Question

Consider a game in which, simultaneously, player 1 selects any real number \(x\) and player 2 selects any real number \(y\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=2 x-x^{2}+2 x y \\ &u_{2}(x, y)=10 y-2 x y-y^{2} \end{aligned} $$ (a) Calculate and graph each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibria of the game. (c) Determine the rationalizable strategy profiles for this game.

Short answer

The Nash Equilibrium is (3, 2); this is also the only rationalizable strategy profile.

Step-by-step solution

Determine Player 1's Best-Response Function

To find Player 1's best-response function, we maximize Player 1's payoff function \( u_1(x, y) = 2x - x^2 + 2xy \) with respect to \(x\). Taking the derivative with respect to \(x\), we get \( \frac{du_1}{dx} = 2 - 2x + 2y \). Setting the derivative equal to zero, we solve \( 2 - 2x + 2y = 0 \) for \(x\), yielding \( x = 1 + y \). So, Player 1's best-response function is \( x = 1 + y \).

Determine Player 2's Best-Response Function

Similarly, to find Player 2's best-response function, we maximize Player 2's payoff \( u_2(x, y) = 10y - 2xy - y^2 \) with respect to \(y\). Taking the derivative with respect to \(y\), we have \( \frac{du_2}{dy} = 10 - 2x - 2y \). Setting \( \frac{du_2}{dy} = 0 \), we obtain \( 10 - 2x - 2y = 0 \). Solving for \(y\), we find \( y = 5 - x \). Thus, Player 2's best-response function is \( y = 5 - x \).

Graph the Best-Response Functions

We now have the best-response functions: \( x = 1 + y \) for Player 1 and \( y = 5 - x \) for Player 2. To graph these functions, plot \( x = 1 + y \) as a line with slope -1, intercepting the y-axis at -1. Plot \( y = 5 - x \) as a line with slope -1 intercepting the y-axis at 5. The point where these lines intersect will be the Nash Equilibrium.

Find the Nash Equilibrium

The Nash Equilibrium occurs where the best-response lines intersect. Solving the equations \( x = 1 + y \) and \( y = 5 - x \) simultaneously, substitute \( x = 1 + y \) into \( y = 5 - x \). This yields \( y = 5 - (1 + y) \), simplifying to \( 2y = 4 \), hence \( y = 2 \). Substituting \( y = 2 \) back into \( x = 1 + y \) gives \( x = 3 \). Thus, the Nash Equilibrium is \((x, y) = (3, 2)\).

Determine Rationalizable Strategy Profiles

A strategy profile is rationalizable if each player's strategy is a best response to some belief about the other player's strategy. Given the linear nature of the best-response functions, players will choose strategies along their best-response functions. Thus, any mix where Player 1 chooses \( x = 1 + y \) and Player 2 chooses \( y = 5 - x \) is rationalizable. However, the only consistent mutual best responses as observed in the game is the Nash Equilibrium, so \((3, 2)\) is the only rationalizable strategy profile.

Key concepts

Game Theory

Game Theory is a fascinating field of study that examines strategic interactions between rational decision-makers. It is used to analyze situations where the outcome for each participant, or player, depends not solely on their own actions but also on the actions of others. In our exercise, the players choose numbers, and their rewards, or payoffs, are determined by these choices.
This approach allows us to predict the behavior in competitive scenarios through mathematical models.

• Players: The decision-makers in the game (Player 1 and Player 2 in this exercise).
• Strategies: The set of all possible actions a player can take (Choosing any real number).
• Payoffs: The reward received based on the chosen strategies (Defined by the payoff functions).

Game theory also explores different concepts like Nash Equilibrium, which provides insights into stable strategy outcomes where no player has an incentive to unilaterally deviate. Understanding these models helps to analyze various real-world scenarios such as economics, politics, and social sciences.

Best-Response Function

The best-response function is a key concept in game theory. It outlines a player's optimal strategy in response to the other player's choice. Essentially, it's about finding what action yields the best outcome for a player, assuming the other player's strategy is given.

In our exercise, we calculated the best-response functions for each player:
- For Player 1: Maximizing their payoff function resulted in the best-response function: \( x = 1 + y \). This means that for any strategy \( y \) chosen by Player 2, the optimal \( x \) for Player 1 to choose is \( 1 + y \).
- For Player 2: Their best-response, calculated similarly, is \( y = 5 - x \), indicating how Player 2 should choose \( y \) based on Player 1's \( x \).

• These functions help players determine their most lucrative responses.
• The intersection of these functions often reveals a Nash Equilibrium.

Understanding best-response functions simplifies strategic decision-making in competitive environments.

Rationalizable Strategies

Rationalizable strategies are those strategies that can be logically justified as a best response to some belief about the opponent’s strategy. In other words, they are plausible choices a rational player would pick, assuming others are also making rational choices.

In our specific game, given the linear nature of best-response functions, strategies fall along these lines.

• Player 1 will align their choice of \( x \) with \( x = 1 + y \).
• Player 2 will choose \( y \) based on \( y = 5 - x \).

While various combinations exist, the mutual best outcome observed is the Nash Equilibrium: \((3, 2)\).
In this context, any deviation wouldn't be rational as it doesn't yield a better payoff. The concept of rationalizable strategies, therefore, streamlines possible strategy choices by focusing on logical decisions influenced by perceived rational actions of others.

Payoff Function

The payoff function is a crucial element in game theory, quantifying the benefits each player receives from various strategic choices. It provides a numerical value representing the outcome or reward for choosing specific actions.

In our exercise, the payoff functions are defined as: - For Player 1: \( u_{1}(x, y) = 2x - x^{2} + 2xy \) - For Player 2: \( u_{2}(x, y) = 10y - 2xy - y^{2} \)
These functions describe how much utility each player gains based on their chosen strategy and their opponent’s choice.

• This quantification helps in evaluating the 'goodness' of an action.
• The maximization of these functions allows players to determine their best responses.

Understanding payoff functions is essential to deciphering player's motivations and the strategic landscape of the game.