Chapter 5, Problem 5.6P

a)

To determine

The mixed strategy Nash equilibrium

Answer to Problem 5.6P

In the mixed strategies of Nash Equilibrium, Wife is playing with probability (ballet) 2/3 and (boxing) 1/3. Husband is playing with probability (ballet) 1/3 and (boxing) 2/3.

Explanation of Solution

In the study of game theory, the payoff is the numeric value that is involved with a possible outcome of a game. It represents the motivation of the players. A Strategy is the plan of action that provides the best payoff in a game.Nash equilibrium is one of the strategies and solutions for games.

If all the playoffs are doubles, the table of battle of sexes will be,

	Husband
Wife		Ballet	Boxing
	Ballet	4, 2	0, 0
	Boxing	0, 0	2, 4

Wife prefers the Ballet having the probability of w

  $\mathrm{Pr}obability(Ballet)=4w+(1-w)0=4w$

Wife prefers the boxing having the probability of (1-w)

  $\mathrm{Pr}obability(Boxing)=0w+(1-w)2=2-2w$

Equating both the equation, we get,

  $\begin{array}{l}4w=2-2w\\ \Rightarrow 6w=2\\ \Rightarrow w=\frac{1}{3}\end{array}$

Husband prefers the Ballet having the probability of h

  $\mathrm{Pr}obability(Ballet)=2h+(1-h)0=2h$

Husband prefers the boxing having the probability of (1-h)

  $\mathrm{Pr}obability(Boxing)=0h+(1-h)4=4-4h$

Equating both the equation, we get,

  $\begin{array}{l}2h=4-4h\\ \Rightarrow 6h=4\\ \Rightarrow h=\frac{4}{6}=\frac{2}{3}\end{array}$

From the above calculation, doubling of payoff never changes the mixed strategies of Nash Equilibrium. Wife is playing with probability (ballet) 2/3 and (boxing) 1/3. Husband is playing with probability (ballet) 1/3 and (boxing) 2/3.

Economics Concept Introduction

Introduction: The game theory is preferable to oligopoly for better understanding. It is the detailed study of interactions between the players, business firms. The aim of this strategic decision is to deduce the responses to actions.

b)

To determine

The mixed strategy Nash equilibrium

Answer to Problem 5.6P

In the mixed strategies of Nash Equilibrium, Wife is playing with probability (ballet) 1/5 and (boxing) 4/5. Husband is playing with probability (ballet) 2/3 and (boxing) 1/3.

The new diagram is shown.

Explanation of Solution

In the study of game theory, the payoff is the numeric value that is involved with a possible outcome of a game. It represents the motivation of the players. A Strategy is the plan of action that provides the best payoff in a game.Nash equilibrium is one of the strategies and solutions for games.

If all the playoffs are doubles, the new playoff table will be,

	Husband
Wife		Ballet	Boxing
	Ballet	4, 1	0, 0
	Boxing	0, 0	1, 2

Wife prefers the Ballet having the probability of w

  $\mathrm{Pr}obability(Ballet)=4w+(1-w)0=4w$

Wife prefers the boxing having the probability of (1-w)

  $\mathrm{Pr}obability(Boxing)=0w+(1-w)1=1-w$

Equating both the equation, we get,

  $\begin{array}{l}4w=1-w\\ \Rightarrow 5w=1\\ \Rightarrow w=\frac{1}{5}\end{array}$

Husband prefers the Ballet having the probability of h

  $\mathrm{Pr}obability(Ballet)=h+(1-h)0=h$

Husband prefers the boxing having the probability of (1-h)

  $\mathrm{Pr}obability(Boxing)=0h+(1-h)2=2-2h$

Equating both the equation, we get,

  $\begin{array}{l}h=2-2h\\ \Rightarrow 3h=2\\ \Rightarrow h=\frac{4}{6}=\frac{2}{3}\end{array}$

From the above calculation, doubling of payoff never changes the mixed strategies of Nash Equilibrium. Wife is playing with probability (ballet) 1/5 and (boxing) 4/5. Husband is playing with probability (ballet) 2/3 and (boxing) 1/3.

  

Economics Concept Introduction

Introduction: The game theory is preferable to oligopoly for better understanding. It is the detailed study of interactions between the players, business firms. The aim of this strategic decision is to deduce the responses to actions.

c)

To determine

The mixed strategy Nash equilibrium

Answer to Problem 5.6P

In the mixed strategies of Nash Equilibrium, Wife is playing with probability (ballet) 1/4 and (boxing) 3/4. Husband is playing with probability (ballet) 2/3 and (boxing) 1/4.

The new diagram is shown.

Explanation of Solution

In the study of game theory, the payoff is the numeric value that is involved with a possible outcome of a game. It represents the motivation of the players. A Strategy is the plan of action that provides the best payoff in a game.Nash equilibrium is one of the strategies and solutions for games.

If changes in the preferred activity from 0 to 1/2, the new playoff table will be,

	Husband
Wife		Ballet	Boxing
	Ballet	2, 1	1/2, 0
	Boxing	1/2, 0	1, 2

Wife prefers the Ballet having the probability of w

  $\mathrm{Pr}obability(Ballet)=2w+(1-w)1/2=\frac{4w-1-w}{2}$

Wife prefers the boxing having the probability of (1-w)

  $\mathrm{Pr}obability(Boxing)=\frac{1}{2}w+(1-w)1=\frac{w+2-2w}{2}$

Equating both the equation, we get,

  $\begin{array}{l}\frac{4w+1-w}{2}=\frac{w+2-2w}{2}\\ \Rightarrow 8w+2-2w=2w+4-4w\\ \Rightarrow 8w=2\\ \Rightarrow w=\frac{2}{8}=\frac{1}{4}\end{array}$

Husband prefers the Ballet having the probability of h

  $\mathrm{Pr}obability(Ballet)=h+(1-h)0=h$

Husband prefers the boxing having the probability of (1-h)

  $\mathrm{Pr}obability(Boxing)=0h+(1-h)2=2-2h$

Equating both the equation, we get,

  $\begin{array}{l}h=2-2h\\ \Rightarrow 3h=2\\ \Rightarrow h=\frac{4}{6}=\frac{2}{3}\end{array}$

From the above calculation, doubling of payoff never changes the mixed strategies of Nash Equilibrium. Wife is playing with probability (ballet) 1/4 and (boxing) 3/4. Husband is playing with probability (ballet) 2/3 and (boxing) 1/4.

  

The above diagram is the new diagram

Economics Concept Introduction

Introduction: The game theory is preferable to oligopoly for better understanding. It is the detailed study of interactions between the players, business firms. The aim of this strategic decision is to deduce the responses to actions.