Evolutionary Games with Time Constraints Vlastimil Krivan Biology Center and Faculty of Science University of South Bohemia Ceske Budejovice Czech Republic vlastimil.krivan@gmail.com www.entu.cas.cz/krivan Padova, 2018
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Evolutionary game theory There are many individuals of the same species that interact pair-wise 1 There is a finite number of different strategies in the population 2 Payoffs are obtained through games animals play 3 Each individual is selfish, i.e., maximizes its own benefit which leads to the Nash 4 equilibrium All interactions take the same time independently from the strategy individuals 5 play (Typically, one interaction per unit of time) Pairs are formed instantaneously and randomly 6
Hawk-Dove game (Maynard Smith and Price, 1973)
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Payoffs for two-strategy games when all interactions take the same time Payoff matrix (entries are payoffs per interaction): � e 1 e 2 � e 1 π 11 π 12 e 2 π 21 π 22 Interaction time matrix when all interactions take single unit of time: � e 1 e 2 � e 1 1 1 e 2 1 1 n 11 − number of e 1 e 1 pairs n 12 − number of e 1 e 2 pairs n 22 − number of e 2 e 2 pairs N 1 = 2 n 11 + n 12 − total number of individuals playing strategy e 1 N 2 = 2 n 22 + n 12 − total number of individuals playing strategy e 2 N = N 1 + N 2 − total number of individuals
Fitnesses are frequency dependent but density independent Assumption: Pairs are formed instantaneously and randomly, i.e., the equilibrium distribution of pairs is given by Hardy-Weinberg distribution � 2 N 2 2 � N 1 2 = N 1 n 12 = N 1 N 2 n 22 = N 2 n 11 = 2 N , , 2 N . N N 2 n 11 - the probability an e 1 strategist is paired with another e 1 strategist 2 n 11 + n 12 n 12 - the probability an e 1 strategist is paired with an e 2 strategist 2 n 11 + n 12 Fitness of the first phenotype, defined as the expected payoff per interaction is 2 n 11 n 12 π 12 = N 1 N π 11 + N 2 W 1 = π 11 + N π 12 = p 1 π 11 + p 2 π 12 2 n 11 + n 12 2 n 11 + n 12 and similar expression W 2 holds for the fitness of the e 2 strategists. Observation The expected payoffs (fitnesses) are frequency dependent but density independent.
Fitnesses are frequency dependent but density independent Assumption: Pairs are formed instantaneously and randomly, i.e., the equilibrium distribution of pairs is given by Hardy-Weinberg distribution � 2 N 2 2 � N 1 2 = N 1 n 12 = N 1 N 2 n 22 = N 2 n 11 = 2 N , , 2 N . N N 2 n 11 - the probability an e 1 strategist is paired with another e 1 strategist 2 n 11 + n 12 n 12 - the probability an e 1 strategist is paired with an e 2 strategist 2 n 11 + n 12 Fitness of the first phenotype, defined as the expected payoff per interaction is 2 n 11 n 12 π 12 = N 1 N π 11 + N 2 W 1 = π 11 + N π 12 = p 1 π 11 + p 2 π 12 2 n 11 + n 12 2 n 11 + n 12 and similar expression W 2 holds for the fitness of the e 2 strategists. Observation The expected payoffs (fitnesses) are frequency dependent but density independent.
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