Guess-½-of-Average Sealed-Bid Lottery (Version #1 for Homework #5)

1)      n bidders as players: 1,2,…, n for a large n (say n > 10000)

2)      For each player i, the set of strategies StrategySeti = { bi  | bi is an integer in the range of 1 to 128.}

3)      Rules of the lottery:

a)       Each player i submit his/her bid bi  in a sealed envelope to the auction. No one other than player i knows what the bid bi is by player i until the envelope is opened.

b)      After all the players submit their bids to the auction, the auctioneer opens the envelopes and calculates ½ of the average of all the submitted numbers. The winner is the player whose number bi  is closest to the ½ of the average of all the submitted numbers and the award is a non-negative number of  128 - bi .

c)       Note about ties: To make it simple, the auctioneer attaches arbitrarily small distinct epsilons (as fixed distinct IDs associated with the payers as described in Homework 4A and Homework 4B) to the bids of the players. In other words, all strategy sets StrategySetis are disjoint, all bids are distinct, and there is no tie just like in Homework 4A and Homework 4B.

4)      Strategy profile: The strategy profile of the auction outcome is simply the vector < b1, b2,…, bn >, recording the strategies (the bids) used by each player.

5)      The happiness payoffs: For each player i, the payoff is simply the amount of money rewarded by the lottery.

6)      Pay-off functions:  For each player i, the pay-off function πi  (b1, b2,…, bn ) simply describes the payoff (utility) player i gets given any strategy profile < b1, b2,…, bn > according to the description of  the happiness payoffs in #5 and rules of the lottery in #3 above.

Questions:

1.       For a player i, does 127 (as a strategy) weakly dominate 128?  If so, prove it. If not, give a scenario of b-i  in which the payoff of the strategy profile (bi=128, b-i ) is better than that of  the strategy profile ( bi=127, b-i ).

2.       For a player i, does 126 (as a strategy) weakly dominate 127?  If so, prove it. If not, give a scenario in which the payoff of the strategy profile (bi=127, b-i ) is better than that of  the strategy profile ( bi=126, b-i ).

3.       For a player i, does 64 (as a strategy) weakly dominate 65?  If so, prove it. If not, give a scenario in which the payoff of the strategy profile (bi=65, b-i ) is better than that of  the strategy profile (bi=64, b-i ).

4.       For a player i, does 63 (as a strategy) weakly dominate 64?  If so, prove it. If not, give a scenario in which the payoff of the strategy profile (bi=64, b-i ) is better than that of  the strategy profile (bi=63, b-i ).

5.       For a player i, does 62 (as a strategy) weakly dominate 63?  If so, prove it. If not, give a scenario in which the payoff of the strategy profile (bi=63, b-i ) is better than that of  the strategy profile (bi=62, b-i ).

6.       For each player i, does 1 (as a strategy) weakly dominate all other strategies? If so, prove it. If not, give a scenario in which the payoff of the strategy profile (bi=1, b-i ) is not as good as that of  the strategy profile (bi=n, b-i ) for some n > 1.

7.       For each player i, is there a strategy that weakly dominates all other strategies? If so, prove it. If not, explain why not.

8.       For each player i, what are the weakly dominated strategies (i.e. each of them is weakly dominated by at least one other strategy)? Are there reasons that a rational player may want to use any of these weakly dominated strategies?

9.       After the reasoning above, a rational player has an understanding about dominated strategies as “bad ideas” to avoid and may assume other rational players have the same understanding. If the players then repeat a similar process of reasoning from #1 to #8 again and again, what eventually would their strategy (strategies) be for the lottery?

10.   Think about #9 above again in the context of #1 and #2 in the summary of chapter 4 of Strategies and Games: Theory and Practice. Is there an IEDS solution for the Guess-½-of-Average Sealed-Bid Lottery? If so, what is the IEDS solution?

Guess-½-of-Average Sealed-Bid Lottery (Version #2)

In this version, the award for the winner in the lottery game is changed from (128 – bi) to bi instead.

What would happen to your answers above? (No need to answer them in the Homework just do it as an exercise on your own.)

 To ensure the reward is non-negative, the reward for the winner actually should be (maximum of the epsilons) + 128 - bi  depending on the epsilons used in c), but this is essentially 128 - bi  since we can make all the epsilons arbitrarily small.