Models of Mechanisms: The Case of the Replicator Dynamics - Mechanism and Causality in Biology and Economics - page 90

would have received had they chosen some other action, on the basis of knowledge
of their own payoff matrices and observations of their opponents actions.
This approach reinterprets EGT in general, and strategy selection in particular,
as a theory of individual mental processes. Under this interpretation, all references
to payoffs of others in a given environment are understood counterfactually as the
payoffs that one would get in that environment if one adopted the other's strategy.
For example, if a player knows the payoffs of each strategy profile, and knows the
frequency with which strategies are played in the population, she can compare the
expected payoffs of these strategies based solely on her own preferences. Having
compared the strategies according to her own preferences, she can then choose that
strategy that is either better than the current strategies or a best reply to her belief
about the frequencies in the population. Variants of such models have been pro-
posed by Sugden ( 1986 ), in Kandori et al.'s ( 1993 ) 'stochastic fictitious play' and in
Young's ( 1993 ) 'adaptive play'.
Take, for example, Young's ( 1993 ) model. He defines play at time t as the
strategy-tuple s ( t ) , s n ( t )), consisting of each player's strategy choice at

time t . At period t , each player samples the past play h of a certain number of past
periods. From this sample, the player constructs strategy-tuple s h by weighing the
past play in some way. Strategy-tuple s h constitutes her estimate how other players
will play in the next period. Thus, for the next period, agent i chooses s i as the best
reply to s h . By choosing s i , the player replaces the history of past play h with a new
history h 0 , in which the earliest period is removed and the most recent play added.
This yields a process

¼ ( s i ( t ),

...

P hh 0 ¼ Π

p i s i j ðÞ h (5.4)

Where P hh 0 is the probability of moving from h to h 0 , determined as the product of
the player's probabilities of choosing s i given sample h . Young calls this process
adaptive play .
Young's model is an example of what I call a mental play interpretation of EGT.
What is relevant for a certain strategy to be selected no longer is the effect of actual
interaction in a real population, but rather the consequence of an individual player
evaluating various options, based on her subjective value criteria and her beliefs
what her opponents will play. She forms these beliefs from her perception of and
through reasoning about others' past play. She chooses her strategy by mentally
representing her various options in the anticipated environment, figuring out the
consequences of these counterfactual scenarios and choosing the one with the
outcomes she values better or best.
Consequently, because the causal relation is between interaction and individuals'
mental attitudes, no interpersonal payoff comparison is necessary. Players only
observe their own payoffs from past play, and this affects only their own attitudes
towards future play. Effects on aggregate properties are not directly modelled.
If noise is introduced into models of fictitious play, the expected motion of
fictitious play becomes a form of noisy replicator dynamic (Hopkins 2002 , p. 2149).