Question

[Nash's demand game \({ }^{6}\) ] Compute and describe the Nash equilibria of the following static bargaining game. Simultaneously and independently, players 1 and 2 make demands \(m_{1}\) and \(m_{2}\). These numbers are required to be between 0 and 1 . If \(m_{1}+m_{2} \leq 1\) (compatible demands, given that the surplus to be divided equals 1), then player 1 obtains the payoff \(m_{1}\) and player 2 obtains \(m_{2}\). In contrast, if \(m_{1}+m_{2}>1\) (incompatible demands), then both players get 0 . In addition to describing the set of equilibria, offer an interpretation in terms of bargaining weights as in the standard bargaining solution.

Short answer

The Nash equilibria are all pairs \((m_1, m_2)\) on the line \(m_1 + m_2 = 1\). Players split the surplus exactly, based on their demands.

Step-by-step solution

Identify Strategy Possibilities

In this game, both players will simultaneously choose their demands \( m_1 \) and \( m_2 \), which have to be between 0 and 1. The sum of these demands \( m_1 + m_2 \) can either be less than or equal to 1, or greater than 1. If it's less than or equal to 1, players receive their demands as payoffs. If greater than 1, both get 0. Thus, each player's strategy is any number between 0 and 1.

Determine Best Response by Player 1

Given player 2's demand \( m_2 \), player 1's best response will be to demand \( m_1 = 1 - m_2 \). This ensures the sum \( m_1 + m_2 = 1 \) stays within the boundary allowing both players to get their demands. If \( m_2 \) is more than \( 1 \), player 1 should choose any \( m_1 \) since both players will receive a payoff of 0 regardless.

Determine Best Response by Player 2

Similarly, given player 1's demand \( m_1 \), player 2's best response will be to demand \( m_2 = 1 - m_1 \). This ensures that \( m_1 + m_2 = 1 \), allowing both players to receive their demands. If \( m_1 \) is more than \( 1 \), player 2 can choose any \( m_2 \) as both will receive 0.

Characterizing Nash Equilibrium

A Nash equilibrium occurs when both player's strategies are best responses to each other's strategies. Thus, the Nash equilibria are any pair \((m_1, m_2)\) such that \( m_1 + m_2 = 1 \), with each player choosing demands compatible with this condition. This means the set of Nash equilibria is the entire boundary line of the strategy space, represented as \( (m_1, 1-m_1) \).

Interpretation in Terms of Bargaining Weights

In terms of bargaining weights, the Nash equilibria imply that each player receives a portion of the surplus determined by the combination of their demands such that \( m_1 + m_2 = 1 \). This suggests players split the total surplus according to their demands, much like how bargaining weights determine the division in the Nash bargaining solution, where the surplus is split perfectly according to the demands.

Key concepts

Bargaining Theory

Bargaining theory is a fundamental part of economics and game theory. It explores how individuals or groups negotiate to divide a resource or achieve an agreement that satisfies all parties. In the context of the Nash demand game, bargaining involves two players making simultaneous demands on a shared surplus.
The surplus in this game is a resource or payoff, represented by the number 1. Players must decide on their share without any outside guidance, ensuring that their combined demands do not exceed the available surplus.
In practical terms, bargaining theory helps us understand how parties can collaborate to reach a mutually beneficial outcome. Each player's strategy is shaped by their expectations of what the other will demand, leading to a balance of interests. Through bargaining, players aim to maximize their share while avoiding the risk of leaving the table empty-handed, which happens if demands are incompatible.

Static Game

A static game is a scenario in which all players make their decisions or choose their strategies simultaneously. There is no opportunity to revise decisions based on others' choices, creating a unique strategic environment.
In the Nash demand game, players decide their demands at the same time without knowing what the other has chosen. This simultaneous decision-making is a hallmark of static games, where each player's result hinges on not only their action but the actions of others.
Due to the simultaneous nature, players rely on their expectations and predictions of others' behavior. Therefore, understanding static games involves assessing how players determine their strategies without real-time feedback, often using calculated guesses or past experiences.

Game Theory

Game Theory is a mathematical framework designed for analyzing situations in which multiple players make decisions that affect each other's outcomes. It provides powerful tools for understanding strategic interactions in economics, politics, and other fields.
The Nash demand game is a classic example of using game theory to comprehend how players negotiate and share resources. Each player aims to maximize their payoff, aware that their success depends on their own and their counterpart's decisions. Held within game theory, this game displays a foundational principle: the Nash equilibrium.
In this context, Nash equilibrium represents a stable state where neither player can benefit from deviating unilaterally, given the strategy of the other. Such insight allows players to forecast the outcomes of strategic encounters and adjust their plans accordingly.

Payoffs

Payoffs in game theory are the rewards or outcomes that players receive as a result of their chosen strategies and interactions with other players. Payoffs quantify the satisfaction or utility derived from a particular outcome.
In the Nash demand game, each player's payoff is based directly on their demand, provided their combined demands do not exceed the total surplus available, which is set at 1. This outcome structure creates a boundary condition for payoffs — if both players demand too much, they receive nothing.
The concept of payoffs is central to understanding negotiations and decision-making in game theory. Players aim to choose strategies that maximize their payoffs while avoiding negative scenarios, such as incompatible demands resulting in zero payoffs. Thus, analyzing payoffs allows players to strategically plan and refine their approaches in engaging with others.