In a sequential game, a Proposer must offer a split of $10 to a Responder. The Proposer can make one of two offers:

1. A 'lopsided' offer: Proposer gets $9, Responder gets $1.
2. An 'even' offer: Proposer gets $5, Responder gets $5.

The Responder can then either Accept the offer, in which case they get the proposed amounts, or Reject it, in which case both players receive $0. Assuming both players are perfectly rational and seek only to maximize their own monetary payoff, what is the Proposer's best strategy?

Analyzing a Sequential Game

Rationality vs. Reality in a Sequential Game

In a sequential game, a Proposer first chooses how to split $10 with a Responder. The Proposer can make an 'Even' offer ($5 for Proposer, $5 for Responder) or a 'Lopsided' offer ($8 for Proposer, $2 for Responder). After seeing the offer, the Responder can either 'Accept' or 'Reject' it. If the Responder rejects, both players receive $0. Match each component of this game with its correct structural description.

Consider a sequential game where a Proposer offers a Responder a split of $20. The Proposer can offer either a 'Lopsided' split ($18 for the Proposer, $2 for the Responder) or an 'Even' split ($10 for each). The Responder can then accept or reject the offer. If the offer is rejected, both players receive $0. Assume it is known that this particular Responder will always reject any offer that gives them less than 25% of the total amount. Given this information, the Proposer's best strategy to maximize their own payoff is to make the 'Lopsided' offer.

Analyzing a Responder's Decision in a Sequential Game

A Proposer and a Responder are playing a sequential game where the Proposer first makes an offer on how to split a sum of money, and the Responder then chooses to either accept or reject that offer. Arrange the following events in the correct chronological order as they would unfold in this game.

In a sequential bargaining game, a Proposer makes an offer to a Responder on how to split $12. The Proposer can offer a ($6, $6) split or a ($10, $2) split. The Responder can then accept or reject the offer. A unique rule in this specific game is that if the Responder rejects any offer, they must pay a $1 penalty, while the Proposer receives $0. In the game tree that models this interaction, the final payoff for the Responder, if they choose to reject the ($10, $2) offer, is ____.

Modifying a Game Tree Structure

A sequential game is described as follows: A Proposer decides how to split $10, choosing between 'Offer A' ($7 for Proposer, $3 for Responder) and 'Offer B' ($5 for each). The Responder then sees the offer and can either 'Accept' or 'Reject'. If the offer is rejected, both players receive $0. An analyst attempts to model this with a game tree, but makes one error. Review the analyst's described tree and identify the mistake:

• The game starts with the Proposer's decision between 'Offer A' and 'Offer B'.
• Following 'Offer A', the Responder can 'Accept' (payoff: $7, $3) or 'Reject' (payoff: $0, $0).
• Following 'Offer B', the Responder can 'Accept' (payoff: $5, $5) or 'Reject' (payoff: $3, $7).