Pricing Partially Compatible Products

1
Pricing Partially Compatible Products

• David Kempe, University of Southern California
• Adam Meyerson, University of California, Los
  Angeles
• Nainesh Solanki, MediaDefender Inc.
• Ramnath Chellappa, Emory University
• This paper to appear in ACM Conference on
  Electronic Commerce (EC) 2007.

2
Two Companies Selling Equivalent Software Products

• Word Processor
• Web Browser
• Operating System
• Spreadsheet
• Web Page Editor
• MP3 Software

• Word Processor
• Web Browser
• Operating System
• Spreadsheet
• Web Page Editor
• MP3 Software

3
Products Differ in Price and Quality
Customer Prefers

• Word Processor
• Q5 P3
• Web Browser
• Q4 P0
• Operating System
• Q9 P4
• Spreadsheet
• Q3 P2
• Web Page Editor
• Q5 P4
• MP3 Software
• Q3 P4
• Total Q29 and P17

• Word Processor
• Q3 P2
• Web Browser
• Q8 P3
• Operating System
• Q4 P0
• Spreadsheet
• Q2 P0
• Web Page Editor
• Q6 P6
• MP3 Software
• Q3 P2
• Total Q26 and P13

4
Customers Strategy Mix and Match?

• Word Processor
• Q5 P3
• Web Browser
• Q4 P0
• Operating System
• Q9 P4
• Spreadsheet
• Q3 P2
• Web Page Editor
• Q5 P4
• MP3 Software
• Q3 P4

• Word Processor
• Q3 P2
• Web Browser
• Q8 P3
• Operating System
• Q4 P0
• Spreadsheet
• Q2 P0
• Web Page Editor
• Q6 P6
• MP3 Software
• Q3 P2

Why not? Not everything is compatible. Using
incompatible products is okay, but it incurs some
additional costs. It may be better to purchase an
inferior product to avoid these added costs.
5
Our Model

• We have a set of n products labeled 1n. Each
  product has two different versions, one made by
  each company.
• Each product has a price for each companys
  version p1j is company ones price for product
  j, and p2j is company twos price.
• Each product has a quality rating for each
  companys version ?1j is company ones quality
  for j, and ?2j is company twos quality.
• Each ordered pair of products (i,j) has an
  incompatibility penalty ?ij which applies if we
  purchase product i from company 1 and product j
  from company 2.
• The customer selects a set of products S to
  purchase from company 1, purchasing the rest from
  company 2.

6
Our Results

• We consider each of the following problems
• Customers Selection Problem Which products
  should the customer buy from each company? We
  give a polynomial-time algorithm.
• Budgeted Improvement Problem If company 1 can
  improve product qualities by some total amount B,
  how should the improvement be allocated? We prove
  this is NP-Hard and equivalent to MaxSBCC.
• Pricing Problem How should company 1 price its
  products in order to maximize revenue? We give a
  polynomial-time algorithm.
• Compatibility Alteration Problem How should
  company 1 set compatibilities to maximize
  revenue? We prove this is NP-Hard and give a PTAS
  by reduction to Knapsack.
• Compatibility Improvement Problem. If company 1
  can only improve compatibilities, how can it
  maximize revenue? We prove this is NP-Hard and
  hard to approximate, but give algorithms for some
  special cases via duality.

7
Previous Work on Compatibility

• A number of previous results in Economics deal
  with compatibility.
• N. Economides. Desirability of compatibility in
  the absence of network externalities. American
  Economic Review, 791165-1181, 1989.
• J. Farrell and G. Saloner. Standardization,
  compatibility, and innovation. Rand Journal of
  Economics, 1670-83, 1985.
• J. Farrell and G. Saloner. Converters,
  compatibility, and control of interfaces. Journal
  of Industrial Economics, 409-36, 1992.
• M. Katz and C. Shapiro. Network externalities,
  competition, and compatibility. American Economic
  Review, 73424-440, 1985.
• C. Malutes and P. Regibeau. Mix and Match
  Product compatibility without network
  externalities. Rand Journal of Economics,
  19221-234, 1988.
• C. Malutes and P. Regibeau. Compatibility and
  bundling of complementary goods in a duopoly. The
  Journal of Industrial Economics, 4037-54, 1992.

8
Our Work vs. Previous Work in Economics

• Previous results considered the situation where
  products are either fully compatible or fully
  incompatible. This makes sense for mechanical
  components (for example) where they will simply
  work together or not.
• We are the first to consider the possibility of
  partial incompatibility, which makes sense in a
  software application, where incompatibility often
  manifests itself as an annoyance to the
  end-user, who must convert file types, purchase
  emulators, or check that documents import
  properly from one application to another.
• We also consider the scenario of digital goods,
  which has become standard in the computer science
  literature (for example on mechanism design) but
  differs from the model in previous economic
  results (including all the previous work on
  incompatibility).

9
Customer Selection Problem

• Customers Selection Problem
• Budgeted Improvement Problem
• Pricing Problem
• Compatibility Alteration Problem
• Compatibility Improvement Problem

10
Customer Selection Problem

• We consider a graph with vertices corresponding
  to the various products along with special
  designated source and sink nodes s,t. We will
  find an s-t cut S, where those vertices in S are
  the products purchased from company one (along
  with the source). We prove a correspondence
  between our problem and minimum cut.

Three kinds of edges Edges to sink capture
p1,?2 Edges from source p2,?1 Edges between
products ? Min Cut ? Customer Selection
OS
S
p1WP?2WP
WP
t
s
?WPWB
WB
p2MP?1WP
MP
11
Budgeted Improvement Problem

• Customers Selection Problem
• Budgeted Improvement Problem
• Pricing Problem
• Compatibility Alteration Problem
• Compatibility Improvement Problem

12
Budgeted Improvement

• We are allowed to improve the quality of company
  1s products in order to maximize company 1s
  revenue. Obviously if we just increase the
  quality to infinity, all products will be
  purchased from company 1. The more interesting
  problem is, if we can improve the products by a
  total of at most B, how much revenue can we
  generate?
• In the cut formulation of the customers problem,
  we are trying to increase the weight of edges
  from s by at most B so as to increase the S size
  of the minimum cut.

Min cut
t
s
Increase
13
Facts about Budgeted Improvement

• Claim Let G be a graph with capacities ce where
  S is the minimum s-t cut, of capacity C. Select
  some other cut S with capacity CC. By
  increasing the capacities out of s by a total of
  at most C-C, we can ensure that there is a new
  minimum cut S with S?S.
• Proof For every node in S-S, we add an edge
  from the source with capacity C-C. This
  guarantees the desired cut property. Since
  minimum cut equals maximum flow and the capacity
  of cut S is unchanged, there is now a maximum
  flow in this graph of value C. There is an
  augmenting flow from the original maximum flow to
  this new flow, of value C-C. We reduce the
  capacity of the additional edges to equal the
  amount of augmenting flow placed upon them.

14
Budgeted Improvement and MaxSBCC

• We can write the following IP for budgeted
  improvement.
• Maximize ?vpvxv
• Subject to xs1 and xt0
• For all e(u,v) we have yexv-xu
• ?eyece C B
• xv,ye?0,1
• This is exactly a weighted version of
  Maximum-Size Bounded Capacity Cut. In this
  problem, the goal is to find a cut with the
  largest possible number of nodes on the s side,
  while maintaining a bounded cut capacity. The
  problem is known to be NP-Hard, and there is an
  algorithm which exceeds the capacity increase (B)
  by at most O(log2 n) while obtaining optimum
  weighted value of nodes on the s side (Feige
  and Krauthgamer, Siam Journal on Computing 2002).
  No results without increasing the budget are
  known.

15
Pricing Problem

• Customers Selection Problem
• Budgeted Improvement Problem
• Pricing Problem
• Compatibility Alteration Problem
• Compatibility Improvement Problem

16
Pricing Integer Program

• We can write a very similar program for the
  pricing problem. It looks like this
• Maximize ?vpvxv - R
• Subject to xs1 and xt0
• For all e(u,v) we have yexv-xu
• ?eyece C R
• xv,ye?0,1
• Here we can think of setting the prices really
  high, and then offering a rebate of R. Reducing
  the price is equivalent to increasing the
  quality, but it also reduces our revenue. We note
  that at optimality, the red constraint must be
  tight.

17
Modifying the Pricing Program

• Maximize ?vpvxv - ?eyece
• Subject to xs1 and xt0
• For all e(u,v) we have yexv-xu
• xv,ye?0,1
• But this simply requires us to find an s-t cut S
  which maximizes
• ?i?Spi - ?i?S,j?Sc(i,j) which is the same as
  minimizing
• ?i?Spi ?i?S,j?Sc(i,j)
• This is just a minimum-cut problem, equivalent to
  reducing the capacity of all edges (i,t) by pi.
  This proves the following lemma
• Lemma With the optimal price setting, company 1
  sells exactly the same set S of products as if it
  gives away all its products for free.

18
Coming up with Optimum Prices

• We now know how to determine (in polynomial time)
  the set S of products company 1 sells with the
  optimum pricing policy. Now we want to determine
  the prices.
• We again use max-flow and min-cut duality. We
  draw the graph corresponding to zero prices for
  company 1 and find the maximum flow. We then
  increase the (i,t) edges to some large capacity
  and find the maximum augmenting flow. If we
  change these new edge capacities to equal the
  augmenting flow on them, we have found the
  optimum prices.
• Theorem There is a polynomial-time algorithm for
  selecting profit-maximizing best-response prices.

19
Compatibility Alteration Problem

• Customers Selection Problem
• Budgeted Improvement Problem
• Pricing Problem
• Compatibility Alteration Problem
• Compatibility Improvement Problem

20
Compatibility Alteration

• Suppose company 1 can unilaterally alter
  compatibilities. In the optimum solution, let S
  be the set of products the customer purchases
  from company 1. Suppose we were to set
  incompatibilities to be zero for pairs (u,v) with
  u?S and v?V-S, and set incompatibility to be
  infinite for all other pairs. Clearly this does
  not change the minimum cut from (S, V-S) to
  something else.
• We observe that the customer can always be forced
  to buy from only one company by setting infinite
  incompatibility. If this means buying all from
  company 1 then its the optimum solution. So we
  assume this means buying all from company 2.
• Now S can be the set of products sold by company
  1 if any only if S is a minimum cut under the
  capacity assignment described.

21
Possible Minimum Cuts

• The only finite-capacity cuts under this
  assignment are S, s, and V-t. We know that
  s is a smaller cut than V-t.

S
0
8
8
s
t
0
8
22
Defining Minimum Cut

• It follows that S can be the set of products sold
  by company 1 if and only if the capacity of cut S
  is smaller than the capacity of cut s. This
  means
• ?i?S(?2ip1i) ?i?S(?1ip2i) ?i(?1ip2i)
• ?i?S(?1ip2i-?2i-p1i) 0
• We define d(i) ?1ip2i-?2i-p1i and observe that
  the optimum solution S will always include all i
  such that d(i)0. We can then define a value
  D?d(i)gt0d(i). Then the optimum solution is the
  set S maximizing the total price pS subject to
  ?i?S, d(i)lt0(-d(i)) D.
• This is exactly Knapsack, with values pi,
  weights -d(i), and weight bound D. As a result,
  the problem is NP-Hard but has a PTAS.

23
Compatibility Improvement Problem

• Customers Selection Problem
• Budgeted Improvement Problem
• Pricing Problem
• Compatibility Alteration Problem
• Compatibility Improvement Problem

24
Compatibility Improvement

• We consider the case where company 1 can improve
  compatibilities arbitrarily, but cannot worsen
  them. This corresponds to looking for new
  capacities ce on each edge not incident with the
  source or sink, such that cece and the value of
  pS (where S is the minimum cut) is as large as
  possible.
• We will call such a set of capacities valid
  capacities, and we call a set S which is a
  minimum s-t cut for some set of valid capacities
  a minimizable cut.
• Note that we can assume for any vertex i, that i
  does not have both an edge to the source and the
  sink. Otherwise we can reduce the capacity of
  these edges by an equal amount without effecting
  the cuts.

25
Properties of Minimizable Cuts

• Suppose there is some cut S which is minimizable.
  Thus there exist some capacities cece (but
  cece for edges adjoining source or sink) such
  that S is a minimum cut.
• Now consider increasing the capacity of edges
  which do not cross S so that cece. This does
  not increase the capacity of cut S, and does not
  decrease the capacity of any cut, so S is still a
  minimum cut.
• Now consider decreasing the capacity of edges
  which cross S (and do not have the source or sink
  as an endpoint) such that ce0. This decreases
  the capacity of cut S by ce and cannot decrease
  the capacity of any other cut by more than ce,
  so S remains a minimum cut.
• So if S is minimizable, then it is the min cut
  under c capacities.

26
S-Exclusive Flows

• We will say that a flow F is S-Exclusive for some
  set S containing the source but not the sink if
• F(i,t)c(i,t) for all i?S
• F(i,j)0 for all j?S with j?t.
• Theorem Let S be a partition with c(s,j)0 for
  all j?S. Then S is minimizable if and only if
  there exists an S-Exclusive flow F.
• Proof If S minimizable, then let F be the
  maximum flow with capacities ce. We can see F
  is S-Exclusive. If there exists a flow F which is
  S-Exclusive, then it will also be a feasible flow
  with capacities ce, and its value is equal to
  the value of cut S under those capacities, which
  implies S is a minimum cut for ce and thus
  minimizable.

27
A Simple Special Case

• Suppose all edges into t have capacity 0 or 1,
  and all the prices are uniform. This corresponds
  to almost equivalent products. In this case we
  can compute the best minimizable cut (and
  therefore solve the compatibility improvement
  problem) in polynomial time.
• Proof Just compute an integral maximum s-t flow.
  We can modify the flow such that any node with
  positive incoming flow will saturate its
  (capacity zero or one) edge to the sink without
  effecting the total flow value. After this
  modification, the flow is S-Exclusive for some
  set S, and S is the number of nodes with
  c(i,t)0 plus the value of the flow. No larger
  minimizable set exists.

28
An Approximation

• Suppose all prices are 1 and the edges into t
  have integral capacities in the range 0,C for
  some C. We consider the following greedy
  approximation algorithm
• Start with S?i c(i,t)0
• Repeat
• Among all i?S such that S?i is minimizable, let
  i be the one minimizing c(i,t).
• Set S?S?i
• Until S?i not minimizable for any i
• Return S

29
Proof of Approximation

• Suppose the greedy algorithm finds set S, and the
  optimum solution is set S. We claim that
  SS/C.
• Proof There exist corresponding exclusive flows
  F and F. For any set R including source but not
  sink, define ?(R) to be the maximum value s-t
  flow that does not use any edges (i,j) with j?R
  unless jt. We observe that R is minimizable if
  and only if ?(R)?i?Rc(i,t).
• Let A be a non-empty node set disjoint from S. We
  prove that
• ?(S?A) ?(S) ?i?Ac(i,t) - A
• We observe that the flow for the union can be
  formed by augmenting the flow for S and therefore
  saturates edges from S to t. We can then view the
  flow as giving us a DAG, and progressively
  reroute flow to saturate earlier DAG vertices
  sink edges.

30
Proof of Approximation

• The argument is that if ?(S?A) was larger, we
  could keep pushing flow back through the DAG
  until we saturate some nodes edges to t where
  the nodes only incoming flow is from S. This
  node then could be added to S, implying the
  greedy algorithm would not terminate.
• Now we consider the set AS-S. Applying the
  inequality along with the maximum capacity of C
  gives us the desired bound.

31
Hardness Results

• Even if all prices and all capacities into the
  sink are from 0,1, the maximum-price
  minimizable cut cannot be approximated to better
  than ?(n1-?) for any ?gt0.
• Even if all prices are 1, the maximum-size
  minimizable cut cannot be approximated to better
  than ?(n(1/3)-?) for any ?gt0.
• Both these results are based on reduction from
  independent set.

32
Idea of Independent Set Reduction

• We want to approximate independent set on graph G.

• We create a capacitated graph with a node for
  each vertex and each edge in G, plus s and t.
  Edge nodes have p0, vertex nodes p1.

s
a
b
a,c
b,c
a,b
b,d
c,d
c
d
c
a
b
d
3
3
2
2
t
33
Proof Sketch

• An S-exclusive flow which includes the node for a
  must saturate the edge (a,t). Since the capacity
  of this edge is degree(a), we have to send all
  flow from nodes representing edges including a
  through a. This prevents us from selecting two
  adjacent nodes, giving us a correspondence
  between S-exclusive flows, minimizable cuts, and
  independent sets in G.

• We can transform the vertex nodes as follows.

a
a
8
?(a)
t
a
t
8
a
34
Open Problems

• Combining the problems. What if company 1 can
  modify any of price, compatibility, quality? How
  do we combine the results?
• Improving MaxSBCC approximation.
• What if theres more than two companies? Even
  customers selection problem is NP-Hard
  (multi-way cut) but there are good
  approximations. How do companies strategize here?
• What if different customers have different
  quality ratings for products?
• What about pricing games between companies?
• What about auctions/mechanisms between customers
  and companies?