A Fuzzy-Logic Based Bidding Strategy
for Autonomous Agents in Continuous
Double Auctions
Minghua He, Ho-fung Leung, and Nicholas R. Jennings
Abstract—Increasingly, many systems are being conceptualized, designed, and implemented as marketplaces in which autonomous
software entities (agents) trade services. These services can be commodities in e-commerce applications or data and knowledge
services in information economies. In many of these cases, there are both multiple agents that are looking to procure services and
multiple agents that are looking to sell services at any one time. Such marketplaces are termed continuous double auctions (CDAs).
Against this background, this paper develops new algorithms that buyer and seller agents can use to participate in CDAs. These
algorithms employ heuristic fuzzy rules and fuzzy reasoning mechanisms in order to determine the best bid to make given the state of
the marketplace. Moreover, we show how an agent can dynamically adjust its bidding behavior to respond effectively to changes in the
supply and demand in the marketplace. We then show, by empirical evaluations, how our agents outperform four of the most prominent
algorithms previously developed for CDAs (several of which have been shown to outperform human bidders in experimental studies).
Index Terms—Intelligent agents, service marketplaces, continuous double auction, fuzzy logic, e-commerce.
æ
1INTRODUCTION
T
HE advent of global network structures, such as the
Internet, has facilitated the development of many large-
scale, open distributed systems in a wide range of industrial,
commercial, and educational domains. In many cases, these
systems can be viewed using a service-oriented metaphor in
which various entities offer services to one another in some
form of marketplace [1], [2]. For example, in deregulated
electricity markets, power generators compete with one
another to provide the service of supplying electricity for
consumers [3], in digital libraries, information services aim to
discover relevant content from providers who offer a variety
of document and archive services [4], and in grid computing,
high performance applications seek to procure the necessary
computational resource services to run [5]. In all of these
applications, and in many others besides, the marketplace in
which the service producers and the service consumers
interact is some form of online auction.
1
The reason for this is
that auctions are a very efficient and effective method of
allocating goods/services, in dynamic situations, to the
entities that value them most highly [6]. While there are
many different types of auction [7], the most common forms
are the simple single sided varieties (e.g., English, first-price
ascending; Dutch, first-price descending; First-Price-Sealed
Bid and Vickrey, second-price sealed-bid) in which there is a
single seller and multiple buyers or a single buyer and
multiple sellers (a reverse auction). However, in many
applications, these simple auction protocols
2
are inadequate
because there are multiple sellers and multiple buyers that
want to trade simultaneously. This can occur, for example,
because entities may want to resell services they have
procured or because delivery of a service requires the
provider to procure subsequent component subservices from
others. Such auctions are called double auctions [9] and they
allow sellers to indicate the services they offer at various
prices (called asks) and buyers to indicate the services they
desire and the price they are willing to pay (called bids). The
most common variety of double auction is the continuous
double auction (CDA)
3
which permits trade at any time in a
IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2003 1345
. M. He and N.R. Jennings are with the Department of Electronics and
Computer Science, The University of Southampton, Southampton, United
Kingdom. E-mail: {mh00r, nrj}@ecs.soton.ac.uk.
. H.-f. Leung is with the Department of Computer Science and Engineering,
The Chinese University of Hong Kong, Shatin, Hong Kong SAR, P.R.
China. E-mail: lhf@cse.cuhk.edu.hk.
Manuscript received 18 May 2001; revised 27 Nov. 2001; accepted 12 Apr.
2002.
For information on obtaining reprints of this article, please send e-mail to:
tkde@computer.org, and reference IEEECS Log Number 114173.
1. In the domain of e-commerce, for example, it is estimated that there
are currently more than 2,000 auction sites on the Internet (http://
www.internetauctionlist.com).
2. In economic and game theory, interactions consist of two components:
a protocol and a strategy [8]. The former defines the valid behavior of the
agents during the interaction (e.g., who can say what to whom at what
time). The later is the method the agents employ to achieve their negotiation
objectives within the specified protocol. The protocol is set at design time by
the marketplace owner and is publicly known to all the participants. The
strategy is designed by each individual participant and is private (divulging
it may leave them open to exploitation). Moreover, the effectiveness of the
strategy is very much determined by the protocol; an optimal strategy for
one protocol may well perform very poorly for other protocols.
3. Although CDAs all conform to this basic protocol, there are several
ways in which they can differ. These variants relate to factors such as
whether the identity of bidders is revealed, whether bids and asks are for
single or multiple units, and whether unaccepted offers are queued or
replaced by better offers [10]. In our case, we do not reveal the identity of
bidders, we allow bids for single units only (one outstanding bid and one
outstanding ask), and unaccepted offers are erased as soon as there is a
more favorable bid (ask). The most restrictive of these assumptions is the
fact that we deal with single unit trades (as do most of the algorithms
against which we benchmark our agent’s performance). However, even
with this restriction in place, the CDA has shown to be a highly efficient
protocol [11].
1041-4347/03/$17.00 ß 2003 IEEE Published by the IEEE Computer Society
trading period (cf. trades only being allowed at discrete time
points) and which allows buyers and sellers to continuously
update their bids and asks at any time throughout the trading
period [12]. CDAs are widely used in the nononline world to
trade stocks, agricultural commodities, metals, currencies,
and derivative instruments [10].
Against this background, this paper develops and evalu-
ates a new algorithm that autonomous software agents [13]
can employ to submit bids and asks in CDAs. We believe this
is an important step in the development of large-scale open
and distributed systems because the existence of effective
strategies means that CDAs can be more readily deployed as
the marketplace protocol. Without such strategies, there has
been some reluctance to choose CDAs in the online world
even though they are the most obvious protocol in many
cases. Furthermore, we believe that, in the long term, software
agents will be more effective than human bidders in these
more complex auction settings. Preliminary evidence for this
is contained in [14] which shows that agents, employing the
algorithms against which we benchmarked our algorithm,
outperformed their human counterparters in CDAs.
In more detail, the bidding algorithms we develop are
heuristic methods that exploit fuzzy logic techniques [15],
especially fuzzy rules, to undertake their reasoning. The
reason for this choice is that, in CDAs, there is no optimal
bidding strategy [9]. This is because an agent’s decision
making about bidding involves uncertainty, multiple
factors, and nondeterminism that are affected by the
attitudes toward risk of its opponents, the nature of the
market supply (demand), and the preferences of the other
bidders. Since no agent can have all this information in
advance (it is, after all, a competitive environment), the best
that can be achieved is a satisfying strategy [16]. We chose
to adopt a fuzzy logic-based approach, in particular,
because we wish to develop a practicable agent that can
cope with the uncertainties in a timely manner and fuzzy
techniques have proven to be successful in a wide range of
domains with these characteristics (e.g., fuzzy control to
drive car-like vehicles [17], making medical diagnosis [18],
vehicle dispatching [19] and emergency electric power
distribution [20]). See Section 6 for details of the other
alternatives we considered.
The specific contributions of this paper are as follows:
First, we develop a novel fuzzy logic-based bidding
strategy—the FL strategy—for agents that participate in
CDAs. Second, we present the design, implementation, and
evaluation of this strategy for buyer and seller agents. This
strategy is shown, via empirical studies, to outperform the
main strategies that have previously been proposed for
CDAs. Third, we enhance the basic strategy so that it can
adapt its behavior to the supply (demand) in the market
(this revised strategy is called the adaptive FL-strategy). We
then show how this revised strategy leads to a further
improvement in the performance of both the individual
agents (buyers and sellers) and of the overall marketplace.
The remainder of this paper is organized as follows:
Section 2 formalizes a CDA and outlines the basics of our
fuzzy reasoning mechanism. Section 3 presents the
FL-strategy. In Section 4, the behaviors of our FL-agents are
analyzed in a range of experiments. Section 5 discusses the
adaptive FL-agents and their evaluation. Section 6 discusses
the related work. Finally, Section 7 concludes this paper and
discusses the future work.
2PRELIMINARIES
This section outlines the basis of our FL-strategy—present-
ing a formal account of our CDA protocol and describing
the fuzzy reasoning mechanism we employ.
2.1 Continuous Double Auctions
According to the parameterization of CDAs given in [10],
we deal with the situation in which there are more than two
goods in the market; two-way traders and the numbers of
buyers and sellers are greater than three; single indivisible
units are to be traded (thus, at any one time, there is one
outstanding bid and one outstanding ask); the preferences
of the traders are the reservation prices of the goods; and
traders have incomplete information of the market. The
CDA terminates after a specified period of inactivity.
In more detail, there are agents that are willing to sell
goods (s-agents) and agents that are willing to buy goods
(b-agents). A given agent can be either a buyer or a seller in
a given context. Specifically, an ask a is the amount
submitted by an s-agent willing to sell a unit of good. The
lowest ask in the market is called the outstanding ask,
denoted a
o
. Similarly, a bid b is the amount submitted by a
b-agent willing to buy a unit of good. The highest bid in the
market is called the outstanding bid, denoted b
o
. A CDA can
thus be described as a place where s-agents submit asks to
decrease a
o
, while b-agents submit bids to increase b
o
, until
b
o
is not less than a
o
[11]. At this moment, the s-agent that
submits a
o
and the b-agent that submits b
o
can make a
transaction, and the price of the transaction is called the
transaction price. Formally, we have:
Definition 1. The descriptor of a CDA is
P
CDA
¼<g;B; S;V
b
;C
s
;
price
;t
round
>;
where:
1. g is the good to be auctioned.
2. B¼fb
1
; ;b
n
g is the finite set of identifiers of
b-agents, where n is the number of b-agents.
3. S¼fs
1
; ;s
m
g is the finite set of identifiers of
s-agents, where m is the number of s-agents.
4. V
b
¼ðV
!
1
; ;V
!
n
Þ, where V
!
i
ðv
i1
;v
i2
; ;v
in
i
Þ is a
vector of unit valuations of b-agent b
i
. Here, n
i
is the
number of units of g that b
i
requires, and v
ij
is the
valuation value for the jth unit acquired.
5. C
s
¼ðC
!
1
; ;C
!
m
Þ, where C
!
i
ðc
i1
; ;c
im
i
Þ is a vector
of unit costs of s-agent s
i
. Here, m
i
is the number of
units that s
i
wants to sell, and c
ij
is the cost of the jth
unit.
6.
price
is the minimum price step required in the
auction. That is, a b-agent (s-agent) must increase
(decrease) its bid (ask) at n
price
, where n is a
nonnegative integer.
1346 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2003
7. t
round
is used for defining the condition for terminating
the CDA; that is, if there are no new asks or bids
during a time period t
round
, the CDA terminates.
4
Definition 2. A round in a CDA is the time period between two
successive deals or the period from the beginning of the CDA to
the time when the first deal takes place. If a round is the rth
(r 2 IN
þ
) round of the CDA, then r is called the round
number. A CDA usually consists of multiple rounds.
Definition 3. For a CDA that has lasted r (r>0) rounds, let
p
i
(1 i r) denote the price of the ith transaction. A
history H
l
in a CDA is the set of transaction prices during
the last l rounds,
H
l
¼fp
rÿlþ1
; ;p
i
; ;p
r
g;
where p
i
(r ÿ l þ 1 i r) is the transaction price of round i,
and l (l r) is called the history length.
5
The following is the formal definition of the valid
behaviors of agents during a CDA.
Definition 4. A CDA protocol with the descriptor P
CDA
consists of the following steps:
1. r=0.
2. A new round of the CDA starts, r ¼ r þ 1, a
o
¼1,
and b
o
¼ 0.
3. Several situations might arise during a round:
a. When an s-agent submits an ask a,
i. if a a
o
then a is an invalid ask;
ii. if b
o
<a<a
o
, then a
o
is updated to a;
iii. if a b
o
, then this s-agent makes a deal at b
o
;
goto 2.
b. When a b-agent submits a bid of b,
i. if b b
o
, then b is an invalid bid;
ii. if b
o
<b<a
o
, then b
o
is updated to b;
iii. if b a
o
, then this b-agent makes a deal at a
o
;
goto 2.
4. Step 3 repeats until no new bids (asks) are submitted
during a time period t
round
.
As can be seen, the outstanding ask and outstanding bid
define the bid-ask spread ½b
o
;a
o
[11] and only bids and asks
that fall within this region are considered valid.
2.2 Fuzzy Reasoning Mechanisms
The fuzzy reasoning inference mechanism employed in this
paper is based on the Sugeno controllers [21], [22]. Consider
the following block of fuzzy IF-THEN rules:
R
1
:ifx is A
1
and y is B
1
then z
1
¼ c
1
also
R
2
:ifx is A
2
and y is B
2
then z
2
¼ c
2
also
.
.
.
also
R
n
:ifx is A
n
and y is B
n
then z
n
¼ c
n
fact : x is x
0
and y is y
0
consequence : z
0
;
where A
1
; ;A
n
and B
1
; ;B
n
are fuzzy sets, and
z
1
; ;z
n
are real numbers. The firing level
i
of the rules
R
i
is computed by the Min operator. That is,
i
¼ minfA
i
ðx
0
Þ;B
i
ðy
0
Þg; ð1Þ
where A
i
ðxÞ and B
i
ðyÞ are the membership functions of the
corresponding fuzzy sets A
i
and B
i
, respectively. If the
output of the individual rule is denoted as z
i
,then
according to the Sugeno controller definition, the crisp
control action of the rule base is obtained by:
z
0
¼
P
n
i¼1
i
z
i
P
n
i¼1
i
: ð2Þ
The extension principle [15] is one of the main means of
fuzzifying a formula with crisply defined numbers. In
particular, we extend (2) to the situation where these real
numbers z
i
(1 i n) are changed to triangular fuzzy
numbers. We made this change because in developing our
rules, we felt unable to estimate the action using a single
real value chosen from within a predefined range. Rather,
we found it easier to estimate a parameter with fuzzy values
and this led us to use triangular fuzzy numbers [23]. Also,
by the extension principle, arithmetic operations on
trapezoidal fuzzy numbers have already been obtained
[24], [25] and fuzzy triangular numbers are special cases of
fuzzy trapezoidal numbers [26]. Thus, the arithmetic
operations on fuzzy triangle numbers can be obtained from
the arithmetic operations on fuzzy trapezoidal numbers.
Given all this, in our inference mechanism, the output of
each rule is a triangular fuzzy number defined with the
following triple:
~aa ¼ðm; ; Þ;
where m is called the center, and and are called
the left and right spreads, respectively [27] (Fig. 1). For
two triangular fuzzy numbers
~
aa
1
¼ðm
1
;
1
;
1
Þ, and
~
aa
2
¼
ðm
2
;
2
;
2
Þ (~aa
1
; ~aa
2
> 0) and k 2 IR , the following formu-
lae hold [24], [25]:
~
aa
1
þ
~
aa
2
¼ðm
1
þ m
2
;
1
þ
2
;
1
þ
2
Þ;
~
aa
1
ÿ
~
aa
2
¼ðm
1
ÿ m
2
;
1
þ
2
;
2
þ
1
Þ;
~
aa
1
~
aa
2
¼ðm
1
m
2
;m
1
2
þ m
2
1
ÿ
1
2
;m
1
2
þ m
2
1
þ
1
2
Þ;
k
~
aa
1
¼ðkm
1
;k
1
;k
1
Þ:
From the above formulae, (2) can be extended to the
following in the situation where
~
zz
i
¼ðm
i
;
i
;
i
Þ (1 i n):
HE ET AL.: A FUZZY-LOGIC BASED BIDDING STRATEGY FOR AUTONOMOUS AGENTS IN CONTINUOUS DOUBLE AUCTIONS 1347
4. Note that, by this definition, we exclude from this paper CDAs that
last infinite periods of time (such as stock markets). To model this, t
round
can
be set to infinity.
5. Through experiments where both the history length (l) and the value
(cost) of the goods that the agents trade varied, the performance of the
agents with different history lengths was investigated. The results showed
that the behavior of FL-agents with a long history length (l>20) was
similar to or worse than that of an agent with a history length ranging from
3to20. This result shows that agents with short or intermediate history
lengths can react more rapidly to changes in a CDA market. When the
history length varied from 3 to 20, we found that 10 was a reasonable
history length where almost all the agents achieve their highest profit. Thus,
this is the value selected for all the experiments in the rest of this paper.
~
z
0
z
0
¼
P
n
i¼1
ð
i
~
zz
i
Þ
P
n
i¼1
i
¼
P
n
i¼1
ð
i
ðm
i
;
i
;
i
ÞÞ
P
n
i¼1
i
¼
P
n
i¼1
ð
i
m
i
Þ
P
n
i¼1
i
;
P
n
i¼1
ð
i
i
Þ
P
n
i¼1
i
;
P
n
i¼1
ð
i
i
Þ
P
n
i¼1
i
:
ð3Þ
Thus, the reasoning mechanism becomes:
R
1
:ifx is A
1
and y is B
1
then
~
zz
1
is
~
cc
1
also
R
2
:ifx is A
2
and y is B
2
then
~
zz
2
is
~
cc
2
also
.
.
.
also
R
n
:ifx is A
n
and y is B
n
then
~
zz
n
is
~
cc
n
fact : x is x
0
and y is y
0
consequence :
~
zz
0
Having defined the protocol and the reasoning mechan-
ism, we can now turn to the FL-strategy itself.
3THE FL-STRATEGY
Building on the foundations of the previous section, this
section describes our FL-strategy and demonstrates how it
works in an exemplar scenario.
3.1 Basic Notation and Concepts
In order to detail the FL-strategy, we first need to introduce
a number of underpinning notations and concepts.
Definition 5. A situation s
during the course of a CDA is a
6-tuple,
s
¼<r; B; S;a
o
;b
o
;H
l
>;
where r is the current round number, B and S are the sets
of b-agents and s-agents; a
o
and b
o
are the outstanding ask
and the outstanding bid, respectively, and H
l
is the history
of the last l rounds.
6
Definition 6. Given a situation s
, the valid bids set (D
b
) is the
set of the valid bids that a b-agent could submit:
D
b
¼fb j b
o
<b minða
o
;v
ij
Þg; ð4Þ
where b is the price at which a b-agent submits a bid and v
ij
is
the valuation of the jth unit of the good by buyer i.
Definition 7. Given a situation s
, the valid asks set (D
s
)is
the set of valid asks that an s-agent could submit:
D
s
¼fa j maxðb
o
;c
ij
Þa<a
o
g; ð5Þ
where a is the price at which an s-agent submits an ask and c
ij
is the cost of the jth unit of the good for seller i.
The prices of previous transactions are stored as history
and may be referred to by the agents in the subsequent
rounds. Generally speaking, CDA markets produce very
efficient allocations and prices [28], and the transaction
prices often converge to a competitive equilibrium price
7
while the CDA is in progress. Thus, the transaction prices
in a CDA provide an important point for reference. To
reflect this fact, we define the reference price P
R
in the
situation s
as the median of the ordered price history.
8
A
reference price, as its name suggests, provides a reference
point that an agent can use to guide its subsequent
bidding behavior. Formally, we have:
Definition 8. Let r be the current round number (r>0).
Suppose the price history is a series of prices
H
l
¼fp
rÿl
; ;p
i
; ;p
rÿ1
g;
where p
i
(r ÿ l i r ÿ 1) is the price in round i. Let their
ordered series be denoted as
p
ð1Þ
p
ðiÞ
p
ðlÞ
: ð6Þ
Then, the reference price, P
R
, is given by
P
R
¼ p
ðb
lþ1
2
cÞ
: ð7Þ
To summarize, when an agent submits its next ask (bid),
it will consider the outstanding ask, the outstanding bid, the
cost (valuation) of the current unit of good, and the
reference price. The way in which these values are used is
described in the next section.
3.2 Fuzzy Reasoning in the FL-Strategy
The FL-strategy is based on a number of heuristic rules and
the fuzzy reasoning mechanism outlined in Section 2.2. The
relation of P
R
, a
o
, and b
o
during a round in a CDA falls into
one of the cases below:
1. P
R
b
o
<a
o
,
2. b
o
<a
o
P
R
, and
3. b
o
P
R
a
o
.
In the first two cases, we use some heuristic rules (given
below); the bidding issue in the third case, which is more
complicated, is handled through the fuzzy reasoning
mechanism on a rule base (described at the end of this
section). Fig. 2 describes all the fuzzy sets used in the
heuristic rules. The heuristic rules applied in the first two
cases for s-agents are:
. When P
R
b
o
<a
o
, the heuristic rule is:
1348 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2003
6. Recall that l is the remembered history length of an agent and, thus, l is
not necessarily equal to r ÿ 1.
7. The equilibrium price is determined by the intersection of the supply
and demand curves of the market, and it is the point where the quantity
supplied is equal to the quantity demanded [29].
8. Originally, both the mean and the median of the ordered price history
were used; however, experimental results showed that the median is more
effective in providing a reference price. This is because the mean price can
be overly influenced by a too high (low) price offered by an irrational agent.
In contrast, the median of the ordered price history is less susceptible to
such bias.
Fig. 1. Triangular fuzzy number
~
aa ¼ðm; ; Þ, where m is the center, is
left spread, and is right spread.
ðSR
1
Þ IF b
o
is much bigger than P
R
THEN accept b
o
ELSE ask is ða
o
ÿ
s;1
;;Þ:
. When b
o
<a
o
P
R
, the heuristic rule is:
ðSR
2
Þ IF a
o
is much smaller than P
R
THEN no new ask
ELSE ask is ða
o
ÿ
s;2
;;Þ:
Intuitively, SR
1
states that when the outstanding bid b
o
is
much_bigger than the reference price P
R
, it is already very
profitable for an s-agent to accept the current outstanding
bid. The relation “b
o
is much_bigger than P
R
”canbe
expressed as fuzzy set A
1
. Let the threshold be
s;1
, that is,
if A
1
ðb
o
Þ
s;1
, the s-agent will accept b
o
. At this point, a
transaction takes place between the s-agent and the b-agent
which submits the outstanding bid. Otherwise, the s-agent
will decrease the outstanding ask a
o
to a fuzzy number ða
o
ÿ
s;1
;;Þ (see Section 2.2), where a
o
ÿ
s;1
is the center of the
new ask, and and are the left and right spread.
s;1
shows
how much the agent would like to decrease its ask and this is
decided by the agents’ attitude to risk (to be discussed in
Section 4.2). SR
2
is applied when a
o
is much smaller than P
R
.
At this moment, an s-agent is in an unfavorable position and
it should be reluctant to decrease a
o
. Thus, the s-agent only
decreases a
o
by a small step. The relationship “a
o
is
much_smaller than P
R
” is expressed as a fuzzy set A
2
. Let
s;2
be the threshold, that is, if A
2
ða
o
Þ
s;2
, the agent
believes the current ask is much smaller than P
R
. In this case,
the s-agent will not submit a new ask.
Similar heuristic rules also apply to b-agents:
. When b
o
<a
o
P
R
, the heuristic rule is:
ðBR
1
Þ IF a
o
is much smaller than P
R
THEN accept a
o
ELSE bid is ðb
o
þ
b;1
;;Þ:
. When P
R
b
o
<a
o
, the heuristic rule is:
ðBR
2
Þ IF b
o
is much bigger than P
R
THEN no new bid
ELSE bid is ðb
o
þ
b;2
;;Þ:
The relationship “a
o
is much_smaller than P
R
” can be
expressed as a fuzzy set A
3
. Let
b;1
be the threshold, that is,
if A
3
ða
o
Þ
b;1
, a
o
is regarded as being much smaller than
P
R
, and a b-agent will accept a
o
; otherwise, a b-agent will
increase b
o
to a fuzzy number ðb
o
þ
b;1
;;Þ. The fuzzy set
A
4
defines the relationship “b
o
is much_bigger than P
R
.” Let
b;2
be the threshold for this rule, that is, if A
4
ðb
o
Þ
b;2
,a
b-agent will not submit a new bid because b
o
is already high
enough and no profit can be made according to its
preference; otherwise, it will increase b
o
to a fuzzy number
ðb
o
þ
b;2
;;Þ. In the above, P
1
, P
2
, P
3
, and P
4
are the
parameters of the fuzzy sets (see Fig. 2) and they are
decided by human intuition and experience according to
the range of the cost and valuation of the goods. The fuzzy
number produced by these heuristic rules is dealt with in
the same way as the fuzzy number produced by the
reasoning mechanism (which we will discuss below).
Now, for the third case (b
o
P
R
a
o
), the fuzzy reason-
ing on a rule base is required. First, the rule bases for the s-
agents and b-agents are presented in Tables 1 and 2,
respectively. Again, the fuzzy numbers are all triangular
fuzzy numbers as described in Section 2.2; the distance
between a
o
(or b
o
) and P
R
is expressed using the fuzzy
linguistic terms: far
from, medium to, and close to, which
are defined in Fig. 3, and or corresponds to operator Max.
s;1
; ;
s;4
and
b;1
; ;
b;4
are parameters decided by the
risk attitude of the agent (see Section 4.2). Based on these
rule bases, we can perform inference through the fuzzy
reasoning mechanism presented in Section 2.2. The overall
output of our fuzzy reasoning is a fuzzy number, i.e., a set
of asks (bids) with membership degrees. For example,
~
zz
may equal ð2:0; 0:02; 0:04Þ, where 2:0 is the center, 0:02 is the
left spread, and 0:04 is the right spread, and its membership
degree might be given by:
HE ET AL.: A FUZZY-LOGIC BASED BIDDING STRATEGY FOR AUTONOMOUS AGENTS IN CONTINUOUS DOUBLE AUCTIONS 1349
Fig. 2. Fuzzy sets in heuristic rules. (a) Outstanding bid is
much_bigger then P
R
. (b) Outstanding ask is much_smaller the
P
R
. (c) Outstanding ask is much_smaller then P
R
. (d) Outstanding
bid is much_bigger than P
R
.
TABLE 1
Fuzzy Rule Base for s-agents
