On the representation of informational structures in games

Shravan Luckraz; Shravan Luckraz

doi:10.3934/math.2022556

AIMS Mathematics

2022, Volume 7, Issue 6: 9976-9988. doi: 10.3934/math.2022556

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Research article

On the representation of informational structures in games

Shravan Luckraz ^,

School of Public Finance and Taxation, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang 310018, China

Received: 02 November 2021 Revised: 01 March 2022 Accepted: 11 March 2022 Published: 21 March 2022
MSC : 91A18, 91A40

We consider two well-known mathematical representations of informational structures in game theory. The first is based on the set of historical sequences of information encountered in the game while the second is given by the informational digraph of the game. We show that while the latter can always be embedded into the former under some mild technical condition, the converse does not hold in general. We give a necessary and sufficient condition for an informational digraph to be equivalent to a sequence theoretic informational structure and discuss the relevance of this result to game theory.
- game theory,
- information structures,
- digraphs
Citation: Shravan Luckraz. On the representation of informational structures in games[J]. AIMS Mathematics, 2022, 7(6): 9976-9988. doi: 10.3934/math.2022556

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Abstract

We consider two well-known mathematical representations of informational structures in game theory. The first is based on the set of historical sequences of information encountered in the game while the second is given by the informational digraph of the game. We show that while the latter can always be embedded into the former under some mild technical condition, the converse does not hold in general. We give a necessary and sufficient condition for an informational digraph to be equivalent to a sequence theoretic informational structure and discuss the relevance of this result to game theory.

References

[1]	R. Aumann, S. Hart, M. Perry, The Absent-Minded driver, Games Econ. Behav., 20 (1997), 102–116. https://doi.org/10.1006/game.1997.0577 doi: 10.1006/game.1997.0577
[2]	J. Bang-Jensen, G. Gutin, Diagraphs: Theory, algorithms and applications (2nd ed.), London: Springer, 2009. https://doi.org/10.1007/978-1-84800-998-1
[3]	T. Başar, G. Olsder, Dynamic Noncooperative Game Theory, SIAM Series in Classics in Applied Mathematics, Philadelphia, 1999.
[4]	E. Dockner, S. Jørgensen, V. Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.
[5]	M. Kaneko, J. Kline, Inductive Game Theory: a Basic Scenario, J. Math. Econ., 44 (2008), 1332–1363. https://doi.org/10.1016/j.jmateco.2008.07.009 doi: 10.1016/j.jmateco.2008.07.009
[6]	M. Kaneko, J. Kline, Information Protocols and Extensive Games in Inductive Game Theory, Game Theory and Applications, 13 (2008), 57–83.
[7]	M. Kaneko, J. Kline, Partial Memories, Inductively Derived Views, and their Interactions with Behavior, Econ. Theor., 53 (2013), 27–59. https://doi.org/10.1007/s00199-010-0519-0 doi: 10.1007/s00199-010-0519-0
[8]	J. Kline, T. Lavendhomme, S. Waltener, From memories to inductively derived views: a constructive approach, Econ. Theor., 68 (2019), 403–420. https://doi.org/10.1007/s00199-018-1129-5 doi: 10.1007/s00199-018-1129-5
[9]	J. Kline, S. Luckraz, A note on the relationship between graphs and information protocols, Synthese, 179 (2011), 103–114. https://doi.org/10.1007/s11229-010-9853-9 doi: 10.1007/s11229-010-9853-9
[10]	J. Kline, S. Luckraz, Equivalence between graph-based and sequence-based extensive form games, Econ. Theor. B., 4 (2016), 85–94. https://doi.org/10.1007/s40505-016-0094-z doi: 10.1007/s40505-016-0094-z
[11]	H. Kuhn, Extensive Games and the Problem of Information, Contributions to the Theory of Games II, H. Kuhn and A. Tucker eds, (2016), 193–216, Princeton University Press.
[12]	M. Osborne, A. Rubinstein, A Course in Game Theory, MIT Press, Cambridge, 1994.
[13]	M. Piccione, A. Rubinstein, On the Interpretation of Decision Problems with Imperfect Recall, Games Econ. Behav., 20 (1997), 3–24. https://doi.org/10.1006/game.1997.0536 doi: 10.1006/game.1997.0536
[14]	P. Streufert, Equivalences among five game specifications, including a new specification whose nodes are sets of past choices, Int. J. Game Theory, 48 (2019), 1–32. https://doi.org/10.1007/s00182-018-0652-8 doi: 10.1007/s00182-018-0652-8
[15]	J. von Neumann, O. Morgenstern, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.
[16]	B. von Stengel, F. Forges, 2008. Extensive-Form Correlated Equilibrium: Definition and Computational Complexity, Math. Oper. Res., 33 (2008), 1002–1022. https://doi.org/10.1287/moor.1080.0340 doi: 10.1287/moor.1080.0340

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