A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by quality and sellers have private reserve prices for their items. Sellers quote prices strategically, inducing a knapsack game. The buyer's problem is to select a subset of maximal quality. We propose a buying mechanism which can be viewed as a game theoretic extension of Dantzig's greedy heuristic for the classic knapsack problem. We use Monte Carlo simulations to analyse the performance of our mechanism. Finally, we discuss how the mechanism can be applied to award R&D subsidies.Auctions, Subsidies, Market Design, Knapsack Problem

Ludwig Ensthaler

Thomas Giebe

English

Research Papers in Economics

Deutsches Institut für 
Wirtschaftsforschung
www.diw.de
Ludwig Ensthaler ￿ Thomas Giebe
Berlin, April 2009
Subsidies, Knapsack Auctions and 
Dantzig‘s Greedy Heuristic
880
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http://papers.ssrn.com/sol3/JELJOUR_Results.cfm?form_name=journalbrowse&journal_id=1079991 
 Subsidies, Knapsack Auctions and Dantzig’s
Greedy Heuristic 1
Ludwig Ensthaler 2 Thomas Giebe 3
April 28, 2009
1Many thanks to Ulrich Kamecke and Elmar Wolfstetter for helpful comments.
Financial support from the Deutsche Forschungsgemeinschaft, SFB Transregio 15:
“Governance and the Eﬃciency of Economic Systems”, is gratefully acknowledged.
2Institute of Economic Theory I, Humboldt University at Berlin, Spandauer Str.
1, 10099 Berlin, Germany, and Graduate Center of Economic and Social Research,
DIW Berlin, email: lensthaler@diw.de, Tel: +49-30-2093-1594, Fax: +49-30-2093-
5619
3Institute of Economic Theory I, Humboldt University at Berlin, Spandauer Str.
1, 10099 Berlin, Germany, email: thomas.giebe@wiwi.hu-berlin.de, Tel: +49-30-
2093-5773, Fax: +49-30-2093-5619Abstract
A budget-constrained buyer wants to purchase items from a shortlisted set.
Items are diﬀerentiated by quality and sellers have private reserve prices for
their items. Sellers quote prices strategically, inducing a knapsack game.
The buyer’s problem is to select a subset of maximal quality. We propose
a buying mechanism which can be viewed as a game theoretic extension of
Dantzig’s greedy heuristic for the classic knapsack problem. We use Monte
Carlo simulations to analyse the performance of our mechanism. Finally,
we discuss how the mechanism can be applied to award R&D subsidies.
JEL classiﬁcations: D21, D43, D44, D45
Keywords: Auctions, Subsidies, Market Design, Knapsack Problem1. introduction
Considerabuyerwhohasaﬁxedbudgettospendonitemsfromashortlisted
set. The items diﬀer in quality. The quality of a subset of items is the
sum of the individual qualities of its elements. A subset of higher quality is
preferred to one of lower quality. Subsets of the same quality are considered
as perfect substitutes. Each seller has private information about his reserve
price for his item. The buyer’s problem is to select a subset of items of
maximal quality subject to his budget constraint.
Under complete information, the buyer faces a binary knapsack problem
with qualities corresponding to values and reserve prices corresponding to
weights in the standard notation. In the realm of incomplete information,
any buying mechanism induces a knapsack game where sellers choose the
weight of their item (i.e. the price they quote) strategically. In this paper we
extend Dantzig’s greedy heuristic (Dantzig (1957)) for the knapsack problem
to this game–theoretic setting. More precisely, we design an auction mech-
anism with the following property: In equilibrium (in dominant strategies)
every item that is chosen has better quality relative to its price than every
item that is not chosen.
For an important application of this problem, consider government funds
to subsidize R&D activities by private businesses. Firms apply for subsidies
by submitting a research proposal. Proposals vary both in quality and the
stated level of required funding. A selection committee chooses an alloca-
tion that maximizes total quality. Since costs are private information, ﬁrms
might overstate the amount they need, leading to suboptimal allocations.
The problem was ﬁrst studied by Giebe, Grebe, and Wolfstetter (2006) who
give an economic analysis of the R&D subsidies program for small and
medium-sized companies in Germany (see Binks, Lockett, Siegel, and Wess-
ner (2003) for an account of the US and UK programs). Aggarwal and Hart-
line (2006) study knapsack auctions with applications to advertisement and
broadband bandwidth markets.
This paper proceeds as follows. Section 2 introduces the problem in a for-
mal setting. In section 3, we present our mechanism. We discuss several
properties and give examples. In particular, we outline its connection to
Dantzig’s algorithm. Section 4 employs a Monte Carlo simulation to com-
pare the performance of the mechanism to theoretical benchmarks. Section
5 brieﬂy debates some aspects of applying the mechanism to award R&D
subsidies.
2. the model
2.1 The formal setting
Assume we have n potential sellers numbered 1;:::;n. We use i to represent
a typical seller and N to represent the whole set of sellers, i.e. N  f1;:::;ng.
Each seller wants to sell an indivisible item for which he has a private reserve
1price zi. Items diﬀer in quality. Let qi denote the quality of seller i’s item
relative to an item of highest quality, so that 1  qi for all i 2 N (This
requires the possibility to compare items’ qualities quantitatively). Sellers
are faced by a single buyer who has a ﬁxed and ﬁnite budget B to spend
on these items. The buyer’s objective is to purchase a subset of items of
maximal quality, where the quality of a set of items is the sum of qualities
of its elements.
2.2 Complete Information: The Knapsack Problem
The 0-1 (binary) knapsack problem is one of the most studied problems
in combinatorial optimization (see e.g. Korte and Vygen (2005, ch. 17)). It
reads as follows:
Given a set of items each with a weight and a value, determine a subset of
items so that the total weight is less than a given capacity and the total value
is as large as possible.
Note that for the special case of complete information, (i.e. if all zi are
known) the buyer’s problem is a knapsack problem, with qualities corre-
sponding to values and reserve prices corresponding to weights. Indeed,
the buyer merely has to solve the optimization problem and pay every seller
in the chosen allocation his reserve price. More formally, the buyer’s prob-
lem reduces to ﬁnding
argmax
x
n X
i1
qixi s:t:
n X
i1
zixi  B; x 2 f0;1gn (1)
The binary knapsack problem is NP-hard. Dantzig proposed the following
simple heuristic algorithm:
Dantzig’s greedy heuristic. Sort items in increasing order of weight–per–
value. Insert items in this order until the cap binds.
The heuristic can perform arbitrarily bad relative to the optimal solution.
However, one can derive a relative bound:
Lemma 2.2.1 (Dantzig (1957)) Suppose that the ﬁrst i   1 items of the list ﬁt
into the knapsack. Then the absolute error is bounded by the value of the ith
item (the critical item).
2.3 Incomplete information: A knapsack game
In the presence of incomplete information (i.e. private reserve prices) any
buyingmechanisminducesaknapsackgamewheresellerschoosetheweight
of their item (the price they quote) strategically. The special case of com-
plete information poses a natural benchmark for any buying mechanism for
this general case. In the next section we develop a simple and very practi-
cable auction mechanism that performs remarkably well (compared to this
2benchmark) in simulations. Our mechanism can be viewed as an extension
of Dantzig’s greedy heuristic to this game theoretic knapsack problem in
the way that, in equilibrium, it chooses those items which oﬀer the best
price-per-quality ratio.
3. the mechanism
In this section we present our main results. In 3.1 we introduce the Dantzig
auction for the buyer’s problem and discuss some properties. We give two
examples in 3.2.
3.1 The Dantzig auction
Consider the following open descending clock auction for the buyer’s prob-
lem introduced in section 2: Each potential seller makes an initial oﬀer (or
bid). Should the sum of all bids exceed the buyer’s budget, the buyer tells
each bidder to lower its bid by a certain decrement (this might be zero or
a positive number). If a bidder refuses to lower his bid by the required
amount, he is eliminated from the auction. If he accepts, he stays active
for the next round (if there is one). This procedure continues until the bud-
get exceeds the sum of all active bids, i.e. until there is no excess demand.
Those bidders who are still active at the end of the last round receive their
ﬁnal bids in return for their items.
A stopping rule izi is the lowest bid that bidder i is willing to make
depending on his private reserve price zi.
Lemma 3.1.1 izi  zi is a weakly dominant strategy for all i  1;:::;n
Proof Assume bidder i chooses i > zi. If i wins at price izi or higher
there is no diﬀerence. But if i loses there is either no diﬀerence or i would
have won (and preferred to win) by playing zi instead. The argument is
analogous for zi > izi. 
Thus, 1;:::;n  z1;:::;zn is an equilibrium in (weakly) dominant
strategies. Whenever we speak of an equilibrium of the auction game, we
will implicitly refer to this particular equilibrium.
Next, we want to specify how the decrements are calculated. To this end,
we propose the following Dantzig clock rule: Each bidder starts at his max-
imal bid. This might simply be the budget B for all bidders. Alternatively,
there may be predeﬁned individual maximal bids for the bidders. A price-
per-quality bidding-clock starts at the highest initial bid-per-quality ratio.
As long as there is excess demand, this price-per-quality clock is lowered by
one unit every round and all active bidders are informed if and by how much
they have to lower their bids to meet the new ratio. As an example, consider
a bidder whose item has quality q 
1
2 and assume that his current bid is
b  100, giving him a current price-per-quality ratio of 200. Now, suppose
3the price-per-quality clock is lowered to 199. The bidder has to lower his
bid to 99:5 to stay active.
Deﬁnition 3.1.1 A descending clock auction as above with a Dantzig clock
rule is a Dantzig auction.
The next proposition shows that this name is indeed justiﬁed.
Proposition 3.1.1 In equilibrium, the Dantzig auction leads to an allocation
A with the following Dantzig–like property: Let i;j 2 N, with i 2 A and
j 2 NnA. Let bw
i be i’s winning bid. Then
bw
i
qi <
zj
qj, i.e. every item in the
ﬁnal allocation has a better price–quality ratio than every item that is not in
that allocation.
Proof As j is not in the ﬁnal allocation, there exists a round r in which j
drops out of the auction. Suppose the price-per-quality clock shows x after
round r  1. Then, in equilibrium, x 
zw
j
qj > x ", where " is the decrement.
But i is still active after round r, thus x   " 
br
i
qi (br
i being i’s bid in round
r). Since qi is constant and bids can only decrease from round to round, the
result follows. 
In other words: Take the complete set of information generated during the
auction, that is the set of winning bids and the ﬁnal bids of all other bidders.
Then the Dantzig auction yields the same allocation as the Dantzig heuristic
applied to this information set.
Instead of allowing for a multiround open bidding procedure, the buyer
could design an equivalent automated mechanism where bidders simply
submit a stopping rule to an algorithm.1
Let B be the budget, c  c1;:::;cn the vector of individual maximal bids
(again, ci  B for all i 2 N is possible), q  q1;::::qn the vector of quality
ratios relative to the best project, " > 0 the size of the decrement and  
1;:::;n the vector of chosen stopping rules. Additionally, let b denote
an arbitrary set of bids and A  N an arbitrary subset of items. With this
notation, the following proxy algorithm is equivalent to the Dantzig auction.
Algorithm 3.1.1
INPUT: B, N, ", c, q;
INITIALIZE: bc, A  f1;:::;ng
WHILE: jbj > B
• Let r  max
n
b1
q1;:::;
bjAj
qjAj
o
• For all i with
bi
qi  r
1Of course, for equivalence the algorithm must not exploit the collected information to
the bidders’ disadvantage.
4• IF bi > i
 bi   bi   "qi // lower the bid
• ELSE
 A   Anfig, b   bnfbig // eliminate from the auction
 rearrange indices appropriately
RETURN: A, b
It should be clear that the algorithm terminates as the sum of active bids
decreases every round.
3.2 Examples
Consider the following examples
1. Let B  100, ci  B for all i and suppose there are four items.
Items 1 2 3 4
qi 1
1
2
1
2
1
4
zi 50 20 20 10
ci
qi 100 200 200 400
zi
qi 50 40 40 40
Consider the Dantzig auction with this input. The price-per-value clock
starts at 400 (
c4
q4). The auction ends when the clock is at 49, once 1
dropped out. 2, 3 and 4 win at prices 24:5, 24:5 and 12:25 yielding a
total quality
1
2 
1
2 
1
4 
5
4. Assuming sellers play their equilibrium
strategies, thebuyerlearnedthatz1  50, z2;z3  24:5andz4  12:25.
Exploiting this information, the buyer would chose 1, 2 and 3 at a total
cost of 99, yielding a quality of 2. Under complete information the
buyer would select the ﬁrst best allocation and purchase all the items
at their reserve prices, giving him quality of
9
4 at a total cost 100.
2. Let B  100, and suppose there are two projects, 1 and 2, with q1  q2.
Let z1; z2  51. Then the algorithm chooses the empty set. This
shows that the algorithm can perform arbitrarily bad relative to the
optimal solution.
4. simulation
In this section we analyse the performance of the Dantzig auction using a
Monte Carlo simulation. For this, we assume that sellers play their equi-
librium strategies. We compare the resulting allocation, D, to two natural
benchmarks. The ﬁrst benchmark, D0, is the optimal allocation that results
5if one considers all bids that are made in the auction (recall that the chosen
allocation ignores players who drop out before the auction ends). The sec-
ond benchmark, CI, is the optimal choice under complete information (see
Section 2).
We chose the following simulation parameters:
• Each item has one of three quality values: qA;qB;qC 

1;
1
2;0:333

• budget: B  10;000
• zi uniformly distributed with support 0;1000
• ci uniformly distributed with support zi;1000
• number of projects: n=100
• number of simulated auctions: 500
Note that every project assumes one of only three possible quality values.
Also, we included individual caps for the bidders. Both of these assumptions
were introduced as they become relevant when applying our mechanism to
award R&D subsidies (see section 5). In the following we present the results
of this simulation.
Our theoretical results imply that the total quality improvement from D to
D0 cannot exceed the value of the most valuable item qA  1. This is con-
ﬁrmed by the simulation results. Figure 1 shows the empirical distribution
of absolute improvements for all simulated auctions. Moreover, we see that
for more than 60 per cent of cases an improvement was impossible. In
about 80 per cent of all cases the improvement does not exceed the value
of qB  0:5. Figure 2 shows the relative quality gain that one obtains if one
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
1.0
Figure 1: cdf of absolute welfare gain after move from D to D0
moves from D to D0. In only a few cases, the quality gain was more than
four per cent. The above simulation results suggest that applying Dantzig’s
heuristic to the collected information during the auction is indeed a good
60.01 0.02 0.03 0.04
0.2
0.4
0.6
0.8
1.0
Figure 2: cdf of relative quality gain after move from D to D0 (0.01 = 1 per
cent)
0.75 0.80
0.2
0.4
0.6
0.8
1.0
Figure 3: cdf of relative total quality of D and D0 as compared to the ﬁrst–
best allocation (100%)
approximation to a maximization over all allocations that are feasible given
that information.
Figure 3 compares the quality (welfare) of D and D0 as compared to the
complete–information benchmark, CI, by computing the empirical distribu-
tions of quality. For most of the observations, allocations D and D’ achieve
between 70 and 80 percent of the ﬁrst best quality.
5. application to r&d subsidies
In this section we brieﬂy sketch a possible application of the Dantzig mech-
anism.
As mentioned above, this work is motivated by Giebe, Grebe, and Wolfstetter
(2006). The authors analyse the German R&D subsidy programs for small
and medium-sized private businesses (which is similar in spirit to the SBIR
program in the US). Firms from all sectors of the economy are encouraged
7to submit their research proposals. Proposals are evaluated by a panel of
experts and ranked into one of three quality categories A, B or C (experts
frequently point out that a ﬁner grading is hard if not impossible to achieve).
Successful applicants receive a subsidy of 50 per cent of their stated person-
nel cost. Current practice allocates subsidies as follows: As many A projects
as possible are funded. If there is money left, it is used to fund as many B
projects as possible etc. Giebe et al outline the two major ﬂaws of this pro-
cedure. First, it puts relatively cheap C and B projects at disadvantage to
expensive A projects even though they might promise much better research
results relative to their cost. Second it is highly likely that some applicants
receive more money than they require to carry out the project (if they agree
to do it for 50 per cent of their personnel cost it is a fair guess that some
would do it for less). The proposed Dantzig mechanism addresses both
these problems. Introducing a price-per-quality clock enforces competition
across the quality classes and the open bidding procedure brings the subsi-
dies closer to the ﬁrms’ reserve prices. The 50 per cent personnel cost serve
as the ﬁrms’ individual maximal bids.
There is one design issue that we haven’t addressed yet, namely the concept
of quality ratios. In current practice quantitative ratios between the diﬀer-
ent quality classes exist only implicitly in the sense that qA >> qB >> qC,
i.e. any A project is always preferred to any B project is always preferred to
any C project independent of the costs. To apply the Dantzig mechanism
one has to explicitly attach numerical equivalence ratios to the three cate-
gories (The ratios of qB  0:5 and qC  0:33 for our simulation were chosen
arbitrarily).
References
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Binks, M., A. Lockett, D. Siegel, and C. Wessner (2003): “Policies pro-
moting innovation in small ﬁrms: evidence from the UK and US,” Small
Business Economics, 20, 121–127.
Dantzig, G. B. (1957): “Discrete–Variable Extremum Problems,” Operations
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Giebe, T., T. Grebe, and E. Wolfstetter (2006): “How to allocate R&D (and
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Combinatorial Optimization: Theory and Algorithms. 3th edition,

Discrete–Variable Extremum Problems,”

How to allocate R&D (and other) subsidies: An experimentally tested policy recommendation,”

Policies promoting innovation in small ﬁrms: evidence from

Subsidies, Knapsack Auctions and Dantzig's Greedy Heuristic

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Subsidies, Knapsack Auctions and Dantzig's Greedy Heuristic

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