) 394-400.
  21. Heller, Y. 2010b. Comment on distribution equilibria. http://www.tau.ac .il/ ~helleryu/distribution.pdf.
  22. Karlin, S. 1959. Mathematical Methods and Theory in Games, Program- ming and Economics, Vol. 2. Addison-Wesley, Reading, MA.
  23. Kreps, M. D., R. Wilson. 1982. Sequential equilibria. Econometrica 50(4) 863-894.
  24. Krishna, V., J. Morgan. 1997. An analysis of the war of attrition and the all-pay auction. J. Econom. Theory 72(2) 343-362.
  25. Kurano, M., M. Yasuda, J. Nakagami. 1980. Multi-variate stopping prob- lem with a majority rule. J. Oper. Res. Soc. Japan 23 205-223.
  26. Laraki, R., E. Solan. 2005. The value of zero-sum stopping games in continuous time. SIAM J. Control Optim. 43(5) 1913-1922.
  27. Maliath, J. G., L. Samuelson, A. Shaked. 1997. Correlated equilibria and local interactions. Econom. Theory 9(3) 551-556.
  28. Mamer, J. W. 1987. Monotone stopping games. J. Appl. Probab. 24(2) 386-401.
  29. Mashiah-Yaakovi, A. 2009. Subgame perfect equilibria in stopping games. Working paper, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel, http://www.math.tau.ac.il/ ~ayalam/ publication.files/general stopping game 3.pdf.
  30. Maynard Smith, J. 1974. The theory of games and the evolution of animal conflicts. J. Theoret. Biol. 47 209-221.
  31. Morimoto, H. 1986. Non-zero-sum discrete parameter stochastic games with stopping times. Probab. Theory Related Fields 72(1) 155-160.
  32. Moulin, H., J. P. Vial. 1978. Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon. Internat. J. Game Theory 7(3-4) 201-221.
  33. Myerson, R. B. 1986a. Multistage games with communication. Economet- rica 54(2) 323-358.
  34. Myerson, R. B. 1986b. Acceptable and predominant correlated equilibria. Internat. J. Game Theory 15(3) 133-154.
  35. Nalebuff, B., J. G. Riley. 1985. Asymmetric equilibria in the war of attri- tion. J. Theoret. Biol. 113(3) 517-527.
  36. Neumann, P., D. Ramsey, K. Szajowski. 2002. Randomized stopping times in Dynkin games. J. Appl. Math. Mech. 82(11-12) 811-819.
  37. Neveu, J. 1975. Discrete-Parameters Martingales. North-Holland, Amsterdam.
  38. Nikkinen, J., M. Omran, P. Sahlstrom, J. Aijo. 2006. Global stock market reactions to scheduled U.S. macroeconomic news announcements. Global Finance J. 17(1) 92-104.
  39. Nowak, A. S., K. Szajowski. 1999. Nonzero-sum stochastic games. M. Bardi, T. E. S. Raghavan, T. Parthasarathy, eds. Stochastic and Differential Games. Birkhäuser, Boston, 297-342.
  40. Ohtsubo, Y. 1991. On a discrete-time non-zero-sum Dynkin problem with monotonicity. J. Appl. Probab. 28(2) 466-472.
  41. Ohtsubo, Y. 1995. Pareto optimum in a cooperative Dynkin's stopping problem. Nihonkai Math. J. 6(2) 135-151.
  42. Ohtsubo, Y. 1996. Core in a cooperative Dynkin's stopping problem. RIMS Kokyuroku 947(1) 13-21.
  43. Ohtsubo, Y. 1998. Pareto optima in a multi-person cooperative stopping problem. RIMS Kokyuroku 1043 184-191.
  44. Osborne, M. J., A. Rubinstein. 1994. A Course in Game Theory. MIT Press, Cambridge, MA.
  45. Ramsey, D., D. Cierpial. 2009. Cooperative strategies in stopping games. V. Gaitsgory, P. Bernhard, O. Pourtallier, eds. Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control. Annals of the International Society of Dynamic Games, Vol. 10. Birkhäuser, Boston, 415-430.
  46. Ramsey, D., K. Szajowski. 2008. Selection of a correlated equilibrium in Markov stopping games. Eur. J. Oper. Res. 184(1) 185-206.
  47. Ramsey, D. M. 2007. Correlated equilibria in n-player stopping games. Scientiae Mathematicae Japonicae 66(1) 149-164.
  48. Ramsey, F. 1930. On a problem of formal logic. Proc. London Math. Soc. 30(1) 264-286.
  49. Rosenberg, D., E. Solan, N. Vieille. 2001. Stopping games with random- ized strategies. Probab. Theory Related Fields 119(3) 433-451.
  50. Rubinstein, A. 1991. Comments on the interpretation of game theory. Econometrica 59(4) 909-924.
  51. Selten, R. 1965. Spieltheoretische behandlung eines oligopolmodells mit nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft 121 301-324.
  52. Selten, R. 1975. Reexamination of the perfectness concept for equilibrium points in extensive games. Internat. J. Game Theory 4(1) 25-55.
  53. Shmaya, E., E. Solan. 2004. Two-player nonzero-sum stopping games in discrete time. Ann. Probab. 32(3B) 2733-2764.
  54. Shmida, A., B. Peleg. 1997. Strict and symmetric correlated equilibria are the distributions of the ESS's of biological conflicts with asymmetric roles. W. Albers, W. Guth, P. Hammerstein, B. Moldovanu, E. van Damme, eds. Understanding Strategic Interaction. Springer-Verlag, Berlin, 149-170.
  55. Solan, E., N. Vieille. 2001. Quitting games. Math. Oper. Res. 26(2) 265-285.
  56. Solan, E., R. V. Vohra. 2001. Correlated equilibrium in quitting games. Math. Oper. Res. 26(3) 601-610.
  57. Solan, E., R. V. Vohra. 2002. Correlated equilibrium payoffs and public signaling in absorbing games. Internat. J. Game Theory 31(1) 91-121.
  58. Sorin, S. 1998. Distribution equilibrium I: Definition and equilibrium. Papers 9835, Paris X-Nanterre, U.F.R. de science economiques ges- tion, mathematiques et informatique, Paris.
  59. Sorin, S. 2002. First Course on Zero-Sum Repeated Games. Springer, Paris.
  60. Szajowski, K. 2002. On stopping games when more than one stop is pos- sible.
  61. V. F. Kolchin, V. Y. Kozlov, V. V. Mazalov, Y. L. Pavlov, Y. V. Prokhorov, eds. Probab. Methods Discrete Math. Proc. 5th Inter- nat. Petrozavodsk Conf. International Science Publishers, Leiden, The Netherlands, 57-72.
  62. Szajowski, K., M. Yasuda. 1997. Voting procedure on stopping games of Markov chain. A. H. Christer, S. Osaki, L. C. Thomas, eds. UK- Japanese Res. Workshop Stochastic Modeling in Innovative Manu- facturing, July 21-22, 1995. Lecture Notes in Economics and Math- ematical Systems, Vol. 445. Springer, Berlin, 68-80.
  63. Taylor, T. E., E. L. Plambeck. 2007. Supply chain relationships and con- tracts: The impact of repeated interaction on capacity investment and procurement. Management Sci. 53(10) 1577-1593.
  64. Urbano, A., J. E. Vila. 2002. Computational complexity and commu- nication: coordination in two-player games. Econometrica 70(5) 1893-1927.
  65. Yasuda, M., K. Szajowski. 2002. Dynkin game and its extension to a multiple stopping model Bull. Japan Soc. Indust. Math. 12(3) 17-28 (Japanese).
  66. Yasuda, M., J. Nakagami, M. Kurano. 1982. Multi-variate stopping problem with a monotone rule. J. Oper. Res. Soc. Japan 25(4) 334-350.

Related papers

Sequential correlated equilibrium in stopping games
Yuval Heller

Operations Research, 2012

In many situations, such as trade in stock exchanges, agents have many opportunities to act within a short interval of time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium.

View PDFchevron_right
Correlated equilibria in Markov stopping games. The numerical methods and examples
David Ramsey

2004

View PDFchevron_right
Correlated equilibria in Markov stopping games
David Ramsey
View PDFchevron_right
On the value of optimal stopping games
stephane villeneuve

The Annals of Applied Probability, 2006

We show, under weaker assumptions than in the previous literature, that a perpetual optimal stopping game always has a value. We also show that there exists an optimal stopping time for the seller, but not necessarily for the buyer. Moreover, conditions are provided under which the existence of an optimal stopping time for the buyer is guaranteed. The results are illustrated explicitly in two examples.

View PDFchevron_right
Selection of a correlated equilibrium in Markov stopping games
David Ramsey

European Journal of Operational Research, 2008

This paper deals with an extension of the concept of correlated strategies to Markov stopping games. The Nash equilibrium approach to solving nonzero-sum stopping games may give multiple solutions. An arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players' decisions according to some optimality criterion. This is a form of equilibrium selection. Examples of correlated equilibria in nonzero-sum games related to the best choice problem are given. Several concepts of criteria for selecting a correlated equilibrium are used.

View PDFchevron_right

Related topics

  • Applied Mathematics
  • Mathematical Economics
  • Social Science Research Network
  • Business and Management
  • Coordination Game
  • Extensive Form Game
  • Sequential equilibrium
  • Sequential Game
  • correlated equilibrium
    • 580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved