Question

You play the following bargaining game. Player \(A\) moves first and makes Player \(B\) an offer for the division of \(\$ 100 .\) (For example, Player \(A\) could suggest that she take \(\$ 60\) and Player \(B\) take \(\$ 40 .\) ) Player \(B\) can accept or reject the offer. If he rejects it, the amount of money available drops to \(\$ 90,\) and he then makes an offer for the division of this amount. If Player \(A\) rejects this offer, the amount of money drops to \(\$ 80\) and Player \(A\) makes an offer for its division. If Player \(B\) rejects this offer, the amount of money drops to 0 Both players are rational, fully informed, and want to maximize their payoffs. Which player will do best in this game?

Short answer

Player A will do best in this game because he gets to make the first offer and the structure of the game allows him to ensure that he gets at least as much as he would in the second stage.

Step-by-step solution

Interpret Player B's Optimal Strategy

We start from the back (or the smallest bargain) to find the optimal strategy. Under the assumption that Player A will reject any offer where Player B gets more than half of the money, Player B should therefore offer Player A the smallest amount that A will accept. Assuming A will not accept less than $40 at the $80 stage, Player B should offer $40 to A and keep $50 for himself.

Determine Player A's Optimal Strategy at the $90 Stage

At the $90 stage, Player A will want to obtain at least as much as they would get at the $80 stage. Knowing that Player B will choose optimally in the next round and offer them $40, Player A should make an offer that Player B will accept but no more. Therefore, Player A should offer Player B $50 and keep $40 for himself.

Determine Player A's Optimal Strategy at the $100 Stage

At the $100 stage, Player A will want to obtain at least as much as they would get at the $90 stage. Therefore, Player A should offer Player B $60 and keep $40 for himself.

Key concepts

Game Theory

Game theory is a mathematical framework that is used to study strategic interactions between different decision-makers, called players. In the context of the bargaining game described in the exercise, game theory helps us understand how Player A and Player B can make decisions to maximize their respective payoffs.

• In game theory, each player considers the possible actions of others to decide their optimal strategy.
• The focus is often on "rational" players, assuming they act logically to achieve the best results according to their preferences.
• Strategic games are represented by their rules, payoffs, and strategies available to the players.

Here, the players need to anticipate each other's actions and potential offers to maximize their earnings from the available sum of money. At each step, they need to weigh up not just the immediate outcome, but also the potential consequences in future rounds.

Nash Equilibrium

The Nash Equilibrium is a key concept within game theory, referring to a state where no player can gain by changing their strategy while the other players keep theirs unchanged. In the bargaining game, the Nash Equilibrium represents the point at which both Player A and Player B have no incentive to alter their offers or responses.

• The equilibrium occurs when both players have selected strategies that maximize their personal gain given the strategy chosen by the other player.
• This condition does not promise maximum possible payoff, but stability in strategy selection, as any deviation would not benefit the deviating player.
• Each player's optimal strategy takes into account the optimal strategies of the other player.

For instance, at the $90 stage, Player A's offering of $50 to Player B is part of the Nash Equilibrium. If Player A attempted to offer less, knowing B's rational response, it would reduce A's final payoff. Similarly, deviating from this would not benefit Player B.

Rational Choice Theory

Rational choice theory is the idea that individuals make decisions by considering all available options and selecting the one that maximizes their utility. This theory assumes that people are rational and act in their best interest. In the bargaining game, both Player A and Player B are rational agents.

• Players will consider future consequences to make informed current decisions.
• This approach assumes individuals have consistent preferences and understand the possible outcomes of their choices.
• In the bargaining game scenario, players understand the diminishing returns of rejecting a deal, thus making offers that are likely to be accepted by the other player.

Given the structured steps in the solution, Player A and Player B consciously make offers that balance between maximizing their immediate and long-term gains. They utilize information about the other's rational responses to each offer—Player B planning to settle for a better share at the $80 step, and Player A preparing offers that still secure a favorable outcome at earlier stages.