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X-Machines for Agent-Based Modeling: FLAME Perspectives
6.5.3 Evolutionary Stable State
An important concept here is the ESSs, which differs from the strict definition of the Nash equilibrium (Table 6.4). ESSs study the strategies adopted
by the players on a population level. This involves studying a large number
of players. The ESS is achieved through the frequency of the players in the
population. Thus ESS is a frequency-dependent concept which has been propagated by the selection mechanisms of evolution [186]. ESS allows for a given
set of behaviors (conserved over time) to determine an optimal strategy for
everyone in the system. At this time no other mutant behavior can enter the
system and survive. This means that behavior is adopted by the individuals
in the population, and no other behavior will invade the population under
natural selection. Suppose the main population plays strategy x ∈ S and mutants can play some strategy y ∈ S. Given that the mutants in the population
are a very small proportion, then the probability a mutant is drawn for the
population is very small probability ǫ by evolutionary selection. The payoff
function for the strategies is determined by u(x). An evolutionary stable state
occurs when no mutant population can invade the main population. This can
be true in the two conditions (Equation 6.8 and 6.9).
u(y, x) ≤ u(x, x)
(6.8)
u(y, x) = u(x, x) ⇒ u(y, y) < u(x, y)∀y = x
(6.9)
The concepts of ESS favor the analysis of dynamical systems which is why
it is extensively used in biological systems [45].
6.5.4 Game Theory versus Evolutionary Game Theory
An important advantage of using ESS compared to Nash equilibrium is
that Nash equilibrium can only be achieved using rational decisions and discrete payoffs in a game, whereas ESS does not depend on rational decisions. It
is rather based on the behavioral aspects of the individuals. Despite this difference, there are some games of an altruistic nature in which the two definitions
can be related. Prisoner’s dilemma game is an example in which only when
all players cooperate a Nash equilibrium is reached, benefitting all players in
the game. These games use rational or discrete utilities as payoffs.
Silverberg [178] favors the use of replicator dynamics to model economic
evolution. Replicator dynamics uses frequency-dependent fitness to depict the
most used strategy in an economic scenario. Frequency-dependent fitness is
different from using a payoff function for the fitness because the payoff function
assesses the performance of a strategy and the strategy fitness is based on the
performance rather than its frequency in the population.
X-Machines for Agent-Based Modeling: FLAME Perspectives
6.5.3 Evolutionary Stable State
An important concept here is the ESSs, which differs from the strict definition of the Nash equilibrium (Table 6.4). ESSs study the strategies adopted
by the players on a population level. This involves studying a large number
of players. The ESS is achieved through the frequency of the players in the
population. Thus ESS is a frequency-dependent concept which has been propagated by the selection mechanisms of evolution [186]. ESS allows for a given
set of behaviors (conserved over time) to determine an optimal strategy for
everyone in the system. At this time no other mutant behavior can enter the
system and survive. This means that behavior is adopted by the individuals
in the population, and no other behavior will invade the population under
natural selection. Suppose the main population plays strategy x ∈ S and mutants can play some strategy y ∈ S. Given that the mutants in the population
are a very small proportion, then the probability a mutant is drawn for the
population is very small probability ǫ by evolutionary selection. The payoff
function for the strategies is determined by u(x). An evolutionary stable state
occurs when no mutant population can invade the main population. This can
be true in the two conditions (Equation 6.8 and 6.9).
u(y, x) ≤ u(x, x)
(6.8)
u(y, x) = u(x, x) ⇒ u(y, y) < u(x, y)∀y = x
(6.9)
The concepts of ESS favor the analysis of dynamical systems which is why
it is extensively used in biological systems [45].
6.5.4 Game Theory versus Evolutionary Game Theory
An important advantage of using ESS compared to Nash equilibrium is
that Nash equilibrium can only be achieved using rational decisions and discrete payoffs in a game, whereas ESS does not depend on rational decisions. It
is rather based on the behavioral aspects of the individuals. Despite this difference, there are some games of an altruistic nature in which the two definitions
can be related. Prisoner’s dilemma game is an example in which only when
all players cooperate a Nash equilibrium is reached, benefitting all players in
the game. These games use rational or discrete utilities as payoffs.
Silverberg [178] favors the use of replicator dynamics to model economic
evolution. Replicator dynamics uses frequency-dependent fitness to depict the
most used strategy in an economic scenario. Frequency-dependent fitness is
different from using a payoff function for the fitness because the payoff function
assesses the performance of a strategy and the strategy fitness is based on the
performance rather than its frequency in the population.