Deterministic multi-player Dynkin games. (English) Zbl 1081.91004
Summary: A multi-player Dynkin game is a sequential game in which at every stage one of the players is chosen, and that player can decide whether to continue the game or to stop it, in which case all players receive some terminal payoff.
We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than 1.
We prove that a subgame-perfect \(\epsilon\)-equilibrium in Markovian strategies exists. If the game is not degenerate this \(\epsilon\)-equilibrium is actually in pure strategies.
We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than 1.
We prove that a subgame-perfect \(\epsilon\)-equilibrium in Markovian strategies exists. If the game is not degenerate this \(\epsilon\)-equilibrium is actually in pure strategies.
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 91A06 | \(n\)-person games, \(n>2\) |
| 91A20 | Multistage and repeated games |
| 91A60 | Probabilistic games; gambling |
References:
| [1] | Dynkin, E. B., Game variant of a problem on optimal stopping, Soviet Mathematics Doklady, 10, 270-274 (1969) · Zbl 0186.25304 |
| [2] | Fine, C. H.; Li, L., Equilibrium exit in stochastically declining industries, Games and Economic Behavior, 1, 40-59 (1989) · Zbl 0755.90010 |
| [3] | Flesch, J.; Thuijsman, F.; Vrieze, K., Cyclic Markov equilibria in stochastic games, International Journal of Game Theory, 26, 303-314 (1997) · Zbl 0879.90200 |
| [4] | Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA.; Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. · Zbl 1339.91001 |
| [5] | Ghemawat, P.; Nalebuff, B., Exit, RAND Journal of Economics, 16, 184-194 (1985) |
| [6] | Kiefer, Y. I., Optimal stopped games, Theory of Probability and its Application, 16, 185-189 (1971) · Zbl 0238.90085 |
| [7] | Kilgour, D. M., The sequential truel, International Journal of Game Theory, 4, 151-174 (1975) · Zbl 0337.90074 |
| [8] | Kilgour, D. M., Equilibrium points in infinite sequential truels, International Journal of Game Theory, 6, 167-180 (1977) · Zbl 0375.90088 |
| [9] | Kilgour, D. M.; Brams, S. J., The truel, Mathematics Magazine, 70, 315-325 (1997) · Zbl 0934.91003 |
| [10] | Neveu, J., 1975. Discrete-Parameter Martingales. North-Holland, Amsterdam.; Neveu, J., 1975. Discrete-Parameter Martingales. North-Holland, Amsterdam. · Zbl 0345.60026 |
| [11] | Rosenberg, D.; Solan, E.; Vieille, N., Stopping games with randomized strategies, Probability Theory and Related Fields, 119, 433-451 (2001) · Zbl 0988.60037 |
| [12] | Rosenberg, D., Solan, E., Vieille, N., 2002. On the maxmin value of stochastic games with imperfect monitoring. Discussion Paper 1337. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University.; Rosenberg, D., Solan, E., Vieille, N., 2002. On the maxmin value of stochastic games with imperfect monitoring. Discussion Paper 1337. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University. · Zbl 1101.91309 |
| [13] | Shmaya, E., Solan, E., 2002. Two-player non-zero-sum stopping games in discrete time. Discussion Paper 1347. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University.; Shmaya, E., Solan, E., 2002. Two-player non-zero-sum stopping games in discrete time. Discussion Paper 1347. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University. · Zbl 1079.60045 |
| [14] | Solan, E., Three-person absorbing games, Mathematics of Operations Research, 24, 669-698 (1999) · Zbl 1064.91500 |
| [15] | Solan, E., 2002. Subgame perfection in quitting games with perfect information and differential equations. Discussion Paper 1356. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University.; Solan, E., 2002. Subgame perfection in quitting games with perfect information and differential equations. Discussion Paper 1356. The Center for Mathematical Studies in Economics and Management Science, Kellogg School of Management, Northwestern University. |
| [16] | Solan, E.; Vieille, N., Quitting games, Mathematics of Operations Research, 26, 265-285 (2001) · Zbl 1082.91008 |
| [17] | Vieille, N., Equilibrium in 2-person stochastic games. II. The case of recursive games, Israel Journal of Mathematics, 119, 93-126 (2000) · Zbl 0978.91007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
