2nd price sealed-bid auction (Version #1, for Homework #4A)

1)      (Revised 03/22) n bidders as players: 1,2,, n. Each player i has a unique decimal ID number i.

2)      (Revised 03/22) For each player i, the set of strategies StrategySeti = { bi | bi as in the decimal format of k.i where k is a positive integer and the unique ID i.} For example, for player 28 the set of strategies is StrategySet28 = { b28 | b28 is in the decimal format of k.28 where k is a positive integer.} In other words, player 28 can only bid 1.28, 2.28, 3.28, 4.28 and so forth. This ensures that all the sets StrategySetis are disjoint and no two players can happen to submit the same number as their bids.

3)      Rules of the auction:

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 arranges them as a list in descending order of the bids. The player on the top of the list (i.e. the one offering the highest bid) gets the auctioned item and need to pay the second price on the bid list.

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. For convenience, the notation < bi, b-i > is used to denote the strategy profile where bi is player is bid while b-i refers to the list of all the bids from payers other than player i.

5)      (Revised 03/22) 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 below:

For each player i, there is a true value vi in his/her mind where vi must be in the decimal format of k.i where k is a positive integer. For example, for player 28 the true value v28 must be something in the decimal format of k.28 for some positive integer k.

a)      If player i does get the auction item and pay no more than the a true value vi for the item, the player is so excited and his/her happiness level is very high, which is represented by a payoff of 10.

b)      If player i does not get the auction item, his/her happiness level is neutral, which is represented by a payoff of 0.

c)       If player i does get the auction item but pay more than the a true value vi for the item, the player is very upset and his/her happiness level is very low, which is represented by a payoff of -10.

6)      (Revised 03/22) Note: The revisions in 1), 2), and 5) above ensures that there is no tie since no two players can happen to submit the same number as their bids. In addition, player i is the only person who can possibly bid the number vi, the true value vi for player i.

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2nd price sealed-bid auction (Version #2 as a separate variant for Homework #4B)

In this version, the playoff function for each player is slightly changed to better model how people feel about the auction result.

5)      New 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 below:

For each player i, there is a true value vi in his/her mind.

a)      If player i does get the auction item and pay no more than the a true value vi for the item, the player is happy and his/her payoff is vi minus the amount of money player i needs to pay. This is closer to how people feel about the auction result.

b)      If player i does not get the auction item, his/her happiness level is neutral, which is represented by a payoff of 0.

c)       If player i does get the auction item but pay more than the a true value vi for the item, the player is very upset and his/her happiness level is very low, which is represented by a payoff of -10.