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# Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium

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## Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium Presentation Transcript

• Beyond Nash Equilibrium Correlated Equilibrium and Evolutionary Equilibrium Jie Bao 2003-05-05 Iowa State University
• Equilibrium in Games
• Pure strategy Nash equilibrium
• Mixed strategy Nash equilibrium
• Correlated equilibrium
• Evolutionary equilibrium
• Bayesian Nash equilibrium
• NE is too strict on what is “ration”…
• Correlated Equilibrium
• Aumann 1974
• A generalization of “rational” solution
• In a CE
• the action played by any player is a best response (in the expected payoff sense) to the conditional distribution over the other players given that action,
• and thus no player has a unilateral incentive to deviate from playing their role in the CE.
• Example: Traffic Signal- a single bit of shares information allows a fair split of waiting times.
• “ running a light” can’t bring greater expected payoff
• The actions of players are “correlated”
View slide
• Example : BoS
• Bach or Stravinsky Game (or Battle of Sex)
• NE: strategy profile -> payoff profile
• (Bach, Stravinsky) ->(2 ,1)
• (Stravinsky, Bach) ->(1 ,2)
• (1/3 Bach, 1/3 Bach) -> (2/3, 2/3)
• Another equilibrium: the player observe the outcome of a public coin toss, which determines which of the two pure strategy Nash equilibria they play.->(3/2,3/2)
? 1,2 1,0 Stravinsky 0,1 2,1 Bach Stravinsky Bach View slide
• BoS: Mixed Strategy   (b) Agent 2’s expected utility, blue line is it’s best response to given p (agent 1’s strategy) (d) Look from the top, (1/3,1/3) is the only mixed NE. pure NEs include (0,1) and (1,0) (a) Agent 1’s expected utility, red line is it’s best response to given q (agent 2’s strategy) (c) Overlap of a and b
• BoS: CE - Toss Coin Equ.
• <N,(A i ),(u i )>= <{1,2} ,{Bach , Stravinsky}, payoff matrix U>
• Probability space ( Ω,π )
• Ω = {x,y}
• π(x) = π(y) = ½
• information partition of each agent
• P1 = P2 = {{x},{y}}
• For each player {1,2} action function σ i : Ω->A i ,
• σ 1 (x) = σ 2 (x) = Bach
• σ 1 (y) = σ 2 (y) = Stravinsky
• Payoff profile: ½(B,B), ½(S,S) -> (3/2, 3/2)
• Compared with (for agent 1):
• ½(S,B), ½(S,S) ->1/2
• ½(B,B), ½(B,S) ->1
• ½(S,B), ½(B,S) ->0
• CE: formal definition
• A strategic game <N,(A i ),(u i )>
• A finite probability space ( Ω,π )
• Ωis a set of states and π is a probability measure on π
• For each player i∈N, a partition P i of Ω(player i’s information partition)
• For each player i∈N, a function σ i : Ω->A i , with σ i (w)= σ i (w’) whenever w∈ P i , and w’∈ P i for some P∈ P i , (σ i is player i’s strategy)
• such that for every I∈N and every function τ i : Ω->A i for which τ i (w)=τ i (w’) whenever w∈ P i , and w’∈ P i for some P∈ P i (i.e. for every strategy of player i) we have
• Note that we assume the players share a common belief about the probabilities with which the states occur.
• CE contains Mixed NE
• For every mixed strategy Nash equilibriumαof a finite game <N,(A i ),(u i )>, there is a correlated equilibrium <( Ω,π ), P i ,σ i > in which for each player i∈N, the distribution on A i induced by σ i is α i .
• Construct CE from Mixed NE
• Ω= A=(X j ∈N A j ) – strategy profiles
• π(α)=Π j ∈N α j (a j ) – prob. Of the strategy profiles
• P i (b i )={a ∈A: a i = b i }, P i consist of the |A i | sets P i (b i )
• σ i (a)=a i
• CE is a more general concept than Mixed NE and Pure NE
NE CE Mixed NE
• Convex Combination of CE->CE
• Let G=<N,(A i ),(u i )> be a strategic game. Any convex combination of correlated equilibrium payoff of G is a correlated equilibrium payoff of G
• Interpret: first a public random device determines which of the K correlated equilibria is to played, and then the random variable corresponding to the k th correlated equilibrium is realized.
• CE: <(Ω k ,π k ), P i k ,σ i k >
• CE payoff profiles: U 1 ,.. U k
• c 1 ,…c k , all c i >=0, and Σ c i =1
• Construct a new CE
• Ω = union of all Ω k
• π(w) = c k π k (w) , if w in Ω k
• P i =union (on k) of P i k
• σ i (w) = σ i k (w)
• Payoff profiles Σ c k U k
• Example
• Pure NE payoff profile: (7,2) (2,7)
• Mixed NE payoff profile: (4 2 / 3 ,4 2 / 3 )
• CE: Ω={x,y,z}, π(x)= π(y)= π(z)=1/3, P1={{x},{y,z}}, P2={{x,y},{z}}, σ 1 (x)=B, σ 1 (y)= σ 1 (z)=T, σ 2 (x)= σ 2 (y)=L, σ 2 (z)=R, -> (5,5)
• The CE is outside the convex hull of Pure / Mixed NE payoff profiles
0,0 7,2 B 2,7 6,6 T R L 1 2 - x B z y T R L 1 2
• States and outcomes in CE
• Let G=<N,(A i ),(u i )> be a strategic game. Every probability distribution over outcomes that can be obtained in a CE of G can be obtained in a CE’ in which
• the set of state Ω’ is A and
• for each i∈N, player i’s information partition P i ’ consists of all sets of the form {a∈A i : a i =b i } for some action b i ∈A i
• [π’(a) = π k ({w ∈ Ω: σ(w)=a}) σ i ’ (a) = σ i k (a i )]
• This theorem allows us to confine attention to equilibria in which the set states is the set of outcomes
• If players hold different beliefs, additional equilibrium payoff profiles are possible.
• Nash equilibrium is a special case of CE in which we demand that πbe a product distribution for some distribution π i , so every player acts independently of all others
• It’s possible to compute CE via linear programming in polynomial time, while NE is exponentially complex!
• Evolutionary Equilibrium
• ESS: Maynard Smith & Price, 1972
• A steady state in which all organism take this action and not mutant can invade the population .
• Example: the sex ratio in bee population is 1(male):3(female)
Selfish Selfish Selfish Selfless Selfless Selfless Selfless Selfless Selfless Selfish
• ESS: Definition:
• Let G=<{1,2},(B,B),(u i )>be a symmetric strategic game, where u 1 (a,b)=u 2 (b,a)=u(a,b) for some function u.
• An evolutionarily stable strategy(ESS) of G is an action
• b*∈B for which (b*,b*) is a NE of G
• and u(b,b)<u(b*,b) for every best response b∈B to b* with b≠b*.
• Example: Hawk--Dove
• Choose to be Hawk or dove?
• Pure NE: (D,H) & (H,D)
• Mixed NE: (0.5D/0.5H, 0.5D/05H)
• If the players have the freedom to choose to be hawk or dove in a repeated game, and utility will be used to reproduce their offspring, what’ll be the optimal strategy?
(1-c/)2, (1-c/)2 1,0 H 0,1 ½, ½ D H D
• HD Game – mixed strategy (a) Agent 1’s expected utility, red line is it’s best response to given q (agent 2’s strategy) (b) Agent 2’s expected utility, blue line is it’s best response to given p (agent 1’s strategy) (c) Overlap of a and b (d) Look from the top, (0.5,0.5) is the only mixed NE. pure NEs include (0,1) and (1,0) With c = 2, Action = {'Hawk' , 'Dove'} ;
• HD Game – ESS invading
• In all those games, ESS (half-dove-half-hawk mixed strategy (0.5,0.5)) starts from a percentage of 0.1 in the population
• Population: 200, Game Round = 1000
• Reproduce: proportional to total utility of each type
• Note that Dove is not completely eliminated
ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
• HD Game – ESS being invaded
• In all those games, ESS (half-dove-half-hawk mixed strategy (0.5,0.5)) starts from a percentage of 0.9 in the population
• Setting is same to the last slide
• ESS can successfully defend the invasion of mutants, although it may not be a completely expelling
• The experiment shows that ESS can be taken to be the set of mixed strategy over some finite set of actions
ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
• Not all NE are ESS
• A strict equilibrium b* is an ESS
• (b*,b*) if a symmetric NE
• and no strategy other than b* is a best response to b*
• A nonstrict equilibrium may not be an ESS
• Mixed NE (1/3, 1/3, 1/3) expected payoff t/3 Can be invaded by any pure strategy
• Receives expected utility t/3 when it encounters MixedNE
• Receives expected t when it encounters another pure strategy
Example: ESS Mixed NE t,t -1,1 1,-1 1,-1 t,t -1,1 -1,1 1,-1 t,t