The Gjerstad Dickhaut (GD) Auction Strategy as presented in the paper: “Price Formation in Double Auctions” by Steven Gjerstad and John Dickhaut Presented by Marek Marcinkiewicz

GD – The main idea Use previous bids to determine the probability that future bids will be accepted Combine this probability with profit to estimate how to place bids to maximize expected profit

History The algorithm requires previous bids made by all traders Since the agent has limited memory, a memory length L is specified L last trades and any shouts between them are stored

Frequency of Takes If there were many bids made at each price point than the probability could simply be the number of shouts accepted at a particular price point In a more likely sparse market this is not possible though since there are few bids But not only bids at some price point provide information to us

What do bids reveal? TBL – taken bids lower than some price would also be taken at this price AL – asks lower than some price would match a bid at this price RBG – rejected bids greater than some price make it less likely that this price will be accepted because if they are rejected then why take this even lower offer?

Probability of bid P(b) = TBL(b) + AL(b) TBL(b) + AL(b) + RBG(b) TBL(b) + AL(b) + RBG(b)

Spread reduction rule All bids must be higher than the last outstanding bid and all asks must be lower than last outstanding ask Minimum price is 0 Maximum price is M (a value that nobody is willing to pay)

Interpolation Since probability is only defined at shout points that where already made, we have to interpolate these into the real space P(b k ) = calculated P(b k ) P(b k+1 ) = calculated P (b k+1 ) P’(b k ) = 0 P’(b k+1 ) = 0

Interpolated Probability P(b) = α 3 b 3 + α 2 b 2 + α 1 b 1 + α 0 Use previous 4 equations to solve for α.

Expected Profit Bid = max(p(b) * (b – private value)) Use the same type of strategy for asks