Chapter 8, Problem 8.1P

a

To determine

To find:

Pure Strategy Nash equilibria.

Explanation of Solution

In a two-player game, $(s_1^{*},s_2^{*})$ is Nash equilibrium, if $s_1^{*}$ and $s_2^{*}$ are mutual best responses against each other, that is,

$u_1(s_1,s_2^{*})\ge u_1(s_1,s_2^{*})forall{s}_1\in S_1$

$u_2(s_2^{*},s_1^{*})\ge u_2(s_2,s_1^{*})forall{s}_2\in S_2$

Solve the following game for pure strategy Nash equilibria:

Player 1	Player 2
		D	E	F
	A	7,6	5,8	0,0
	B	5,8	7,6	1,1
	C	0,0	1,1	4,4

To find the pure strategy Nash equilibria,one will use the underlining the “best response payoffs” method.

Step 1:

Underline the payoffs corresponding to player 1’s best responses. Player 1’s best response when Player 2 plays strategy D is A; one should underline the payoff corresponds to it. Player 1’s best response when Player 2 plays strategy E is B; one should underline the payoff corresponds to it. Player 1’s best response when Player 2 plays strategy F is C; one should underline the payoff corresponds to it. The matrix will be as follows:

Player 1	Player 2
		D	E	F
	A	7,6	5,8	0,0
	B	5,8	7,6	1,1
	C	0,0	1,1	4,4

Step 2:

One should follow the same procedure for Player 2’s responses. One should underline the payoffs corresponding to player 2’s best responses. Player 2’s best response when Player 1 plays strategy A is E; one should underline the payoff corresponds to it. Player 2’s best response when Player 1 plays strategy B is D; one should underline the payoff corresponds to it. Player 2’s best response when Player 1 plays strategy C is F; one should underline the payoff corresponds to it. The matrix will be as follows:

Player 1	Player 2
		D	E	F
	A	7,6	5,8	0,0
	B	5,8	7,6	1,1
	C	0,0	1,1	4,4

Step 3:

Now, one should look for the box where the responses of both the Players are underlined. It is the cell (C,F). This box corresponds to Nash equilibrium The given payoff is (4,4).

Economics Concept Introduction

Introduction:

Nash equilibrium is a stable state in which different participants interact each other, in which no participant gains unilaterally, if strategy of other remains unchanged.

b)

To determine

To find:

Mixed strategy Nash equilibrium.

Explanation of Solution

One should have to find the mixed strategy. Nash equilibrium for the firdt two strategies of both the players.

Player 1	Player 2
		D	E
	A	7,6	5,8
	B	5,8	7,6

When a player doesnot have a dominant strategy, she plays a mixed strategy. Here, to get the mixed strategy, Nash equilibrium one should assume that Player 1 plays the strategy A with probability p and strategy B with probability (1-p). Player 2 plays the strategy D with probability q and strategy E with probability (1-q).

Step 1:

Here, the expected payoff of player 1 for strategy A is given by multiplying each of the payoffs corresponding to S by their respective probabilities and then summing them over. This way the expected payoff from strategy is:

$\begin{array}{l}E(A)=7p+5(1-p)\\ =2p+5\end{array}$

The expected pay off from strategy B is:

$\begin{array}{l}E(B)=5p+7(1-p)\\ =-2p+7\end{array}$

These expected payoffs must be equal. Therefore:

$\begin{array}{l}4p=2\\ p=\frac{1}{2}\end{array}$

Therefore, player 2 plays both of his strategy with equal probability of ½.

Step 2:

The expected payoff from strategy D is:

$\begin{array}{l}E(D)=6q+8(1-q)\\ =-2q+8\end{array}$

The expected payoff from strategy B is:

$\begin{array}{l}E(B)=8q+6(1-q)\\ =2q+6\end{array}$

These expected payoffs must be equal.

$\begin{array}{l}4q=2\\ q=1/2\end{array}$

Therefore, player 2 plays both of his strategy with equal probability of ½

Hence, the mixed strategy Nash equilibrium for the player 1 and player 2 is (0.5, 0.5)

Economics Concept Introduction

Introduction:

Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.

c)

To determine

To ascertain:

Player’s expected payoffs

Explanation of Solution

The expected payoff of the player for a given strategy in a mixed strategy game is given by summing over the actual probability multiplied by their respected probability.

In pure strategy equilibrium of the game described above is (4,4). That is the payoff of player A is 4 and that of B is also 4.

If player 1 choses strategy B, then the player 2 will play either of the strategy D or E with probability 0.5. Then for strategy A the expected payoff of player 1 is:

$(0.5\times 7)+(0.5\times 5)=6$

For player 2:

If player 2 choses strategy E, then the player 1 will play either of the strategy A or B with probability 0.5. Then for strategy E, the expected payoff of player 2 is:

$(0.5\times 8)+(0.5\times 6)=7$

Economics Concept Introduction

Introduction:

Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.

d)

To determine

To ascertain:

Extensive form of the game.

Explanation of Solution

The extensive form of a game corresponds to the game tree; where the action proceeds from left to right. The first move in this game belongs to player 1; he must chose whether to pay strategy A,B OR C. Then player 2 makes his decision. Payoffs are given at the end of the tree.

Economics Concept Introduction

Introduction:

Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.