# American Institute of Mathematical Sciences

July  2014, 1(3): 507-535. doi: 10.3934/jdg.2014.1.507

## Strong approachability

 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, Israel

Received  February 2013 Revised  October 2013 Published  July 2014

We introduce the concept of strongly approachable sets in two-player repeated games with vector payoffs. A set in the payoff space is strongly approachable by a player if the player can guarantee that from a certain stage on the average payoff will be inside that set, regardless of the strategy that the other player implements. We provide sufficient conditions that ensure that a closed convex approachable set is also strongly approachable in the expected deterministic version of the game.
Citation: Barak Shani, Eilon Solan. Strong approachability. Journal of Dynamics and Games, 2014, 1 (3) : 507-535. doi: 10.3934/jdg.2014.1.507
##### References:
 [1] R. J. Aumann and M. Maschler, Repeated games of incomplete information: A survey of recent results, in Reports of the U.S. Arms Control and Disarmament Agency ST-116, Washingtom, D.C., Chapter III, 1967, 287-403. [2] R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, 1995. [3] D. Blackwell, An analog of the minmax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1. [4] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546921. [5] D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games Economics Behavior, 21 (1997), 40-55. doi: 10.1006/game.1997.0595. [6] M. A. Goberna, E. Gonzalez, J. E. Martinez-Legaz and M. I. Todorov, Motzkin decomposition of closed convex sets, Journal of Mathematical Analysis and Applications, 364 (2010), 209-221. doi: 10.1016/j.jmaa.2009.10.015. [7] S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153. [8] T. F. Hou, Approachability in a two-person game, The Annals of Mathematical Statistics, 42 (1971), 735-744. doi: 10.1214/aoms/1177693422. [9] E. Kohlberg, Optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399. [10] E. Lehrer, Approachability in infinitely dimensional spaces, International Journal of Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115. [11] E. Lehrer, The game of normal numbers, Mathematics of Operations Research, 29 (2004), 259-265. doi: 10.1287/moor.1030.0087. [12] S. Mannor and V. Perchet, Approachability, Fast and slow, JMLR Workshop and Conference Proceedings, 30 (2013), 474-488. [13] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, 2013. doi: 10.1017/CBO9780511794216. [14] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [15] D. Rosenberg, E. Solan and N. Vieille, Stochastic games with a single controller and incomplete information, SIAM Journal on Control and Optimization, 43 (2004), 86-110. doi: 10.1137/S0363012902407107. [16] S. Sorin, Zero-sum repeated games: recent advances and new links with differential games, Dynamic Games and Applications, 1 (2011), 172-207. doi: 10.1007/s13235-010-0006-z. [17] X. Spinat, A necessary and sufficient condition for approachability, Mathematics of Operations Research, 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333.

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##### References:
 [1] R. J. Aumann and M. Maschler, Repeated games of incomplete information: A survey of recent results, in Reports of the U.S. Arms Control and Disarmament Agency ST-116, Washingtom, D.C., Chapter III, 1967, 287-403. [2] R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, 1995. [3] D. Blackwell, An analog of the minmax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1-8. doi: 10.2140/pjm.1956.6.1. [4] N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546921. [5] D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games Economics Behavior, 21 (1997), 40-55. doi: 10.1006/game.1997.0595. [6] M. A. Goberna, E. Gonzalez, J. E. Martinez-Legaz and M. I. Todorov, Motzkin decomposition of closed convex sets, Journal of Mathematical Analysis and Applications, 364 (2010), 209-221. doi: 10.1016/j.jmaa.2009.10.015. [7] S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica, 68 (2000), 1127-1150. doi: 10.1111/1468-0262.00153. [8] T. F. Hou, Approachability in a two-person game, The Annals of Mathematical Statistics, 42 (1971), 735-744. doi: 10.1214/aoms/1177693422. [9] E. Kohlberg, Optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399. [10] E. Lehrer, Approachability in infinitely dimensional spaces, International Journal of Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115. [11] E. Lehrer, The game of normal numbers, Mathematics of Operations Research, 29 (2004), 259-265. doi: 10.1287/moor.1030.0087. [12] S. Mannor and V. Perchet, Approachability, Fast and slow, JMLR Workshop and Conference Proceedings, 30 (2013), 474-488. [13] M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, 2013. doi: 10.1017/CBO9780511794216. [14] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. [15] D. Rosenberg, E. Solan and N. Vieille, Stochastic games with a single controller and incomplete information, SIAM Journal on Control and Optimization, 43 (2004), 86-110. doi: 10.1137/S0363012902407107. [16] S. Sorin, Zero-sum repeated games: recent advances and new links with differential games, Dynamic Games and Applications, 1 (2011), 172-207. doi: 10.1007/s13235-010-0006-z. [17] X. Spinat, A necessary and sufficient condition for approachability, Mathematics of Operations Research, 27 (2002), 31-44. doi: 10.1287/moor.27.1.31.333.
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