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 \(\left(d_{1}, d_{n}\right) .\) If \(d_{4}+d_{n} \leq 100\) you can keep the figure you name and I'll take the remainder. If \(d_{A}+d_{B} > 100,\) I'll keep the \(\$ 100 .^{\prime \prime}\) Is there a unique Nash equilibrium in this game of continuous strategies?

Short answer

Yes, there is a unique Nash equilibrium in this game. It occurs when both players A and B choose to split the money evenly, with each player naming $50 as their desired amount. In this case, the sum of their demands is equal to $100, and both players keep the amount they named without the passerby taking any money. This strategy ensures that neither player can unilaterally change their strategy and expect a higher payoff, fulfilling the conditions for a Nash equilibrium.

Step-by-step solution

Representation of the problem

We are given player A and player B who found \(100 and need to decide how to split it. Let \)d_A\( and \)d_B$ represent the amounts that player A and player B wish, respectively. The passerby suggests the following rule: - If \(d_A + d_B \leq 100\), then each player keeps the amount they named, and the passerby takes the remainder of the $100 - If \(d_A + d_B > 100\), the passerby takes all the $100. Our task is to determine whether there is a unique Nash equilibrium for this game.

Understanding Nash Equilibrium

A Nash equilibrium is a strategy profile in which no player can unilaterally change their strategy and expect to have a higher payoff. In this case, it would mean that neither player A nor player B could change their named amount (\(d_A\) or \(d_B\), respectively) and expect to get more money.

Setting up the Payoff Functions

For player A, the payoff function would look like this: - \(p_A(d_A, d_B) = d_A\) when \(d_A + d_B \leq 100\) - \(p_A(d_A, d_B) = 0\) when \(d_A + d_B > 100\) For player B, the payoff function would look like this: - \(p_B(d_A, d_B) = d_B\) when \(d_A + d_B \leq 100\) - \(p_B(d_A, d_B) = 0\) when \(d_A + d_B > 100\)

Deducing Optimal Strategies

We will now deduce the optimal strategies for both players A and B. 1. If player A predicts that player B will choose an amount \(d_B \leq 100 - d_A\), then player A's best response is to choose \(d_A = 100 - d_B\), so the passerby doesn't take any money. 2. If player A predicts that player B will choose an amount \(d_B > 100 - d_A\), then player A's best response is to choose \(d_A = 0\), so player B would get nothing, and the passerby takes all the money. 3. Similarly, player B's optimal strategy would be: a. Choose \(d_B = 100 - d_A\) when player A chooses \(d_A \leq 100 - d_B\). b. Choose \(d_B = 0\) when player A chooses \(d_A > 100 - d_B\).

Checking for Nash Equilibrium

Now that we know the optimal strategies for player A and player B, we can check for a Nash equilibrium. 1. If both player A and player B choose \(d_A = 100 - d_B\) and \(d_B = 100 - d_A\) (respectively), then we get \(d_A = d_B = 50\), and since \(d_A + d_B =100\), this satisfies the passerby's condition, and each player keeps the amount they named. This is a Nash equilibrium. 2. If one player chooses their optimal strategy and the other player chooses their second strategy, none of the players will want to unilaterally change their strategy, so no Nash equilibrium exists here. In conclusion, there is a unique Nash equilibrium in this game, which is when player A and player B choose \(d_A = d_B = 50\). This strategy profile results in both players keeping the named amounts and the passerby not taking any money.

Key concepts

Understanding Microeconomic Theory

Microeconomic theory is the study of how individuals and firms make decisions to allocate scarce resources, typically in markets where goods and services are bought and sold. It involves analyzing the behavior of key economic agents, with an emphasis on how these behaviors affect the distribution of resources and the well-being of individuals involved.

In the textbook problem, the concept of resource allocation is illustrated through a game where players A and B decide how to split \(100 they found. The decisions made by each player regarding how much to claim reflect the core principle of microeconomics: individuals act in their self-interest, attempting to maximize their welfare, in this case, the amount of money they end up with.

In relation to the problem, a fundamental lesson from microeconomic theory is the concept of marginality and trade-offs. Here, the decision each player makes carries a potential marginal benefit — the extra amount they could secure — against a marginal cost — the risk that the passerby would take the entire sum if their combined claims exceed \)100.

Game Theory Essentials

Game theory is a framework for understanding strategic interactions among rational decision-makers. It provides tools for analyzing situations where the outcomes depend critically on the actions of multiple agents, each trying to maximize their payoff in the face of uncertainty about the others’ actions.

Considering Nash equilibrium, a fundamental concept in game theory mentioned in the textbook exercise, it occurs when no player can benefit by unilaterally changing their strategy, assuming the other players' strategies remain unchanged. The equilibrium represents a state of balanced expectations where each player's decision is optimal, given the decisions of others.

In our textbook example, the game involves continuous strategies since the amounts players can choose are not limited to a set of discrete options but can range continuously up to \(100. When A and B simultaneously choose an amount that sums to \)100 or less, they are at a Nash equilibrium if neither can improve their outcome by unilaterally changing the amount they claim.

Continuous Strategies in Game Theory

Continuous strategies in game theory involve decisions where players can choose from an entire range of options, as opposed to discrete strategies where choices are limited to specific, isolated alternatives. This type of strategy opens a broad spectrum of possible actions that can be finely tuned in response to the expected strategies of other players.

In the exercise, players A and B can name any monetary amount up to $100, which reflects the continuous nature of their strategies. The passerby's game demonstrates how continuous strategies require a deeper analysis since the infinite possibilities mean the players must consider not just a single response but the best response to any possible choice of the other player.

When examining games with continuous strategies, one must not only pinpoint an exact strategy but also understand how a player's payoffs change with incremental changes in the opponent's strategies. This is closely related to the concept of a 'best response function' where each player's optimal strategy (how much to claim) constantly adjusts as a direct consequence of the strategy pursued by the other player. Hence, the establishment of a Nash equilibrium in such games is a delicate balance, which, once achieved, implies no incentivized move for either player.