On the hardness of game equivalence under local isomorphism

Joaquim Gabarró; Alina García; Maria Serna

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2013)

  • Volume: 47, Issue: 2, page 147-169
  • ISSN: 0988-3754

Abstract

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We introduce a type of isomorphism among strategic games that we call local isomorphism. Local isomorphisms is a weaker version of the notions of strong and weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695]. In a local isomorphism it is required to preserve, for any player, the player’s preferences on the sets of strategy profiles that differ only in the action selected by this player. We show that the game isomorphism problem for local isomorphism is equivalent to the same problem for strong or weak isomorphism for strategic games given in: general, extensive and formula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] this implies that local isomorphism problem for strategic games is equivalent to (a) the circuit isomorphism problem for games given in general form, (b) the boolean formula isomorphism problem for formula games in general form, and (c) the graph isomorphism problem for games given in explicit form.

How to cite

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Gabarró, Joaquim, García, Alina, and Serna, Maria. "On the hardness of game equivalence under local isomorphism." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 47.2 (2013): 147-169. <http://eudml.org/doc/273051>.

@article{Gabarró2013,
abstract = {We introduce a type of isomorphism among strategic games that we call local isomorphism. Local isomorphisms is a weaker version of the notions of strong and weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695]. In a local isomorphism it is required to preserve, for any player, the player’s preferences on the sets of strategy profiles that differ only in the action selected by this player. We show that the game isomorphism problem for local isomorphism is equivalent to the same problem for strong or weak isomorphism for strategic games given in: general, extensive and formula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] this implies that local isomorphism problem for strategic games is equivalent to (a) the circuit isomorphism problem for games given in general form, (b) the boolean formula isomorphism problem for formula games in general form, and (c) the graph isomorphism problem for games given in explicit form.},
author = {Gabarró, Joaquim, García, Alina, Serna, Maria},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {game isomorphism; succinct representations; strategic games; formula games; computational complexity; circuit isomorphism; boolean formula isomorphism; graph isomorphism; Boolean formula isomorphism},
language = {eng},
number = {2},
pages = {147-169},
publisher = {EDP-Sciences},
title = {On the hardness of game equivalence under local isomorphism},
url = {http://eudml.org/doc/273051},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Gabarró, Joaquim
AU - García, Alina
AU - Serna, Maria
TI - On the hardness of game equivalence under local isomorphism
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 2
SP - 147
EP - 169
AB - We introduce a type of isomorphism among strategic games that we call local isomorphism. Local isomorphisms is a weaker version of the notions of strong and weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695]. In a local isomorphism it is required to preserve, for any player, the player’s preferences on the sets of strategy profiles that differ only in the action selected by this player. We show that the game isomorphism problem for local isomorphism is equivalent to the same problem for strong or weak isomorphism for strategic games given in: general, extensive and formula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] this implies that local isomorphism problem for strategic games is equivalent to (a) the circuit isomorphism problem for games given in general form, (b) the boolean formula isomorphism problem for formula games in general form, and (c) the graph isomorphism problem for games given in explicit form.
LA - eng
KW - game isomorphism; succinct representations; strategic games; formula games; computational complexity; circuit isomorphism; boolean formula isomorphism; graph isomorphism; Boolean formula isomorphism
UR - http://eudml.org/doc/273051
ER -

References

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  1. [1] M. Agrawal and T. Thierauf, The formula isomorphism problem. SIAM J. Comput.30 (2000) 990–1009. Zbl0970.68068MR1778296
  2. [2] C. Àlvarez, J. Gabarro and M. Serna, Equilibria problems on games : Complexity versus succinctness. J. Comput. Syst. Sci.77 (2011) 1172–1197. Zbl1230.91006MR2858016
  3. [3] D. Bergemann and S. Morris, Robust implementation in general mechanisms. Games Econ. Behav.71 (2011) 261–281. Zbl1208.91043MR2814851
  4. [4] E. Bonzon, M.-C. Lagasquie-Schiex, J. Lang and B. Zanuttini, Boolean games revisited, in ECAI 2006, 17th European Conference on Artificial Intelligence (2006) 265–269. 
  5. [5] B. Borchet, D. Ranjan and F. Stephan, On the computational complexity of some classical equivalence relations on boolean functions. Theory Comput. Syst.31 (1998) 679–693. Zbl0916.68059MR1642457
  6. [6] P. Borm, A classification of 2 × 2 bimatrix games. Cahiers du C.E.R.O29 (1987) 69–84. Zbl0634.90095MR919046
  7. [7] J. Gabarro, A. Garcia and M. Serna, On the complexity of game isomorphism, in Mathematical Foundations of Computer Science 2007, 32nd International Symposium, MFCS 2007. Lect. Notes Comput. Sci. 4708 (2007) 559–571. Zbl1147.91303MR2539457
  8. [8] J. Gabarro, A. Garcia and M. Serna, The complexity of game isomorphism. Theor. Comput. Sci.412 (2011) 6675–6695. Zbl1227.91016MR2885088
  9. [9] A. Garcia, The Complexity of Angel-Daemons and Game Isomorphism. Ph.D. thesis, Universitat Politècnica de Catalunya (Barcelona Tech) (2012). 
  10. [10] F. Germano, On some geometry and equivalence classes of normal form games. Inter. J. Game Theory4 (2006) 561–581. Zbl1154.91309MR2302144
  11. [11] D. Kilgour and N. Fraser, A taxonomy of all ordinal 2 × 2 games. Theory Decis.24 (1988) 99–117. MR931042
  12. [12] J. Kobler, U. Schoning and J. Torán, The Graph Isomorphism Problem : Its Structural Complexity. Birkhauser (1993). Zbl0813.68103MR1232421
  13. [13] M. Mavronicolas, B. Monien and K.W. Wagner, Weighted boolean formula games, in Internet and Network Economics, Third International Workshop, WINE 2007. Lect. Notes Comput. Sci. 4858 (2007) 469–481. Zbl1331.68112
  14. [14] J. Nash, Non-Cooperative Games, in Classics in Game Theory (1997) 14–26. Zbl0045.08202
  15. [15] M.J. Osborne, An Introduction to Game Theory. Oxford University Press (2003). 
  16. [16] M.J. Osborne and A. Rubinstein, A Course in Game Theory. MIT Press (1994). Zbl1194.91003MR1301776
  17. [17] G. Schoenebeck and S.P. Vadhan, The computational complexity of Nash equilibria in concisely represented games, in ACM Conf. Electr. Commer. (2006) 270–279. Zbl1322.68110
  18. [18] K. Siorpaes and M. Hepp, Ontogame : Weaving the semantic web by online games, in ESWC-2008. Lect. Notes Comput. Sci. 5021 (2008) 751–766. 
  19. [19] K. Siorpaes and M. Hepp, Games with purpose for the semantic web. IEEE Intell. Syst.23 (2008) 50–60. 
  20. [20] L. von Ahn, Games with a purpose. Comput.39 (2006) 92–94. 
  21. [21] M. Voorneveld, Best-response potential games. Econ. Lett.66 (2000) 289–295. Zbl0951.91008MR1737523
  22. [22] J. Williams, The Complet Strategyst. Dover (1986). Zbl1227.91001MR868665

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