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IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 109 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
USING AN ANALYTIC NETWORK PROCESS MODEL
IN COMBINATORIAL AUCTIONS 1
Petr Fiala
Department of Econometrics, University of Economics
Praha, 130 67, Czech Republic
E-mail: pfiala@vse.cz
ABSTRACT
Auctions are important market mechanisms for the allocation of goods and services.
Auctions are often preferred to other common processes because they are open, quite fair,
and easy to understand by participants, and lead to economically efficient outcomes.
Design of auctions is a multidisciplinary effort made of contributions from economics,
operations research, informatics, and other disciplines. Combinatorial auctions are those
auctions in which bidders can place bids on combinations of items, called “packages,”
rather than just individual items. The advantage of combinatorial auctions is that the
bidder can more fully express his preferences. This is particularly important when items
are complements. The multiple evaluation criteria can be used. There are dependencies
among sellers, buyers, criteria and bundles of items. A variety of feedback processes
creates complex system of items. The whole structure seems to be very appropriate to the
Analytic Network Process (ANP) approach. The ANP method makes it possible to deal
systematically with all kinds of dependence and feedback in the system of items. By
using the ANP approach, the preferences of bundles of items can be evaluated. Dynamic
Network Process (DNP) as an extension of ANP can deal with time-dependent priorities
in combinatorial auctions.
Keywords: combinatorial auctions, preference elicitation, Analytic Network Process,
Dynamic Network Process
Introduction
Auctions are important market mechanisms for the allocation of goods and services.
Many modern markets are organized as auctions. Combinatorial auctions are those
auctions in which bidders can place bids on combinations of items, so called bundles. The
advantage of combinatorial auctions is that the bidder can more fully express his
preferences. This is particularly important when items are complements. The auction
designer also derives value from combinatorial auctions. Allowing bidders more fully to
express preferences often leads to improved economic efficiency and greater auction
overall valuation. However, alongside their advantages, combinatorial auctions raise a
host of questions and challenges (Cramton et al. 2006, Rothkopf et al. 1998).
Auction theory has caught tremendous interest from both the economic side as well as the
Internet industry. An auction is a competitive mechanism to allocate resources to buyers
based on predefined rules. These rules define the bidding process, how the winner is
1 The paper is supported by the Grant Agency of Czech Republic (grant 402/07/0166
“Combinatorial auctions – modeling and analysis“).
Rob
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http://dx.doi.org/10.13033/ijahp.v1i2.47
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 110 Vol. 1 Issue 1 2009 ISSN 1936-
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determined, and the final agreement. In electronic commerce transactions, software
agents that negotiate on behalf of buyers and sellers conduct auctions. The popularity of
auctions and the requirements of e-business have led to growing interest in the
development of complex trading models (see Bellosta et al. 2004, Bichler 2000, Oliveira
et al. 1999).
Classification of auctions is based on some specific characteristics as:
• the numbers of sellers and buyers,
• the number of items,
• traded items (indivisible, divisible, pure commodities, structured commodities),
• participants’ roles in auctions (one-sided, multilateral auctions),
• preferences of the participants,
• the form of the private information participants have about preference,
• objectives of auctions (optimization, allocation rules, pricing rules),
• evaluating criteria,
• complexity of bids (simply, related bids),
• organization of auctions (single-round, multi-round, sequential, parallel, price
schemes).
Many specific factors of combinatorial auctions can be modeled and analyzed by
Analytic Network Process and Dynamic Network Process (ANP/DNP) methods.
1. Multicriteria combinatorial auctions
For our model, we propose to use multidimensional auctions. These auctions can be
classified:
• multi-unit auction,
• multi-item auction,
• multi-criteria auction,
Multi-unit auctions contain multiple units of items and make possible volume discount
auction. Multi-item auctions can place bids on combinations of items, so called
combinatorial auctions. The combinatorial auctions can define multiple criteria as:
• revenue maximization - the seller should extract the highest possible price,
• efficiency - the buyers with the highest valuation get the goods,
• collusion possibility,
Auctions with complex bid structures are also called multicriteria auctions because they
address multiple attributes of the items (quality, quantity, price) in the negotiation space.
Multicriteria optimization can be helpful for detailed analysis of combinatorial auctions.
Buyers can specify aspiration levels that express their desired values on the attributes of
the items to be purchased and reservation levels that represent the minimal values
required. There are possible combinations of the multidimensional characteristics as
multi-criteria, multi-unit, multi-item auctions.
Multicriteria auctions require several key components to automate the process:
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 111 Vol. 1 Issue 1 2009 ISSN 1936-
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• a preference model,
• a multicriteria optimization model,
• a negotiation model.
The preference model is used to let the buyer express his preferences. The multicriteria
optimization model selects the best offer for the buyer. The negotiation model helps to
find a consensus in auctions (Fiala, 1997).
2. Winner determination problem
The problem, called the winner determination problem, has received considerable
attention in the literature. The problem is formulated as: Given a set of bids in a
combinatorial auction, find an allocation of items to bidders to maximize the seller's
revenue. It has introduced many important ideas, such as the mathematical programming
formulation of the winner determination problem, the connection between the winner
determination problem and the set packing problem, as well as the issue of complexity.
There are some different approaches to solve winner determination problems. Main
approaches for solving the problem are exact or approximate algorithms. The algorithms
have some pros and cons. We propose to use an iterative approach for solving winner
determination problems. The iterative approach is based on a primal-dual algorithm.
Problem formulation
Many types of combinatorial auctions can be formulated as mathematical programming
problems. From different types of combinatorial auctions we present an auction of
indivisible items with one seller and several buyers. Let us suppose that one seller offers
a set G of m items, j = 1, 2, …, m, to n potential buyers. Items are available in single
units. A bid made by buyer i, i = 1, 2, …, n, is defined as
Bi = {S, vi(S)},
where
S ⊆ M, is a combination of items,
vi(S), is the valuation by buyer i for the combination of items S, the valuation is made by
multiple criteria.
The objective is to maximize the overall valuation of the seller given the bids made by
buyers. Constraints establish that no single item is allocated to more than one buyer and
that no buyer obtains more than one combination.
Bivalent variables are introduced for model formulation:
xi(S) is a bivalent variable specifying if the combination S is assigned to buyer i (xi(S) =
1).
The winner determination problem can be formulated as follows
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 112 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
∑
=
n
i 1
∑
⊆MS
vi(S) xi(S) → max
subject to
∑
⊆MS
xi(S) ≦ 1, ∀ i, i = 1, 2, …, n,
∑
=
n
i 1
∑
⊆MS
xi(S) ≦ 1, ∀ j ∊ M, (1)
xi(S) ∊ {0, 1}, ∀ S ⊆ M, ∀ i, i = 1, 2, …, n.
The objective function expresses the overall valuation. The first constraint ensures that no
bidder receives more than one combination of items. The second constraint ensures that
overlapping sets of items are never assigned.
The winner determination problem, i.e., determining the items that each bidder wins, is
not difficult in the case of non-combinatorial auctions. It would take O(nm) time where n
is the number of bidders and m is the number of items. But in the case of combinatorial
auctions, the winner determination problem is much more complex. The payment
determination problem also is an important and non-trivial problem.
Complexity of the problem
Complexity is a fundamental question in combinatorial auction design. There are some
types of complexity:
• computational complexity
• valuation complexity
• strategic complexity
• communication complexity
Computational Complexity covers such questions as: How much computation is
expected of the mechanism to compute an outcome given the bid information of the
bidders? This is an extremely important question because winner determination problem
is an NP-complete optimization problem. The winner determination problem turns out to
be an instance of a weighted set packing problem. The weighted set packing problem is a
problem of finding a disjoint collection of weighted subsets of a larger set with maximal
total weight. Weighted set packing is a classical NP-complete problem.
Valuation complexity deals with such questions as: How much computation is required
to provide preference information within a mechanism? Estimating every possible bundle
of items requires exponential space and hence exponential time. Bidders need to
determine valuations for 2m -1 possible bundles.
Strategic complexity concerns such questions: Which of the 2m -1 bundles to bid on?
What is the best strategy for bidding? Must bidders model behavior of other bidders and
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 113 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
solve problems to compute an optimal strategy? For instance, in a sealed bid
combinatorial procurement scenario, sellers must not only take their valuation of the
bundles into consideration but also the bidding behavior of their competitors. This
requires sophisticated bidding logic.
Communication complexity concerns such questions as: How much communication is
required between bidders and auctioneer until an equilibrium price is reached the
mechanism to compute an outcome? The amount of communication between the bidders
and the auctioneer can become quite high. For instance, in an iterative combinatorial
auction, where individual valuations are revealed progressively in an iterative manner,
the communication costs could be high if the auction were conducted in a distributed
manner over space and/or time. The problem of communication complexity can be
addressed through the design of careful bidding languages that provide expressive but
concise bids.
Solving the problem
The algorithms proposed for solving the winner determination problem fall into two
classes:
• exact algorithms
• approximate algorithms
Exact approaches to solving the winner determination problem require algorithms that
generate both good lower and upper bounds on the maximum objective function value. In
general, the upper bound on the optimal solution value is obtained by solving a relaxation
of the optimization problem. There are two standard relaxations for winner determination
problem: Lagrangean relaxation and the linear programming relaxation. By Lagrangean
relaxation the feasible set is usually required to maintain 0-1 feasibility, but many if not
all of the constraints are moved to the objective function with a penalty term. By LP
relaxation only the integrality constraints are relaxed, the objective function remains the
original function. Exact methods come in three varieties: branch and bound, cutting
planes and a hybrid called branch and cut. Integer programming techniques can be used
to handle winner determination in combinatorial auctions with a small enough number of
items.
There are a few special cases where the winner determination problem can be solved
exactly using polynomial time algorithms. For example, paper (de Vries and Vohra.
2003) looks at polynomially solvable instances of the winner determination problem in
terms of the structure of the constraint matrix: totally unimodular matrices, balanced
matrices, and perfect matrices. A matrix is said to be totally unimodular if the
determinant of every square submatrix is 0, 1, or −1. A 0-1 matrix is called balanced if it
has no submatrix of odd order with exactly two 1’s in each row and column. A matrix is
said to be perfect if it is the vertex-clique adjacency matrix of a perfect graph. In all these
cases, it is shown that the winner determination problem can be solved as a linear
program.
Approximate algorithms have also emerged as a major approach to solving the allocation
problem in combinatorial auctions. One way of dealing with hard integer programs is to
give up on finding the optimal solution. An approximation algorithm is a polynomial time
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 114 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
algorithm with a provable performance guarantee. Approximate algorithm seeks a
feasible solution fast and hopes that it is near optimal. Fast approximation algorithms for
the winner determination problem can be obtained by translating results for combinatorial
problems related to winner determination, mainly the weighted set packing problem and
the weighted stable set problem to the context of winner determination (Sandholm 2002).
Approximate algorithms and heuristics include greedy algorithms, simulated annealing,
genetic algorithms and others.
3. Preference elicitation
The key feature that makes combinatorial auctions most appealing is the ability for
bidders to express complex preferences over bundles of items, involving
complementarity and substitutability. Items are complements when a set of items has
greater utility than the sum of the utilities for the individual items. Items are substitutes
when a set of items has less utility than the sum of the utilities for the individual items.
Two items A and B are complementary, if it holds
v({A, B}) > v({A}) + v({B}).
Two items A and B are substitute, if it holds
v({A, B}) < v({A}) + v({B}).
Different elicitation algorithms may require different means of representing the
information obtained by bidders. Sandholm and Boutilier (2006) describe a general
method for representing an incompletely specified valuation functions. A constraint
network is a labeled directed graph consisting of one node for each bundle b representing
the elicitor's knowledge of the preferences of a bidder. A directed edge (a, b) indicates
that bundle a is preferred to bundle b. Figure 1 represents an example of a constraint
network for bundles of three items (A,B,C).
Figure 1 Constraint network.
{A,B,C}
{A,B} {A,C}
{A}
{B,C}
{B} {C}
∅
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 115 Vol. 1 Issue 1 2009 ISSN 1936-
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The constraint network representation is useful conceptually, and can be represented
explicitly for use in various elicitation algorithms. But its explicit representation is
generally tractable only for small problems, since it contains 2m nodes. Preference
elicitation of bundles in a constraint network can be used Analytic Network Process.
The Analytic Hierarchy Process (AHP) is a method for setting priorities (Saaty, 1996). A
priority scale based on reference is the AHP way to standardize non-unique scales in
order to combine multiple performance measures. The AHP derives ratio scale priorities
by making paired comparisons of elements on a common hierarchy level by using a 1 to 9
scale of absolute numbers. The absolute number from the scale is an approximation to the
ratio wj/wk and then is possible to derive values of wj and wk. The AHP method uses the
general model for synthesis of the performance measures in the hierarchical structure.
j
1
n
i jk
j
u v w
=
= ∑ .
The Analytic Network Process (ANP) is the method (Saaty, 2001) that makes it possible
to deal systematically with all kinds of dependence and feedback in the performance
system. The well-known AHP theory is a special case of the Analytic Network Process
that can be very useful for incorporating linkages in the system.
The structure of the ANP model is described by clusters of elements connected by their
dependence on one another. A cluster groups elements that share a set of attributes. At
least one element in each of these clusters is connected to some element in another
cluster. These connections indicate the flow of influence between the elements (see
Figure 2).
Sellers
Criteria Buyers
Bundles
Figure 2 Clusters and connections in multicriteria combinatorial auctions.
The clusters in multicriteria combinatorial auctions can be sellers, buyers, bundles of
items, and evaluating criteria. Paired comparisons are inputs for preference elicitation in
combinatorial auctions. A supermatrix is a matrix of all elements by all elements. The
weights from the paired comparisons are placed in the appropriate column of the
supermatrix. The sum of each column corresponds to the number of comparison sets. The
weights in the column corresponding to the cluster are multiplied by the weight of the
cluster. Each column of the weighted supermatrix sums to one and the matrix is column
stochastic. Its powers can stabilize after some iteration to the limit supermatrix. The
columns of each block of the matrix are identical in many cases, though not always, and
we can read off the global priority of the elements.
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 116 Vol. 1 Issue 1 2009 ISSN 1936-
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Recent work has focused on the question of how to limit the amount of valuation
information provided by bidders by adaptively limiting the precision of the bids that are
specified.
4. Iterative combinatorial auctions
In the iterative approach, there are multiple rounds of bidding and allocation and the
problem is solved in an iterative and incremental way. Iterative combinatorial auctions
with multiple criteria are proposed in this paper as complex trading models.
One way of reducing some of the computational burden in solving the winner
determination problem is to set up a fictitious market that will determine an allocation
and prices in a decentralized way. In the iterative approach, there are multiple rounds of
bidding and allocations and the problem is solved in an iterative and incremental way.
Iterative combinatorial auctions are attractive to bidders because they learn about their
rivals' valuations through the bidding process, which could help them to adjust their own
bids. For analysis of iterative combinatorial auctions, this paper proposes to use dynamic
version of ANP.
Auctions have emerged as a particularly interesting tool for negotiations. Combinatorial
auctions provide a mechanism for negotiation between buyers and sellers. Various
concepts of negotiation models can be used for modeling of combinatorial auctions.
There is a connection between efficient auctions for many items and duality theory. The
Vickrey auction can be taken as an efficient pricing equilibrium, which corresponds to
the optimal solution of a particular linear programming problem and its dual. The simplex
algorithm can be taken as a static approach to determining the Vickrey outcome.
Alternatively, the primal-dual algorithm can be taken as a decentralized and dynamic
method of determining the pricing equilibrium.
A primal-dual algorithm usually maintains a feasible dual solution and tries to compute a
primal solution that is both feasible and satisfies the complementary slackness conditions.
If such a solution is found, the algorithm terminates. Otherwise the dual solution is
updated towards optimality, and the algorithm continues with the next iteration.
The fundamental work (Bikhchandani and Ostroy 2002) demonstrates a strong
interrelationship between the iterative auctions and the primal-dual linear programming
algorithms. A primal-dual linear programming algorithm can be interpreted as an auction
where the dual variables represent item prices. The algorithm maintains a feasible
allocation and a price set, and terminates as the efficient allocation and competitive
equilibrium prices are found.
For the winner determination problem, we will formulate the LP relaxation and its dual.
Consider the LP relaxation of the winner determination problem (1):
∑
=
n
i 1
∑
⊆MS
vi(S) xi(S) → max
subject to
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 117 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
∑
⊆MS
xi(S) ≦ 1, ∀ i, i = 1, 2, …, n,
∑
=
n
i 1
∑
⊆MS
xi(S) ≦ 1, ∀ j ∊ M, (2)
xi(S) ≥ 0, ∀ S ⊆ M, ∀ i, i = 1, 2, …, n.
The corresponding dual to problem (2)
∑
=
n
i 1
p(i) + ∑
∈Sj
p(j) → min
subject to
p(i) + ∑
∈Sj
p(j) ≥ vi(S) ∀ i, S, (3)
p(i), p(j) ≥ 0, ∀ i, j,
The dual variables p(j) can be interpreted as anonymous linear prices of items, the term
∑
∈Sj
p(j) is then the price of the bundle S and p(i) =
S
max [vi(S) −∑
∈Sj
p(j)] is the
maximal utility for the bidder i at the prices p(j).
Several auction formats based on the primal-dual approach have been proposed in the
literature. Although these auctions differ in several aspects, the general scheme can be
outlined as follows:
1. Choose minimal initial prices.
2. Announce current prices and collect bids. Bids have to be higher than or equal to the
prices.
3. Compute the current dual solution by interpreting the prices as dual variables. Try to
find a feasible allocation, an integer primal solution that satisfies the stopping rule. If
such solution is found, stop and use it as the final allocation. Otherwise update prices and
go back to step 2.
Concrete auction formats based on this scheme can be implemented in different ways.
The most important design choices are the following:
• pricing scheme,
• price update rule,
• way of computing a feasible primal solution in each iteration,
• stopping rule,
• type of information feedback.
The combinatorial auctions can be classified into the auctioneer-side allocation auctions
and the bidder-side allocation auctions. The bidder-side allocation auctions were
developed for small problems where bidders can cooperate in order to find a better
IJAHP ARTICLE: Fiala/Using an Analytic Network Process Model in Combinatorial Auctions
International Journal of the 118 Vol. 1 Issue 1 2009 ISSN 1936-
6744 Analytic Hierarchy Process
allocation by themselves at each step in the iteration. In the auctioneer-side allocation
auctions the auctioneer solves the winner determination problem after the bids are
collected. The auctioneer then provides some kind of feedback to support the bidders in
improving their bids in the next iteration. Usually the bidder’s current winning bids and
item prices are used as the feedback. The key challenge in the iterative combinatorial
auction design is to provide information feedback to the bidders after each step in the
iteration. Assigning prices to items and/or item bundles was adopted as the most intuitive
mechanism of providing feedback.
The AHP and ANP have been static, but for today’s world, analyzing is very important
time dependent decision making. The DHP/DNP (Dynamic Hierarchy Process/ Dynamic
Network Process) methods were introduced by Saaty (Saaty, 2003). There are two ways
to study dynamic decisions: structural, by including scenarios, and functional by
explicitly involving time in the judgment process. For the functional dynamics, there are
analytic or numerical solutions. The basic idea with the numerical approach is to obtain
the time dependent principal eigenvector by simulation.
The Dynamic Network Process seems to be the appropriate instrument for analyzing
dynamic network effects (Fiala, 2006). The method is appropriate also for the specific
features of multicriteria combinatorial auctions. The method computes time dependent
weights for bundles of items of weights of bidders (Figure 3).
Figure 3 Time dependent weights.
In the multicriteria combinatorial auction model, we take into account auctioneer,
bidders, criteria and packages as clusters and different types of connections in the system.
There are also some dependencies and feedback among elements and clusters also. The
dynamic version of the model is tested.
Time
Weights
Winner
Loser
Battle zone
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We used the alpha version of the ANP software Super Decisions developed by Creative
Decisions Foundation (CDF) for some experiments testing the possibilities of the
expression and evaluation of the multicriteria combinatorial auction models (Figure 4).
Figure 4. Multicriteria Combinatorial Auction Model.
5. Conclusions
For electronic auctions, we propose to use multicriteria iterative combinatorial auctions.
Combinatorial auction is the important subject of an intensive economic research.
Iterative process helps the bidders express their preferences. Combinatorial auctions
promise to increase efficiency and reduce exposure risk in an economic environment
where synergy is significant. The winner determination problem is by far the most
researched issue in combinatorial auctions. Winner determination problem is an NP-
complete optimization problem. Decentralized way for iterative combinatorial auctions
based on primal-dual algorithm seems to be very promising for solving the winner
determination problem. There are many variations how to use this approach. Multicriteria
optimization can be helpful for detailed analysis of combinatorial auctions. The
combination of such approaches can give more complex views on auctions. A possible
flexible approach is presented. The approach is based on the Dynamic Network Process.
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