American Institute of Mathematical Sciences

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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

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 & 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. Google Scholar

[2]

R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[4]

N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546921.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. F. Hou, Approachability in a two-person game, The Annals of Mathematical Statistics, 42 (1971), 735-744. doi: 10.1214/aoms/1177693422.  Google Scholar

[9]

E. Kohlberg, Optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399.  Google Scholar

[10]

E. Lehrer, Approachability in infinitely dimensional spaces, International Journal of Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115.  Google Scholar

[11]

E. Lehrer, The game of normal numbers, Mathematics of Operations Research, 29 (2004), 259-265. doi: 10.1287/moor.1030.0087.  Google Scholar

[12]

S. Mannor and V. Perchet, Approachability, Fast and slow, JMLR Workshop and Conference Proceedings, 30 (2013), 474-488. Google Scholar

[13]

M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, 2013. doi: 10.1017/CBO9780511794216.  Google Scholar

[14]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[2]

R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[4]

N. Cesa-Bianchi and G. Lugosi, Prediction, Learning and Games, Cambridge University Press, 2006. doi: 10.1017/CBO9780511546921.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. F. Hou, Approachability in a two-person game, The Annals of Mathematical Statistics, 42 (1971), 735-744. doi: 10.1214/aoms/1177693422.  Google Scholar

[9]

E. Kohlberg, Optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 4 (1975), 7-24. doi: 10.1007/BF01766399.  Google Scholar

[10]

E. Lehrer, Approachability in infinitely dimensional spaces, International Journal of Game Theory, 31 (2002), 253-268. doi: 10.1007/s001820200115.  Google Scholar

[11]

E. Lehrer, The game of normal numbers, Mathematics of Operations Research, 29 (2004), 259-265. doi: 10.1287/moor.1030.0087.  Google Scholar

[12]

S. Mannor and V. Perchet, Approachability, Fast and slow, JMLR Workshop and Conference Proceedings, 30 (2013), 474-488. Google Scholar

[13]

M. Maschler, E. Solan and S. Zamir, Game Theory, Cambridge University Press, 2013. doi: 10.1017/CBO9780511794216.  Google Scholar

[14]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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