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) Player 1's best-response: \( x = \frac{8}{y+2} \), Player 2's: \( y = \frac{8}{x+2} \). (b) Nash equilibrium: \((x, y) = (2, 2)\). (c) Largest strategy set for Player 1: \([0, 3]\).

Step-by-step solution

Understand the Payoff Functions

The given payoff functions are \( u_1(x, y) = \frac{16x}{y+2} - x^2 \) for player 1 and \( u_2(x, y) = \frac{16y}{x+2} - y^2 \) for player 2. These functions represent how a player's payoff depends on their own choice and the choice of the other player.

Setup First Order Conditions for Best Response

To find the best-response function, we differentiate each payoff function with respect to the player's own strategy. For player 1, we partially differentiate \( u_1(x, y) \) with respect to \( x \), yielding \( \frac{\partial u_1}{\partial x} = \frac{16}{y+2} - 2x \). Similarly, for player 2, differentiate \( u_2(x, y) \) with respect to \( y \), yielding \( \frac{\partial u_2}{\partial y} = \frac{16}{x+2} - 2y \).

Solve for Best Response Functions

Set the derivative from step 2 equal to 0 for optimization. For player 1, \( \frac{16}{y+2} = 2x \), solving gives player 1's best-response function: \( x = \frac{8}{y+2} \). For player 2, \( \frac{16}{x+2} = 2y \), solving gives player 2's best-response function: \( y = \frac{8}{x+2} \).

Find the Nash Equilibrium

To find the Nash equilibrium, set the best-response functions equal to each other: \( x = \frac{8}{y+2} \) and \( y = \frac{8}{x+2} \). Solving these simultaneously, assume \( x = y \). Substituting into either equation gives \( x = \frac{8}{x+2} \). Solving \( x(x+2) = 8 \), we expand and use the quadratic formula to find solutions. The solutions to the quadratic are \( x = 2 \) and \( x = -4 \). Given \( x \in [0,6] \), \( x = 2 \) and accordingly \( y = 2 \).

Discuss Strategy Space When Rationality is Not Common Knowledge

When rationality is known but not common knowledge, player 1 will assume player 2 will behave rationally but player 2 knows that player 1 has this assumption. The largest set of rational expectations consistent strategies for player 1 are those which consider player 2's perception of rationality, so \([0, 6]\) remains valid. However, player 1 might stick closer to perceived Nash strategies, leading to \([0, 3]\) based on best responses.

Key concepts

Game Theory

Game theory is an area of mathematics that explores strategic decision-making among different players. It enables us to analyze situations where the outcome for each participant depends not only on their own actions but also on the actions of others. Consider the exercise at hand where two players, player 1 and player 2, each simultaneously choose a number within the range 0 to 6. This scenario is a classic illustration of game theory involving strategic interactions.

In such games, each player's objective is to maximize their own payoff, and the decisions they make can be immensely complex, as they depend on what they believe the other player will do. Understanding how players can best respond to one another within this matrix of choices is pivotal in game theory.

• This includes determining what strategies lead to optimal outcomes (also known as Nash Equilibrium).
• Game theory applications range from economics to evolutionary biology, illustrating its versatility.

Best Response Function

A best response function defines the strategy that yields the highest payoff for a player, given the strategy chosen by the other player involved in the game. In our exercise, the players are looking for a best-response strategy to maximize their payoffs based on the other player's actions.

For example, if player 2 selects a strategy based on their expectation that player 1 picks a certain number, player 2's best-response function calculates the optimal choice. The exercise demonstrates this through the differentiation of payoff functions, resulting in clear functions like:

• For player 1: their best-response function is given by \(x = \frac{8}{y+2}\).
• For player 2: the response is \(y = \frac{8}{x+2}\).

These responses adjust dynamically to the strategies chosen by their opponent, offering a strategy that leads to the best possible outcome given the other's decision.

Rationality

Rationality in game theory refers to the assumption that players will strive to maximize their payoffs and make decisions based on this objective. Each player will choose the strategy that provides them with the highest possible payoff, assuming that the other player will do the same.

In the exercise, rationality plays a crucial role when determining optimal strategies, especially when considering the possibility of no external coordination. When players understand that the others are rational but don't have a mutual understanding of each other's rationality, the range of potential strategies broadens significantly. This scenario illustrates how rational thinking leads to strategic forecasts about the opponent's moves, narrowing down the most plausible set of strategies.

• Rationality ensures that each choice logically relates to maximizing utility.
• It doesn't just involve instantaneous gains but also considers future implications.

Payoff Function

A payoff function is a mathematical representation that assigns a numerical value to each possible outcome a player can experience in the game. This function reflects the utility or satisfaction a player derives from particular results.

In our example, player 1's payoff is represented as \(u_1(x, y) = \frac{16x}{y+2} - x^2\), while player 2's payoff is \(u_2(x, y) = \frac{16y}{x+2} - y^2\). These functions show how changes in their own and the other player's choices impact their payoff.

• The payoff function guides players in selecting their strategy, aiming at outcomes that offer the highest payoff.
• By modeling choices through payoffs, players can evaluate potential outcomes and make informed decisions.

These functions are vital, as they form the basis for calculating strategies like the Nash Equilibrium, allowing players to consider the most beneficial response given various scenarios by the opponent.