Applying Evolutionary Game Theory to Auction
Mechanism Design
Andrew Byde
Hewlett-Packard Laboratories
Commodity Trading Using An Agent-Based
Iterated Double Auction
Chris Preist
Hewlett Packard Laboratories
1
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.
2
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.
3
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

4
Terms and Notation
 In
this representation, assuming twicedifferentiability 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 riskseeking
5
Terms and Notation
6
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.
7
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.
8
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)
9
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.
10
Context Parameterization
 In
this paper they examine a onedimensional 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 secondprice auction, and all other values of w2
correspond to non-standard auction types.
11
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.
12
Context Parameterization
 Utility

function:
αis zero for risk-neutral agents, negative
for risk-averse agents and positive for riskseeking agents.
13
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.
14
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.
15
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
}
}
16
Results

The expected-utilitymaximizing genome
is (0, 0.25, 0.5, 0.75,
1.0).
17
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.
18
Results
19
Results
20
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.
21
Commodity Trading Using
An Agent-Based Iterated
Double Auction
Chris Preist
22
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.
23
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.
24
Measurement
 Smith
introduce a measure of
convergence on this equilibrium price. This
measure, which referred to as Smith’s
alpha, is defined as:

25
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.
26
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.
27
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.
28
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.
29
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.
30
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.
31
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.
32
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.
33
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
34
The Agent Algorithm
 For
Sellers:
If Smin > Bmax then
target = Smin – δ
If Smin ≤ Bmax then
target = Bmax + δ
35
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].
36
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.
37
Agent Performance
38
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.
39
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.
40