Question

Consider a game in which, simultaneously, player 1 selects a number \(x \in[0,6]\) and player 2 selects a number \(y \in[0,6]\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=\frac{16 x}{y+2}-x^{2} \\ &u_{2}(x, y)=\frac{16 y}{x+2}-y^{2} \end{aligned} $$ (a) Calculate each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibrium of the game. (c) Suppose that there is no social institution to coordinate the players on an equilibrium. Suppose that each player knows that the other player is rational, but this is not common knowledge. What is the largest set of strategies for player 1 that is consistent with this assumption?

Short answer

(a) Best responses: \(x = \frac{8}{y+2}\), \(y = \frac{8}{x+2}\). (b) Nash Equilibrium: \((2, 2)\). (c) Largest set \(x \in [0,6]\) consistent with player's rationality, but exact set needs belief assumptions.

Step-by-step solution

Understand Player 1's Best Response

Calculate player 1's best response given player 2’s choice. The best response is the value of \(x\) that maximizes \(u_1(x, y) = \frac{16x}{y+2} - x^2\) for a given \(y\). Take the derivative of \(u_1\) with respect to \(x\), set it to zero, and solve for \(x\). The derivative is \(\frac{d}{dx}\left(\frac{16x}{y+2} - x^2\right) = \frac{16}{y+2} - 2x\). Set this to zero to find the critical points: \(x = \frac{8}{y+2}\). This is player 1's best response to player 2 choosing \(y\).

Understand Player 2's Best Response

Calculate player 2's best response given player 1’s choice. The best response is the value of \(y\) that maximizes \(u_2(x, y) = \frac{16y}{x+2} - y^2\) for a given \(x\). Take the derivative of \(u_2\) with respect to \(y\), set it to zero, and solve for \(y\). The derivative is \(\frac{d}{dy}\left(\frac{16y}{x+2} - y^2\right) = \frac{16}{x+2} - 2y\). Set this to zero for the critical points: \(y = \frac{8}{x+2}\). This is player 2's best response to player 1 choosing \(x\).

Find the Nash Equilibrium

The Nash equilibrium occurs when each player's strategy is a best response to the other’s. Set the best response functions equal to each other to find \(x\) and \(y\). Equate \(x = \frac{8}{y+2}\) and \(y = \frac{8}{x+2}\). Solving these simultaneously, we substitute \(x = \frac{8}{y+2}\) into \(y = \frac{8}{x+2}\): \(y = \frac{8}{\frac{8}{y+2} + 2}\). Solve to find \(y = 2\). Substitute \(y = 2\) back into \(x = \frac{8}{y+2}\), finding \(x = 2\). Hence, the Nash equilibrium is \((x, y) = (2, 2)\).

Determine Largest Set of Rational Strategies for Player 1

Without a coordinating social institution and with rationality as common knowledge, the largest set of strategies for player 1 includes all rational strategies consistent with player 2 being rational but not common knowledge. This means player 1 could consider a broader range than just the best-response strategy. Player 1 knows the opponent is rational, so one approach could include any \(x\) that satisfies player 1's belief in rationality, consistent with the range of \(x\) in \[0, 6\]. The precise largest set of rational strategies consistent with these beliefs, however, requires more game-theoretic assumptions about belief structures.

Key concepts

Best Response

In game theory, the best response is a fundamental concept. It refers to the strategy that yields the most favorable outcome for a player, given the strategies chosen by the other players.
In our exercise, player 1's payoff is maximized by choosing the best response to player 2's choice. Player 1's utility function is given by \( u_1(x, y) = \frac{16x}{y+2} - x^2 \). To find the best response for player 1:

• Take the derivative of \( u_1 \) with respect to \( x \), which results in \( \frac{16}{y+2} - 2x \).
• Set the derivative equal to zero to find the critical point: \( x = \frac{8}{y+2} \). This expression tells us how player 1 chooses the best response to player 2's choice of \( y \).

Similarly, player 2 determines their best response by maximizing their utility function \( u_2(x, y) = \frac{16y}{x+2} - y^2 \). The procedure is analogous to player 1's:

• Differentiate \( u_2 \) with respect to \( y \), resulting in \( \frac{16}{x+2} - 2y \).
• Set the derivative to zero for the critical point: \( y = \frac{8}{x+2} \).

This formula offers player 2's best response to player 1's choice.

Nash Equilibrium

The Nash equilibrium represents a stable state in a game where no player can gain by unilaterally changing their strategy. Each player's strategy is optimal, given the strategies of the other players. It's like a balance where each player is doing their best in response to the others.
To find the Nash equilibrium for our exercise, equate the best response functions for player 1 and player 2. This means solving the equations:

• For player 1: \( x = \frac{8}{y+2} \).
• For player 2: \( y = \frac{8}{x+2} \).

Substituting one equation into the other allows us to solve simultaneously. In this case:

• Substitute \( x = \frac{8}{y+2} \) into player 2's equation.
• After solving, we find \( y = 2 \).
• Use \( y = 2 \) back in player 1's equation to find \( x = 2 \).

Thus, the Nash equilibrium for this game is \((x, y) = (2, 2)\). Each player is maximizing their payoff, given the other player's strategy, and neither can improve by changing their strategy alone.

Rational Strategies

Rational strategies in game theory imply that players will make decisions that maximize their payoffs, assuming all players are rational. Rationality ensures that players act logically, considering the available information and the likely strategies of their opponents.
In this exercise, players need to decide what numbers to choose to maximize their payoffs based on the choices of the opponent. Both players calculate their best responses, as outlined in the solutions. However, if there is no mechanism to coordinate these choices and not all information is shared, it hinges on rationality but lacks the full assurance of common knowledge.
This means that without certain coordination, player 1 may have to consider a set of strategies that take into account the rational behavior of player 2 without assuming player 2's strategy is known with certainty. The players will consider strategies consistent with rational behavior across a range of possible scenarios.

Simultaneous Game

Simultaneous games are scenarios where players make decisions at the same time, without knowing the choices of their opponents. This type of game models real-world decisions where timing and secrecy in decision-making play a crucial role.
In our game:

• Each player chooses a number between 0 and 6 simultaneously.
• The payoff functions provided determine the outcome for each player based on the chosen numbers.

Because decisions are made simultaneously, each player's strategy must account for the entire set of possible responses from their opponent. This complicates things because each player has to contend with uncertainty regarding the other's actions. The payoff ultimately depends on assumptions about rationality and the best responses, which are principles grounded in game theory's analysis of strategic interaction.