The aim of the work is to investigate how agents can become more intelligent using contemporary methods for decision making in electronic commerce auctions. The idea is to implement fuzzy logic for agents' evaluations of market conditions and sort their preferences using difference combinations of fuzzy input data to make rational choice which is the best behavior under given market conditions. The task is to prove that a family of decision making algorithms, created in the Bulgarian Academy of Sciences, can be embedded in existing prototype of multi-agent simulator for online auctions for change of agents' bidding strategy.
The behavior of agents in an auction is based on their bidding strategies. A strategy is a methodology which the agent implements to achieve its goals while following the auction rules. Strategies are private and are chosen by auction participants (agents’ owners). Various protocols for bidding are used in practice, so there is no universal strategy for successful negotiation. A given strategy can be effective under one protocol and ineffective under another. Creation of an optimal strategy for continuous double auction (CDA) is a complex task that still challenges electronic commerce researchers. The aim is creating strategies that would pick the “right” deal sides, so that the CDA efficiency would be maximized and there would be quick deal price convergence toward the equilibrium price. The goal of this work is to investigate two alternative algorithm families for multi-criteria analysis with fuzzy logic for bidding strategy selection in CDA. The described methods for decision making can be used in electronic auctions not only for preliminary selection of the most suitable strategy from a given set of agent strategies, but also for changing the used strategy during auction. For detailed investigation of intelligent agents’ behavior as participants in electronic commerce, various auction models have been simulated. The suggested methods for bidding strategies evaluation are based on two different approaches. The first one of them is essentially a comparison between strategies’ efficiency in a static agent population (the strategies are selected in advance and cannot be changed once an auction has begun). The second approach investigates agent populations in which change of the used strategy after the beginning of an auction is allowed using dynamics replicator.
Anthony and Jennings, who follow the first approach, generate biddings that depend
on the following parameters: time left until the end of the auction; number of open
auctions; agent’s intention towards deal and agent’s attitude towards risk. The
combination of these four indices through relative weighting coefficients, defined by
the user, gives a bidding strategy. Later, the two authors suggest a genetic algorithm
for search of an effective strategy from the solution set defined by the specific market
conditions [
In the current work, the applicability of two algorithms with fuzzy logic with a
common purpose of solving the task of multi-criteria ordering of bidding strategies in
an auction is investigated - algorithm with Fuzzy Techniques and Negotiable
Attitudes (FTNA) [
The input data for decision making procedure are: n - a number of alternatives; m - a number of criteria; l – a number of bidding agents.
During the experiment, we use the next parameter values: Five strategies as alternatives for agents’ bidding in CDA: A1 – snipping strategy (Snipping), A2 – strategy with fixed markup (L), A3 - zero intelligence with budget constraints (ZIC), A4 - zero intelligence plus (ZIP) and A5 – risk based strategy (RB).
These strategies are the most cited in literature sources.
For example, we can use the next three criteria for strategies’ evaluation: C1 – time complexity, C2 – price prediction, C3 – risk attitude.
All criteria are maximizing. Five bidding agents participate in the experiments. 3.1
Algorithm with Fuzzy Techniques and Negotiable Attitudes (FTNA)
Step 1: The decision maker (DM) determines the relative weighting coefficients of
the criteria for each strategy by using the method of analytical hierarchic process. The
values of the weights depend on the degree of importance of the given criterion. The
fuzzy relations (matrices for comparison) of the criteria are completed according to
the degree of importance of the paired criteria. Evaluations vary in the range from 1 to
9: 1-insignificantly important; 3-more important; 5-equally important; 7-substantially
more important; 9-absolutely more important, and ranks 2,4,6,8 represent values that
are between the given ones. The weight of each criterion is given by the formula of
geometric mean of the corresponding row of the comparison matrix. If we let w to be
the set of weights and w={w1,w2,..,wm}, then here we will also have wi∈[
To fulfill the matrices we use experts’ knowledge of evaluation of bidding strategies features.
Step 2: The agent’s attitude towards the bidding strategies and the criteria for their evaluation are determined. Here “attitude” represents the preference of agent k to choose a strategy i with criterion j. The evaluations of the attitudes aijk (i=1,2,..,n, j=1,2,..,m, k=1,2,..,l) are presented through linguistic terms such as “very low”, “low”, “average”, “high”, and “very high”. The fuzzy agent relations towards strategies and Ак = (a ) k criteria are completed in the attitude matrices ij nxm (Figure 2). To fulfill the matrices we use historical data from auction deals until the present moment.
Step 3: Each attitude matrix is aggregated into attitude vector Ai (1=1,2,..,n) as follows: Aik = w1k aik1 + w2k aik2 + ... + w a k k n in (Figure 3).
Step 4: We assume that all the agents are equally important and calculate the normalized vector of the fuzzy solution r. The normalized weight of the agents Dk (k=1,2,...,l) is denoted with (ν 1, ν 2 ,...,ν l):
a11 a12 .. an1 (r1, r2 ,..,rn ) = (υ1,υ 2 ,..,υl )a.1.2 a..22 .... a..n2 a1l a2l .. anl
Step 5: The elements of the normalized vector of the fuzzy solution ri are positive
triangular fuzzy numbers and belong to the interval [
Step 6: To determine the rank of each strategy the coefficient of closeness is calculated with the formula: CCi= 12 (di++ (1 - di- ) for i=1,2,...,n.
The strategy with the largest coefficient of closeness is the most appropriate one for bidding at that moment (Figure 6, left – before, right – after sort). 3.2 Algorithm with Fuzzy Numbers as Alternatives Evaluations and Real Numbers as Weighted Coefficients (ARAKRI1) Step 1: The evaluations of the alternatives (strategies) by the criteria are fuzzy triangular numbers (Figure 7).
According to [
~ ~ ~
F ( A)= kF1 ( A+) (1 - k)F2 ( A), k ∈[
2 F2 ( A~) = a4 + S A~ sin[R, f (x)=] a4 - (a 4- a1 +)2 (a3 - a2 ) × 1 1
1 [(a4 - a3 )2 + 1] 2 and
1 [(a2 - a1 )2 + 1] 2 (Figure 8).
Step 2: As the criteria evaluation can be expressed in different measurement units, for their unification is used the following transforming function: 1 μ k (ai , a j ) = 0.5 +
xik - x jk 2(maix{xik } - miin{xik }) ако i = j ако i ≠ j where xik , x jk , i, j = 1,2,..., n, k = 1,2,..., m are the evaluations of the alternatives ai and a j , i, j = 1,2,..., n according to the criterion ck , k = 1,2,..., m . The obtained fuzzy relations are Rk , k = 1,2,..., m . If a certain criterion, ck, is minimizing, in order for the alternatives to be sorted in a descending order, the complement to the relation Rk ' =1 - Rk is calculated, in other words, for this relation new membership function is calculated using the formula μk '(ai , a j ) = 1 - μk (ai , a j ) (Figure 9).
Step 3: All relations R1, R2 ,..., Rm are combined so that an aggregated relation R with
the following matrix can be obtained with the membership function
μ (ai , a j ) = Agg{μ 1 (ai , a j ), μ 2 (ai , a j ),..., μ m (ai , a j )}
Each element from matrix R is calculated by the aggregation operators’ formula with
weighting coefficients. The following operators are used: WMean, WGeom,
WMaxMin and WMinMax with weighting coefficients-real numbers. If w is the set of
average criteria weights from the FTNA algorithm and w={w1,w2,...,wm}, then
wk∈[
k=1 The calculations with the operators WMean, WGeom, WМaxMin and WМinMax for degree of membership to each of the aggregated relations R of the pair на двойката (ai , a j ) are as follows:
m μ (ai , a j ) = WMean {μ1(ai , a j ),μ 2 (ai, a j ),...,μ m (ai , a j )} = ∑k=1 wkμ k (ai , a j )
m μ (ai ,a j ) = WGeom{μ1(ai , aj ),μ 2(ai , a j ),...,μ m (ai ,a j )} = ∏[μ k (ai , aj )]wk
k=1
μ (ai , a j ) = WMaxMin {μ1(ai , a j ),μ 2 (ai , a j ),..., μ m (ai , a j )} = mkax{min( μ k (ai , a j ), wk )}
μ (ai , a j ) = WMinMax {μ1(ai , a j ),μ 2 (ai , a j ),..., μ m (ai , a j )} = mkin{max(μ k (ai , a j ), wk )}
For the last two operators, WMaxMin and WMinMax the weights of the criteria are
recalculated so that they belong to the interval [
k=1 Four obtained aggregated matrices are shown in Figure 10.
Step 4: The four matrices from Figure 10 are from type R. Each of these matrices is recalculated so that matrices R' are obtained in the following way: if μ (ai , a j ) ≥ μ (a j , ai ) , then μ '(ai , a j ) = μ (ai , a j ) and μ '(a j , ai ) = 0 . Every asymmetric fuzzy rearrangement R' of R is fuzzy partial order. R' can be rearranged into a triangular matrix. After the triangular matrix R' is rearranged, a relation is obtained which represents a fuzzy linear arrangement. A non-fuzzy order of the alternatives is the same as their order in the title raw of the obtained table and is the solution to the problem of multi criteria arrangement (Figure 11). 4 Results Comparison from the Multi Criteria Analysis with the FTNA and ARAKRI1 algorithms
The proposed method for bidding strategies selection is appropriate for the development and modeling of economic objects. The work can serve as a basis for the practical orientation of teaching in the following fields: Electronic Commerce, Electronic Business etc. As a result, this would sharpen the motivation and interest of students in those fields and would provide them with practical instruments. On the other hand, realization of these types of methods with fuzzy logic broadens the practical applicability of related mathematical tasks and will supply additional information for future theoretical research.
Acknowledgments. The work reported in this paper is partially supported by the projects No. RS11-FESS-0xx/30.05.2011 „Intelligent methods for business information processing”.