Question

The mixed-strategy Nash equilibrium for the Battle of the Sexes game described in Example 10.4 may depend on the numerical values of the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by

Short answer

Answer: In the given Battle of the Sexes game, the Nash equilibrium in mixed strategies is \(p^* = \frac{1}{2}\) for Player 1 (row player) and \(q^* = \frac{1}{K+1}\) for Player 2 (column player). Player 1's equilibrium strategy is independent of the value of \(K\), always choosing both strategies with equal probability. However, Player 2's equilibrium strategy depends on the value of \(K\), with the probability of choosing the first strategy decreasing as \(K\) increases, making them more likely to select the second strategy.

Step-by-step solution

Define the Battle of the Sexes game and its payoff matrix

The Battle of the Sexes game is a two-player game with two possible strategies for each player. The players are often referred to as Player 1 (the row player) and Player 2 (the column player). The payoff matrix \(A\) for the row player, in this case, is given by: $$ A = \begin{pmatrix} K & 0 \\ 0 & 1 \end{pmatrix} $$ Let \(p\) be the probability of Player 1 choosing the first strategy, and \(1-p\) be the probability of Player 1 choosing the second strategy. Similarly, let \(q\) be the probability of Player 2 choosing the first strategy, and \(1-q\) be the probability of Player 2 choosing the second strategy. Our goal is to find the mixed-strategy Nash equilibrium by finding \(p^*\) and \(q^*\) that satisfy the equilibrium conditions.

Calculate the best response functions

In order to find the Nash equilibrium in mixed strategies, we need to find the best response functions for both players. These functions tell us what each player should optimally do, given the other player's strategy. In this case, we need to find Player 1's best response to Player 2's strategy \(q\), denoted by \(BR_1(q)\), and Player 2's best response to Player 1's strategy \(p\), denoted by \(BR_2(p)\). Given Player 2's strategy \(q\), Player 1's expected payoff from playing the first strategy is \(Kq\), and the expected payoff from playing the second strategy is \(1 - q\). The best response function for Player 1, \(BR_1(q)\), is given by: $$ BR_1(q) = \begin{cases} 1, & \text{if } Kq > 1 - q \\ 0, & \text{if } Kq < 1 - q \\ p, & \text{otherwise} \end{cases} $$ Similarly, given Player 1's strategy \(p\), Player 2's expected payoff from playing the first strategy is \(p\), and the expected payoff from playing the second strategy is \(1 - p\). The best response function for Player 2, \(BR_2(p)\), is given by: $$ BR_2(p) = \begin{cases} 1, & \text{if } p > 1 - p \\ 0, & \text{if } p < 1 - p \\ q, & \text{otherwise} \end{cases} $$

Find the Nash equilibrium in mixed strategies

It can be seen from the best response functions for both players that a Nash equilibrium in mixed strategies occurs when neither player has an incentive to change their mixed strategy if the opponent keeps their strategy unchanged. This is true when both best response functions hold simultaneously, i.e., when \(BR_1(q^*) = p^*\) and \(BR_2(p^*) = q^*\). When we look at the best response functions, we can see the Nash equilibrium in mixed strategies occurs when: $$ Kq^* = 1 - q^* \text{ and } p^* = 1 - p^* $$ From these conditions, we can find the equilibrium strategies for both players: $$ q^* = \frac{1}{K+1} \text{ and } p^* = \frac{1}{2} $$

Analyze the dependence on \(K\)

From our solution, we can see that the mixed-strategy Nash equilibrium depends on the value of \(K\) as follows: - For Player 1 (row player), their equilibrium probability of playing the first strategy, \(p^*\), is always \(\frac{1}{2}\), independent of the value of \(K\). - For Player 2 (column player), their equilibrium probability of playing the first strategy, \(q^*\), depends on \(K\) through the expression \(\frac{1}{K+1}\). As \(K\) increases, \(q^*\) decreases, making Player 2 more likely to choose the second strategy. In conclusion, the mixed-strategy Nash equilibrium in the Battle of the Sexes game depends on the value of \(K\). Specifically, the row player's equilibrium strategy is always to choose both strategies with equal probability, while the column player's equilibrium strategy depends on the value of \(K\). As \(K\) increases, the column player becomes more likely to choose the second strategy.

Key concepts

Battle of the Sexes game

The Battle of the Sexes game encapsulates a classic scenario in game theory where two players, often called Player 1 and Player 2, face a decision problem with interdependent payoffs. Each player has a set of two possible strategies, and the game mirrors situations where two parties wish to cooperate but have different preferences over the outcome, such as deciding on a joint activity.
A typical story used to describe this game involves a couple that wants to spend the evening together, but one prefers to attend a sports event while the other prefers a concert. Their satisfaction depends on whether they can agree and act together, despite their individual preferences. The dilemma encourages exploring mixed strategies, where players might opt for a balance between personal preference and predictive coordination.
In summary, the Battle of the Sexes game illustrates how players can strategically mix their strategies to find a compromise in situations where aligning decisions is beneficial for both, despite differing individual priorities.

Payoff matrix

In the context of game theory, a payoff matrix is a table that describes the potential outcomes of a strategic interaction between players. Each cell of the matrix reflects the payoff, which is the benefit or loss, that a player gains depending on the chosen strategies by themselves and the opponent.
In the Battle of the Sexes, the payoff matrix is used to represent the preferences of both players. For example, a typical payoff matrix with Player 1 as the row player and Player 2 as the column player might look like this:

• If both attend the sports event, Player 1 gets a higher payoff because they prefer this option.
• If both attend the concert, Player 2 receives a higher payoff for similar reasons.
• If they attend different events, both receive lower payoffs.

Payoff matrices reveal valuable insights into possible strategies each player might adopt, making it easier to analyze scenarios leading to either cooperative outcomes or conflicts.

Best response functions

Best response functions are crucial tools in game theory analysis, representing a player's optimal strategy given the other player's choice. These functions help players determine their best possible action in reaction to the expected decision of their counterpart.
For example, if Player 1 expects Player 2 to attend the sports event, Player 1's best response might be to join as well if that maximizes their own payoff. In these calculations, each player considers the opponent’s actions and then decides their optimal strategy based on expected payoffs.
The best response functions, denoted by symbols like \(BR_1(q)\) and \(BR_2(p)\), are mainly used to pinpoint Nash equilibria in games. By solving these functions, players can identify mixed strategies that might seem counterintuitive – like randomizing their choices – but actually lead to rational decision-making.

Game theory analysis

Game theory analysis is an essential framework for understanding strategic interactions in which multiple decision-makers affect each other's outcomes. It provides a systematic approach to examining scenarios involving competition and cooperation.
In the Battle of the Sexes game, game theory helps players evaluate possible strategies using mixed-strategies Nash equilibrium. Players calculate equilibrium conditions where neither has an incentive to deviate from their chosen strategies, given what they expect from the other. This often involves balancing between decisions for higher individual payoff versus achieving a successful coordination with another player.
By analyzing various games through the lens of game theory, players can predict outcomes, strategize effectively, and sometimes achieve more optimal results, even in complex interdependent scenarios. Thus, game theory not only clarifies potential strategies and outcomes but also enhances decision-making through mathematical rigor and logical consistency.