Group Strategyproofness and
No Subsidy via LP-Duality
By Kamal Jain and Vijay V. Vazirani
Abstract
In this article Jain and Vazirani introduce a fairness criterion on
service providers, which they call “No Subsidy”
In their second result they give budget balanced and group
strategyproof cost sharing method for cost function that is neither
submodular nor supermodular
Definitions

Cost function C(S) denotes the cost incurred by
the company to serve the users in S. S  U, U – all
the customers (users) of the company.
 Each user, i, has utility ui’ for receiving the news.
User i enjoys benefit of ui’ – xi. Each user may
misreport this utility as some other number ui For
the rest of discussion, utility of user i is ui
Definitions (cont.)

A cost sharing mechanism determines
which user receive service and at what
price.
 A cost sharing method is a function ξ
which distributes the cost of the service
between users receiving it.
The cost sharing method is cross monotonic if
iQ,
Q  R  ξ(Q,i)  ξ(R,i)
It is weakly cross monotonic if
Q  R  iQ ξ(Q,i)  iQ ξ(R,i)
Definitions (cont.)
More formally, ξ takes two arguments, a set of users Q and user
i, and returns a non-negative real number, satisfying
1. If i  Q, then ξ(Q,i) = 0
2. Else, iQ ξ(Q,i) = C(Q) , where C(Q) represents the cost of
serving the Q (not necessary the optimal cost)
The Model
Jain and Vazirani use example of service provider company that
broadcasts news on the net.
Consider graph G = (V, E), with edge weights ce and marked node
root (service provider). Other nodes are users.
Messages are sent using multicasting. Each message can be
duplicated at any node at no cost. Edge e charges ce to transfer
message from one end to other. The cost of broadcasting a message
is total price charged by the edges.
The Model (cont.)
Each edge assumed to be of infinite capacity, so the
message can be sent through the shortest path.
Hence we can assume that ce satisfy the triangle
inequality
All users report their utilities ui and now provider
should decide about three things:

A set Q of users, selected to receive a
message
 A tree T containing Q to broadcast the
message
 For each user i, the price xi to be charged as
a cost of delivering the message
Computational constraint

Decisions should be made in polynomial
time
Economic constraints

Optimum
 No Positive Transfers (NPT)
 Voluntary Participation (VP)
 Consumer Sovereignty (CS)
 Budget-balance (BB)
 Efficiency
 Group Strategyproof
But…
The first constraint can’t be met unless P = NP
It is also impossible to find strategyproof mechanism that is both
budget balanced and efficient, so we drop efficiency.
Other constraints can be captured by a cross monotonic sharing
method (Moulin and Shenker)
For every cross monotonic cost sharing method Moulin and
Shenker give a mechanism M(ξ) which computes Q and xi =
ξ(Q,i) :
1. Q is initialized to U
2. If user iQ and ui < ξ(Q,i) , then drop i from Q. Keep
repeating this step in arbitrary order until no such user found
3. Set xi = ξ(Q,i)
Theorem 1 (Moulin and Shenker)
For any cross monotonic cost sharing method ξ , the mechanism
M(ξ) is budget balanced, meets NPT, VP, CS and is group
strategyproof.
Theorem 2 (Moulin and Shenker)
Suppose U is the set of all users and C: 2U  R+ is a submodular
cost function. If a cost sharing mechanism is budget balanced,
meets NPT, VP, CS and is group strategyproof then it is equivalent
to M(ξ) for some cross monotonic cost sharing method, ξ.
No Subsidy condition
If Q is the set of users selected to receive the service and xi is the
cost of service asked from user i, then for any R  Q,
iR xi C(R)
Where C(R) is the cost of serving R
No Subsidy (cont.)
Budget balanced, NPT, VP, CS and group strategyproof cost
sharing mechanism is not necessary no subsidy :
Users a and b with utilities ua and ub.
Cost function: C() = 0; C(a) = C(b) = 1; C(a,b) = 3
Fractional set
A fractional set Sf is a set S and function f : S  [0,1]
(for each element e of S, f tells “which part of e is in S”)
Union of two fractional sets S1f1  S2f2 is S1  S2 with
function f = min (f1 + f2, 1)
Fractional sets S1, S2, … Sn cover S if
S1  S2  …  Sn = S
Covering property
Cost function C is said to exhibit the covering property if for any
set S of users and for any covering of S = j fj Sj
C(S)  j fj C(Sj)
Where each Sj is set of users
The theorem
Suppose U is the set of all users and C: 2U  R+ is a cost
function. There is a cost sharing mechanism which satisfies
budget balance, meets NPT, VP, CS and also satisfies the No
Subsidy condition if and only if C exhibits the covering property.
Proof


Lemma 1: Suppose U is the set of all users and C: 2U 
R+ is a cost function. There is a budget balanced, NPT,VP,
CS and No Subsidy cost sharing mechanism if and only if
there is a weakly cross monotonic cost sharing method
Lemma 2: Suppose U is the set of all users and C: 2U 
R+ is a cost function. There exist a weakly cross monotonic
cost sharing method for C if and only if C exhibits the
covering property.
1. The theorem follows from the two lemmas
2. From lemma 1 and from Moulin and Shenker theorem 2 follows
that if C is submodular function and a cost sharing mechanism is
BB, NPT, VP, CS and is group strategyproof, then it satisfies No
Subsidy condition too.
3. From the theorem and from Moulin and Shenker theorem 1
follows that submodular functions are subclass of functions that
exhibit the covering property.
The cost sharing method
In this section, a costsharing method for multicasting is presented. It achieves
budget balance and is crossmonotone, and for any set Q of users chosen for
distribution, finds a tree of cost at most twice the optimal Steiner tree
containing the root and users Q.
The method utilizes two facts:
•
If the edge costs satisfy the triangle inequality, the cost of a minimum
spanning tree on the set of required vertices is within twice the cost of an
optimal Steiner tree containing all required vertices. (Kou, Markowsky,
Berman)
•
There is an exact linear programming relaxation for the minimum spanning
tree problem, i.e., a relaxation that always has optimal integral solutions.
(Edmonds)
In minimum branching problem, we are given a directed graph with nonnegative costs on the directed edges, and one of the vertices is marked as root.
The problem is to find a minimum cost tree containing all vertices and
directed into the root. The transformation from the minimum spanning tree
problem in an undirected graph to the minimum branching problem is
straightforward. Simply replace each undirected edge e of G by two directed
edges each of cost of e , and ask for a minimum cost branching directed into
the root. Let us denote this directed graph by H = (V, E’)
S  V is valid if it is non-empty and does not contain root
Let (S) = { (uv)  E’ | u  S and v  S} and
F(S) = { (uv)  E’ | F  E’, u  S and v  S}
LP relaxation
LP relaxation:
DLP:
Minimize eE’ cexe
Maximize  (valid set S) yS
Subject to  e| e (S) xe  1,  valid
set S
Subject to  S | e (S) yS  ce , e  E’
xe  0, e  E’
yS  0,  valid set S
Edge e feels dual yS if yS > 0 and e (S). Edge e is tight if the total
amount of dual it feels equals its cost. The dual program is trying to
maximize the sum of the dual variables yS subject to the condition that no
edge feels more dual than its cost, i.e., no edge is overtight.
Set S  V is unsatisfied if it is valid and F(S) = 
Any minimal unsatisfied set is said to be active
Algorithm
(minimum branching)
1.
(Initialization) F  ; for each S  V, yS  0
2.
(Edge augmentation) While there exists an unsatisfied set do: Find all active
sets w.r.t. F . For each such set S, raise its dual variable yS until some edge e
goes tight; F  F  {e}
3.
Let e1, e2, … el be the ordered list of edges in F.
4.
(Reverse delete) For j = l downto 1 do:
If there are no unsatisfied sets w.r.t. F – {ej}, then F  F  {ej}
5.
Return F
For every valid S such that yS > 0 | F(S) | = 1
 eF ce =  (valid set S) yS
Given a set Q of recipients, the Algorithm is used to find a minimum
spanning tree containing Q and the root. The costsharing method, ξ,
simply distributes each raised dual equally among all vertices
contained in this dual. So, the amount charged to user i is
ξ (Q, i) =  (valid set S) ( yS / |S|)
For this costsharing method ξ, the mechanism M(ξ) is budget
balanced, meets NPT, VP, CS, is group strategyproof, and satisfies
No Subsidy.
Appendix 1

Cost function is submodular if
1. C() = 0
2. Q1 , Q2 (subsets of users):
C(Q1) + C(Q2)  C(Q1Q2) + C(Q1Q2)
 Cost function is supermodular if the
second condition is reversed
Appendix 2

Cost sharing mechanism is budget balanced if the
total amount it charges from the receivers S is
same as C(S).
 It is efficient if it maximizes over all possible S,
the sum of the utilities of users in S minus C(S)
 It is strategyproof if the dominant strategy of
each user is to reveal the true value of his utility.
 It is group strategyproof if it holds for coalitions.
Appendix 3
LP is
min <c, x>
subject to Ax  b
x0
c, x – vectors in Rn
b – vector in Rm
A – matrix n x m
DLP is
max <b, y>
subject to yA  c
y0
Appendix 3 (cont.)
Duality theorem
If LP has a solution then DLP has a solution
and:
min <c,x> = max <b,y>