Unformatted text preview: agents follow a myopic bestresponse bidding strategy, bidding for bundles that maximize
their utility given the prices in each round. 4.2 Linear Programming Theory
First, I provide a brief review of basic results in linear programming. See Papadimitriou
& Steiglitz PS82 for a text book introduction, and Chandru's excellent survey papers
CR99b, CR99a for a modern review of the literature on linear programming and integer
programming.
Consider the linear program:
max cT x P s:t: Ax b
x0
where A is a m n integer matrix, x 2 Rn is a nvector, and c and b are n, and
mvectors of integers. vectors are columnvectors, and notation cT indicates the transpose
of vector c, similarly for matrices. The primal problem is to compute a feasible solution
for x that maximizes the value of the objective function.
The dual program is constructed as:
min bT y D s:t: AT y c
y0
where y 2 Rm is a mvector. The dual problem is to compute a feasible solution for y
that minimizes the value of the objective function.
Let VLP x = cT x, the value of feasible primal solution x, and VDLP y = bT y, the
value of feasible dual solution y.
The weak duality theorem of linear programming states that the value of the dual always
dominates the value of the primal:
87 weakduality. Given a feasible primal solution x with value VLP x
and a feasible dual solution y with value VDLP y, then VLP x VDLP y.
Proof. Solution x is feasible, so Ax b. Solution y is feasible, so AT y c.
Therefore, x AT b and y Ac, and cT x cT AT b = bT AC bT y, and P D.
Theorem 4.1 The strong duality theorem of linear programming states that primal and dual solutions
are optimal if and only if the value of the primal equals the value of the dual:
strongduality. Primal solution x and dual solution y are a pair of
optimal solutions for the primal and dual respectively, if and only if x and y are feasible
satisfy respective constraints and VLP x = VDLP y .
Theorem 4.2 The strongduality theorem of linear programming can be restated in terms of
complementaryslackness conditions CS for short. Complementaryslackness conditions
expresses logical relationships between the values of primal and dual solutions that are
necessary and su cient for optimality.
complementaryslackness Complementaryslackness conditions constrain pairs of primal and dual solutions. Primal CS conditions state xT AT y , c = 0, or
in logical form:
Definition 4.1 xj 0 Aj y = cj PCS where Aj denotes the j th column of A written as a row vector to avoid the use of
transpose. Dual CS conditions state yT Ax , b = 0, or in logical form: yj 0 Ai x = bi DCS where Ai denotes the ith row of A.
The strongduality theorem can be restated as the complementaryslackness theorem:
complementaryslackness. A pair of feasible primal, x, and dual solutions, y, are primal and dual optimal if and only if they satisfy the complementaryslackness
conditions.
Theorem 4.3 88 PCS i xT AT y , c = 0, and DCS i yT Ax , b = 0. Equating, and observing that xT AT y = yT Ax, we have PCS and DCS i xT c = yT b, or cT x = bT y. The LHS
is the value of the primal, VLP x, and the RHS is the value of the dual, VDLP y. By the
strong duality theorem, VLP x = VDLP y is a necessary and su cient condition for the solutions to be optimal.
Proof. 4.2.1 PrimalDual Algorithms
Primaldual is an algorithmdesign paradigm that is often used to solve combinatorial
optimization problems. A problem is rst formulated both as a primal and a dual linear
program. A primaldual algorithm searches for feasible primal and dual solutions that
satisfy complementaryslackness conditions, instead of searching for an optimal primal or
dual solution directly. Primaldual can present a useful algorithmdesign paradigm for
combinatorial optimization problems. Instead of solving a single hard primal solution, or
a single hard dual solution, a primaldual approach solves a sequence of restricted primal
problems. Each restricted primal problem is often much simpler to solve than the full
primal or dual problem PS82 .
Primaldual theory also provides a useful conceptual framework for the design of iterative combinatorial auctions. Prices represent a feasible dual solution, and bids from
agents allow a search for a primal solution that satis es complementaryslackness conditions. If the current solution is suboptimal there is enough information available to
adjust dual prices in the right direction. Complementaryslackness conditions provide the
key to understanding how it is possible to compute and verify optimal solutions without complete information: it is su cient to just verify that a feasible solution satis es
CS conditions. Primaldual algorithms are consistent with the decentralized information
inherent in distributed agentbased systems. Optimality reduces to a test of feasibility
and complementaryslackness, which is available from agent bids, rather than the direct
solution of a primal problem, which requires information about agent valuation functions.
A standard primaldual formulation maintains a feasible dual solution, y, and computes
a solution to a restricted primal problem, given the dual solution. The restricted primal is
formulated to compute a primal solution that is both feasible and satis es CS conditions
with the dual solution. In general this is not possible until the dual solution is optimal,
89 and a relaxed solution is computed. The restricted primal problem is typically formulated
to compute this relaxed solution in one of two ways:
1. Compute a feasible primal solution x0 that minimizes the violation" of
complementaryslackness conditions with dual solution y.
2. Compute a primal solution x0 that satis es complementary slackness conditions with
dual solution y, and minimizes the violation" of feasibility constraints.
Method 1 is more useful in the context of iterative auction design because it maintains
a feasible primal solution, which becomes the provisional allocation in the auction, i.e. a
tentative allocation that will be implemented only when the auction terminates. The
restricted primal problem can be solved as a winnerdetermination problem. I show that
computing the allocation that maximizes revenue given agent bids the solution to winnerdetermination is a suitable method to minimize the violation of CS conditions between
the prices and the provisional allocation in each round iBundle. Prices in each round of
an auction de ne the feasible dual solution, and agent bestresponse bids provide enough
information to test for complementaryslackness and adjust solutions towards optimality.
As discussed in the introduction to this chapter, I rst assume myopic bestresponse,
but later justify this assumption with an extension to compute Vickrey payments at the
end of the auction in addition to the e cient allocation see Chapters 6 and 7.
Initial
Dual
Solution
(prices) Adjust Dual
Solution
(prices) NO
YES
Terminate y Do x’and y
satisfy CS
conditions? Receive
Bids Compute
Feasible
Primal Solution
(allocation) x’ Figure 4.1: A primaldual interpretation of an auction algorithm.
A primaldual based auction method has the following form see Figure 4.1:
90 1. Maintain a feasible dual solution prices".
2. Compute a feasible primal solution provisional allocation" to minimize violations
with complementaryslackness conditions given agents' bids.
3. Terminate if all CS conditions are satis ed are the allocation and prices in competitive equilibrium?"
4. Adjust the dual solution towards an optimal solution, based on CS conditions and
the current primal solution increase prices based on agent bids" 4.3 Allocation Problems
Let us consider the particular form of an allocation problem, in which there are a set of
discrete items to allocate to agents, and the goal is to maximize value. We assume quasilinear preferences, and use utility to refer to the di erence between an agent's value for a
bundle and the price. The primal and dual allocation problems can be stated as follows:
allocation problem: primal The primal allocation problem is to allocate items to agents to maximize the sum value over all agents, such that no item is
allocated to more than one agent.
Definition 4.2 allocation problem: dual The dual allocation problem is to assign
prices to items, or bundles of items, to minimize the sum of i each agents' maximum
utility given the prices, over all possible allocations; and ii the maximum revenue over
all possible allocations given the prices.
Clearly, without information on agents' values the auctioneer cannot compute an optimal primal or an optimal dual because of term i in the dual. However, under a
reasonable assumption about agents' bidding strategies myopic bestresponse the auctioneer can verify complementaryslackness conditions between primal and dual solutions,
and adjust prices and the allocation towards optimal solutions.
An auction interpretation of the complementaryslackness conditions can be stated as
follows:
Definition 4.3 allocation problem: CS conditions The CS between a feasible primal
solution to an allocation problem, x, and a feasible dual solution, prices p, are:
Definition 4.4 91 CS1 Agent i receives bundles Si in the provisional allocation if and only if the bundle
maximizes its utility given the prices, and has nonnegative utility.
CS2 The provisional allocation S = S1 ; : : : ; SI is the revenuemaximizing allocation
given the prices.
Left deliberately vague at this stage is the exact structure of the prices. In a combinatorial allocation problem these might need to be nonlinear and nonanonymous prices
to support the optimal allocation. Similarly, the revenuemaximization concept must be
de ned with respect to a particular linear program formulation. Note also that CS2 is
not automatically satis ed with a provisional allocation computed to maximize revenue
given agents' bids. CS2 makes a stronger claim, that the provisional allocation must
maximize revenue over all possible allocations given the current ask prices, not just over
all allocations consistent with bids.
Primaldual auction analysis requires the following assumption about agent strategies:
myopic bestresponse A myopic bestresponse bidding strategy is to
bid for all items or bundles of items that maximize utility at the current prices.
Definition 4.5 Bestresponse bids provide enough information to test CS1, because the bestresponse
of an agent is precisely those bundles that maximize an agent's utility given the current
prices. For any feasible primal solution, the auctioneer can test CS2 because that only
requires price information.
The restricted primal has a natural auction interpretation:
auction restrictedprimal problem Given best response bids from each
agent allocate bundles to maximize revenue, breaking ties in favor of including more agents
in the provisional allocation.
Note well that a bundle is only allocated to an agent in the restricted primal problem if the agent bids for that bundle. This restriction ensures that CS1 is satis ed for
that agent, given the de nition of myopic bestresponse. CS2 is satis ed with careful
priceadjustment rules, such that prices are increased slowly enough" that the revenuemaximizing allocation can always be computed from agent bids.
Given myopic bestresponse, the termination condition, which tests for complementaryslackness between the provisional allocation and the prices, must check that CS1 holds
Definition 4.6 92 for every agent. This is achieved when every agent to submit a bid receives a bundle in
the provisional allocation, i.e. in competitive equilibrium.
Our interest is in solving the CAP, which is most immediately formulated as an integer
program see Section 4.4. In order to apply primaldual methods it is essential that we
have a linear program formulation of the CAP. We must be careful enough to use a strong
enough formulation, such that the optimal solution is integral 01 and not fractional. The
ideal situation is illustrated in Figure 4.2. The auction implements a primaldual algorithm
for a linear program that is strong enough to compute the optimal integer solution.
Complementaryslackness conditions
satisfied. Value V
DLP * V
IP
V(integral)
LP
Auction Round Figure 4.2: An auctionbased primaldual algorithm in which the linear program formulation is
strong enough to eliminate all fractional solutions. In comparison, consider Figures 4.3 a and b, which illustrate a primaldual algorithm and iterative auction method for a linear program that is not strong enough, and
admits optimal fractional solutions. The primaldual algorithm algorithm terminates with
a fractional primal solution and value greater than the value of the best possible integer solution. The auction always maintains an integral primal solution solving winnerdetermination to compute the provisional allocation, but can terminate with a primal
solution that does not satisfy complementaryslackness conditions. Although the primal
solution is perhaps optimal, its optimality cannot be assessed without CS information. 4.3.1 Price Adjustment
Left unde ned at the moment, and the challenging part of primaldual auction design, are
the precise rules used to de ne price updates. The goal is to use information from agents'
bids, and the current provisional allocation, to adjust prices towards an optimal dual
solution that will support an optimal primal solution. Primaldual methods traditionally
93 Complementaryslackness holds V
DLP Value Value V
DLP V*
IP
V
LP V*
IP
V(integral)
LP
Auction Round Auction Round b a Figure 4.3: Primaldual algorithm a and Primaldual auction method b in which the linear
program relaxation is too weak, and VLPR VIP . use the dual of the restricted primal to adjust the dual solution across iterations. A simpler
method in allocation problems is to increase prices on overdemanded items, or bundles of
items. The method can be explained both in terms of its e ect on complementaryslackness
conditions and in terms of its e ect on the value of the dual solution.
The idea is to increase prices to: a maintain CS2 in the next round and b move
towards satisfying CS1 for all agents.
progress. Progress is made towards satisfying CS1 and CS2 with
the provisional allocation and the ask prices if: 1 the auctioneer increases prices on one
or more bundles that receive bids in each round; and 2 the auctioneer increases prices
by a small enough increment that bestresponse bids from agents continue to maximize
revenue in the next round.
Proposition 4.1 CS1 holds whenever every agent that bids receives a bundle in the provisional allocation. This is trivially achieved for high enough prices because no agent will bid, but we
need to achieve this condition in combination with CS2. The trick is to increase prices
just enough to maintain revenuemaximization from bids CS2 across all rounds. This is
achieved in iBundle by ensuring that myopic agents continue to bid for bundles at the new
prices, i.e. increasing price on overdemanded bundles.
An alternative interpretation is that increasing prices on overdemanded items will
reduce the value of the dual, making progress towards the optimal solution, see Figure 4.4.
Recall that the value of the dual is the sum of the auctioneer's maximal revenue and each
94 agent's maximal utility at the current prices. A price increase will decrease the value of
the dual if the increase in maximal revenue from the price increase is less than the decrease
in total maximal utility summed across agents.
The auctioneer can achieve this e ect of increasing revenue by less than the decrease in
agent utility by selecting overdemanded items, or bundles of items, on which to increase
the price. Suppose that two agents bid for bundle S1 , and that both agents have at least
0 more utility for that bundle than any other bundle at the current prices. Increasing
the price on overdemanded bundle S1 by will decrease the maximal utility of both agents
by , for a decrease in dual value of 2 . However, increasing the price on this one bundle
by can increase the auctioneer's maximal revenue by at most . The result is that the
net change in utility must a decrease of at least .
Complementaryslackness holds Value V
DLP * V
IP V
LP Auction Round Figure 4.4: Primaldual interpretation of an ascendingprice auction. 4.3.2 Competitive Equilibrium
The optimal primal and dual solutions in an allocation problem correspond to a classic
statement of competitive equilibrium.
Definition 4.7 competitive equilibrium Allocation S and prices p are in competitive equilibrium when:
a every agent receives a bundle in its bestresponse utility maximizing set
b the allocation maximizes the revenue for the auctioneer at the prices
The allocation in competitive equilibrium is e cient, by equivalence between competitive equilibrium and primaldual optimality:
Theorem 4.4 competitive equilibrium e ciency. An allocation S is e cient if and
95 only if there exists competitive equilibrium prices p, for an appropriate type of prices e.g.
linear, bundle, nonanonymous. In the context of the combinatorial allocation problem Bikchandani & Ostroy BO99
have characterized the structure on prices required for the existence of competitive equilibrium and equivalently for integral solutions to linear program formulations of CAP.
These formulations are introduced in Section 4.4 and discussed at length.
In some problems it is necessary that prices are both nonlinear bundle prices and
nonanonymous di erent prices for the same bundle to di erent agents to support a
competitive equilibrium solution.
Wurman & Wellman WW99, WW00 propose an alternative de nition of competitive
equilibrium, which is essentially complementary slackness condition CS1 without CS2.
This relaxed condition is su cient for the existence of equilibrium prices even without
nonanonymous prices, but too weak to be able to claim that equilibrium prices imply an
e cient allocation. 4.3.3 Example: The English Auction
The standard English auction illustrates the primaldual framework for auction design. The
English auction is an ascendingprice auction for single items, where the price increases as
long as more than one agent bids at the current price.
Let vi denote agent i's value for the item. The singleunit resource allocation problem
is:
max s:t: Xv x Xx 1
i i ii IPsingle i xi 2 f0; 1g where xi = 1 if and only if agent i is allocated the item, i.e. the goal is to allocate the
item to the agent with the highest value. This can be solved as a linear program, LPsingle ,
relaxing the integral constraint
96 max s:t: Xv x X x i1
i ii LPsingle i xi 0
and VLP = VIP , i.e. there is always an integral optimal solution to the relaxed problem.
The dual formulation, DLPsingle , is min DLPsingle s:t: vi ; 8i
0
The complementaryslackness conditions are Xx 0 = vi ; 8 i
Xx = 1
0
i
i The complementaryslackness conditions can be interpreted in terms of competitive
equilibrium conditions on the allocation and the prices. An allocation and prices in a
singleitem auction are in competitive equilibrium, and the allocation is e cient, when:
i the item is sold to an agent, that agent bids for the item at the price, and no other
agent bids for the item at the price.
or ii the item is sold to no agent, the price is zero, and no agent bids for the item.
It is straightforward to understand e ciency in these cases: in i the agent with the
highest value receives the item; in ii no agent has a positive value for the item.
The English auction maintains price p on the item, initially p = 0. Agent i bids
whenever p vi , and the provisional allocation sets xj = 1 for one of the agents that bids
in each round, and increases the price p whenever more than one agent bids.
Let the provisional allocation de ne a feasible primal solution, and the price de ne
P
dual solution = i maxf0; vi , pg + p. This is feasible, maxf0; vi , pg + p vi for
all agents i.
97 Assume that agents follow a myopic bestresponse bidding strategy, bidding for the item
at the ask price whenever the price is below their value. The optimality of the English
auction can be understood in two di erent ways:
The English auction terminates with primal and dual solutions that satisfy CS1 and
CS2.
Clearly, CS2 is satis ed throughout the auction because the item is always allocated
to one of the agents. CS1 is satis ed when the auction terminates. Let j indicate
the only agent that bids at price p. Therefore vi , p 0 for all agents i 6= j and
vj , p 0 for agent j because agents follow bestresponse bidding strategies, and
P
= i maxf0; vi , pg + p = maxf0; vj , pg + p = vj .
The value of the dual strictly decreases in each round of the auction. Let m 1
equal the number of agents that bid in each round of the auction except the nal
round. For price increment , the sum maximal utility to the agents decreases by m
and the maximal revenue to the auctioneer increases by , for a net change in of
,m , 1 .
In fact, the nal price in the English auction approaches the Vickrey payment i.e. the
secondhighest value as the bid increment ! 0. It follows that myopicbest response
is a rational sequential strategy for an agent, in equilibrium with myopic bestresponse
strategies from other agents see Chapter 7 for a full discussion of the incentive properties
of iterative Vickrey auctions. 4.4 Linear Program Formulations for the Combinatorial Allocation Problem
Primaldual based auction methods require linear programming formulations of allocation
problems. Bikchandani & Ostroy BO99 have formulated a hierarchy of linear programs for
the problem, introducing additional constraints to remove fractional solutions. Although
it is always possible to add enough constraints to a linear program relaxation to make the
optimal solution integral Wol81a, Wol81b, TW81 , the particular formulations proposed by
Bikchandani & Ostroy are interesting because the constraints have natural interpretations
as prices in the dual.
98 The hierarchy of linear program formulations, LP1 , LP2 , and LP3 , all retain the
set of integer allocations but prune additional fractional solutions. Each formulation introduces new constraints into the primal, with the dual problems DLP1 , DLP2 , and
DLP3 containing richer price structures. For example, in DLP1 the prices on a bundle
P
are linear in the price of items, i.e. pS = j 2S pj , where pj is the price of item j
in bundle S . Moving to DLP2 , the price on a bundle can be nonlinear in the price on
items, and in DLP3 the price on a bundle can be di erent to di erent agents. Bikchandani & Ostroy prove that LP3 solves all CAP instances, and demonstrate the existence of
competitive equilibrium prices, even though they must sometimes be both nonlinear and
nonanonymous.
Solving the CAP with the highlevel linear program formulations is likely to be less
e cient computationally than direct searchbased methods applied to the integer program
formulation. Formulations LP2 and LP3 introduce an exponential number of additional
primal constraints, and dual variables, e ectively enumerating all possible solutions to the
CAP. In comparison, search methods, such as branchandbound with LPbased heuristics,
solve the problem with implicit enumeration and pruning.
However the formulations are very useful in the context of mechanism design and
decentralized CAP problems. In Section 4.6 I present CombAuction, a primaldual
algorithm for the CAP, which
a computes optimal primal and dual solutions without complete information about
agent valuation functions.
b computes optimal primal and dual solutions without complete enumeration of all
primal constraints and or dual variables.
In fact most of the computation within CombAuction occurs in winner determination,
which solves the restricted primal problem in each round, and winnerdetermination itself
is solved with a branchandbound search algorithm. 4.4.1 Integer Program Formulation
Introducing xi S to indicate that agent i receives bundle S the straightforward integer
program, IP , formulation of the combinatorial allocation problem is:
max X X x Sv S xiS S i 99 i i IP X x S 1;
i
S
X X x S 1; 8i i S 3j i IP1
8j s:t: IP2 xiS 2 f0; 1g; 8i; S where S 3 j indicates a bundle S that contains item j . The objective is to compute the
allocation that maximizes value over all agents, without allocating more than one bundle
to any agent IP1 and without allocating a single item multiple times IP2. Let VIP
denote the value of the optimal allocation. 4.4.2 Firstorder LP Formulation
LP1 is a direct linear relaxation, which replaces the integral constraints xi S 2 f0; 1g with
nonnegativity constraints, xi S 0.
max X X x Sv S xiS S X x S 1; i
s:t:
i
S
X X x S 1;
S 3j i i xiS 0;
min i LP1 8i LP1 1 8j LP1 2 8i; S X pi + X pj DLP1 8i; S DLP1 1 pi;pj i s:t: i X
pi + pj v S ;
i j 2S pi; pj 0; j 8i; j Prices pj on items jo G de ne a feasible dual solution, with the substitution pi =
n
Pj2S pj 2.
maxS vi S ,
rstorder dual. The value of the rstorder dual is the sum of the
maximal utility to each agent plus the total price over all items this is the auctioneer's
maximal revenue.
Proposition 4.2 100 A B AB Agent 1 0 0
Agent 2 2 0
Agent 3 0 2 3
2
2 Table 4.1: Problem 1. AB C AB BC AC ABC Agent 1 60 50 50 200 100 110
Agent 2 50 60 50 110 200 100
Agent 3 50 50 75 100 125 200 250
255
250 Table 4.2: Problem 2. P The dual variables de ne linear prices, the price for bundle S G is pS = j 2S pj .
From De nition 4.7 the optimal dual solution de nes competitive equilibrium prices if and
only if a partition of items exists at the prices that allocates each agent a bundle in its
utilitymaximizing set and allocates every item with positive price exactly once.
Problem 1 in Table 4.1 can be solved with LP1 ; VLP1 = VIP = 4. The optimal
allocation is x2 A = 1 and x3 B = 1, indicated by . To see that VLP1 4, notice that
dual prices pA = pB = 1:6 gives a dual solution with value VDLP1 = 0+0:4+0:4+3:2 =
4. Remember that VLP1 VDLP1 for all dual solutions by the weakduality theorem of
linear programming. These are one set of competitive equilibrium prices.
However, in general the value VLP1 VIP and the optimal primal solution makes
fractional assignments to agents. As an example of when LP1 fails, consider Problem 2
in Table 4.2. In this problem VLP1 = 300 VIP = 275. The primal allocates fractional
solution x1 AB = 0:5; x2 BC = 0:5 and x3 AC = 0:5, which satis es constraints LP1 PP
1 because S 3 j i xi S 1 for all items j 2 G . Prices pA = pB = pC = 100
solve the dual problem DLP1 .
Kelso & Crawford KC82 prove that grosssubstitutes GS preferences are a su cient
condition for the existence of linear competitive equilibrium prices, such that VLP1 = VIP .
To de ne grosssubstitutes preferences, let Di p de ne the demand set of agent i at
prices p, i.e. the set of bundles that maximize its utility value  price.
grosssubstitutes GS For all price vectors p; p0 such that p0 p, and
all S 2 Di p, there exists T 2 Di p0 such that fj 2 S : pj = p0j g T .
Definition 4.8 101 In words, an agent has GS preferences if an agent continues to demand items with the
same price as the price on other items increases. If preferences are also monotonic, such
that vi S 0 vi S for all S 0 S , then GS implies submodular preferences.
Definition 4.9 all S; T G , submodular preferences Valuation function vi S is submodular if for vi S + vi T vi S T + vi S T Submodularity is equivalent to a generalized statement of decreasing returns:
decreasing returns Valuation function vi S has decreasing marginal
returns if for all S T G and all j 2 G ,
Definition 4.10 vi T , vi T n fj g vi S , vS n fj g
In other words, the value of an item increases as it is introduced to larger sets of items.
Subadditivity implies that the value for any bundle is no greater than the minimal sum of
values for a partition of the bundle.
In fact, grosssubstitutes preferences de ne the largest set of preferences that contain
unitdemand preferences see De nition 4.14 for which the existence of linear competitive
equilibrium prices can be shown GS99 .
The rest of this section introduces two alternative linear program formulations of CAP,
LP2 and LP3 , due to Bikchandani & Ostroy BO99 . 4.4.3 Secondorder LP Formulation
Introducing new constraints to the rstorder linear program relaxation LP1 of IP gives
a secondorder linear program LP2 with dual DLP2 . The corresponding dual variables
to the new primal constraints are interpreted as bundle prices within an auctionbased
primaldual algorithm. 102 max s:t: X X x Sv S xiS ;yk S i X x S 1; 8i
i
S
X x S X yk;
i
i
X yk k1 S
i i 3 k xiS ; yk 0;
min LP2 1 8S pi;pS ; i s:t: pi + pS viS ;
X pS 0; 8k 8i; S
, pi; pS ; 0; LP2 2
LP2 3 8i; S; k X pi + S 2k LP2 DLP2
DLP2 1
DLP2 2 8i; S where k 2 K is a partition of items in set K , and k 3 S indicates that bundle S is represented in partition k. A partition is a feasible bundling" of items, e.g. A; B; C or AB; C ,
etc., and K is the set of all possible partitions, e.g. K = f A; B; C ; AB; C ; A; BC ; : : : ;
ABC g in Problem 2 Table 4.2.
Constraints LP2 2 and LP2 3 replace constraints LP1 1, and ensure that no more
than one unit of every item is allocated. The dual DLP2 has variables pi, pS and ,
which correspond to constraints LP2 1, LP2 2 and LP2 3, and constraints DLP2 1
and DLP2 2 correspond to primal variables xiS and yk.
Dual variables pS can be interpreted as bundle prices, and with substitution pi =
max fvi S , pS g, i.e. the maximal utility to agent i at prices pS , and
S
P
= maxk2K S2k pS , i.e. the maximal revenue to the auctioneer at prices pS .
secondorder dual. The value of the dual is the sum of the maximal
utility to each agent with bundle prices pS plus the auctioneer's maximal revenue over
all feasible and nonfractional allocations at the prices.
Proposition 4.3 103 A B AB Agent 1 0 0
Agent 2 2 2 3
2 Table 4.3: Problem 3.
The dual variables correspond to bundle prices, pS , and the optimal dual solution
de nes competitive equilibrium prices by De nition 4.7 if there is an allocation that gives
each agent a bundle in its utilitymaximizing set at the prices, and maximizes revenue to
the auctioneer over all possible allocations.
With the additional constraints LP2 solves Problem 2. Allocation x1 AB = x2 BC
= x3 AC = 0:5 is not feasible in LP2 because it is not possible to allocate yk1 =
yk2 = yk3 = 0:5 for k1 = AB; C ; k2 = AC; B and k3 = AB; C without violating
constraint LP2 3 and without this we violate constraints LP2 2. LP2 solves Problem
2, with VLP2 = VIP = 275. An optimal dual solution is given by bundle prices p =
50; 60; 75; 190; 200; 200; 255, with total agent maximal utility 10 + 0 + 0 and maximal
auctioneer revenue 75 + 190 = 265, i.e. VDLP2 = 275.
However, Problem 3 is an example that LP2 does not solve. The value of the optimal
primal solution is VLP2 = 3:5, which is greater than the value of the optimal feasible
allocation, VIP = 3. The primal allocates fractional bundles x1 AB = 0:5 and x2 A =
x2 B = 0:5, which satis es constraints LP2 2 and LP2 3 with yk1 = yk2 = 0:5 for
partitions k1 = AB; ; and k2 = A; B . Prices pA = 1:5; pB = 1:5; pAB = 3 solves
the dual problem DLP2 . 4.4.4 Thirdorder LP Formulation
Introducing new constraints to the secondorder linear program relaxation LP2 of IP
gives a thirdorder linear program LP3 with dual DLP3 . The corresponding dual variables to the new primal constraints are interpreted as nonanonymous, or discriminatory
bundle prices, with di erent prices for the same bundle to di erent agents. 104 max s:t: X X x Sv S xiS ;yk S i X x S 1; 8i
i
S
X yk;
xiS
X yk k1 i;S
i i 3 k LP3
LP3 1 8i; S LP3 2
LP3 3 xiS ; yk 0; 8i; S; k
min X pi + pi;pi S ; i s:t: pi + pi S viS ; 8i; S
X p S 0; 8k
,
i
i;S 2k DLP3
DLP3 1
DLP3 2 pi; pi S ; 0; 8i; S
where k 3 i; S indicates that agentpartition k 2 K 0 contains bundle S designated
for agent i. Variable yk in LP3 corresponds to an agentpartition k, where the set
of agentpartitions in Problem 3 is K 0 = f 1; A; 2; B ; 1; B ; 2; A ; 1; AB ; 2; ; ;
1; ;; 2; AB . It is important to note that each agent can receive at most one bundle in
a particular agentpartition.
The dual variables pi S that correspond to primal constraints LP3 2 are interpreted
as nonanonymous bundle prices, price pi S is the price to agent i for bundle S . As before,
substitutions pi = max fvi S , pi S g, i.e. the maximal utility to agent i at individual
S
P
prices pi S , and = maxk2K i;S 2k pi S , i.e. the maximal revenue to the auctioneer
at prices pi S given that it can allocate at most one bundle at prices pi S to each agent
i.
0 thirdorder dual. The value of the dual to LP3 is the sum of
the maximal utility to each agent with bundle prices pi S plus the auctioneer's maximal
revenue over all feasible allocations at the prices. In this case an allocation is feasible if it
allocates no more than one bundle to each agent.
Proposition 4.4 105 The dual variables correspond to nonanonymous bundle prices, pi S , and the optimal
dual solution de nes competitive equilibrium prices if there is an allocation of items that
simultaneously gives each agent a bundle in its utilitymaximizing set and maximizes the
auctioneer's revenue, over all possible allocations that sell at most one bundle to each
agent.
Bikchandani & Ostroy BO99 prove this important theorem:
integrality. The optimal solution to linear program LP3 is always
integral, and therefore an optimal solution to CAP, with VLP3 = VDLP3 = VIP .
Theorem 4.5 Therefore, there are always competitive equilibrium bundles prices for CAP, although
these prices must be nonanonymous in some problems.
Consider Problem 3. Allocation x1 AB = 0:5 and x2 A = x3 B = 0:5 is not feasible
in LP3 because yk1 = yk2 = yk3 = 0:5 for k1 = 1; AB ; 2; ; ; k2 = 1; A; 2; B
and k3 = 1; B ; 2; A violates constraint LP3 3, but without this constraints LP3
2 are violated. In this problem VLP3 = VIP = 3. To see this, consider bundle prices
p1 = 0; 0; 2:5 and p2 = 2; 2; 2, for which the value of the dual is 0:5 + 0 + 2:5 = 3. This
proves that VLP3 3 by the weakduality theorem of linear programming.
I will return to this hierarchy of linearprogram formulations of the CAP in Section
4.6, when I introduce the CombAuction primaldual algorithm. CombAuction constructs feasible primal and dual solutions to an appropriate linear program formulation,
and adjusts the solution until complementaryslackness conditions are also satis ed. 4.5 Tractable Combinatorial Allocation Problems
The CAP is equivalent to the maximum weighted set packing problem SPP, a wellstudied problem in the operations research literature. In SPP there are a set of items,
and a set of subsets each with nonnegative weights, and the goal is to pack the items
into sets to maximize total value, without using any item more than once. CAP can
be reduced to SPP by introducing an additional dummy item" for the XOR bids from
each agent. de Vries & Vohra dVV00 also note two closely related problems, the set
partitioning problem SPA, in which the goal is to select a set of subsets with minimal
106 cost that include all items at most once, and the set covering problem SCP, in which
the goal is to select a set of subsets with minimal cost that include all items at least once.
Set covering problems nd applications in railway crewscheduling and airline scheduling,
where items are ights trains, and bundles represent possibility sets for individual workers.
A considerable amount is known about the complexity of this class of problems.
A classic technique in combinatorial optimization theory is to relax an integer program
to a linear one. Many tractable special cases follow by considering the conditions on the
natural relaxation of the integer program that provide integer solutions. For example, one
su cient condition is that the linear program is integral, such that all extremal feasible
points are integral, i.e. 01. In this case the integrality requirement can be dropped and
the problem solved as a linear program in polynomial time. Restrictions on the constraint
matrix, corresponding to restrictions on the kinds of subsets permitted in CAP, can provide
this integrality property dVV00 .
Additional restrictions, for example on the size of bids, or on the valuation structure of
bids, can also lead to tractable special cases. Given the connection with linear programming
relaxations this is a good place to review known tractable specialcases in the literature.
The results here are drawn from Rothkopf et al. RPH98 , de Vries & Vohra dVV00 ,
Nisan Nis00 , and earlier work due to Kelso & Crawford KC82 .
It is important to understand the characteristics of tractable specialcases of CAP
because this knowledge can be leveraged within mechanism design, achieving tractable
and strategyproof solutions see Section 3.2.1 in Chapter 3. Restrictions on Structure of Bundles
Table 4.4 presents tractable instances of CAP that follow from restrictions on the types
of bundles on which agents can submit bids. de Vries & Vohra note that the linearordering or consecutive ones condition implies that the constraint matrix satis es total
unimodularity,1 and that the nestedhierarchical structure implies that the constraint matrix is balanced.2 Nisan Nis00 provides a proofbyinduction that the linear program has
integral solutions in these cases, and also describes a method to combine two bid structures
with the integral property into a single structure that retains the property.
A matrix satis es total unimodularity if the determinant of every square submatrix is 0, 1, or 1.
A 01 matrix is balanced if it has no square submatrix of odd order with exactly two 1's in each row
and column.
1 2 107 linearorder ordering = 1 2
RPH98
n
every bid is for a contiguous sequence
circular ones
also allow bids of form n 1 2 , etc.
RPH98
nestedhierarchical
for every two subsets of items 1 2
RPH98
that appear as part of any bid they are either
disjoint or one contains the other
orsingletons
bids for singleitems
singleitem bids
one item
bids for pairs of items
cardinality constraint on size of bids
RPH98
multiunit, decreasing returns identical items, each agent has decreasing value for Nis00
each additional item
G g ;g ;::: ;g ggg S ;S Table 4.4: Tractable structure on bids
nondecreasing and supermodular increasing returns"
twotypes of agents
grosssubstitutes
decreasingreturns"
unitdemand
agents only want one item
linearadditive
agents have linear values across items dVV00
KC82
Kuh55
CK81 Table 4.5: Constraints on valuation functions Restrictions on Values on Bundles
Table 4.5 presents tractable instances of CAP that follow from restrictions on the value
structure of agents bids. de Vries & Vohra dVV00 note that the nondecreasing and supermodular preferences condition again provides the linear program relaxation of the CAP
with integral solutions. Grosssubstitutes were de ned earlier in De nition 4.8 and have
an intuitive interpretation as decreasingreturns, and also imply submodular preferences.
Definition 4.11 all S; T G , supermodular preferences Bid function bi S is supermodular if for bi S + bi T vi S T + viS T The equivalence of supermodularity and increasing returns is wellknown in the literature GS99 .
increasing returns Bid function bi S has increasing marginal returns
if for all S T G and all j 2 G ,
Definition 4.12 biT , vi T n fj g bi S , vS n fj g
108 Note carefully that we can have any number of di erent types of submodular valuation functions, one from each agent, but only at most two di erent types of supermodular
functions if the CAP problem is to be tractable. It is easier to solve a maximization problem, such as the CAP, with submodular convex objective functions than supermodular
concave objective functions. Exact Solutions
Rothkopf et al. RPH98 also suggest a dynamic programming algorithm for CAP, which
has runtime complexity independent of the number of bids actually placed, but quickly
becomes intractable for large numbers of items, with scaling property O3m in the number
of items m. Branchandbound search methods, either with AIbased heuristics San99,
FLBS99 , or with linearprogram based heuristics ATY00 have also been studied for
general CAP instances. Approximate Solutions
The CAP is di cult to approximate, at least within a worstcase multiplicative factor.
There is no polynomial time algorithm with a reasonable worstcase guarantee Has99 .
Approximation algorithms in the literature without this guarantee include a localsearch approach HB00 , a simple relax and round" method Nis00 , and iterative methods
FLBS99 . CombAuction can itself be viewed as an approximate algorithm for CAP.
CombAuction provides a worstcase bound on the di erence between the value of its
solution and the value of the optimal solution. This errorterm increases linearly with the
minimal bid increment, which de nes the rate at which prices are increased across rounds,
while the number of rounds in the auction is inverselyproportional to the minimal bid
increment. A larger bid increment reduces the number of rounds in the auction, reducing
the number of winnerdetermination problems the auction must solve, in return for a loss
in worstcase e ciency. Experimental results in Section 5.5.1 show the e ectiveness of this
approach. 109 ...
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