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Beyond Nash Equilibrium Correlated Equilibrium and Evolutionary Equilibrium Jie Bao 2003-05-05 Iowa State University
Equilibrium in Games ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Correlated Equilibrium ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example : BoS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],? 1,2 1,0 Stravinsky 0,1 2,1 Bach Stravinsky Bach
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. ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
CE: formal definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
CE contains Mixed NE ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],NE CE Mixed NE
Convex Combination of CE->CE ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example ,[object Object],[object Object],[object Object],[object Object],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 ,[object Object],[object Object],[object Object],[object Object],[object Object]
Notes about CE ,[object Object],[object Object],[object Object]
Evolutionary Equilibrium ,[object Object],[object Object],[object Object],Selfish Selfish Selfish Selfless Selfless Selfless Selfless Selfless Selfless Selfish
ESS: Definition: ,[object Object],[object Object],[object Object],[object Object]
Example: Hawk--Dove ,[object Object],[object Object],[object Object],[object Object],(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 ,[object Object],[object Object],[object Object],[object Object],ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
HD Game – ESS being invaded ,[object Object],[object Object],[object Object],[object Object],ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
Not all NE are ESS ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Example: ESS Mixed NE t,t -1,1 1,-1 1,-1 t,t -1,1 -1,1 1,-1 t,t
More about ESS ,[object Object],[object Object],[object Object],[object Object],[object Object]

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BEYOND NASH

  • 1. Beyond Nash Equilibrium Correlated Equilibrium and Evolutionary Equilibrium Jie Bao 2003-05-05 Iowa State University
  • 2.
  • 3.
  • 4.
  • 5. 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
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.
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  • 16. 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'} ;
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  • 20.