Introduction

1. Applying Evolutionary Game Theory to Auction Mechanism DesignAndrew BydeHewlett-Packard LaboratoriesCommodity Trading Using An Agent-Based Iterated Double AuctionChris PreistHewlett Packard Laboratories

2. Introduction • Auctions are an important class of mechanism for resolving multi-agent allocation problems of various types. • Researchers have begun investigating how to design autonomous agents capable of participating in auctions.

3. Introduction • This paper examines a space of auction mechanisms that includes the standard first- and second-price auctions, uses GAs applied to a multi-agent system to evolve good players for each mechanism under consideration.

4. Terms and Notation • The risk preferences of agents are differentiated by use of a von Neumann-Morgenstern utility function u, so that an agent strictly prefers a selection of possible outcomes xi with corresponding probabilities pi, over a second selection of possible outcomes xj with corresponding probabilities qj, if and only if

5. Terms and Notation • In this representation, assuming twice-differentiability of u, and agent for which u’’(x)=0 is known as risk-neutral; • If u’’(x)<0, the agent is risk-averse. • If u’’(x)>0, the agent is known as risk-seeking

6. Terms and Notation

7. Terms and Notation • The value of a good to an agent can be independent of the value of the good to other agents, or it can be derived from information about how other agents value the good. (private value & common value) • They treat both these cases by postulating that each bidder receives a “signal”, and that the value of the good to the agent is some specified function of all the agents’ signals.

8. Revenue Equivalence Theorem It states that if • There is a fixed number of bidders, known to everyone • All agents are risk-neutral • All bidders’ signals are picked from a common, known distribution and if • In equilibrium, the good always goes to the bidder with the highest signal • Any bidder whose signal is the lowest possible expects to make nothing Then the expected revenue to the seller is the same, independent of the mechanism.

9. Revenue Equivalence Theorem • This rather surprising result means that, subject to these hypotheses, it doesn’t matter what type of auction a seller runs, he should expect to make the same amount of money whatever the mechanism. • But of course there are many different auction mechanisms in use, because at least one of the hypotheses on which the theorem rests is often violated. (eg. Most people are not risk-neutral)

10. Context Parameterization • They chose to investigate a space of mechanisms very similar to the first- and second-price bid auctions. • Definition: Let w=(w1,…,wn) be a vector of n real numbers. A w-price auction is a sealed bid auction in which the highest bidder wins the good, and pays • Where N is the minimum of n and the number of bidders, and bid1, bid2,… are the bids, ordered highest to lowest.

11. Context Parameterization • In this paper they examine a one-dimensional sub-space of w-price auctions, namely those of type w=(1-w2,w2). In this parameterization, w2=0 is a standard first price auction, w2=1 is a standard second-price auction, and all other values of w2 correspond to non-standard auction types.

12. Context Parameterization • In the experiment, they allowed variable group size, variable risk preference, and correlated bidders’ signals. In addition, they allowed the degree of commonality in values to be altered. • The signals (t1,…,tn) of a group (a1,…,an) of bidders were chosen to be a weighted sum of a shared random signal and a sequence of independent random signals, with each such signal coming from a uniform distribution on [0,1]. Thus independent variables S, X1,…, Xn were generated, and the signal ti for agent ai was chosen to be cS+ (1-c)Xi, where c is in [0,1] parameterizes the degree of correlation between agents’ signals.

13. Context Parameterization • Utility function: • αis zero for risk-neutral agents, negative for risk-averse agents and positive for risk-seeking agents.

14. Context Parameterization • To model common value, they assumed that the monetary value to agent ai of winning the good was given by d·(Σjvj)/n+(1-d)vi, where d is a parameter controlling the degree of common value, with d=0 representing purely private values, and d=1 purely common values.

15. Strategy Optimization • Each agent in a population of bidders is equipped with a bidding function which can be modified through evolution to adapt to the necessities of the game. • The main drawback is that it can neither be guaranteed that the population will evolve a good strategy within a reasonable period of time, nor that the solution on which the population eventually converges is a global rather than local optimum. So they run the entire process of evolution many times independently, and reduce the effect of mutation as time goes by, so as to encourage convergence.

16. Strategy Optimization • The evaluation of a population of genomes was according to the following algorithm: • For each of a large number of iterations { while (not all agents have played in this round) { select some as-yet-unplayed agents to play a game generate random signals for the agents get bids for each agent, according to their genome select a winner and determine payments accumulate the corresponding utility rewards } }

17. Results • The expected-utility-maximizing genome is (0, 0.25, 0.5, 0.75, 1.0).

18. Results • The two lines in grey, above and below the plotted curve of average revenue, are plus and minus one standard deviation relative to the average, and give an indication of the magnitude of experimental uncertainty.

19. Results

20. Results

21. Conclusions • They have demonstrated that this technique can be used to explore a space of auction mechanisms. • The advantages of such a method for exploring auction design issues are clear: the agents discover good bidding strategies by evolution, without the need for complicated, possibly intractable, and certainly fragile mathematical analysis.

22. Commodity Trading Using An Agent-Based Iterated Double Auction Chris Preist

23. Introduction • The two main market institutions used for trading commodities are the continuous double auction and the call auction. • With the advent of the Internet, the creation of global marketplaces has become far cheaper and easier than it once was. • Agent technology will play an increasingly important role in this revolution.

24. Introduction • Agents can have another role in electronic trading. We can use agents to design new market institutions. Agents can be used to negotiate rapidly and anonymously on behalf of their owners, resulting in frictionless markets that trade at a fair market price and are less open to fraudulent behavior. • The author present the agent-based iterated double auction, it uses agent technology to combine the best features of CDA and the call auction.

25. Measurement • Smith introduce a measure of convergence on this equilibrium price. This measure, which referred to as Smith’s alpha, is defined as:

26. Call Auction Call Auction: • There is a central auctioneer who plays an active role in calculating which trades take place. • All trades take place at the same price. • In the call auction, trades do not publicly announce bids or offers. Instead, they privately prepare information about how many units they would like to buy or sell at a given price. • The auctioneer finds the intersection point of the supply and demand curves and announces this price, and then all trades take place at the equilibrium price.

27. Drawbacks of CDA and Call Auction CDA: • In the early stages the differences can be quite significant. Traders who could trade at equilibrium may fail to make a trade. • The negotiations in the CDA take time. Call Auction: • It relies on a central auctioneer. The auctioneer may enter into collusion with some of the participants, and manipulate the market in their favor.

28. The Agent-Based Iterated Double Auction • Smith has shown that if the auction is repeated several times, with participants trading goods with the same values each time, then trades rapidly converge to the equilibrium price as participants respond to market conditions.

29. The Agent-Based Iterated Double Auction • This suggests an approach that can allow the double auction to be used to produce trades at equilibrium. Participants engage in a series of mock double auctions and then carry out a final double auction where the trades are actually made.

30. The Agent-Based Iterated Double Auction • In practice, this approach will not work. Participants may attempt to manipulate the market during the mock auctions by refusing to agree trades that in reality they would accept. • However, this can be overcome if we use agents to trade on behalf of the participants.

31. The Agent-Based Iterated Double Auction The agent-based iterated double auction proceeds as follows: • 1. Prior to the auction, all participants receive a copy of the agent. The agent is inspectable, but cannot be altered. Participants can make as many copies as they wish. • 2. Participants privately prepare information about how many units they would like to buy or sell at a given price, as in the call auction. • 3. Traders then enter appropriate reservation prices into their agents.

32. The Agent-Based Iterated Double Auction • 4. The agents enter the marketplace and begin trading. When a buyer and seller agree a trade, they no longer participate in this iteration of the auction, but remain in the marketplace to observe. Once an agent has entered the marketplace it is unable to receive communications from its owner until all iterations of the auction are over. • 5. Stage 4 is repeated, with the marketplace measuring the standard deviation of trade prices agreed in each auction. When the standard deviation falls below a previously agreed value, the trade agreed by the agents can be considered binding. • 6. The owner of each agent is informed of any trades it has agreed to, and exchange of goods takes place.

33. The Agent-Based Iterated Double Auction • Therefore, the new institution combines the best properties of the call auction and the continuous double auction. • They also introduce an anonymizing service between the participants and the marketplace to satisfy the security requirement.

34. The Agent Algorithm • Let Bmax be the highest bid at the beginning of this round, and Smin be the lowest offer. Let δbe a small random value. The target value t for agents to adjust towards are determined as follows: • For Buyers: If Smin > Bmax then target = Bmax + δ If Smin ≤ Bmax then target = Smin - δ

35. The Agent Algorithm • For Sellers: If Smin > Bmax then target = Smin – δ If Smin ≤ Bmax then target = Bmax + δ

36. The Agent Algorithm If the target is Bmax + δ then δ=r1 Bmax +r2 If the target is Smin – δ then δ=r1 Smin +r2 • Where r1 and r2 are independent random variables identically distributed in the range [0, 0.2].

37. The Agent Algorithm • Given the target value, the agent does not jump straight to that value, but moves towards it at a rate determined by the learning rule.

38. Agent Performance

39. Conclusions • The agent-based iterated double auction is faster and fairer than the standard CDA. Unlike the call auction, it does not require a trusted auctioneer and disclosure of supply/demand information.

40. Conclusions However, there are two special circumstances where its behavior is inadequate: • When there is a price tunnel; in other words, when the equilibrium price is not a single value, but is a range instead. In this case, the institution will converge on this range, but the stopping criteria would never be satisfied. • Certain supply and demand curves, such as box markets, result in very slow convergence to equilibrium both in markets of agents and of humans.