On the hardness of game equivalence under local isomorphism
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 2, pp. 147-169.

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.

DOI : 10.1051/ita/2012024
Classification : 68Q17
Mots-clés : game isomorphism, succinct representations, strategic games, formula games, computational complexity, circuit isomorphism, boolean formula isomorphism, graph isomorphism 
@article{ITA_2013__47_2_147_0,
     author = {Gabarr\'o, Joaquim and Garc{\'\i}a, Alina and Serna, Maria},
     title = {On the hardness of game equivalence under local isomorphism},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {147--169},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     doi = {10.1051/ita/2012024},
     mrnumber = {3072315},
     zbl = {1272.68141},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita/2012024/}
}
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
SP  - 147
EP  - 169
VL  - 47
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita/2012024/
DO  - 10.1051/ita/2012024
LA  - en
ID  - ITA_2013__47_2_147_0
ER  - 
%0 Journal Article
%A Gabarró, Joaquim
%A García, Alina
%A Serna, Maria
%T On the hardness of game equivalence under local isomorphism
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2013
%P 147-169
%V 47
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita/2012024/
%R 10.1051/ita/2012024
%G en
%F ITA_2013__47_2_147_0
Gabarró, Joaquim; García, Alina; Serna, Maria. On the hardness of game equivalence under local isomorphism. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 2, pp. 147-169. doi : 10.1051/ita/2012024. http://archive.numdam.org/articles/10.1051/ita/2012024/

[1] M. Agrawal and T. Thierauf, The formula isomorphism problem. SIAM J. Comput. 30 (2000) 990-1009. | MR | Zbl

[2] C. Àlvarez, J. Gabarro and M. Serna, Equilibria problems on games : Complexity versus succinctness. J. Comput. Syst. Sci. 77 (2011) 1172-1197. | MR | Zbl

[3] D. Bergemann and S. Morris, Robust implementation in general mechanisms. Games Econ. Behav. 71 (2011) 261-281. | MR | Zbl

[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] 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. | MR | Zbl

[6] P. Borm, A classification of 2 × 2 bimatrix games. Cahiers du C.E.R.O 29 (1987) 69-84. | MR | Zbl

[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. | MR | Zbl

[8] J. Gabarro, A. Garcia and M. Serna, The complexity of game isomorphism. Theor. Comput. Sci. 412 (2011) 6675-6695. | MR | Zbl

[9] A. Garcia, The Complexity of Angel-Daemons and Game Isomorphism. Ph.D. thesis, Universitat Politècnica de Catalunya (Barcelona Tech) (2012).

[10] F. Germano, On some geometry and equivalence classes of normal form games. Inter. J. Game Theory 4 (2006) 561-581. | MR | Zbl

[11] D. Kilgour and N. Fraser, A taxonomy of all ordinal 2 × 2 games. Theory Decis. 24 (1988) 99-117. | MR

[12] J. Kobler, U. Schoning and J. Torán, The Graph Isomorphism Problem : Its Structural Complexity. Birkhauser (1993). | MR | Zbl

[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.

[14] J. Nash, Non-Cooperative Games, in Classics in Game Theory (1997) 14-26. | Zbl

[15] M.J. Osborne, An Introduction to Game Theory. Oxford University Press (2003).

[16] M.J. Osborne and A. Rubinstein, A Course in Game Theory. MIT Press (1994). | MR | Zbl

[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.

[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] K. Siorpaes and M. Hepp, Games with purpose for the semantic web. IEEE Intell. Syst. 23 (2008) 50-60.

[20] L. Von Ahn, Games with a purpose. Comput. 39 (2006) 92-94.

[21] M. Voorneveld, Best-response potential games. Econ. Lett. 66 (2000) 289-295. | MR | Zbl

[22] J. Williams, The Complet Strategyst. Dover (1986). | MR | Zbl

Cité par Sources :