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Fixation conditions in finite population games
Freitag, 02.11.2007 11:15 - 12:15
Immanuel Bomze
Wien, Institut für Statistik und Decision Support Systems

Fixation conditions in finite population games

In a recent article "Finite population always choose optimal
languages" (Theor.Pop.Biol., to appear 2007), Pawlowitsch employed a firstorder
condition (as selection intensity becomes small) for fixation of a strategy
in finite populations. The theory draws upon earlier results by Nowak et al.
(Nature, 2004) who modeled contests with 2 x 2 base games in a finite (large)
population by means of a frequency-dependent Moran process. In essence,
this condition was claimed by Nowak et al. (2004) to be equivalent to fixation
with a probability exceeding that of equidistribution for large population
numbers.
While being correct for a large class of games (i.e. generically), this condition
is not necessary for an important subclass of doubly symmetric base games.
We investigate first- and second-order conditions and arrive at interesting
local interpretations, as well as a nice characterization of degeneracy in terms
of the base game for the case where the first-order condition fails. Exactly in
this situation, we show that fixation never holds if the strict inequality specified
by Nowak et al. (2004) is weakened. These results become relevant in highly
structured games like sender-receiver games treated in Pawlowitsch (2007)
which may exhibit inherent degeneracy in the above sense. Fortunately, the
results in Pawlowitsch (2007) are unaffected since her argument only uses
sufficiency, not necessity of the conditions specified by Nowak et al. (2004).
„Games in Communication II“ 2007
http://wwwhomes.uni-bielefeld.de/gjaeger/conferences/gamesincommunication2007
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homepage: wwwhomes.uni-bielefeld.de/gja
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