#GPT-4 Conversation #1102949589251784774

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errant iceBOT
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A seller of an artwork agrees to split the proceeds of the auction with the auctioneer. They agree a bargaining game that works as follows: The auctioneer proposes a division of 100% of the proceeds, then the seller decides whether to accept or reject. If the seller rejects, the seller can propose a division of the proceeds to the auctioneer. The auctioneer can accept or reject this division. If the auctioneer rejects the seller’s proposal, the seller makes another proposal of a division of the proceeds. The auctioneer decides whether to accept or reject this division. If the auctioneer rejects the seller’s second proposal, then the game ends and both the auctioneer and seller are left with payoffs of zero. The auctioneer discounts payoffs in the next period by a, the seller discounts payoffs in the next period by s.

a) Draw the extensive form of this finite sequential game.
b) In terms of the share of the proceeds that each player receives, what are the equilibrium payoffs for each player in this game?

errant iceBOT
# open cipher A seller of an artwork agrees to split the proceeds of the auction with the auct...

a) The extensive form of this finite sequential game can be represented as a tree diagram with the following nodes and decision branches:

  1. Auctioneer (A) proposes a division of proceeds: xA for Auctioneer, and (1 - xA) for Seller (S).
  2. Seller can either Accept (A') or Reject (R') the proposal.
    • If Seller accepts, then the payoffs are (xA, 1 - xA).
    • If Seller rejects, we move to the next node.
  3. Seller proposes a division of proceeds: yS for Seller, and (1 - yS) for Auctioneer.
  4. Auctioneer can either Accept (A') or Reject (R') the proposal.
    • If Auctioneer accepts, then the payoffs are (1 - yS, yS).
    • If Auctioneer rejects, we move to the next node.
  5. Seller proposes another division of proceeds: zS for Seller, and (1 - zS) for Auctioneer.
  6. Auctioneer can either Accept (A') or Reject (R') the proposal.
    • If Auctioneer accepts, then the payoffs are (1 - zS, zS).
    • If Auctioneer rejects, both get a payoff of 0 (0,0).

b) To find the equilibrium payoffs for each player in this game, we can use the concept of subgame perfect equilibrium, and work backwards (backward induction):

  1. In the final node (Seller's second proposal), Auctioneer will accept the proposal if (1 - zS) >= 0, as their payoff from rejection (0) is less than or equal to (1 - zS).
  2. Knowing this, Seller will propose a division in which zS = 1. Thus, the payoffs will be (0, 1) if the game reaches this stage.
  3. In the second node (Seller's first proposal), Auctioneer will accept if (1 - yS) >= a(0), as a(0) is equal to their discounted payoff from rejecting. This inequality implies that Auctioneer prefers the payoff from accepting yS. Since a > 0, the inequality holds for any yS, and Auctioneer will accept any value offered by the Seller.
  4. Knowing this, Seller will propose a division in which yS = 1. Thus, the payoffs would be (0, 1) if the game reaches this stage.
  5. In the first node (Auctioneer's proposal), Seller will accept the proposal if (1 - xA) >= s(1), as s(1)
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is equal to their discounted payoff from rejecting. With this inequality, Seller prefers the payoff from accepting xA to rejecting.
6. Knowing the Seller will reject any offer less than s(1), Auctioneer will propose a division of (1 - s, s).

In this game, the equilibrium payoffs are (1 - s) for the Auctioneer and s for the Seller, where s denotes the discount factor of the Seller.

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