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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.
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