Evolutionary game Evolutionary game dynamics with migration dynamics with migration Hamidou Tembine Hamidou Tembine Outline The Model Joint work with Eitan Altman, INRIA, MAESTRO Group, Sophia Antipolis migration ESS Rachid ElAzouzi and Yezekael Hayel, University of Avignon, LIA/CERI. CCE William H. Sandholm, Department of Economics, University of Wisconsin, hybrid dynamics Madison. Replicator class of games potential games stable games Power control POPEYE/GAMECOMP Workshop, May 21, 2008, Grenoble, France. Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
1 -Outline 1 The Model Evolutionary game migration dynamics with migration ESS Hamidou CCE Tembine hybrid dynamics Outline Replicator The Model migration 2 class of games ESS CCE potential games hybrid dynamics Replicator stable games class of games potential games Power control 3 stable games Power control Multihoming 4 Multihoming 5 extensive extensive conclusions 6 conclusions Hamidou Tembine Evolutionary game dynamics with migration
Association between technologies/interfaces Evolutionary game dynamics with migration Hamidou Tembine Outline The Model migration ESS CCE hybrid dynamics Replicator class of games potential games stable games Power control Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Evolutionary game dynamics with migration Hamidou Tembine Outline The Model Evolutionary Games migration ESS CCE hybrid dynamics Replicator class of games potential games stable games Power control Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Evolutionary Games Consider the following evolutionary game model consisting of : Evolutionary game Several class of players E = { 1 , 2 , . . . } . dynamics with migration A large number of players (to guarantee non-atomicity), Hamidou Many local and simultaneous interactions between some Tembine (possibility random number) of players. Outline Each local interaction is described as follows: The Model region or resource called primary action ( r ∈ R ) migration ESS each player of each class have a finite set of secondary CCE hybrid dynamics actions Replicator class of games A e = { ( r , a ) | r ∈ R , a ∈ A r e } , X r e ( A r e = ∆ m r e ) potential games stable games where A r e is the secondary actions in region r for e . Power control a payoff (reward,fitness) function: Multihoming � � X r → R |A e | , F e ( x ) = ( F r F e : e − e , a ( x )) r , a . extensive conclusions e ∈E r ∈R the system evolves under some evolutionary game dynamics based on revision protocol . Hamidou Tembine Evolutionary game dynamics with migration
Objectives Evolutionary game dynamics with migration Hamidou Solution concepts: Global Equilibrium, Global Neutrally Tembine Equilibrium, Choice Constrained Equilibrium, Global Outline Evolutionarily Stable Strategy. The Model migration Multicomponent evolutionary game dynamics, Dynamics ESS CCE with migration constraints. hybrid dynamics Replicator Convergence ??? local,global, asymptotic, cycle etc, under class of games some class of evolutionary game dynamics with migration potential games stable games for some class of population games. Power control Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Some terms and notations Evolutionary game dynamics with migration migration : If a player changes its actions from region r to Hamidou Tembine r (physically at the same)) or moves from r to ¯ ¯ r (mobility), we say that player migrates from r to ¯ r . Outline The Model migration constraints: we assume that player from class e migration ESS in region r can migrate only in the ”neighboring regions” CCE N e , ( r , a ) ⊆ � r {{ r } × A r hybrid dynamics e } in one-hop. Replicator class of games the neighborhood set N e , ( r , a ) have the property that: potential games stable games (¯ r , b ) ∈ N e , ( r , a ) ⇐ ⇒ ( r , a ) ∈ N e , (¯ Power control r , b ) Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Examples of migration/mobility constraints Evolutionary game dynamics with migration Hamidou Tembine Unidirectional constraint : the player can change only one of the two components of strategy. Outline The Model � migration A r N e , ( r , a ) = ( R × { a } ) ESS e CCE hybrid dynamics Replicator N e , ( r , a ) = R × A e (free migration). class of games N e , ( r , a ) = A r potential games e ( standard, no migration). stable games Power control Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Concept of solution: refinement of equilibria Evolutionary game dynamics with ESS: Evolutionary Stable Strategy (Maynard Smith & Price, migration 1973). Hamidou Tembine Definition Outline (GESS) A strategy x is global evolutionary stable if for every The Model strategy mut � = x , there exists ǫ mut > 0 such that migration ESS ∀ e , � � a ( x r e , a − mut r e , a ) F r e , a ( ǫ mut + (1 − ǫ ) x ) > 0 , ∀ ǫ ∈ CCE r hybrid dynamics (0 , ǫ mut ) Replicator class of games potential games Robustness, refinement and equilibrium selection: ESS stable games = ⇒ proper equilibrium(Myerson), Nash equilibrium, the Power control Multihoming converse is not always true. See Maynard Smith(1982), extensive Weibull(1995), Hofbauer & Sigmund(1998). conclusions Hamidou Tembine Evolutionary game dynamics with migration
Choice constrained equilibrium Evolutionary game dynamics with migration Hamidou Tembine A strategy x is a choice constrained equilibrium (CCE) if for all e , r and a such that x r e , a > 0 one has, Outline The Model F r F ¯ r migration e , a ( x ) = max e , b ( x ) ESS (¯ r , b ) ∈N e , ( r , a ) CCE hybrid dynamics Replicator This corresponds to ”local equilibrium” under migration class of games constraints. If x r potential games e , a = 0 , we say that ( r , a ) is not used in class e . stable games Power control Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Evolutionary Game Dynamics with migration Let β (¯ r , b ) Evolutionary e , ( r , a ) ( . ) be a rule of actions’ choice called revision game dynamics with protocol (Sandholm,2007). β depends on state of all the migration population and the payoffs. β (¯ r , b ) e , ( r , a ) conditional switch rate from Hamidou Tembine the secondary action b in region ¯ r to the secondary action a in Outline region r . The Model Inflow the action ( r , a ) of class e migration ESS CCE � e , b β (¯ r , b ) x ¯ r e , ( r , a ) ( x , F ( x )) hybrid dynamics Replicator (¯ r , b ) class of games potential games stable games Outflow of the action ( r , a ) of class e : Power control Multihoming � β ( r , a ) x r r , b ) ( x , F ( x )) extensive e , a e , (¯ (¯ r , b ) conclusions Hamidou Tembine Evolutionary game dynamics with migration
Evolutionary game Let dynamics with migration � � V ( r , a ) e , b β (¯ r , b ) β ( r , a ) Hamidou x ¯ r e , ( r , a ) ( x , F ( x )) − x r e , F ( x ) = r , b ) ( x , F ( x )) Tembine e , a e , (¯ (¯ r , b ) (¯ r , b ) Outline The Model Assume that the revision protocol satisfies migration ESS CCE β ( r , a ) r , b ) ( x , F ( x )) > 0 = ⇒ (¯ r , b ) ∈ N e , ( r , a ) . hybrid dynamics e , (¯ Replicator class of games The game dynamics is then given by potential games stable games Power control d e , a ( t ) = V ( r , a ) dt x r e , F ( x ( t )) . Multihoming extensive conclusions Hamidou Tembine Evolutionary game dynamics with migration
Evolutionary game dynamics with migration: examples Evolutionary Replicator dynamics (Taylor & Jonker, 1978) game dynamics with β (¯ r , b ) e , ( r , a ) = x r e , a max(0 , F r e , a − F ¯ r migration e , b ) Hamidou Brown-von Neumann-Nash (BNN)(1950) dynamics Tembine β (¯ r , b ) e , a − � e , b F ¯ e , ( r , a ) = max(0 , F r r , b ) x r r e , b ) Outline (¯ The Model Smith dynamics β (¯ r , b ) e , ( r , a ) = max(0 , F r e , a − F ¯ r e , b ) migration ESS CCE Other dynamics: fictitious play, gradient, projection, differential hybrid dynamics Replicator inclusion, best response BR: class of games BR e ( x ) = arg max y { � ( r , a ) y r e , a F r e , a ( x ) } . BR dynamics: potential games stable games x e ∈ BR e ( x ) − x e ˙ Power control Folk theorem (evolutionary version): Every GNE is a Multihoming stationary of the theses dynamics (in particular for GESS). extensive Every stationary point of BNN, BR or Smith dynamics is GNE conclusions of the local game. Hamidou Tembine Evolutionary game dynamics with migration
Recommend
More recommend
