Canceled-Class Assignment

Please answer the following questions and send the answers by email to me until Monday, September 29.

Please read also the Shubik Auction page. I will have a question on that in the test on Tuesday to check whether you have read it.

The questions refer to this auction game, where you can and should simulate the game, but unfortunately you have to play both roles. The payoff is the sum of money left after the auction and value of the pictures obtained. The two players are either totally neutral to each other, or slightly hostile. Note that in both cases the players don't care what the other gets as payoff, they just want to maximize their own payoff. But in the end, if a player sees that he or she will not get a certain painting, he or she may think "Ok, you get it, but I make you pay a high price for it" in the slightly hostile version.

Please try to answer these questions (1, 2, 3a, 3b, 4a, 4b, 5a, 5b, 6, 7) as precisely as you can. Tell me exactly how far both will bid on the first painting and who gets it, and what will happen with the second one, and also the payoffs for both players. Play the game repeatedly to get a feeling for it.

1. What would happen if the first picture is worth $7500 and the second one only $3500, and both players have initially $4000? Which player will get higher payoff? How much will each get in the slightly hostile case?
2. What would happen if the first picture is worth $7500 and the second one only $3500, and both players have initially $5000? Which player will get higher payoff? How much will each get in the slightly hostile case?
3. What would happen if the first picture is worth only $3500 and the second one as much as $7500, and both players have initially $4000? Which player will get higher payoff? How much will each get?
   1. In the neutral case
   2. Will the slightly hostile case be any different? How?
4. You may use the graph to the right for your analysis (where we assume, quite reasonably, I would say, that Beth will not bet $8000 for the first painting). It doesn't show the full game tree, since it is too large, but just the "pruned" one for the auction of the first painting. The nine vertices that look like end vertices are not really end vertices, but rather the beginning of the auction of the second painting. In the five vertices labeled by "Ann", A has the first move for the second painting (beting $1000 or not), and owns presently still all money, since she wasn't successfull with the first painting. At these five vertices except the second one, Beth owns now the first painting, and has less money than before. At the four vertices below labeld by "Beth" the situation is just the other way---Ann owns the first painting but has less money left, and Beth has still all money and decides whether or not to bet $1000 for the second painting.
   Can you find the payoffs Ann and Beth would expect to get at the end when facing these nine positions? Can you find them by reasoning, not by backward induction? Since, if you can, you can finish the analysis on the shown graph using just ordinary backward induction.
   1. Try to find these expected values for the nine vertices for case (3), a worth of $3500 for the first and a worth of $7500 for the second painting, and start money of $4000 for both players.
   2. Try to find these expected values for the nine vertices for case (2), a worth of $7500 for the first and a worth of $3500 for the second painting, and start money of $5000 for both players.
5. What would happen if the first picture is worth only $3500 and the second one as much as $7500, and both players have initially $5000? Which player will get higher payoff? How much will each get?
   1. In the neutral case
   2. Will the slightly hostile case be any different? How?
6. Based on (1), (2), (3), and (5), can you draw any conclusion of whether having an odd or even amount of money (in 1000s) benefits Ann or Beth? Would it depend on whether the second painting being much more valuable, or the first one being much more valuable? Explain.
7. In case you observed a pattern in the previous question, do another combination of values to see whether it confirms the pattern you saw.