Question

1. Assuming all players use pure strategies, describe the best response strategy for each player and determine the pure strategy Nash Equilibrium for each of the following games. a) Coordination Game: In such games the two players benefit from coordinating their actions. The "Prisoner's Dilemma" game we studied in class was an example of such coordination game. Here is another example known as "Battle of the Sexes." In this game a husband and wife trying to decide how they should spend Friday evening. The husband prefers to go to a football game, whereas the wife would prefer to go to an opera. Both would however like to spend the evening together rather than apart. However, they cannot communicate with each other (let's say they left their cell phone at home!) Wife Football Game 2,1 0,0 Opera 0,0 1,2 Football Game Opera Note: The first payoff in each cell indicates the Husband's payoff. Husband b) Zero-Sum Game: In these games the payoff to one player is the loss of the other player. Consider for example how teams score points from penalty kicks in soccer. Team A is making the penalty kick and Team B is defending the goal. If Team A makes the penalty wins the game. On the other hand if Team A does not make the goal with the penalty kick, Team B wins. Team A can kick left or right. Similarly Team B can defend left or right Team B Defend Left Defend Right 1-1 -1,1 Kick Left Kick Right Note: The first payoff in each cell indicates the Team A's payoff -1,1 1-1 Team A c) Games of Coexistence: This is known as the "Hawk-Dove" game. Such games are used in evolutionary biology to predict the evolutionary path of a given species Consider two grizzly bears that come across a piece of food. They have to decide whether they will fight each other for that piece of food or if they will share it. Fighting is the hawkish strategy where one will win and one will lose. Sharing is a dovish strategy. But this works only when both are being dovish. If one is dovish while the other is hawkish, the hawk gets all the food while the dove gets none. Hawk 2-2 Dove 4,0 2,2 Hawk Dove 0,4 Note: The first payoff in each cell indicates the Row player's payoff.

Answer 1

Hi ,

Nash Equilibrium: Nash Equilibrium occurs when no player has any incentive to deviate from his current strategy.

Note 1: We will be using Underline Method to solve for the Nash equilibrium. We will underline the best response for a player given other player's strategy and try to find a steady state where no player has an incentive to move from one strategy to another.

(a) Coordination Game:

		Wife
		F	O
Husband	F	2,1	0,0
	O	0,0	1,2

Suppose Husband decides to play F then Wife's best response is to play F too and get a pay off of 1 rather than getting a pay off of 0 by playing O.

Similarly if Husband decides to play O then Wife's best response is to play O too and get a pay off of 2 rather than getting a pay off of 0 by playing F.

Therefore,

		Wife
		F	O
Husband	F	2,1	0,0
	O	0,0	1,2

On the similar basis, let us find the best responses of Husband.

Suppose Wife decides to play F then Husband's best response is to play F too and get a pay off of 2 rather than getting a pay off of 0 by playing O.

Similarly if Wife decides to play O then Husband's best response is to play O too and get a pay off of 2 rather than getting a pay off of 0 by playing F.

Now,

		Wife
		F	O
Husband	F	2,1	0,0
	O	0,0	1,2

Hence, we can say that there are two pure strategy Nash equilibrium in this game, namely, (F,F) & (O,O) as none of the players gain from anything if they deviate from these strategies.

(b) Zero-Sum Game:

		B
		L	R
A	L	-1,1	1,-1
	R	1,-1	-1,1

As shown in part (a), B's best responses are L and R if A plays L and R respectively.

Similarly A's best responses are L and R for if B plays R and L respectively.

Therefore, using underline method.

		B
		L	R
A	L	-1,1	1,-1
	R	1,-1	-1,1

There is no Nash Equilibrium here.

(c) Coexistence Game:

		B
		H	D
A	H	-2,-2	4,0
	D	0,4	2,2

As we can see, when player A is playing H, B's best response is to play D and when A is playing D, B's best response is to play H.

Similarly, when player B is playing H, A's best response is to play D and when B is playing D, A's best response is to play H.

Therefore,

		B
		H	D
A	H	-2,-2	4,0
	D	0,4	2,2

Hence, there are two nash equilibrium, (H,D) & (D.H).

Thanks. Please let me know in the comments below if you need any clarification.

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