SP2019HW5_soln.pdf - Stat 155 Spring 2019 Homework 5...

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Stat 155 Spring 2019Homework 5: SolutionProblem 1 (KP 7.1)Find all Nash equilibria and determine which of the symmetric equilibria are evolutionarilystable in the following games:player IIABplayer IA(4,4)(2,5)B(5,2)(3,3)andplayer IIABplayer IA(4,4)(3,2)B(2,3)(5,5)Solution.First, let’s find the pure equilibria. Leave only the maximal payoff in a column for the first player and in a rowfor the second:(*,*)(*,5)(5,*)(3,3),and(4,4)(*,*)(*,*)(5,5).We see that there is one pure Nash equilibrium in the first game and two in the second. Moreover, they all are symmetric.Next, we search for the fully mixed strategies. The pair of fully mixed strategies is a Nash equilibrium iff the strategiesare both equalizing. Let’s find such equilibria:The first game:4x1+ 2(1-x1) = 5x1+ 3(1-x1)never happens as the left hand side is less than the right hand side.The second game:4x1+ 3(1-x1) = 2x1+ 5(1-x1) =x1= 1/2.Since the game is symmetric, the equalizing strategy of the second player is the same.Note that in both games there is no equalizing pure strategy of any of the players. Thus there are no Nash equilibriawhere the strategy of one of the players is pure and of the other is fully mixed.We’ve just found all the Nash equilibria: in first game there is only one equilibrium((0,1)T,(0,1)T), in the secod gamethere are three:((0.5,0.5)T,(0.5,0.5)T),((0,1)T,(0,1)T),((1,0)T,(1,0)T).Now we need to check, which of the equilibria are evolutionary stable.Recall the definition- the symmetric equilibrium(x, x)is evolutionary stable iff it is a Nash equlibrium and for anypure strategyzs.t.zTAx=xTAxholdszTAz < xTAx, whereAis the payoff matrix of the first player.First, note that all the pure equilibria in this game are stable.Indeed, the conditionzTAx=xTAxnever holds fordifferent pure strategiesx, zin any of the games, as such equation means thatxis an equalizing strategy, and we saw thatthere are no pure equalizing strategies.Thus, we only need to check the equilibrium((0.5,0.5)T,(0.5,0.5)T). Sincex= (0.5,0.5)Tis an equalizing strategy, theconditionzTAx=xTAxholds for any purez, so we need to check if there is such purez

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