Skip to main content Accessibility help
×
Home
Hostname: page-component-7ccbd9845f-6pjjk Total loading time: 0.327 Render date: 2023-01-31T00:39:20.151Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

On the hardness of game equivalence under local isomorphism

Published online by Cambridge University Press:  05 November 2012

Joaquim Gabarró
Affiliation:
ALBCOM Research Group, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Ω Building, 08034 Barcelona, Spain. . gabarro@lsi.upc.edu, agarcia@lsi.upc.edu, mjserna@lsi.upc.edu
Alina García
Affiliation:
ALBCOM Research Group, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Ω Building, 08034 Barcelona, Spain. . gabarro@lsi.upc.edu, agarcia@lsi.upc.edu, mjserna@lsi.upc.edu
Maria Serna
Affiliation:
ALBCOM Research Group, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Ω Building, 08034 Barcelona, Spain. . gabarro@lsi.upc.edu, agarcia@lsi.upc.edu, mjserna@lsi.upc.edu
Get access

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.

Type
Research Article
Copyright
© EDP Sciences 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrawal, M. and Thierauf, T., The formula isomorphism problem. SIAM J. Comput. 30 (2000) 9901009. Google Scholar
Àlvarez, C., Gabarro, J. and Serna, M., Equilibria problems on games : Complexity versus succinctness. J. Comput. Syst. Sci. 77 (2011) 11721197. Google Scholar
Bergemann, D. and Morris, S., Robust implementation in general mechanisms. Games Econ. Behav. 71 (2011) 261281. Google Scholar
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.
Borchet, B., Ranjan, D. and Stephan, F., On the computational complexity of some classical equivalence relations on boolean functions. Theory Comput. Syst. 31 (1998) 679693. Google Scholar
Borm, P., A classification of 2 × 2 bimatrix games. Cahiers du C.E.R.O 29 (1987) 6984. Google Scholar
Gabarro, J., Garcia, A. and Serna, M., On the complexity of game isomorphism, in Mathematical Foundations of Computer Science 2007, 32nd International Symposium, MFCS 2007. Lect. Notes Comput. Sci. 4708 (2007) 559571. Google Scholar
Gabarro, J., Garcia, A. and Serna, M., The complexity of game isomorphism. Theor. Comput. Sci. 412 (2011) 66756695. Google Scholar
A. Garcia, The Complexity of Angel-Daemons and Game Isomorphism. Ph.D. thesis, Universitat Politècnica de Catalunya (Barcelona Tech) (2012).
Germano, F., On some geometry and equivalence classes of normal form games. Inter. J. Game Theory 4 (2006) 561581. Google Scholar
Kilgour, D. and Fraser, N., A taxonomy of all ordinal 2 × 2 games. Theory Decis. 24 (1988) 99117. Google Scholar
J. Kobler, U. Schoning and J. Torán, The Graph Isomorphism Problem : Its Structural Complexity. Birkhauser (1993).
Mavronicolas, M., Monien, B. and Wagner, K.W., Weighted boolean formula games, in Internet and Network Economics, Third International Workshop, WINE 2007. Lect. Notes Comput. Sci. 4858 (2007) 469481. Google Scholar
J. Nash, Non-Cooperative Games, in Classics in Game Theory (1997) 14–26.
M.J. Osborne, An Introduction to Game Theory. Oxford University Press (2003).
M.J. Osborne and A. Rubinstein, A Course in Game Theory. MIT Press (1994).
G. Schoenebeck and S.P. Vadhan, The computational complexity of Nash equilibria in concisely represented games, in ACM Conf. Electr. Commer. (2006) 270–279.
Siorpaes, K. and Hepp, M., Ontogame : Weaving the semantic web by online games, in ESWC-2008. Lect. Notes Comput. Sci. 5021 (2008) 751766. Google Scholar
Siorpaes, K. and Hepp, M., Games with purpose for the semantic web. IEEE Intell. Syst. 23 (2008) 5060. Google Scholar
von Ahn, L., Games with a purpose. Comput. 39 (2006) 9294. Google Scholar
Voorneveld, M., Best-response potential games. Econ. Lett. 66 (2000) 289295. Google Scholar
J. Williams, The Complet Strategyst. Dover (1986).

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the hardness of game equivalence under local isomorphism
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

On the hardness of game equivalence under local isomorphism
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

On the hardness of game equivalence under local isomorphism
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *