Overview Preliminaries The replicator dynamics Rationality analysis References
ALGORITHMIC GAME THEORY
FROM MULTI-AGENT OPTIMIZATION TO ONLINE LEARNING
Roberto Cominetti¹ Panayotis Mertikopoulos² ³ ¹Universidad Adolfo Ibáñez
²French National Center for Scientific Research (CNRS) ³Criteo AI Lab
Overview Preliminaries The replicator dynamics Rationality analysis References
ALGORITHMIC GAME THEORY
LEC. 4: POPULATION GAMES AND EVOLUTIONARY DYNAMICS
Roberto Cominetti¹ Panayotis Mertikopoulos² ³ ¹Universidad Adolfo Ibáñez
²French National Center for Scientific Research (CNRS) ³Criteo AI Lab
Overview Preliminaries The replicator dynamics Rationality analysis References
Outline
Overview
Preliminaries
The replicator dynamics Rationality analysis
Overview Preliminaries The replicator dynamics Rationality analysis References
Bird's eye view
Yesterday:
▸ Games, examples, solution concepts (Nash/Wardrop equilibrium,…) ▸ Congestion games (atomic / nonatomic, splittable / non-splittable,…) ▸ Efficiency and stability (PoA, PoS,…)
Today:playing day-by-day
▸ Lecture 4:population games↭evolutionary dynamics ▸ Lecture 5:finite games↭multi-armed bandits
▸ Lecture 6:continuous games↭online convex optimization Caveats
▸ Big picture:Focus on concepts + selected deep dives
Overview Preliminaries The replicator dynamics Rationality analysis References
Bird's eye view
Yesterday:
▸ Games, examples, solution concepts (Nash/Wardrop equilibrium,…) ▸ Congestion games (atomic / nonatomic, splittable / non-splittable,…) ▸ Efficiency and stability (PoA, PoS,…)
Today:playing day-by-day
▸ Lecture 4:population games↭evolutionary dynamics ▸ Lecture 5:finite games↭multi-armed bandits
▸ Lecture 6:continuous games↭online convex optimization
Caveats
▸ Big picture:Focus on concepts + selected deep dives
Overview Preliminaries The replicator dynamics Rationality analysis References
Bird's eye view
Yesterday:
▸ Games, examples, solution concepts (Nash/Wardrop equilibrium,…) ▸ Congestion games (atomic / nonatomic, splittable / non-splittable,…) ▸ Efficiency and stability (PoA, PoS,…)
Today:playing day-by-day
▸ Lecture 4:population games↭evolutionary dynamics ▸ Lecture 5:finite games↭multi-armed bandits
▸ Lecture 6:continuous games↭online convex optimization Caveats
▸ Big picture:Focus on concepts + selected deep dives
Overview Preliminaries The replicator dynamics Rationality analysis References
Today: Learning to play
A typicalonline decision process:
repeat
At each epocht≥
Chooseaction [focal player]
Incurloss/ Receivereward [depends on context]
Getfeedback [depends on context]
untilend
Key considerations
▸ Time:continuous or discrete? ▸ Players:continuous or discrete? ▸ Actions:continuous or discrete?
▸ Payoffs / Losses:determined by other players or “Nature”? ▸ Feedback:full info / observation? payoff-based?
Overview Preliminaries The replicator dynamics Rationality analysis References
Today: Learning to play
A typicalonline decision process:
repeat
At each epocht≥
Chooseaction [focal player]
Incurloss/ Receivereward [depends on context]
Getfeedback [depends on context]
untilend
Key considerations
▸ Time:continuous or discrete? ▸ Players:continuous or discrete? ▸ Actions:continuous or discrete?
▸ Payoffs / Losses:determined by other players or “Nature”? ▸ Feedback:full info / observation? payoff-based?
Overview Preliminaries The replicator dynamics Rationality analysis References
Games: A taxonomy
Actions Players Finite Finite Continuous Continuous Finite Games Population GamesOverview Preliminaries The replicator dynamics Rationality analysis References
Games: A taxonomy
Actions Players Finite Finite Continuous Continuous Atomic non-splittable Nonatomic non-splittableMean Field Games splittableAtomic
Overview Preliminaries The replicator dynamics Rationality analysis References
Games: A taxonomy
Actions Players Finite Finite Continuous Continuous Atomic non-splittable Nonatomic non-splittableMean Field Games splittableAtomic CONGESTION GAMES
Overview Preliminaries The replicator dynamics Rationality analysis References
Games: A taxonomy
Actions Players Finite Finite Continuous Continuous Finite Games Population GamesNonatomic Games Continuous Games
Lecture 4 Lecture 5
Overview Preliminaries The replicator dynamics Rationality analysis References
Relations between classes
Porous boundaries:
Mixed Extensions
Random Matching Multilinear Games Finite Games
Population Games Continuous Games
Mixing = P op. Shar es Mixing = P oint in simple x Important
▸ Random matching⊊population games of interest ▸ (Multi)Linear games⊊continuous games of interest
Overview Preliminaries The replicator dynamics Rationality analysis References
Relations between classes
Porous boundaries:
Mixed Extensions
Random Matching Multilinear Games Finite Games
Population Games Continuous Games
Mixing = P op. Shar es Mixing = P oint in simple x Important
▸ Random matching⊊population games of interest ▸ (Multi)Linear games⊊continuous games of interest
Overview Preliminaries The replicator dynamics Rationality analysis References
Outline
Overview
Preliminaries
The replicator dynamics Rationality analysis
Overview Preliminaries The replicator dynamics Rationality analysis References
Population games
▸ Players:continuous, nonatomicpopulationsi=, . . . ,N [species, types,…]
▸ Actions:finiteaction setAiper population [phenotypes, routes,…] ▸ Payoffs:depend only on the players’distribution [anonymity]
▸ Population shares
xi ai=relative frequency ofai∈Aiin populationi
▸ Population states
xi=(xi ai)ai∈Ai∈Xi∶=∆(Ai) x=(x, . . . ,xN)∈X∶=∏iXi ▸ Payoff functionsui ai∶X→R
ui ai(x)=payoff toai∈Aiwhen the population is at statex∈X
▸ Mean population payoff
Overview Preliminaries The replicator dynamics Rationality analysis References
Population games
▸ Players:continuous, nonatomicpopulationsi=, . . . ,N [species, types,…]
▸ Actions:finiteaction setAiper population [phenotypes, routes,…]
▸ Payoffs:depend only on the players’distribution [anonymity] ▸ Population shares
xi ai=relative frequency ofai∈Aiin populationi
▸ Population states
xi=(xi ai)ai∈Ai∈Xi∶=∆(Ai) x=(x, . . . ,xN)∈X∶=∏iXi ▸ Payoff functionsui ai∶X→R
ui ai(x)=payoff toai∈Aiwhen the population is at statex∈X
▸ Mean population payoff
Overview Preliminaries The replicator dynamics Rationality analysis References
Population games
▸ Players:continuous, nonatomicpopulationsi=, . . . ,N [species, types,…]
▸ Actions:finiteaction setAiper population [phenotypes, routes,…] ▸ Payoffs:depend only on the players’distribution [anonymity]
▸ Population shares
xi ai=relative frequency ofai∈Aiin populationi ▸ Population states
xi=(xi ai)ai∈Ai∈Xi∶=∆(Ai) x=(x, . . . ,xN)∈X∶=∏iXi
▸ Payoff functionsui ai∶X→R
ui ai(x)=payoff toai∈Aiwhen the population is at statex∈X
▸ Mean population payoff
Overview Preliminaries The replicator dynamics Rationality analysis References
Population games
▸ Players:continuous, nonatomicpopulationsi=, . . . ,N [species, types,…]
▸ Actions:finiteaction setAiper population [phenotypes, routes,…] ▸ Payoffs:depend only on the players’distribution [anonymity]
▸ Population shares
xi ai=relative frequency ofai∈Aiin populationi ▸ Population states
xi=(xi ai)ai∈Ai∈Xi∶=∆(Ai) x=(x, . . . ,xN)∈X∶=∏iXi ▸ Payoff functionsui ai∶X→R
ui ai(x)=payoff toai∈Aiwhen the population is at statex∈X
▸ Mean population payoff
Overview Preliminaries The replicator dynamics Rationality analysis References
Examples and more
Example 1:Nonatomic congestion games (“playing the field”) ▸ See Lecture 2
Example 2:Multi-population random matching [Maynard Smith and Price, 1973] ▸ Given: finiteN-player gameΓ≡Γ(N,A,u) [base game] ▸ Given:Nplayer populations, each with action setAi
▸ During play:
▸ Players drawn uniformly at random from each population
▸ Drawn players matched to playΓ [random matching] ▸ Mean payoffs
ui ai(x)=∑a′
∈A⋯∑a′N∈ANx,a′⋯δa′iai⋯xN,a′Nui(a ′
, . . . ,a′N) ▸ Caveat Single-population matching isdifferent(quadratic) [why?] For more:Weibull [1995], Hofbauer and Sigmund [1998], Sandholm [2010, 2015]
Overview Preliminaries The replicator dynamics Rationality analysis References
Examples and more
Example 1:Nonatomic congestion games (“playing the field”) ▸ See Lecture 2
Example 2:Multi-population random matching [Maynard Smith and Price, 1973] ▸ Given: finiteN-player gameΓ≡Γ(N,A,u) [base game] ▸ Given:Nplayer populations, each with action setAi
▸ During play:
▸ Players drawn uniformly at random from each population
▸ Drawn players matched to playΓ [random matching] ▸ Mean payoffs
ui ai(x)=∑a′
∈A⋯∑a′N∈ANx,a′⋯δa′iai⋯xN,a′Nui(a ′
, . . . ,a′N) ▸ Caveat Single-population matching isdifferent(quadratic) [why?] For more:Weibull [1995], Hofbauer and Sigmund [1998], Sandholm [2010, 2015]
Overview Preliminaries The replicator dynamics Rationality analysis References
Examples and more
Example 1:Nonatomic congestion games (“playing the field”) ▸ See Lecture 2
Example 2:Multi-population random matching [Maynard Smith and Price, 1973] ▸ Given: finiteN-player gameΓ≡Γ(N,A,u) [base game] ▸ Given:Nplayer populations, each with action setAi
▸ During play:
▸ Players drawn uniformly at random from each population
▸ Drawn players matched to playΓ [random matching] ▸ Mean payoffs
ui ai(x)=∑a′
∈A⋯∑a′N∈ANx,a′⋯δa′iai⋯xN,a′Nui(a ′
, . . . ,a′N) ▸ Caveat Single-population matching isdifferent(quadratic) [why?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Examples and more
Example 1:Nonatomic congestion games (“playing the field”) ▸ See Lecture 2
Example 2:Multi-population random matching [Maynard Smith and Price, 1973] ▸ Given: finiteN-player gameΓ≡Γ(N,A,u) [base game] ▸ Given:Nplayer populations, each with action setAi
▸ During play:
▸ Players drawn uniformly at random from each population
▸ Drawn players matched to playΓ [random matching] ▸ Mean payoffs
ui ai(x)=∑a′
∈A⋯∑a′N∈ANx,a′⋯δa′iai⋯xN,a′Nui(a ′
, . . . ,a′N) ▸ Caveat Single-population matching isdifferent(quadratic) [why?] For more:Weibull [1995], Hofbauer and Sigmund [1998], Sandholm [2010, 2015]
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (-population):
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (-population):
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (-population):
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (-population):
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (multi-population): u(x)=−u(x)=x⊺Ax
u(x)=x⊺Ax
Overview Preliminaries The replicator dynamics Rationality analysis References
Go-to example: Rock-Paper-Scissors
R eP eR eS R P S ∆{R,P,S} ( , , ) Equilibrium ▸ Players: N={, }. ▸ Actions: Ai={R,P,S},i=, .
▸ Payoff matrix (win, lose−, tie): A= R P S R − P − S − ▸ Mixed strategies: xi∈∆{R,P,S}.
▸ Payoff functions (single-population): u(x)=x⊺Ax
Overview Preliminaries The replicator dynamics Rationality analysis References
Solution concepts redux
▸ Dominated strategies:ai∈Aiisdominatedbya′i∈Ai(ai ≺a′i) if ui ai(x)<ui a′i(x) for allx∈X
[Mixed version:pi≺qi ⇐⇒ ui(pi;x−i)<ui(qi;x−i)for allx∈X]
▸ Nash equilibrium:no player has an incentive to switch strategies ui ai(x
∗)≥u
i a′i(x∗) for allai,a′i∈Aiwithx∗i ai >
▸ Supportofx∗:supp(x∗)={(a, . . . ,aN)∈A∶x∗i ai>for alli}
▸ Interior / Full support equilibria:supp(x∗)=A ▸ Pure equilibria:supp(x∗)=singleton
▸ Strict equilibria:“>” instead of “≥” whena′i∉supp(x∗i) ▸ Examples:
▸ Rock-Paper-Scissors: unique, full support equilibrium
▸ Prisoner’s dilemma(Lecture 1): dominance-solvable, one strict equilibrium ▸ Battle of the Sexes(Lecture 1): one full support equilibrium; two strict equilibria
Overview Preliminaries The replicator dynamics Rationality analysis References
Solution concepts redux
▸ Dominated strategies:ai∈Aiisdominatedbya′i∈Ai(ai ≺a′i) if ui ai(x)<ui a′i(x) for allx∈X
[Mixed version:pi≺qi ⇐⇒ ui(pi;x−i)<ui(qi;x−i)for allx∈X] ▸ Nash equilibrium:no player has an incentive to switch strategies
ui ai(x ∗)≥u
i a′i(x∗) for allai,a′i∈Aiwithx∗i ai >
▸ Supportofx∗:supp(x∗)={(a, . . . ,aN)∈A∶x∗i ai>for alli}
▸ Interior / Full support equilibria:supp(x∗)=A ▸ Pure equilibria:supp(x∗)=singleton
▸ Strict equilibria:“>” instead of “≥” whena′i∉supp(x∗i) ▸ Examples:
▸ Rock-Paper-Scissors: unique, full support equilibrium
▸ Prisoner’s dilemma(Lecture 1): dominance-solvable, one strict equilibrium ▸ Battle of the Sexes(Lecture 1): one full support equilibrium; two strict equilibria
Overview Preliminaries The replicator dynamics Rationality analysis References
Solution concepts redux
▸ Dominated strategies:ai∈Aiisdominatedbya′i∈Ai(ai ≺a′i) if ui ai(x)<ui a′i(x) for allx∈X
[Mixed version:pi≺qi ⇐⇒ ui(pi;x−i)<ui(qi;x−i)for allx∈X] ▸ Nash equilibrium:no player has an incentive to switch strategies
ui ai(x ∗)≥u
i a′i(x∗) for allai,a′i∈Aiwithx∗i ai >
▸ Supportofx∗:supp(x∗)={(a, . . . ,aN)∈A∶x∗i ai>for alli}
▸ Interior / Full support equilibria:supp(x∗)=A ▸ Pure equilibria:supp(x∗)=singleton
▸ Strict equilibria:“>” instead of “≥” whena′i∉supp(x∗i)
▸ Examples:
▸ Rock-Paper-Scissors: unique, full support equilibrium
▸ Prisoner’s dilemma(Lecture 1): dominance-solvable, one strict equilibrium ▸ Battle of the Sexes(Lecture 1): one full support equilibrium; two strict equilibria
Overview Preliminaries The replicator dynamics Rationality analysis References
Solution concepts redux
▸ Dominated strategies:ai∈Aiisdominatedbya′i∈Ai(ai ≺a′i) if ui ai(x)<ui a′i(x) for allx∈X
[Mixed version:pi≺qi ⇐⇒ ui(pi;x−i)<ui(qi;x−i)for allx∈X] ▸ Nash equilibrium:no player has an incentive to switch strategies
ui ai(x ∗)≥u
i a′i(x∗) for allai,a′i∈Aiwithx∗i ai >
▸ Supportofx∗:supp(x∗)={(a, . . . ,aN)∈A∶x∗i ai>for alli}
▸ Interior / Full support equilibria:supp(x∗)=A ▸ Pure equilibria:supp(x∗)=singleton
▸ Strict equilibria:“>” instead of “≥” whena′i∉supp(x∗i) ▸ Examples:
▸ Rock-Paper-Scissors: unique, full support equilibrium
▸ Prisoner’s dilemma(Lecture 1): dominance-solvable, one strict equilibrium ▸ Battle of the Sexes(Lecture 1): one full support equilibrium; two strict equilibria
Overview Preliminaries The replicator dynamics Rationality analysis References
Rest of this lecture
Are game-theoretic solution concepts consistent with evolutionary models?
▸ Evolutionary models;dynamical systems (Lotka-Volterra, replicator, etc.) ▸ Do dominated strategies become extinct?
▸ Is equilibrium play stable/attracting?
Overview Preliminaries The replicator dynamics Rationality analysis References
Rest of this lecture
Are game-theoretic solution concepts consistent with evolutionary models?
▸ Evolutionary models;dynamical systems (Lotka-Volterra, replicator, etc.) ▸ Do dominated strategies become extinct?
▸ Is equilibrium play stable/attracting?
Overview Preliminaries The replicator dynamics Rationality analysis References
Outline
Overview Preliminaries
The replicator dynamics
Overview Preliminaries The replicator dynamics Rationality analysis References
A biologist's viewpoint
▸ Populations arespecies,strategies arephenotypes:
zi ai =absolute population mass of typeai ∈Ai
zi=∑aizi ai =absolute population mass ofi-th species
▸ Utilities measurefecundity / reproductive fitness:
ui ai(x)=per capita growth rate of typeai ▸ Population evolution:
˙
zi ai=zi aiui ai ▸ Evolution of population shares (xi ai=zi ai/zi):
˙ xi ai= d dt zi ai zi = ˙ zi aizi−zi ai∑a′i˙zi a′i z i =zi ai zi ui ai− zi ai zi ∑a′i zi a′i zi ui a′i Replicator dynamics[Taylor and Jonker, 1978]
˙
Overview Preliminaries The replicator dynamics Rationality analysis References
A biologist's viewpoint
▸ Populations arespecies,strategies arephenotypes:
zi ai =absolute population mass of typeai ∈Ai
zi=∑aizi ai =absolute population mass ofi-th species ▸ Utilities measurefecundity / reproductive fitness:
ui ai(x)=per capita growth rate of typeai ▸ Population evolution:
˙
zi ai=zi aiui ai
▸ Evolution of population shares (xi ai=zi ai/zi): ˙ xi ai= d dt zi ai zi = ˙ zi aizi−zi ai∑a′i˙zi a′i z i =zi ai zi ui ai− zi ai zi ∑a′i zi a′i zi ui a′i Replicator dynamics[Taylor and Jonker, 1978]
˙
Overview Preliminaries The replicator dynamics Rationality analysis References
A biologist's viewpoint
▸ Populations arespecies,strategies arephenotypes:
zi ai =absolute population mass of typeai ∈Ai
zi=∑aizi ai =absolute population mass ofi-th species ▸ Utilities measurefecundity / reproductive fitness:
ui ai(x)=per capita growth rate of typeai ▸ Population evolution:
˙
zi ai=zi aiui ai ▸ Evolution of population shares (xi ai=zi ai/zi):
˙ xi ai= d dt zi ai zi = ˙ zi aizi−zi ai∑a′i˙zi a′i z i =zi ai zi ui ai− zi ai zi ∑a′i zi a′i zi ui a′i
Replicator dynamics[Taylor and Jonker, 1978] ˙
Overview Preliminaries The replicator dynamics Rationality analysis References
A biologist's viewpoint
▸ Populations arespecies,strategies arephenotypes:
zi ai =absolute population mass of typeai ∈Ai
zi=∑aizi ai =absolute population mass ofi-th species ▸ Utilities measurefecundity / reproductive fitness:
ui ai(x)=per capita growth rate of typeai ▸ Population evolution:
˙
zi ai=zi aiui ai ▸ Evolution of population shares (xi ai=zi ai/zi):
˙ xi ai= d dt zi ai zi = ˙ zi aizi−zi ai∑a′i˙zi a′i z i =zi ai zi ui ai− zi ai zi ∑a′i zi a′i zi ui a′i Replicator dynamics[Taylor and Jonker, 1978]
˙
Overview Preliminaries The replicator dynamics Rationality analysis References
An economist's viewpoint
▸ Agents receiverevision opportunitiesto switch strategies ρaa′(x)=conditional switch rate fromatoa′
[NB:dropping player index for simplicity]
▸ Pairwise proportional imitation:
ρaa′(x)=xa′[ua′(x)−ua(x)]+
[Imitate with probability proportional to excess payoff (Helbing, 1992; Schlag, 1998)]
▸ Inflow/outflow:
Incoming towarda=∑a′mass(a′;a)=∑a′∈Axa′ρa′a(x) Outgoing froma=∑a′mass(a;a
′)=x
a∑a′∈Aρaa′(x) ▸ Detailed balance:
˙
xa=inflowa(x)−outflowa(x)=⋯=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
An economist's viewpoint
▸ Agents receiverevision opportunitiesto switch strategies ρaa′(x)=conditional switch rate fromatoa′
[NB:dropping player index for simplicity]
▸ Pairwise proportional imitation:
ρaa′(x)=xa′[ua′(x)−ua(x)]+
[Imitate with probability proportional to excess payoff (Helbing, 1992; Schlag, 1998)]
▸ Inflow/outflow:
Incoming towarda=∑a′mass(a′;a)=∑a′∈Axa′ρa′a(x) Outgoing froma=∑a′mass(a;a
′)=x
a∑a′∈Aρaa′(x) ▸ Detailed balance:
˙
xa=inflowa(x)−outflowa(x)=⋯=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
An economist's viewpoint
▸ Agents receiverevision opportunitiesto switch strategies ρaa′(x)=conditional switch rate fromatoa′
[NB:dropping player index for simplicity]
▸ Pairwise proportional imitation:
ρaa′(x)=xa′[ua′(x)−ua(x)]+
[Imitate with probability proportional to excess payoff (Helbing, 1992; Schlag, 1998)]
▸ Inflow/outflow:
Incoming towarda=∑a′mass(a′;a)=∑a′∈Axa′ρa′a(x) Outgoing froma=∑a′mass(a;a
′)=x
a∑a′∈Aρaa′(x)
▸ Detailed balance:
˙
xa=inflowa(x)−outflowa(x)=⋯=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
An economist's viewpoint
▸ Agents receiverevision opportunitiesto switch strategies ρaa′(x)=conditional switch rate fromatoa′
[NB:dropping player index for simplicity]
▸ Pairwise proportional imitation:
ρaa′(x)=xa′[ua′(x)−ua(x)]+
[Imitate with probability proportional to excess payoff (Helbing, 1992; Schlag, 1998)]
▸ Inflow/outflow:
Incoming towarda=∑a′mass(a′;a)=∑a′∈Axa′ρa′a(x) Outgoing froma=∑a′mass(a;a
′)=x
a∑a′∈Aρaa′(x) ▸ Detailed balance:
˙
xa=inflowa(x)−outflowa(x)=⋯=xa[ua(x)−u(x)] (RD)
Overview Preliminaries The replicator dynamics Rationality analysis References
An economist's viewpoint
▸ Agents receiverevision opportunitiesto switch strategies ρaa′(x)=conditional switch rate fromatoa′
[NB:dropping player index for simplicity]
▸ Pairwise proportional imitation:
ρaa′(x)=xa′[ua′(x)−ua(x)]+
[Imitate with probability proportional to excess payoff (Helbing, 1992; Schlag, 1998)]
▸ Inflow/outflow:
Incoming towarda=∑a′mass(a′;a)=∑a′∈Axa′ρa′a(x) Outgoing froma=∑a′mass(a;a
′)=x
a∑a′∈Aρaa′(x) ▸ Detailed balance:
˙
xa=inflowa(x)−outflowa(x)=⋯=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
A learning viewpoint
Evolution of mixed strategies in a finite game: ▸ Agents record cumulative payoff of each strategy
ya(t)=
∫
t ua(τ)dτ
Ô⇒propensityof choosing a strategy [Littlestone and Warmuth, 1994; Vovk, 1995]
▸ Choice probabilities;exponentially proportional to propensity scores xa(t)
▸ Evolution of mixed strategies [Hofbauer et al., 2009; Rustichini, 1999] ˙
xa=⋅ ⋅ ⋅=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
A learning viewpoint
Evolution of mixed strategies in a finite game: ▸ Agents record cumulative payoff of each strategy
ya(t)=
∫
t ua(τ)dτ
Ô⇒propensityof choosing a strategy [Littlestone and Warmuth, 1994; Vovk, 1995] ▸ Choice probabilities;exponentially proportional to propensity scores
xa(t)∝exp(ya(t))
▸ Evolution of mixed strategies [Hofbauer et al., 2009; Rustichini, 1999] ˙
xa=⋅ ⋅ ⋅=xa[ua(x)−u(x)] (RD)
Overview Preliminaries The replicator dynamics Rationality analysis References
A learning viewpoint
Evolution of mixed strategies in a finite game: ▸ Agents record cumulative payoff of each strategy
ya(t)=
∫
t ua(τ)dτ
Ô⇒propensityof choosing a strategy [Littlestone and Warmuth, 1994; Vovk, 1995] ▸ Choice probabilities;exponentially proportional to propensity scores
xa(t)= exp(ya(t)) ∑a′exp(ya′(t))
▸ Evolution of mixed strategies [Hofbauer et al., 2009; Rustichini, 1999] ˙
xa=⋅ ⋅ ⋅=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
A learning viewpoint
Evolution of mixed strategies in a finite game: ▸ Agents record cumulative payoff of each strategy
ya(t)=
∫
t ua(τ)dτ
Ô⇒propensityof choosing a strategy [Littlestone and Warmuth, 1994; Vovk, 1995] ▸ Choice probabilities;exponentially proportional to propensity scores
xa(t)= exp(ya(t)) ∑a′exp(ya′(t))
▸ Evolution of mixed strategies [Hofbauer et al., 2009; Rustichini, 1999] ˙
xa=⋅ ⋅ ⋅=xa[ua(x)−u(x)] (RD) [Check:verify computation]
Overview Preliminaries The replicator dynamics Rationality analysis References
Basic properties
Different viewpoints, same dynamics:
˙
xi ai =xai[ui ai(x)−ui(x)] (RD)
[NB:all viewpoints will be useful later]
Structural properties [Hofbauer and Sigmund, 1998; Weibull, 1995]
▸ Well-posed:every initial conditionx∈Xadmits unique solution trajectory
x(t)that exists for all time
[Assuminguiis Lipschitz]
▸ Consistent:x(t)∈Xfor allt≥
[Assumingx()∈X]
▸ Faces are forward invariant(“strategies breed true”):
xi ai()> ⇐⇒ xi ai(t)> for allt≥
Overview Preliminaries The replicator dynamics Rationality analysis References
Basic properties
Different viewpoints, same dynamics:
˙
xi ai =xai[ui ai(x)−ui(x)] (RD)
[NB:all viewpoints will be useful later] Structural properties [Hofbauer and Sigmund, 1998; Weibull, 1995]
▸ Well-posed:every initial conditionx∈Xadmits unique solution trajectory
x(t)that exists for all time
[Assuminguiis Lipschitz]
▸ Consistent:x(t)∈Xfor allt≥
[Assumingx()∈X]
▸ Faces are forward invariant(“strategies breed true”):
xi ai()> ⇐⇒ xi ai(t)> for allt≥
Overview Preliminaries The replicator dynamics Rationality analysis References
Outline
Overview Preliminaries
The replicator dynamics
Overview Preliminaries The replicator dynamics Rationality analysis References
Dynamics and rationality
Are game-theoretic solution concepts consistent with evolutionary models?
▸ Do dominated strategies die out in the long run? ▸ Are Nash equilibria stationary?
▸ Are they stable? Are they attracting? ▸ Do the replicator dynamics always converge? ▸ What other behaviors can we observe?
Overview Preliminaries The replicator dynamics Rationality analysis References
Dynamics and rationality
Are game-theoretic solution concepts consistent with the replicator dynamics?
▸ Do dominated strategies die out in the long run? ▸ Are Nash equilibria stationary?
▸ Are they stable? Are they attracting? ▸ Do the replicator dynamics always converge? ▸ What other behaviors can we observe?
Overview Preliminaries The replicator dynamics Rationality analysis References
Dynamics and rationality
Are game-theoretic solution concepts consistent with the replicator dynamics?
▸ Do dominated strategies die out in the long run? ▸ Are Nash equilibria stationary?
▸ Are they stable? Are they attracting? ▸ Do the replicator dynamics always converge? ▸ What other behaviors can we observe?
Overview Preliminaries The replicator dynamics Rationality analysis References
Phase portraits
(�� �) (�� �) (�� �) (�� �) ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ���������� �������� �� ��� ���������� �������Overview Preliminaries The replicator dynamics Rationality analysis References
Phase portraits
(�� �) (�� �) (�� �) (�� �) ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ���������� �������� �� ��� ������ �� ��� �����Overview Preliminaries The replicator dynamics Rationality analysis References
Phase portraits
(��-�) (-�� �) (-�� �) (��-�) ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ���������� �������� �� �������� �������Overview Preliminaries The replicator dynamics Rationality analysis References
Phase portraits
(�� �) (�� �) (�� �) (�� �) ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� �� �� ���������� �������� �� � ���������� ����Overview Preliminaries The replicator dynamics Rationality analysis References
Dominated strategies
Supposeai∈Aiisdominatedbya′i∈Ai ▸ Consistent payoff gap:
ui ai(x)≤ui ai′(x)−ε for someε>
▸ Consistent difference in scores: yi ai(t)=
∫
t ui ai(x)dτ≤∫
t [ui a ′ i(x)−ε]dτ=yi a′i(t)−εt ▸ Consistent difference in choice probabilitiesxi ai(t)
xi a′i(t)
=exp(yi ai(t)) exp(yi a′i(t))
≤exp(−εt)
▸ Dominated strategies become extinct[Samuelson and Zhang, 1992]
lim
t→∞xi ai(t)= wheneveraiis dominated
[Check #1:extend toiteratively / mixeddominated strategies] [Check #2:what aboutweaklydominated strategies?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Dominated strategies
Supposeai∈Aiisdominatedbya′i∈Ai ▸ Consistent payoff gap:
ui ai(x)≤ui ai′(x)−ε for someε> ▸ Consistent difference in scores:
yi ai(t)=
∫
t ui ai(x)dτ≤∫
t [ui a ′ i(x)−ε]dτ=yi a′i(t)−εt▸ Consistent difference in choice probabilities xi ai(t)
xi a′i(t)
=exp(yi ai(t)) exp(yi a′i(t))
≤exp(−εt)
▸ Dominated strategies become extinct[Samuelson and Zhang, 1992]
lim
t→∞xi ai(t)= wheneveraiis dominated
[Check #1:extend toiteratively / mixeddominated strategies] [Check #2:what aboutweaklydominated strategies?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Dominated strategies
Supposeai∈Aiisdominatedbya′i∈Ai ▸ Consistent payoff gap:
ui ai(x)≤ui ai′(x)−ε for someε> ▸ Consistent difference in scores:
yi ai(t)=
∫
t ui ai(x)dτ≤∫
t [ui a ′ i(x)−ε]dτ=yi a′i(t)−εt ▸ Consistent difference in choice probabilitiesxi ai(t)
xi a′i(t)
=exp(yi ai(t)) exp(yi a′i(t))
≤exp(−εt)
▸ Dominated strategies become extinct[Samuelson and Zhang, 1992]
lim
t→∞xi ai(t)= wheneveraiis dominated
[Check #1:extend toiteratively / mixeddominated strategies] [Check #2:what aboutweaklydominated strategies?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Dominated strategies
Supposeai∈Aiisdominatedbya′i∈Ai ▸ Consistent payoff gap:
ui ai(x)≤ui ai′(x)−ε for someε> ▸ Consistent difference in scores:
yi ai(t)=
∫
t ui ai(x)dτ≤∫
t [ui a ′ i(x)−ε]dτ=yi a′i(t)−εt ▸ Consistent difference in choice probabilitiesxi ai(t)
xi a′i(t)
=exp(yi ai(t)) exp(yi a′i(t))
≤exp(−εt)
▸ Dominated strategies become extinct[Samuelson and Zhang, 1992]
lim
t→∞xi ai(t)= wheneveraiis dominated
[Check #1:extend toiteratively / mixeddominated strategies] [Check #2:what aboutweaklydominated strategies?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Dominated strategies
Supposeai∈Aiisdominatedbya′i∈Ai ▸ Consistent payoff gap:
ui ai(x)≤ui ai′(x)−ε for someε> ▸ Consistent difference in scores:
yi ai(t)=
∫
t ui ai(x)dτ≤∫
t [ui a ′ i(x)−ε]dτ=yi a′i(t)−εt ▸ Consistent difference in choice probabilitiesxi ai(t)
xi a′i(t)
=exp(yi ai(t)) exp(yi a′i(t))
≤exp(−εt)
▸ Dominated strategies become extinct[Samuelson and Zhang, 1992]
lim
t→∞xi ai(t)= wheneveraiis dominated
[Check #1:extend toiteratively / mixeddominated strategies] [Check #2:what aboutweaklydominated strategies?]
Overview Preliminaries The replicator dynamics Rationality analysis References
Stationarity of equilibria
Nash equilibrium:ui ai(x∗)≥ui a′i(x∗)for allai,a′i∈Aiwithx∗i ai > ▸ Supported strategies have equal payoffs:
ui ai(x ∗)=u
i a′i(x∗) for allai,a′i∈supp(x∗i)
▸ Mean payoff equal to equilibrium payoff: ui(x∗)=ui ai(x
∗) for alla
i∈supp(x∗i) ▸ Replicator field vanishes at equilibria:
x∗i ai[ui ai(x ∗)−u
i(x∗)]= for allai∈Ai ▸ Nash equilibria are stationary:
x()=x∗ ⇐⇒ x(t)=x∗for allt≥ ▸ The converse does not hold(never used inequality)
Overview Preliminaries The replicator dynamics Rationality analysis References
Stationarity of equilibria
Nash equilibrium:ui ai(x∗)≥ui a′i(x∗)for allai,a′i∈Aiwithx∗i ai > ▸ Supported strategies have equal payoffs:
ui ai(x ∗)=u
i a′i(x∗) for allai,a′i∈supp(x∗i) ▸ Mean payoff equal to equilibrium payoff:
ui(x∗)=ui ai(x
∗) for alla
i∈supp(x∗i)
▸ Replicator field vanishes at equilibria: x∗i ai[ui ai(x
∗)−u
i(x∗)]= for allai∈Ai ▸ Nash equilibria are stationary:
x()=x∗ ⇐⇒ x(t)=x∗for allt≥ ▸ The converse does not hold(never used inequality)
Overview Preliminaries The replicator dynamics Rationality analysis References
Stationarity of equilibria
Nash equilibrium:ui ai(x∗)≥ui a′i(x∗)for allai,a′i∈Aiwithx∗i ai > ▸ Supported strategies have equal payoffs:
ui ai(x ∗)=u
i a′i(x∗) for allai,a′i∈supp(x∗i) ▸ Mean payoff equal to equilibrium payoff:
ui(x∗)=ui ai(x
∗) for alla
i∈supp(x∗i) ▸ Replicator field vanishes at equilibria:
x∗i ai[ui ai(x ∗)−u
i(x∗)]= for allai∈Ai
▸ Nash equilibria are stationary:
x()=x∗ ⇐⇒ x(t)=x∗for allt≥ ▸ The converse does not hold(never used inequality)
Overview Preliminaries The replicator dynamics Rationality analysis References
Stationarity of equilibria
Nash equilibrium:ui ai(x∗)≥ui a′i(x∗)for allai,a′i∈Aiwithx∗i ai > ▸ Supported strategies have equal payoffs:
ui ai(x ∗)=u
i a′i(x∗) for allai,a′i∈supp(x∗i) ▸ Mean payoff equal to equilibrium payoff:
ui(x∗)=ui ai(x
∗) for alla
i∈supp(x∗i) ▸ Replicator field vanishes at equilibria:
x∗i ai[ui ai(x ∗)−u
i(x∗)]= for allai∈Ai ▸ Nash equilibria are stationary:
x()=x∗ ⇐⇒ x(t)=x∗for allt≥
▸ The converse does not hold(never used inequality)
Overview Preliminaries The replicator dynamics Rationality analysis References
Stationarity of equilibria
Nash equilibrium:ui ai(x∗)≥ui a′i(x∗)for allai,a′i∈Aiwithx∗i ai > ▸ Supported strategies have equal payoffs:
ui ai(x ∗)=u
i a′i(x∗) for allai,a′i∈supp(x∗i) ▸ Mean payoff equal to equilibrium payoff:
ui(x∗)=ui ai(x
∗) for alla
i∈supp(x∗i) ▸ Replicator field vanishes at equilibria:
x∗i ai[ui ai(x ∗)−u
i(x∗)]= for allai∈Ai ▸ Nash equilibria are stationary:
x()=x∗ ⇐⇒ x(t)=x∗for allt≥ ▸ The converse does not hold(never used inequality)
Overview Preliminaries The replicator dynamics Rationality analysis References
Stability
Are all stationary points created equal?
Definition
x∗is(Lyapunov) stableif, for every neighborhoodUofx∗inX, there exists a neighborhoodU′ofx∗such that
x()∈U′ ⇐⇒ x(t)∈U for allt≥
[Trajectories that start close tox∗remain close for all time]
Overview Preliminaries The replicator dynamics Rationality analysis References
Stability
Are all stationary points created equal?
Definition
x∗is(Lyapunov) stableif, for every neighborhoodUofx∗inX, there exists a neighborhoodU′ofx∗such that
x()∈U′ ⇐⇒ x(t)∈U for allt≥
[Trajectories that start close tox∗remain close for all time]
Overview Preliminaries The replicator dynamics Rationality analysis References
Asymptotic stability
Are all Nash equilibria created equal?
Definition
▸ x∗isattractingiflimt→∞x(t)=x∗wheneverx()is close enough tox∗
▸ x∗isasymptotically stableif it is stable and attracting
Overview Preliminaries The replicator dynamics Rationality analysis References
Asymptotic stability
Are all Nash equilibria created equal?
Definition
▸ x∗isattractingiflimt→∞x(t)=x∗wheneverx()is close enough tox∗
▸ x∗isasymptotically stableif it is stable and attracting
Overview Preliminaries The replicator dynamics Rationality analysis References
The "folk theorem" of evolutionary game theory
Theorem (
Cressman, 2003; Hofbauer and Sigmund, 2003)
In multi-population random matching games:▸ x∗is a Nash equilibrium Ô⇒ x∗is stationary
▸ x∗is the limit of an interior trajectory Ô⇒ x∗is a Nash equilibrium
▸ x∗is stable Ô⇒ x∗is a Nash equilibrium
Overview Preliminaries The replicator dynamics Rationality analysis References
The "folk theorem" of evolutionary game theory
Theorem (
Cressman, 2003; Hofbauer and Sigmund, 2003)
Insingle-population random matching games:▸ x∗is a Nash equilibrium Ô⇒ x∗is stationary
▸ x∗is the limit of an interior trajectory Ô⇒ x∗is a Nash equilibrium
▸ x∗is stable Ô⇒ x∗is a Nash equilibrium
Overview Preliminaries The replicator dynamics Rationality analysis References
Attracting full-support equilibria
The replicator dynamics in “good” RPS (win>loss):
R
Overview Preliminaries The replicator dynamics Rationality analysis References
Convergence in potential games
Potential games[Sandholm, 2001] ui ai =−
∂V ∂xi ai
for somepotential functionV∶X→R NASC (Poincaré):
potential ⇐⇒ ∂ui ai
∂xi a′i =∂ui a′i
∂xi ai
Positive correlation / Lyapunov property: dV
dt ≤ under (RD)
[Check:verify this]
Theorem (
Sandholm, 2001)
▸ In potential games,(RD)converges to its set of stationary points
▸ In random matching potential games, interior trajectories of(RD)converge to Nash equilibrium
Overview Preliminaries The replicator dynamics Rationality analysis References
Convergence in potential games
Potential games[Sandholm, 2001] ui ai =−
∂V ∂xi ai
for somepotential functionV∶X→R NASC (Poincaré):
potential ⇐⇒ ∂ui ai
∂xi a′i =∂ui a′i
∂xi ai Positive correlation / Lyapunov property:
dV
dt ≤ under (RD)
[Check:verify this]
Theorem (
Sandholm, 2001)
▸ In potential games,(RD)converges to its set of stationary points
▸ In random matching potential games, interior trajectories of(RD)converge to Nash equilibrium
Overview Preliminaries The replicator dynamics Rationality analysis References
Convergence in potential games
Potential games[Sandholm, 2001] ui ai =−
∂V ∂xi ai
for somepotential functionV∶X→R NASC (Poincaré):
potential ⇐⇒ ∂ui ai
∂xi a′i =∂ui a′i
∂xi ai Positive correlation / Lyapunov property:
dV
dt ≤ under (RD)
[Check:verify this]
Theorem (
Sandholm, 2001)
▸ In potential games,(RD)converges to its set of stationary points
▸ In random matching potential games, interior trajectories of(RD)converge to Nash equilibrium
Overview Preliminaries The replicator dynamics Rationality analysis References
Non-convergence in zero-sum games
The landscape is very different in zero-sum games:
x∗is full-support equilibrium Ô⇒ (RD) admitsconstant of motion KL divergence: DKL(x∗,x)=∑i∑aix∗i ailog
x∗i ai
xi ai
Theorem (
Hofbauer et al., 2009)
Assume a bilinear zero-sum game admits an interior equilibrium. Then:
▸ Interior trajectories of(RD)do not converge(unless stationary) ▸ Time-averagesx¯(t)=t−
∫
tx(τ)dτconverge to Nash equilibriumOverview Preliminaries The replicator dynamics Rationality analysis References
Non-convergence in zero-sum games
The landscape is very different in zero-sum games:
x∗is full-support equilibrium Ô⇒ (RD) admitsconstant of motion KL divergence: DKL(x∗,x)=∑i∑aix∗i ailog
x∗i ai
xi ai
Theorem (
Hofbauer et al., 2009)
Assume a bilinear zero-sum game admits an interior equilibrium. Then:
▸ Interior trajectories of(RD)do not converge(unless stationary) ▸ Time-averagesx¯(t)=t−
∫
tx(τ)dτconverge to Nash equilibriumOverview Preliminaries The replicator dynamics Rationality analysis References
Non-convergence in zero-sum games
The landscape is very different in zero-sum games:
x∗is full-support equilibrium Ô⇒ (RD) admitsconstant of motion KL divergence: DKL(x∗,x)=∑i∑aix∗i ailog
x∗i ai
xi ai
Theorem (
Hofbauer et al., 2009)
Assume a bilinear zero-sum game admits an interior equilibrium. Then:
▸ Interior trajectories of(RD)do not converge(unless stationary) ▸ Time-averagesx¯(t)=t−
∫
tx(τ)dτconverge to Nash equilibriumOverview Preliminaries The replicator dynamics Rationality analysis References
Convergence of time-averages
The replicator dynamics in a game of Matching Pennies
H1,-1L H-1, 1L H-1, 1L H1,-1L 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x2
Overview Preliminaries The replicator dynamics Rationality analysis References
Poincaré recurrence in zero-sum games
Definition (
Poincaré)
A dynamical system isPoincaré recurrentif almost all solution trajectories return arbitrarily closeto their starting pointinfinitely many times
Theorem (
Piliouras and Shamma, 2014; M et al., 2018)
Overview Preliminaries The replicator dynamics Rationality analysis References
Poincaré recurrence in zero-sum games
Definition (
Poincaré)
A dynamical system isPoincaré recurrentif almost all solution trajectories return arbitrarily closeto their starting pointinfinitely many times
Theorem (
Piliouras and Shamma, 2014; M et al., 2018)
Overview Preliminaries The replicator dynamics Rationality analysis References
What's missing
▸ Evolutionary stability [Maynard Smith and Price, 1973]
▸ Other (classes of) dynamics:
▸ Fictitious play [Brown, 1951; Robinson, 1951]
▸ Best response dynamics [Gilboa and Matsui, 1991]
▸ Imitative / Innovative dynamics [Sandholm, 2010; Weibull, 1995]
▸ Higher-order dynamics [Laraki and M, 2013]
▸ Unexpected / Complex behaviors:
▸ Survival of dominated strategies [?]
▸ Chaos [Sandholm, 2010]
▸ Evolution in the presence of uncertainty
[Fudenberg and Harris, 1992; Imhof, 2005; M & Moustakas, 2010; M & Viossat, 2016]
Overview Preliminaries The replicator dynamics Rationality analysis References
What's missing
▸ Evolutionary stability [Maynard Smith and Price, 1973]
▸ Other (classes of) dynamics:
▸ Fictitious play [Brown, 1951; Robinson, 1951]
▸ Best response dynamics [Gilboa and Matsui, 1991]
▸ Imitative / Innovative dynamics [Sandholm, 2010; Weibull, 1995]
▸ Higher-order dynamics [Laraki and M, 2013]
▸ Unexpected / Complex behaviors:
▸ Survival of dominated strategies [?]
▸ Chaos [Sandholm, 2010]
▸ Evolution in the presence of uncertainty
[Fudenberg and Harris, 1992; Imhof, 2005; M & Moustakas, 2010; M & Viossat, 2016]
Overview Preliminaries The replicator dynamics Rationality analysis References
What's missing
▸ Evolutionary stability [Maynard Smith and Price, 1973]
▸ Other (classes of) dynamics:
▸ Fictitious play [Brown, 1951; Robinson, 1951]
▸ Best response dynamics [Gilboa and Matsui, 1991]
▸ Imitative / Innovative dynamics [Sandholm, 2010; Weibull, 1995]
▸ Higher-order dynamics [Laraki and M, 2013]
▸ Unexpected / Complex behaviors:
▸ Survival of dominated strategies [?]
▸ Chaos [Sandholm, 2010]
▸ Evolution in the presence of uncertainty
[Fudenberg and Harris, 1992; Imhof, 2005; M & Moustakas, 2010; M & Viossat, 2016]
Overview Preliminaries The replicator dynamics Rationality analysis References
What's missing
▸ Evolutionary stability [Maynard Smith and Price, 1973]
▸ Other (classes of) dynamics:
▸ Fictitious play [Brown, 1951; Robinson, 1951]
▸ Best response dynamics [Gilboa and Matsui, 1991]
▸ Imitative / Innovative dynamics [Sandholm, 2010; Weibull, 1995]
▸ Higher-order dynamics [Laraki and M, 2013]
▸ Unexpected / Complex behaviors:
▸ Survival of dominated strategies [?]
▸ Chaos [Sandholm, 2010]
▸ Evolution in the presence of uncertainty
[Fudenberg and Harris, 1992; Imhof, 2005; M & Moustakas, 2010; M & Viossat, 2016]
Overview Preliminaries The replicator dynamics Rationality analysis References
What's missing
▸ Evolutionary stability [Maynard Smith and Price, 1973]
▸ Other (classes of) dynamics:
▸ Fictitious play [Brown, 1951; Robinson, 1951]
▸ Best response dynamics [Gilboa and Matsui, 1991]
▸ Imitative / Innovative dynamics [Sandholm, 2010; Weibull, 1995]
▸ Higher-order dynamics [Laraki and M, 2013]
▸ Unexpected / Complex behaviors:
▸ Survival of dominated strategies [?]
▸ Chaos [Sandholm, 2010]
▸ Evolution in the presence of uncertainty
[Fudenberg and Harris, 1992; Imhof, 2005; M & Moustakas, 2010; M & Viossat, 2016]
Overview Preliminaries The replicator dynamics Rationality analysis References
References I
Brown, George W. 1951. Iterative solutions of games by fictitious play. T. C. Coopmans, ed.,Activity Analysis of Productions and Allocation. 374-376, Wiley.
Cressman, Ross. 2003.Evolutionary Dynamics and Extensive Form Games. The MIT Press.
Fudenberg, Drew, Christopher Harris. 1992. Evolutionary dynamics with aggregate shocks.Journal of Economic Theory57(2) 420–441.
Gilboa, Itzhak, Akihiko Matsui. 1991. Social stability and equilibrium.Econometrica59(3) 859–867. Helbing, Dirk. 1992. A mathematical model for behavioral changes by pair interactions. Günter Haag,
Ulrich Mueller, Klaus G. Troitzsch, eds.,Economic Evolution and Demographic Change: Formal Models in Social Sciences. Springer, Berlin, 330–348.
Hofbauer, Josef, William H. Sandholm. 2011. Survival of dominated strategies under evolutionary dynamics.Theoretical Economics6(3) 341–377.
Hofbauer, Josef, Karl Sigmund. 1998.Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK.
Hofbauer, Josef, Karl Sigmund. 2003. Evolutionary game dynamics.Bulletin of the American Mathematical Society40(4) 479–519.
Hofbauer, Josef, Sylvain Sorin, Yannick Viossat. 2009. Time average replicator and best reply dynamics.
