1 applying evolutionary game theory to auction mechanism design andrew byde hewlett-packard...

11

Applying Evolutionary Game Theory to Auction Applying Evolutionary Game Theory to Auction Mechanism DesignMechanism Design

Andrew BydeAndrew BydeHewlett-Packard LaboratoriesHewlett-Packard Laboratories

Commodity Trading Using An Agent-Based IterCommodity Trading Using An Agent-Based Iterated Double Auctionated Double Auction

Chris PreistChris PreistHewlett Packard LaboratoriesHewlett Packard Laboratories

22

IntroductionIntroduction

Auctions are an important class of Auctions are an important class of mechanism for resolving multi-agent mechanism for resolving multi-agent allocation problems of various types.allocation problems of various types.

Researchers have begun investigating Researchers have begun investigating how to design autonomous agents how to design autonomous agents capable of participating in auctions.capable of participating in auctions.

33


This paper examines a space of auction mThis paper examines a space of auction mechanisms that includes the standard first- echanisms that includes the standard first- and second-price auctions, uses GAs appliand second-price auctions, uses GAs applied to a multi-agent system to evolve good ed to a multi-agent system to evolve good players for each mechanism under considplayers for each mechanism under consideration.eration.

44

Terms and NotationTerms and Notation

The risk preferences of agents are differentiated The risk preferences of agents are differentiated by use of a von Neumann-Morgenstern utility funby use of a von Neumann-Morgenstern utility function u, so that an agent strictly prefers a selectiction u, so that an agent strictly prefers a selection of possible outcomes xi with corresponding pron of possible outcomes xi with corresponding probabilities pi, over a second selection of possiblobabilities pi, over a second selection of possible outcomes xj with corresponding probabilities qj,e outcomes xj with corresponding probabilities qj, if and only if if and only if

55


In this representation, assuming twice-diffeIn this representation, assuming twice-differentiability of u, and agent for which u’’(x)=rentiability of u, and agent for which u’’(x)=0 is known as risk-neutral;0 is known as risk-neutral;

If u’’(x)<0, the agent is risk-averse.If u’’(x)<0, the agent is risk-averse. If u’’(x)>0, the agent is known as risk-seekiIf u’’(x)>0, the agent is known as risk-seeki

ngng

66


77


The value of a good to an agent can be The value of a good to an agent can be independent of the value of the good to other independent of the value of the good to other agents, or it can be derived from information agents, or it can be derived from information about how other agents value the good. (private about how other agents value the good. (private value & common value)value & common value)

They treat both these cases by postulating that They treat both these cases by postulating that each bidder receives a “signal”, and that the each bidder receives a “signal”, and that the value of the good to the agent is some specified value of the good to the agent is some specified function of all the agents’ signals.function of all the agents’ signals.

88

Revenue Equivalence TheoremRevenue Equivalence TheoremIt states that ifIt states that if There is a fixed number of bidders, known to everyoneThere is a fixed number of bidders, known to everyone All agents are risk-neutralAll agents are risk-neutral All bidders’ signals are picked from a common, known All bidders’ signals are picked from a common, known

distribution and ifdistribution and if In equilibrium, the good always goes to the bidder with In equilibrium, the good always goes to the bidder with

the highest signalthe highest signal Any bidder whose signal is the lowest possible expects Any bidder whose signal is the lowest possible expects

to make nothingto make nothingThen the expected revenue to the seller is the same, Then the expected revenue to the seller is the same, independent of the mechanism.independent of the mechanism.

99

Revenue Equivalence TheoremRevenue Equivalence Theorem

This rather surprising result means that, subject tThis rather surprising result means that, subject to these hypotheses, it doesn’t matter what type o these hypotheses, it doesn’t matter what type of auction a seller runs, he should expect to makof auction a seller runs, he should expect to make the same amount of money whatever the mece the same amount of money whatever the mechanism. hanism.

But of course there are many different auction mBut of course there are many different auction mechanisms in use, because at least one of the hyechanisms in use, because at least one of the hypotheses on which the theorem rests is often violpotheses on which the theorem rests is often violated. (eg. Most people are not risk-neutral)ated. (eg. Most people are not risk-neutral)

1010

Context ParameterizationContext Parameterization

They chose to investigate a space of mechanisms very sThey chose to investigate a space of mechanisms very similar to the first- and second-price bid auctions.imilar to the first- and second-price bid auctions.

Definition: Let w=(wDefinition: Let w=(w11,…,w,…,wnn) be a vector of n real number) be a vector of n real numbers. A w-price auction is a sealed bid auction in which the hs. A w-price auction is a sealed bid auction in which the highest bidder wins the good, and paysighest bidder wins the good, and pays

Where N is the minimum of n and the number of bidders, Where N is the minimum of n and the number of bidders, and bidand bid11, bid, bid22,… are the bids, ordered highest to lowest.,… are the bids, ordered highest to lowest.

1111


In this paper they examine a one-In this paper they examine a one-dimensional sub-space of w-price dimensional sub-space of w-price auctions, namely those of type w=(1-auctions, namely those of type w=(1-ww22,w,w22). In this parameterization, w). In this parameterization, w22=0 is a =0 is a standard first price auction, wstandard first price auction, w22=1 is a =1 is a standard second-price auction, and all standard second-price auction, and all other values of wother values of w2 2 correspond to non-correspond to non-standard auction types.standard auction types.

1212


In the experiment, they allowed variable group size, variaIn the experiment, they allowed variable group size, variable risk preference, and correlated bidders’ signals. In adble risk preference, and correlated bidders’ signals. In addition, they allowed the degree of commonality in values dition, they allowed the degree of commonality in values to be altered.to be altered.

The signals (tThe signals (t11,…,t,…,tnn) of a group (a) of a group (a11,…,a,…,ann) of bidders were ) of bidders were chosen to be a weighted sum of a shared random signal chosen to be a weighted sum of a shared random signal and a sequence of independent random signals, with eacand a sequence of independent random signals, with each such signal coming from a uniform distribution on [0,1]. h such signal coming from a uniform distribution on [0,1]. Thus independent variables S, XThus independent variables S, X11,…, X,…, Xnn were generated, were generated, and the signal ti for agent ai was chosen to be cS+ (1-c)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 correlXi, where c is in [0,1] parameterizes the degree of correlation between agents’ signals.ation between agents’ signals.

1313


Utility function:Utility function:

ααis zero for risk-neutral agents, negative fis zero for risk-neutral agents, negative for risk-averse agents and positive for risk-or risk-averse agents and positive for risk-seeking agents.seeking agents.

1414


To model common value, they assumed thTo model common value, they assumed that the monetary value to agent aat the monetary value to agent aii of winnin of winning the good was given by g the good was given by

dd·(·(ΣΣjjvvjj)/n+(1-d)v)/n+(1-d)vii, where d is a parameter c, where d is a parameter controlling the degree of common value, witontrolling the degree of common value, with d=0 representing purely private values, ah d=0 representing purely private values, and d=1 purely common values.nd d=1 purely common values.

1515

Strategy OptimizationStrategy Optimization

Each agent in a population of bidders is equipped with a Each agent in a population of bidders is equipped with a bidding function which can be modified through evolution bidding function which can be modified through evolution to adapt to the necessities of the game.to adapt to the necessities of the game.

The main drawback is that it can neither be guaranteed The main drawback is that it can neither be guaranteed that the population will evolve a good strategy within a that the population will evolve a good strategy within a reasonable period of time, nor that the solution on which reasonable period of time, nor that the solution on which the population eventually converges is a global rather the population eventually converges is a global rather than local optimum. So they run the entire process of than local optimum. So they run the entire process of evolution many times independently, and reduce the evolution many times independently, and reduce the effect of mutation as time goes by, so as to encourage effect of mutation as time goes by, so as to encourage convergence.convergence.

1616

Strategy OptimizationStrategy Optimization

The evaluation of a population of genomes was accordinThe evaluation of a population of genomes was according to the following algorithm:g to the following algorithm:

For each of a large number of iterations {For each of a large number of iterations { while (not all agents have played in this round) {while (not all agents have played in this round) { select some as-yet-unplayed agents to play a gameselect some as-yet-unplayed agents to play a game generate random signals for the agentsgenerate random signals for the agents get bids for each agent, according to their genomeget bids for each agent, according to their genome select a winner and determine paymentsselect a winner and determine payments accumulate the corresponding utility rewardsaccumulate the corresponding utility rewards }} }}

1717

ResultsResults

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

1818

ResultsResults

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

1919

ResultsResults

2020

ResultsResults

2121

ConclusionsConclusions

They have demonstrated that this They have demonstrated that this technique can be used to explore a space technique can be used to explore a space of auction mechanisms.of auction mechanisms.

The advantages of such a method for The advantages of such a method for exploring auction design issues are clear: exploring auction design issues are clear: the agents discover good bidding the agents discover good bidding strategies by evolution, without the need strategies by evolution, without the need for complicated, possibly intractable, and for complicated, possibly intractable, and certainly fragile mathematical analysis.certainly fragile mathematical analysis.

2222

Commodity Trading Using Commodity Trading Using An Agent-Based Iterated An Agent-Based Iterated

Double AuctionDouble Auction

Chris PreistChris Preist

2323


The two main market institutions used for The two main market institutions used for trading commodities are the continuous trading commodities are the continuous double auction and the call auction.double auction and the call auction.

With the advent of the Internet, the With the advent of the Internet, the creation of global marketplaces has creation of global marketplaces has become far cheaper and easier than it become far cheaper and easier than it once was.once was.

Agent technology will play an increasingly Agent technology will play an increasingly important role in this revolution.important role in this revolution.

2424


Agents can have another role in electronic Agents can have another role in electronic trading. We can use agents to design new trading. We can use agents to design new market institutions. Agents can be used to market institutions. Agents can be used to negotiate rapidly and anonymously on behalf of negotiate rapidly and anonymously on behalf of their owners, resulting in frictionless markets that their owners, resulting in frictionless markets that trade at a fair market price and are less open to trade at a fair market price and are less open to fraudulent behavior.fraudulent behavior.

The author present the agent-based iterated The author present the agent-based iterated double auction, it uses agent technology to double auction, it uses agent technology to combine the best features of CDA and the call combine the best features of CDA and the call auction.auction.

2525

MeasurementMeasurement

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

2626

Call AuctionCall Auction

Call Auction:Call Auction: There is a central auctioneer who plays an active role in There is a central auctioneer who plays an active role in

calculating which trades take place.calculating which trades take place. All trades take place at the same price.All trades take place at the same price. In the call auction, trades do not publicly announce bids In the call auction, trades do not publicly announce bids

or offers. Instead, they privately prepare information or offers. Instead, they privately prepare information about how many units they would like to buy or sell at a about how many units they would like to buy or sell at a given price.given price.

The auctioneer finds the intersection point of the supply The auctioneer finds the intersection point of the supply and demand curves and announces this price, and then and demand curves and announces this price, and then all trades take place at the equilibrium price.all trades take place at the equilibrium price.

2727

Drawbacks of CDA and Call Drawbacks of CDA and Call AuctionAuction

CDA:CDA: In the early stages the differences can be quite In the early stages the differences can be quite

significant. Traders who could trade at significant. Traders who could trade at equilibrium may fail to make a trade.equilibrium may fail to make a trade.

The negotiations in the CDA take time.The negotiations in the CDA take time.Call Auction:Call Auction: It relies on a central auctioneer. The auctioneer It relies on a central auctioneer. The auctioneer

may enter into collusion with some of the may enter into collusion with some of the participants, and manipulate the market in their participants, and manipulate the market in their favor.favor.

2828

The Agent-Based Iterated Double The Agent-Based Iterated Double AuctionAuction

Smith has shown that if the auction is Smith has shown that if the auction is repeated several times, with participants repeated several times, with participants trading goods with the same values each trading goods with the same values each time, then trades rapidly converge to the time, then trades rapidly converge to the equilibrium price as participants respond equilibrium price as participants respond to market conditions.to market conditions.

2929


This suggests an approach that can allow This suggests an approach that can allow the double auction to be used to produce the double auction to be used to produce trades at equilibrium. Participants engage trades at equilibrium. Participants engage in a series of mock double auctions and in a series of mock double auctions and then carry out a final double auction where then carry out a final double auction where the trades are actually made.the trades are actually made.

3030


In practice, this approach will not work. In practice, this approach will not work. Participants may attempt to manipulate the Participants may attempt to manipulate the market during the mock auctions by market during the mock auctions by refusing to agree trades that in reality they refusing to agree trades that in reality they would accept.would accept.

However, this can be overcome if we use However, this can be overcome if we use agents to trade on behalf of the agents to trade on behalf of the participants.participants.

3131


The agent-based iterated double auction proceeds as The agent-based iterated double auction proceeds as

follows:follows: 1. Prior to the auction, all participants receive a copy of t1. Prior to the auction, all participants receive a copy of t

he agent. The agent is inspectable, but cannot be altered.he agent. The agent is inspectable, but cannot be altered. Participants can make as many copies as they wish. Participants can make as many copies as they wish.

2. Participants privately prepare information about how 2. Participants privately prepare information about how many units they would like to buy or sell at a given price, many units they would like to buy or sell at a given price, as in the call auction.as in the call auction.

3. Traders then enter appropriate reservation prices into 3. Traders then enter appropriate reservation prices into their agents.their agents.

3232


4. The agents enter the marketplace and begin trading. 4. The agents enter the marketplace and begin trading. When a buyer and seller agree a trade, they no longer When a buyer and seller agree a trade, they no longer participate in this iteration of the auction, but remain in participate in this iteration of the auction, but remain in the marketplace to observe. Once an agent has entered the marketplace to observe. Once an agent has entered the marketplace it is unable to receive communications the marketplace it is unable to receive communications from its owner until all iterations of the auction are over.from its owner until all iterations of the auction are over.

5. Stage 4 is repeated, with the marketplace measuring 5. Stage 4 is repeated, with the marketplace measuring the standard deviation of trade prices agreed in each the standard deviation of trade prices agreed in each auction. When the standard deviation falls below a auction. When the standard deviation falls below a previously agreed value, the trade agreed by the agents previously agreed value, the trade agreed by the agents can be considered binding.can be considered binding.

6. The owner of each agent is informed of any trades it 6. The owner of each agent is informed of any trades it has agreed to, and exchange of goods takes place.has agreed to, and exchange of goods takes place.

3333


Therefore, the new institution combines thTherefore, the new institution combines the best properties of the call auction and the best properties of the call auction and the continuous double auction.e continuous double auction.

They also introduce an anonymizing servicThey also introduce an anonymizing service between the participants and the markete between the participants and the marketplace to satisfy the security requirement. place to satisfy the security requirement.

3434

The Agent AlgorithmThe Agent Algorithm

Let BLet Bmaxmax be the highest bid at the beginning of thi be the highest bid at the beginning of this round, and Ss round, and Sminmin be the lowest offer. Let be the lowest offer. Let δδbe a sbe a small random value. The target value t for agents mall random value. The target value t for agents to adjust towards are determined as follows:to adjust towards are determined as follows:

For Buyers:For Buyers: If If SSmin > min > BBmaxmax then then target = Btarget = Bmaxmax + + δδ If If SSmin min ≤ ≤ BBmax max thenthen target = Starget = Sminmin - - δδ

3535


For Sellers:For Sellers:If If SSmin > min > BBmaxmax then then

target = Starget = Sminmin – – δδ

If If SSmin min ≤ ≤ BBmax max thenthen

target = Btarget = Bmaxmax + + δδ

3636


If the target is If the target is BBmaxmax + + δδ

then then δδ=r=r1 1 BBmaxmax +r +r22

If the target is If the target is SSminmin – – δδ

then then δδ=r=r1 1 SSminmin +r +r22

Where rWhere r11 and r and r22 are independent random variabl are independent random variables identically distributed in the range [0, 0.2].es identically distributed in the range [0, 0.2].

3737


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

3838

Agent PerformanceAgent Performance

3939


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

4040


However, there are two special circumstances However, there are two special circumstances

where its behavior is inadequate: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.

1 applying evolutionary game theory to auction mechanism design andrew byde hewlett-packard...

Documents

wprice auction

neutral slide

notation slide

type w

values of w

type of auction

price auctions

space of auction mechanisms