Question

Consider the following two-player game. First, player 1 selects a real number \(x\), which must be greater than or equal to zero. Player 2 observes \(x\). Then, simultaneously and independently, player 1 selects a number \(y_{1}\) and player 2 selects a number \(y_{2}\), at which point the game ends. Player 1's payoff is $$ u_{1}=y_{1} y_{2}+x y_{1}-y_{1}^{2}-\frac{x^{3}}{3} $$ and player 2 's payoff is $$ u_{2}=-\left(y_{1}-y_{2}\right)^{2} $$

Short answer

Player 1 should choose \( x = 0 \) and \( y_1 = y_2 = 0 \) for a Nash equilibrium.

Step-by-step solution

Understand the Payoff Functions

First, let's dissect the payoff functions for both players. Player 1's payoff function is given by \( u_{1}=y_{1} y_{2}+x y_{1}-y_{1}^{2}-\frac{x^{3}}{3} \). Player 2's payoff function is \( u_{2}=-\left(y_{1}-y_{2}\right)^{2} \), which implies that player 2 wants \( y_{1} \) to be as close to \( y_{2} \) as possible to minimize the loss - maximizing payoff.

Determine Optimal Choice for Player 2

Player 2 seeks to maximize \( u_{2} = -\left(y_{1} - y_{2}\right)^{2} \). The optimal strategy for player 2 is to choose \( y_2 = y_1 \), because this choice maximizes \( -\left(y_{1} - y_{2}\right)^{2} = 0 \). This makes \( u_{2} = 0 \), the best possible outcome for player 2.

Simplify Player 1's Payoff Given Player 2's Strategy

Since player 2 chooses \( y_{2} = y_{1} \), substitute \( y_{2} = y_{1} \) into player 1's payoff function: \( u_{1} = y_{1}^{2} + x y_{1} - y_{1}^{2} - \frac{x^{3}}{3} \), which reduces to \( u_{1} = x y_{1} - \frac{x^{3}}{3} \).

Determine Optimal Choice for Player 1’s \\(y_1\\)

Player 1 maximizes \( u_{1} = x y_{1} - \frac{x^{3}}{3} \) given \( x \). The first derivative of \( u_{1} \) with respect to \( y_{1} \): \( \frac{\partial u_{1}}{\partial y_{1}} = x \). Since the derivative is constant, any positive value of \( x \) gives player 1 an incentive to choose a large \( y_{1} \); however, in a bounded range or context, player 1 would choose \( y_1 \) as some factor or proportion of \( x \).

Determine Best Response for Player 1 with Respect to \\(x\\)

Consider the impact of \( x \) on the payoff: \( \frac{\partial u_{1}}{\partial x} = y_{1} - x^{2} \). Set this derivative to zero to find critical points: \( y_{1} = x^{2} \). Use assessments like second derivative tests to ensure maximizing behavior, and examine when \( x = 0 \) might make sense if non-positive payoffs are possible due to the \( \frac{x^3}{3} \) term subtracting too much value.

Key concepts

Payoff Function

In a game theory scenario, the payoff function is essential as it defines the outcome or reward a player receives based on their actions and those of their opponents. In this exercise, player 1's payoff function is: \[ u_{1} = y_{1} y_{2} + x y_{1} - y_{1}^{2} - \frac{x^{3}}{3} \]This equation captures how player 1's rewards are determined by both their own choices and those of player 2.
The different terms within the function account for respective gains and losses:

• The term \(y_{1} y_{2}\) represents the interaction between player 1 and player 2's decisions.
  
• The term \(x y_{1}\) reflects the added reward that depends on both player 1's initial choice \(x\) and their second choice \(y_1\).


• The negative \(-y_{1}^{2}\) accounts for any costs or diminishing returns of choosing a higher \(y_{1}\).


• Finally, \(-\frac{x^{3}}{3}\) penalizes large initial selections of \(x\), adding a balancing disincentive for excessive choices.

Player 2's payoff function, given by:\[ u_{2} = -\left(y_{1} - y_{2}\right)^{2} \]is indicative of a strategy focused on minimizing the difference between \(y_{1}\) and \(y_{2}\).
This implies that player 2 benefits when their choice is closely aligned with player 1's final decision.

Optimal Strategy

Determining the optimal strategy involves choosing actions that maximize a player's payoff while considering their opponent's potential decisions. For player 2, the optimal strategy is straightforward:
They should choose \(y_2 = y_1\). By doing so, they ensure their payoff function is maximized:
\[ -\left(y_{1} - y_{2}\right)^{2} = 0 \]This strategy effectively neutralizes any penalty, achieving the best possible result for player 2.
For player 1, deriving an optimal strategy is more intricate:
- Player 1 initially determines \(x\), knowing this will influence both their and player 2's subsequent choices.- After observing player 2's strategy, player 1 must re-evaluate and choose \(y_1\) to maximize their payoff \(u_1\).

As long as player 1 considers the interactive elements within their payoff function and selects \(x\) strategically, they can optimize their outcome based on how \(y_{1}\) combines with the variable \(x\) itself.

Nash Equilibrium

The Nash equilibrium is a critical concept in game theory, representing a stable state where no player can benefit by unilaterally changing their strategy. In this exercise, finding the Nash equilibrium involves understanding how each player's strategies intersect and influence payoffs.

Given player 2's strategy of setting \(y_2 = y_1\), player 1 must consider their strategy to maximize their payoff function. By analyzing the possible outcomes, player 1 seeks values of \(x\) and \(y_1\) that make the system balanced.
- A Nash equilibrium in this context exists when player 1's choice of \(x\) and \(y_1\) leads both players to achieve the highest payoff possible, such that neither player benefits from changing their decision.- For player 1, part of reaching equilibrium involves setting \(y_1 = x^2\) based on the first-order condition maximizing behavior.

This state ensures that both players are selecting strategies optimized with respect to one another, making any unilateral changes disadvantageous.

Player Strategy

Understanding player strategy is vital in predicting outcomes in game theory.
Players make strategic decisions based on potential payoff outcomes and those of their opponents.

• Player 1 has a two-step strategy, where they choose \(x\) and later choose \(y_1\), assessing player 2's behavior. This two-phase approach allows player 1 to adapt and optimize their initial choice.
  
• Player 2's strategy, on the other hand, is reactive. They observe player 1's \(x\) and act optimally by setting \(y_2 = y_1\) to maximize their payoff. This ensures alignment with player 1's final decision.
  

This game involves dynamic strategic interactions, where players both anticipate and react to one another's choices. The emphasis lies in understanding how each move impacts the overall payoff structure. By having clear strategies, each player strives to maximize their outcomes given the decisions at hand.