Version #1, for Homework #4A

1^st price sealed-bid auction with billionaires in the auction

1) n bidders as players: 1,2,…, n. Each player i has a unique decimal ID number i.

2) Each player i submits a natural number with his/her ID then automatically attached after the decimal point as the bid. This ensures that no two players can happen to have the same number as their bids and there is no tie in the result. In other words, the set of strategies StrategySet_i is { b | b is 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 = { b | b is in the decimal format of k.28 where k is a positive integer.} In other word, player 28 can bid 1.28, 2.28, 3.28, 4.28 and so forth.

3) Rules of the auction:

a) Each player i submit his/her bid (denoted as bid_i below) in a sealed envelope to the auctioneer. No one other than player i knows what the bid 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 highest price (the 1^st price) on the bidding list.

4) Strategy profile: The strategy profile of the auction outcome is simply the vector < bid₁, bid₂,…, bid_n >, recording the strategies (the bids) used by each player. For convenience, the notation

< bid_i, bid_-i > is often used in game theory to denote the strategy profile where bid_i is player i’s bid while bid_-i refers to the list of all the bids from payers other than player i.

5) Pay-off functions: For each player i, the pay-off function π_i (bid₁, bid₂,…, bid_n ) simply describes the payoff (utility) player i gets given any strategy profile < bid₁, bid₂,…, bid_n > according to the description of the happiness payoffs below:

For each player i, there is some subjective constant value v_i in his/her mind regarding the worth of the item to pay. Note that v_i is in the decimal format of k.i where k is a positive integer since he/she knows the ID will be automatically attached in the end of his/her bid.

a) If player i does get the auction item and pay no more than the value v_i in his/her mind for the item, the player is very happy, which is represented by a payoff of 10. Note that he/she is equally happy whether he/she pays a lot or pays very little as long as and he/she pays no more than the value v_i in his/her mind for the item. This is close to how billionaires may feel about the auction result since the amount of money paid is not an issue as long as they feel it is worthwhile.

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

b) If player i does get the auction item but pay more than the value v_i in his/her mind for the item, the player feels it is not worthwhile and is very upset, which is represented by a payoff of -10.

6) Note: The formulation in 1) and 2) above ensures that there is no tie since no two players can happen to submit the same number as their bids.

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Version #2 as a separate variant for Homework #4B

1^st price sealed-bid auction with ordinary folk in the auction

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

6) New pay-off functions: For each player i, the pay-off function π_i (bid₁, bid₂,…, bid_n ) simply describes the payoff (utility) player i gets given any strategy profile < bid₁, bid₂,…, bid_n > according to the description of the happiness payoffs below:

Again, for each player i, there is some subjective constant value v_i in his/her mind regarding the worth of the item to pay.

a) If player i does get the auction item and pay no more than v_i for the item, the player is happy and his/her payoff is v_i minus the amount of money player i needs to pay. In other words, the payer is happier when paying less for the item. This is closer to how ordinary people may 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 value v_i in his/her mind for the item, the player feels it is not worthwhile and is very upset, which is represented by a payoff of -10.