Strong approachability

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

Strong approachability

Barak Shani ^1, and Eilon Solan ^1,

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

Received February 2013 Revised October 2013 Published July 2014

Abstract
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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.

Keywords: vector payoffs, strong approachability., Approachability, repeated games.

Mathematics Subject Classification: Primary: 91A05, 91A10, 91A2.

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.

show all references

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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