Q. Suppose you are attempting to buy a used bicycle, and you are bargaining with the owner over the sale price. The bike is worth $200 to you and $100 to the owner (he can sell it to another customer at a later date). Assume that bargaining takes place with alternating offers and that each stage of bargaining (an offer and a response) takes a full day to complete. If no agreement is reached after 4 days of bargaining, then the opportunity for sale disappears and both get zero. Suppose that both you and the current owner discount the future according to a discount factor of δ per day. The seller has allowed you to make the first offer. (Denote the buyer as player 1, the seller as player 2, and denote the selling price by x.)
1) Sketch the extensive form of the game, carefully labelling the players that move and the actions they have available. Add the payoffs for both players taking into account the discount factor δ.
2) What is the backward induction solution of this game? Does the buyer (Player 1) buy the bicycle? If yes, at what price and in which period?
3) Suppose δ is small, say δ < ½. Should the buyer (Player 1) make the first offer or better let the current owner (Player 2) make the first offer? (In which case they still play 4 rounds but starting with the owner making the first proposal)
4) Suppose now there are only 3 days to trade, after which the opportunity for sale disappears and both get zero. What is the backward induction solution of the game where again the buyer (Player 1) is the first to make a proposal. How does it compare to your answer in (3)? Explain.