Wednesday, May 26, 2010

Auction Game

I am keen on running a game that has some elements of game theory. The particulars of the game are described below.

The game is an auction game, where each player aims to end up with the most amount of money. The items to be auctioned are bundles of money. As with typical auctions, the highest bid win the auctioned bundle of money. However, there are two important differences. Firstly, the auction system is sealed-bid, meaning that each person places his bid in secret. Secondly, the auction is all-pay, meaning that everyone pays the amount they bid, regardless of whether they won anything. Hence, the auction is an all-pay sealed-bid first-price auction.

Thus, the rules are:

1) Each player starts with the same amount of money, S dollars.
2) The X bundles of money, and their denominations, are announced in advance.
3) Each player may place, secretly, up to one bid per bundle of money, so long as the sum of his bids is equal to or less than S. Each bid must be a positive integer.
4) Everyone pays an amount of money equal to amount they bid.
5) After each player has placed his bids, for each bundle of money, the person(s) with the highest bid(s) share the bundle of money equally.
6) Any remaining unclaimed bundles of money are forfeited.
7) The final score of each player is equal to the total amount of money he has; i.e., the sum of money won and money not spent during bidding.
8) The final score is used to rank the players in the game.
9) No communications are allowed between players before or during bidding.

An exception to rule 9 is that before the bidding begins, each player may issue at most one public statement, viewable to all players. The public statements are displayed on a first-post-first-displayed basis.

10) At any point before the bidding phase, each player is allowed to make at most one public statement which is viewable by all players. The public statement will be displayed immediately.

A trial run of the game is provided below.

Preparation Phase:

There are three players, A, B, and C competing. They have $10 each.

There are 4 items being auctioned. The items are:
#1: $3 note.
#2: $5 note.
#3: $5 note.
#4: $12 note.

Before the bidding begins, it is possible to make public statements. C makes a first public statement, “Hello, I love money.” Everyone is notified of C’s statement.

A also wishes to make a public statement, “I will not make any bids.” Everyone is notified of A’s statement.

B forfeits his right to make statements as he does not wish to make any statements.

Bidding Phase:

The bidding phase begins. Each player independently and secretly makes their bids for the items.

A bids $2 for item #2, $2 for item #3, and $6 for item #4.

B bids $10 for item #4.

C bids $2 for items #1, #2, #3 and #4.

After each player has submitted his bids, the bidding phase ends. Each player pays for his bids. A pays $10 for his bids, and he has $0 left over. B pays $10 for his bid, and he has $0 left over. C pays $8 for his bids, and he has $2 left over.

Settlement Phase:

It is time to settle the auctions, and compute the final scores. Item #1 is won by player C. Item #2 is won and shared by players A and C. Item #3 is won and shared between players A and C. Item #4 is won by player B.

Player A has $5 in the end.

Player B has $12 in the end.

Player C has $10 in the end.

Player B has the most money, and wins.

If there are any willing participants to try this game, please contact me by some means. I will post more details if/when there is sufficient interest.

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