Question

Players \(A\) and \(B\) have found \(\$ 100\) on the sidewalk and are arguing about how it should be split. A passerby suggests the following game: "Each of you state the number of dollars that you wish \(\left(d_{d}, d_{B}\right) .\) If \(d_{A}+d_{B} \wedge 100\) you can keep the figure you name and I'll take the remainder. If \(d_{A}+d_{B}>100,\) I'll keep the \(\$ 100 . "\) Is there a unique Nash equilibrium in this game of continuous strategies?

Short answer

Answer: The unique Nash equilibrium of the game is (50, 50), where both players A and B choose to take $50 each.

Step-by-step solution

Understand the concepts and constraints

In this game, players A and B need to choose the amounts \((d_A, d_B)\) such that \(d_A + d_B \leq 100\), otherwise the passerby takes the entire amount. The Nash equilibrium will be a pair of amounts \((d_A^*, d_B^*)\) such that no player can be better off by changing their strategy, given the other player's strategy.

Setup players' utility functions

The objective of both players is to maximize their utilities. Let's assume that the utility function of player A is \(U_A(d_A, d_B) = d_A\), and the utility function of player B is \(U_B(d_A, d_B) = d_B\). Both players want to maximize their respective utility functions, given the other player's strategy.

Find the Best Response Functions

To find the Nash equilibrium, we need to find the best response functions (BRF) for both players. The BRFs are the amounts that each player will choose to maximize their utility, given the other player's amount. For player A, the BRF is given by the following equation: \(BRF_A(d_B) = \max_{d_A} U_A(d_A, d_B) = \max_{d_A} d_A\) Subject to the constraint: \(d_A + d_B \leq 100\) Similarly, for player B, the BRF is given by the following equation: \(BRF_B(d_A) = \max_{d_B} U_B(d_A, d_B) = \max_{d_B} d_B\) Subject to the constraint: \(d_A + d_B \leq 100\)

Solve for the Nash equilibrium

The Nash equilibrium can be found by solving the BRFs for both players: \((d_A^*, d_B^*) = (BRF_A(d_B^*), BRF_B(d_A^*))\) Using the constraint \(d_A + d_B \leq 100\), we can rewrite the BRFs as follows: \(BRF_A(d_B^*) = 100 - d_B^*\) \(BRF_B(d_A^*) = 100 - d_A^*\) Now, substituting the BRFs for equilibrium amounts \((d_A^*, d_B^*)\), we get: \((d_A^*, d_B^*) = (100 - d_B^*, 100 - d_A^*)\) Since the game is symmetric, i.e., both players face the same constraints and have the same objective, the Nash equilibrium will have both players choosing the same amounts: \(d_A^* = d_B^* = \frac{100}{2} = 50\) Thus, the unique Nash equilibrium of the game is \((d_A^*, d_B^*) = (50, 50)\).

Key concepts

Continuous Strategies

A key concept in this game is the use of continuous strategies. Unlike games where players have a finite set of choices (such as rock, paper, scissors), here players have a continuous range of possible strategies. Players choose a dollar amount they wish to claim, which can be any number between 0 and 100. This allows for infinite possible choices, making the analysis slightly more complex.

With continuous strategies, both players can adjust their strategies in small increments. This means that there isn't a limited set of outcomes; instead, players can explore the numerous possible interactions based on their decisions.

• The range of strategic choices is continuous, not discrete.
• Players can fine-tune their demands to best suit their strategy.
• The analysis of the game often involves calculus and algebra to handle the infinite choices.

Understanding continuous strategies in game theory requires examining the chosen strategy's impact on the payoff, which leads us to further explore utility maximization.

Utility Maximization

Utility maximization is a core component of this particular game scenario. In game theory, utility functions represent a player's preferences and indicate how players value different outcomes. Here, both players, A and B, aim to maximize the amount they keep. Player A's utility function is simply the amount they claim, noted as \(U_A(d_A, d_B) = d_A\). Similarly, player B's function is \(U_B(d_A, d_B) = d_B\).

The goal for each player is to choose their strategy such that their utility is maximized, given the strategy of the other player. The game is structured so that unless both players' claims total 100 dollars, there could be a way for one or both to increase their claim without the passerby taking all the money. This search for the optimal strategy, subject to the constraint \(d_A + d_B \leq 100\), is at the heart of utility maximization.

• Utility represents how desirable a particular outcome is for a player.
• Each player chooses an amount to maximize their utility, within the constraints.
• Utility maximization leads players to the best response functions, pivotal in determining the equilibrium.

Players are driven to maximize utility while being mindful of the potential penalties for overreaching, thus balancing greed and strategic negotiation.

Game Theory

Game theory forms the foundation of analyzing strategic situations like this. It frames and analyzes situations where players make interdependent decisions, as seen in this dollar-splitting scenario. At its core, game theory focuses on predicting and explaining choice behaviors, understanding interactions among competitive agents, and finding stable solutions, known as equilibria.

In our scenario, the Nash equilibrium is the central concept from game theory. It happens when each player has chosen a strategy and cannot benefit by changing it unless the other player changes theirs too. With both players having continuous strategies and aiming for utility maximization, the Nash equilibrium shows a stable strategy pair where both claims total exactly 100 dollars, split evenly.

• Players make decisions based on their expectations of the other player's strategy.
• A Nash equilibrium is a strategy set where no player can gain by unilaterally changing their strategy.
• Game theory helps explain rational decision-making and strategic interactions among players.

Understanding game theory helps elucidate why mutually beneficial decisions are reached and why they form the backbone of strategic thinking in competitive settings.