Question

4.Consider a modified version of the divide the dollar game in problem (3) in which player 2 can make a counteroffer if she does not accept player 1’s offer. After player 2 makes her counteroffer –if she does– player 1 can accept or reject the counteroffer. As before, if there is no agreement after the two rounds of offers, neither player gets anything. If there is an agreement in either round then each player gets the amount agreed to.Represent the game in extensive form using a game tree. Again, you must assign payoffs appropriately at the terminal nodes.

3. (Because the 2 questions are related)- Consider the following game of divide the dollar. There is a dollar to be split between two players. Player 1 can make any offer to player 2 in increments of 25 cents; that is, player 1 can make offers of 0 cents, 25 cents, 50 cents, 75 cents, and $1. An offer is the amount of the original dollar that player 1 would like player 2 to have. After player 2 gets an offer, she has the option of either accepting or rejecting the offer. If she accepts, she gets the offered amount and player 1 keeps the remainder. If she rejects, neither player gets anything. Represent the game in extensive form using a game tree. Note that here you must assign payoffs appropriately at the terminal nodes.

Answer 1

Since Player 2 makes the last counter offer, he will put up an offer of $1 for himself and 0 for player 1.

If Player 1 rejects, neither gets anything. So it is highly probable that Player 1 accepts the offer in spite of him not getting anything. If he rejects, the net payoff will be 0 to both. However, if he accepts, the net payoff will be $1.

Seeing this scenario, in the very first stage, Player 1 can propose $0 for himself and $1 for Player 2 which Player 2 will accept and that will be the end of the game.

Or player 1 can propose $1 for himself and $0 for player 2 which Player 2 will reject and the game will continue. So irrespective of what Player 1 suggests, since there is no discounting, so the game will continue till the last stage. The last person to make the offer gets $1 and the other gets $0. This will be the outcome irrespective of what happens in the stages proceeding the last stage.

If you found this helpful, please rate it so that I can have higher earnings at no extra cost to you. This will motivate me to write more.