Group Strategyproofness and No Subsidy via LPDuality

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Group Strategyproofness and No Subsidy via
LP-Duality

• By Kamal Jain and Vijay V. Vazirani

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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
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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

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Definitions (cont.)

• A cost sharing mechanism determines which user
  receive service and at what price.
• A cost sharing method is a function ? which
  distributes the cost of the service between users
  receiving it.

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The cost sharing method is cross monotonic if
?i?Q, Q ? R ? ?(Q,i) ? ?(R,i) It is
weakly cross monotonic if Q ? R ? ?i?Q
?(Q,i) ? ?i?Q ?(R,i)
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Definitions (cont.)

• More formally, ? takes two arguments, a set of
  users Q and user i, and returns a non-negative
  real number, satisfying
• If i ? Q, then ?(Q,i) 0
• Else, ?i?Q ?(Q,i) C(Q) , where C(Q) represents
  the cost of serving the Q (not necessary the
  optimal cost)

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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.
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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
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• 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

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Computational constraint

• Decisions should be made in polynomial time

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Economic constraints

• Optimum
• No Positive Transfers (NPT)
• Voluntary Participation (VP)
• Consumer Sovereignty (CS)
• Budget-balance (BB)
• Efficiency
• Group Strategyproof

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But
The first constraint cant 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)
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• For every cross monotonic cost sharing method
  Moulin and Shenker give a mechanism M(?) which
  computes Q and xi ?(Q,i)
• Q is initialized to U
• If user i?Q and ui lt ?(Q,i) , then drop i from Q.
  Keep repeating this step in arbitrary order until
  no such user found
• Set xi ?(Q,i)

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Theorem 1 (Moulin and Shenker) For any cross
monotonic cost sharing method ? , the mechanism
M(?) 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(?) for some cross monotonic cost
sharing method, ?.
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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
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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
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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
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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
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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.
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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.

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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.
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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)

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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
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LP relaxation

• LP relaxation
• Minimize ?e?E cexe
• Subject to ? e e? ?(S) xe ? 1, ? valid set S
• xe ? 0, e ? E

• DLP
• Maximize ? (valid set S) yS
• Subject to ? S e? ?(S) yS ? ce , e ? E
• yS ? 0, ? valid set S

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Edge e feels dual yS if yS gt 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
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Algorithm(minimum branching)

• (Initialization) F ? ? for each S ? V, yS ? 0
• (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
• Let e1, e2, el be the ordered list of edges in
  F.
• (Reverse delete) For j l downto 1 do
  
  If there are no unsatisfied sets w.r.t. F ej,
  then F ? F ? ej
• Return F
• For every valid S such that yS gt 0 ?F(S)
  1
• ? e?F ce ? (valid set S) yS

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Given a set Q of recipients, the Algorithm is
used to find a minimum spanning tree containing Q
and the root. The costsharing method, ?, simply
distributes each raised dual equally among all
vertices contained in this dual. So, the amount
charged to user i is ? (Q, i) ? (valid set S) (
yS / S) For this costsharing method ?, the
mechanism M(?) is budget balanced, meets NPT, VP,
CS, is group strategyproof, and satisfies No
Subsidy.
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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

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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.

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Appendix 3

• LP is
• min ltc, xgt
• subject to Ax ? b
• x ? 0
• c, x vectors in Rn
• b vector in Rm
• A matrix n x m

DLP is max ltb, ygt subject to yA ? c y ? 0
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Appendix 3 (cont.)

• Duality theorem
• If LP has a solution then DLP has a solution and
• min ltc,xgt max ltb,ygt