### The Battle of the Sexes: Game Theory RepresentationThe game scenario known as "the Battle of the Sexes" can be illustrated with the following payoff matrix:#### Husband vs. Wife| | **Husband: Ballet** | **Husband: Boxing** ||--------------|---------------------|---------------------|| **Wife: Ballet** | $K, 1$ | $0, 0$ || **Wife: Boxing** | $0, 0$ | $1, K$ |#### Explanation:- **Players**: The two players in the game are the Husband and the Wife.- **Strategies**: Each player has two strategies: choosing between attending Ballet or Boxing.- **Payoffs**: - The numbers in the cells represent the payoffs for Wife and Husband respectively. - $K$ is a payoff multiplier for the player who prefers a specific outcome, and it is given that $K \geq 1$.#### Questions:(a) **Pure-strategy Nash Equilibria**: Find all equilibrium points where both players choose a single strategy.(b) **Mixed-strategy Nash Equilibria**: Determine the equilibrium where players mix their strategies with certain probabilities.This classic example in game theory demonstrates the conflict of interest between two players with differing preferences and the possible solutions to reach mutually beneficial decisions.

(a) If the husband chooses battle and wife will choose ballet as pay-off. If the husband chooses boxing wife will choose boxing as pay-off. Likewise, if wife chooses ballet, then husband chooses ballet pay-off. If the wife chooses boxing, then husband chooses boxing as pay-off. Thus, Nash equilibrium is; (Ballet, Ballet) (Boxing, Boxing) Pay-off table is give below. The pay-off of wife when husband choose ballet with probability q and boxing (1-q), thus, from ballet: qK+(1-q) 0 = Kq and from boxing: qK+ (1-q)1= 1-q From equilibrium point, wife indifferent between boxing and ballet, thus, pay-off from ballet is equal to pay-off from boxing, therefore, Kq = 1-qKq+q=1q=11+K Husband play ballet with probability 11+K and boxing 1-11+K =K1+Kq,1-q=11+K,K1+K (b) If the husband indifferent between ballet and boxing when pay-off from ballet is equal to the pay-off from boxing. Thus from ballet: p.1+0(1-p) = p and from boxing: p.0+(1-p)K= (1-P)K Thus, p=(1-p)K P+pK=Kp=K1+KProbability of wife goes for ballet P=K1+K for boxing =1-p=1-k1+k=11+k Thus, mixed strategy Nash equilibrium is given below. p,1-p =K1+K,11+Kq,1-q =11+K,k1+K

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A general form of "the Battle of the Sexes" can be represented by the following figure. Husband Ballet Вoxing Wife Ballet К, 1 0,0 Вoxing 0, 0 1, к Where K 21. (a) Find all pure-strategy Nash equilibria. (b) Find all mixed-strategy Nash equilibria.

A general form of "the Battle of the Sexes" can be represented by the following figure. Husband Ballet Вoxing Wife Ballet К, 1 0,0 Вoxing 0, 0 1, к Where K 21. (a) Find all pure-strategy Nash equilibria. (b) Find all mixed-strategy Nash equilibria.

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### The Battle of the Sexes: Game Theory Representation

The game scenario known as "the Battle of the Sexes" can be illustrated with the following payoff matrix:

#### Husband vs. Wife

| | **Husband: Ballet** | **Husband: Boxing** |
|--------------|---------------------|---------------------|
| **Wife: Ballet** | $K, 1$ | $0, 0$ |
| **Wife: Boxing** | $0, 0$ | $1, K$ |

#### Explanation:
- **Players**: The two players in the game are the Husband and the Wife.
- **Strategies**: Each player has two strategies: choosing between attending Ballet or Boxing.
- **Payoffs**:
- The numbers in the cells represent the payoffs for Wife and Husband respectively.
- $K$ is a payoff multiplier for the player who prefers a specific outcome, and it is given that $K \geq 1$.

#### Questions:
(a) **Pure-strategy Nash Equilibria**: Find all equilibrium points where both players choose a single strategy.

(b) **Mixed-strategy Nash Equilibria**: Determine the equilibrium where players mix their strategies with certain probabilities.

This classic example in game theory demonstrates the conflict of interest between two players with differing preferences and the possible solutions to reach mutually beneficial decisions.

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