In which currency are paid payoffs in evolutionary games?
In which currency are paid payoffs in evolutionary games? Krzysztof Argasinski Deparetment of Mathematics University of Sussex
Charles Darwin
Population growth equation (Malthus Law)
Payoff function which describe per capita growth rate (number of new produced individuals)
Evolutionary Game Theory
Evolutionarily stable strategies
John Maynard Smith
John von Neumann
Oskar Morgenstern
HAWKDOVE GAME
Opponent
Player
(V-C)/2
V
0
V/2
gołąb
V‐winners benefit C‐cost of conflict
C>V Hawk strategy dominate
C ≤V
Population growth equation
Payoff function which describe per capita growth rate (number of new produced individuals)
By following transition of coordinates:
We obtain following system of equations described on related frequencies Called…
Replicator dynamics:
Population growth equation
N(t) Payoff function which describe per capita growth rate (number of new produced individuals)
Problem with approach related to malthusian parameter with logistic suppression
Solution: turnover of individuals (Argasinski, Kozłowski, 2008)
Form of the nonlinear fitness function in classical evolutionary game theory…
n
W( p , pmean ) p Wj pmean i
j 1
i j
Economical application of game theory… Examples of mixed strategy phenotypes:
buy
P1
sell
buy
sell P2
Biological application of game theory…
Examples of mixed strategy phenotypes:
aggresive
P1
peacefull
aggresive
peacefull
P2
…so classic theory is focused on effects of elementary actions (pure strategies)
Problem of hidden assumptions
Problem of hidden assumptions Mathematical structure bias
Are those assumptions testable?
Problem of hidden assumptions Mathematical structure bias
Are those assumptions testable?
We should compare predictions of the model with empirical data to show that there is something wrong
How can We measure parameters such costs and benefits?
How can We measure parameters such costs and benefits?
None of them will be pregnant in effect of this interaction
Therefore models related on malthusian parameter are not cause‐effect models. What happens when we forget about malthusian parameter?
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
We can imagine case when mortality acts before reproduction or individuals draw opponent that can kill them or mate (or both). This leads to variant called SEX OR VIOLENCE EQUATIONS
We can imagine more complicated cases, for example density dependent selection with two types of mortality pressures, one affecting fertility and second not:
When we apply new approach to the hawk dove game, we have two payoff matrices:
Survival matrix fertility matrix
When we apply new approach to the hawk dove game, we have two payoff matrices:
Survival matrix fertility matrix
New model is totally equivalent to classical Maynard‐Smith approach, however in new terminology we have:
B=W C=d(W+2)
In new model hawks may be stable
GAME OVER
Charles Darwin
Population growth equation (Malthus Law)
Payoff function which describe per capita growth rate (number of new produced individuals)
Evolutionary Game Theory
Evolutionarily stable strategies
John Maynard Smith
John von Neumann
Oskar Morgenstern
HAWKDOVE GAME
Opponent
Player
(V-C)/2
V
0
V/2
gołąb
V‐winners benefit C‐cost of conflict
C>V Hawk strategy dominate
C ≤V
Population growth equation
Payoff function which describe per capita growth rate (number of new produced individuals)
By following transition of coordinates:
We obtain following system of equations described on related frequencies Called…
Replicator dynamics:
Population growth equation
N(t) Payoff function which describe per capita growth rate (number of new produced individuals)
Problem with approach related to malthusian parameter with logistic suppression
Solution: turnover of individuals (Argasinski, Kozłowski, 2008)
Form of the nonlinear fitness function in classical evolutionary game theory…
n
W( p , pmean ) p Wj pmean i
j 1
i j
Economical application of game theory… Examples of mixed strategy phenotypes:
buy
P1
sell
buy
sell P2
Biological application of game theory…
Examples of mixed strategy phenotypes:
aggresive
P1
peacefull
aggresive
peacefull
P2
…so classic theory is focused on effects of elementary actions (pure strategies)
Problem of hidden assumptions
Problem of hidden assumptions Mathematical structure bias
Are those assumptions testable?
Problem of hidden assumptions Mathematical structure bias
Are those assumptions testable?
We should compare predictions of the model with empirical data to show that there is something wrong
How can We measure parameters such costs and benefits?
How can We measure parameters such costs and benefits?
None of them will be pregnant in effect of this interaction
Therefore models related on malthusian parameter are not cause‐effect models. What happens when we forget about malthusian parameter?
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
In effect we obtain generalization of replicator dynamics called
SEX &VIOLENCE EQUATIONS
We can imagine case when mortality acts before reproduction or individuals draw opponent that can kill them or mate (or both). This leads to variant called SEX OR VIOLENCE EQUATIONS
We can imagine more complicated cases, for example density dependent selection with two types of mortality pressures, one affecting fertility and second not:
When we apply new approach to the hawk dove game, we have two payoff matrices:
Survival matrix fertility matrix
When we apply new approach to the hawk dove game, we have two payoff matrices:
Survival matrix fertility matrix
New model is totally equivalent to classical Maynard‐Smith approach, however in new terminology we have:
B=W C=d(W+2)
In new model hawks may be stable
GAME OVER
