Combinatorial betting mechanisms: In a betting mechanism, people take bets on fu
ID: 407130 • Letter: C
Question
Combinatorial betting mechanisms: In a betting mechanism, people take bets on future events. Many such mechanism function by letting people buy or sell securities in a particular event. For example, you may be to buy a security that pays off 100 if basketball team A defeats basketball team B in their next game; let us suppose that this security can currently be bought for 70. If you by this security, then if A wins, you make a profit of 30, but if B wins, you lose 70. We will consider a setup in which bettors propose bets to a party that we will call the auctioneer. A bet from a bettor might be "If A wins you have to pay me 30, but if B wins I will pay you 70". The auctioneer is only interested in accepting combinations of bets that result in a guaranteed profit for her. Hence, the auctioneer will not accept the above bet by itself. However, suppose she has the following bet proposed to her: "If A wins I will pay you 50, but B wins you have to pay me 60." Now, if the auctioneer accepts both of these bets, then no matter which of A and B wins, the auctioneer will have a profit of at least 10. The auctioneer's goal is to accept a subset of the bets that maximizes her guaranteed profit.Explanation / Answer
Answer:
There are m mutually exclusive potential states at maturity time. Exactly one state will be true at maturity. I An auctioneer is prepared to issue contracts on bids as follows:
Each bid should specify a subset of states
The bidder will receive $w if the final state is one of those specified in the bid, and $0 o/w
Each auction participant j (out of n) places orders with the auctioneer, with the order consisting of:
Bid vector Aj = (a1j , a2j ,..., amj)T , with each component either 0 or 1
Number $j , which is the price participant j is willing to pay for 1 unit of this order I Number qj , which is the maximum number of units of this order he/she is willing to buy