Strategyproof Sharing of
submodular costs:
Budget Balance Vs. Efficiency
Liad Blumrosen
May 2001
Motivation
U1 = 2
U2 = 2
U3 = 3
Welfare:
M
• Knows costs
Cost( {1,2} ) = 3
1
2
3


X
2$
1$
0$
ui - Cost({1,2})
=2+2-3=1
Budget Balance:
xi - Cost({1,2 })
=2+1-3=0
Lecture outline
• Introduction
– Budget Balance Vs. Efficiency
• Suggested mechanisms
– Marginal Cost
– Shapley
}
• Multicast networks
• Feasibilty of mechanisms in multicast
networks
• conclusions
Game
theory
}
cs
The Model
• N agents
– Agent can either receive service or not (binary)
• ui - willingness of agent i to pay for the service
• C(S) - cost for providing the service for a set
of users S
The mechanism’s output:
• qi - does agent i receive the service?
– if qi = 1 she receives. if qi = 0, she doesn’t
• xi - the payment of agent i (cost shares)
Submodular cost function
• We will deal with submodular cost functions:
• C is submodular
if  S,T  N
C(T) - C(ST)  C(ST) - C(S)
T
S
• (In our model C is also non-decreasing and
C() = 0)
Mechanism’s desired properties
• No Positive Transfers (NPT)
– Cost shares (payments) are nonnegative:
i xi  0
• Voluntary Participation (VP)
– Welfare level (u - x) of no service at no cost
(qi=0,xi=0) is guaranteed for truthful agents
• Consumer Sovereignty (CS)
– Each agent has ui guaranteeing getting the
service (regardless of the other reported values
u-i)
Mechanism’s desired properties:
Incentive Compatibility
• Strategyproof mecahnsim
– Telling the true ui is a dominant straegy for any
agent
• Group-strategyproof mechanism
– No coalition of agents has an incentive to jointly
misreport their true ui
– Stronger form of Incentive Compatibility.
Model’s desired properties (cont.)
social welfare
is not the sum of
•The
Budget
Balance
the –agent
and doesn’t
xi =surpluses,
C(R)
(when
R is the receivers set)
depend on payments (xi)
• Efficiency
– For any u, the mechanism should maximize the
social welfare: W(N,u) = maxTN[uT -C(T)]
(where uT = iRuj)
• Remark: In our model the utilities are quasilinear (uiqi - xi)
Model’s desired properties
NPT
VP
CS
strategy-proof
Budget-Balance
Efficiency
Budget-balance and Efficiency are
mutual exclusive !!!
Model’s desired properties
NPT
VP
CS
strategy-proof
Efficiency
Marginal
Cost
Budget-Balance
shapley
Cost Sharing Methods
• A Cost Sharing Method f allocates C(S)
among the agents in S
– fi(S) - is the payment of agent i when the
receivers set is S
– fi(S) = C(S) (budget-balance)
• Cost Sharing Function is cross-monotonic if:
ST, i  S
 fi(S)  fi(T)
– Agent can’t pay more when receivers set expands
Cost Sharing Methods (cont.)
• Consider the following allocation algorithm
that uses the Cost Sharing Method f
• S0 = N
• St+1 = { i | ui  fi( St ) }
(proceed untill St is unchanged)
S*(f,u) is the final allocation
• The mechanism that uses f with allocation
S*(f,u) is denoted by M(f)
Theorem 1 (without proof)
• For any cross-monotonic function f, the
mechanism M(f) is budget balanced, group
strategy-proof and meets NPT,VP,CS.
Conversely, for any mechanism M which is
group strategy-proof, budget-balanced and
meets NP,VP,CS, there is a cross monotonic
cost sharing method f such that M(f) is
welfare-equivalent to M
Choosing cost sharing function
• We saw that every cross-monotonic function
defines a mechanism with the desired
properties (except efficiency)
– Which mechanism is the “best”?
• We will choose the method f for which M(f)
minimzes the maximal welfare loss:
– (f) = supu[ bestWelfare(u) - welfareM(f)(u) ]
– where: bestWelfare(u) = maxTN(uT - C(T))
welfareM(f)(u) = (Us*(f,u) - C(s*(f,u))
Shapley’s cost sharing method
• Consider the following cost sharing
function, based on Shapley Value:
|T|!(|S| - |T| - 1)! [C(Ti) - C(T)]
– f*i(S) = TS-i |S|!
• Theorem 2: (without proof)
Among all M(f) derived from crossmonotonic functions, M(f*) has the uniquely
smallest maximal welfare loss
– (f*) < (f)
ff*
Model’s desired properties
NPT
VP
CS
strategy-proof
Efficiency
Marginal
Cost
Budget-Balance
shapley
cross-monotonic
Marginal Cost Mechanism
• The welfare of coalition S is
== maxT  S ( UT - C(T) )
surplusi = uiq*w(S,u)
x*
i
i
• Coalition
is called
if
( w(N,u) -Sw(n
- i,u) efficent
)
us - C(S) = w(N,u)
Marginal cost pricing mechanism:
– The reciever set (q*) is the largest efficent
coalition
– The cost shares (payments) given by VCG:
x*i = uiq*i - ( w(N,u) - w(n - i,u) )
marginal welfare of agent i
Marginal Cost Mechanism
• Theorem 3:
If M is a strategyproof and efficient
mechanism, meeting NPT, VP,
then M is welfare equivalent to MC.
Conversely, the MC mechanism meets NPT,
VP (and CS), and is efficient and strategyproof
• Efficient mechanism is mechanism that select efficient
allocations (not necessarily the largest) for all profiles
(u’s)
• Welfare equivalent means that:
u i uiqi(u) - xi(u) = uiq*i(u) - x*i(u)
Marginal Cost Mechanism: proof
• Let M be any strategyproof and efficient
mechanism (also meets NPT,VP)
– I’ll show that M is welfare equivalent to MC
• strategyproofness + efficiency 
x(u) is: xi(u) = uiqi(u) - [ W(N,u) - hi(u-i) ]
• I’ll prove the following:
– hi(u-i) = W(N-i,u) (as in the MC mechanism)
– if efficient set is not maximal, welfare equivalence
maintains
Marginal Cost Mechanism:proof
• We know xi(u) = uiqi(u) - [ W(N,u) - hi(u-i) ]
I’ll show hi(u-i) = W(N-i,u)
• Consider arbitrary u-i
• u0 - the completion of u-i by u0i = 0
– NPT, VP  xi(u0) = 0
• xi(u0) = uiqi(u0) - [ W(N,u0) - hi(u-i) ]
 hi(u-i) = W(N,u0) = W(N - i,u0) = W(N - i,u)

if S efficient, S-{i} also efficient:
xi(u)
= uiq
[ -W(N,u)
us - C(S)
C(S-{i}) i(u)u-s-{i}
W(N - i,u)]
Marginal Cost Mechanism:proof
• Now we know that M takes the same form as
MC, except R (the receivers set) can be any
efficient allocation
– not necessarily the maximal efficient set
• Lemma (technical, without proof):
if any S,T are efficient, then so is ST
– S is efficient if
us - C(S) = W(N,u)
( = maxTN(uT - C(T) )
– consequence of submodularity of C
•  if S efficient, and S* is largest-efficient
then S  S*
Marginal Cost Mechanism:proof
• If iS*, in both M, MC:
S*
– qi(u) = 0 , xi(u) = 0
• If iS*S, in both M, MC:
– qi(u) = 1,
xi(u) = uiqi(u) - [ W(N,u) - W(N - i,u)]
S
• If iS* - S
– W(N,u) = W(N-i,u) (S N is efficient)
– In M: qi(u) = 0, xi(u) = 0
 Agent i has welfare of: ui*qi - xi = 0
– In MC: qi(u) = 1, xi(u) = ui
 Agent i has welfare of: ui*qi - xi = 0
M and
MC are
welfare
equivalent
Marginal Cost Mechanism
• Theorem 3:
If M is a strategyproof and efficient
mechanism, meeting NPT, VP,
then M is welfare equivalent to MC.

Conversely, the MC mechanism meets NPT,
VP (and CS), and is efficient and strategyproof
Marginal Cost Mechanism:proof
• Strategypoofness and efficiency are known
properties of the VCG mechanism.
• NPT:
W(N,u) = us* - C(S*)  ui + us* - i - C(S* - i)
 ui + W(N-i, u)
 x*i(u) = uiqi(u) - [ W(N,u) - W(N - i,u)] 
ui - [ W(N,u) - W(N - i,u)]  0
• VP:
welfarei = uiqi(u) - xi (u) =
= uiqi(u) - uiqi(u) - [ W(N,u) - W(N - i,u)]  0 =
= welfarei(qi=0, xi = 0 )
Marginal Cost Mechanism:proof
• CS:
lemma: If ui  C( {i} )
then us{i} - C( s{i} )  us - C( s )
proof:
(1) C(S{i})) + C(S{i})  C(S) - C({i}) (submodulaity)
(2) C(S{i})  C(S) - C({i})
(iS, C() = 0)
(3) us-C(S{i}) - C({i})  us -C(S)
us{i} - C( s{i} ) = us + ui - C( s{i} ) 
us + C({i}) - C( s{i} )  us -C(S) 
 for big enough ui (  C(i) ), any largest
efficient set will contain i
Marginal Cost Mechanism
shapley
marginal cost
NPT


VP


CS

(not needed) 
Incentive Compatibility
group
singelton
Budget Balance

X (never surplus)
Efficiency
X (minmax loss)

Lecture outline
• Introduction
– Budget Balance Vs. Efficiency
• Suggested mechanisms
– Marginal Cost
– Shapley
}
• Multicast networks
• Feasibilty of mechanisms in multicast
networks
• conclusions
Game
theory
}
cs
Multicast transmission
7
5
• Pick set of
receivers
4
2
2
1
3
source
3
Multicast transmission
• Pick set of
receivers
• create a tree
connecting the
receivers
• multicast the
movie on the
tree.
source
Multicast transmission model
• (N,L) - an undirected graph
– N - the nodes in the network
– L - links in network
• P - user population (0 or more users in each node)
• C(l) - cost of link lL
–0,
known to nodes on both ends
• R - the receivers set
• T(R) - tree connecting R
– subtree of a given universal tree T(P) covering R !!!
• C( T(R) ) = lT(R)C(l)
(submodular)
Computational model
• An instance of this problem contains 3 parameters:
– n - number of nodes in the multicast tree
– p - number of users (population size)
– m - total size of input : {C(l)}lL{ui}i P
• Desired commnication-complexity properties:
–
–
–
–
Total messages on links (ideally O(n))
Maximal number of messages on link (ideally O(1))
Limited maximal message size
Local computation comlexity
We will ignore
these properties
MC cost sharing feasibility
• Theorem 4:
MC cost sharing requires exactly two messages
per link.
Proof idea:
There is an algorithm that computes the cost
shares by performing one bottom-up traversal on
tree, followed by one top-down traversal.
Theorem 4: proof
• W(u) : welfare from the subtree rooted at 
• W(u) = u + [

W(u) ] - c

child() | W (u)  0
– child() is all the child nodes in the tree
– u is the sum of the utilities of the user in 
– C the cost of the link between  and its parent
root
C

p()
C

Theorem 4: proof
• Following is an algorithm for the
implementation of MC in multicast network
• The allocation (q  {0,1}|P| ):
qi(u) = 1 if W(u)  0 for all nodes  on
the path from user i to the root
Else, qi(u) = 0.
– if the welfare of any subtree on the way to the
root is negative, no broadcast to this subtree !
Theorem 4: proof
• How the algorithm uses 2 messages per link?
– The W(u) can be computed by bottom-up
traversal
– The allocations can be computed by propagating
qi(u) in a top-down traversal
– Computing the cost shares will also be
computed in the same top-down traversal
Theorem 4: proof
• Cost sharing (payments)
according to the VCG formula:
xi(u) = uiqi(u) - [ W(N,u) - W(N-i,u) ]
– Recall that W(N,u) = maxTN[ uT - C( R(T) ) ]
• How can we compute W(N-i,u) ?
Theorem 4: proof
yi(u) :
• Case 1:
min
w(u)
 node on the path
from i to the root
If ui  yi(u)
– Receivers set stays the same when dropping i.
Thus, W(N,u) - W(N-i,u) = ui
 xi(u) = ui - [W(N,u) - W(N-i,u)] = 0
• Case 2:
If ui > yi(u)
– Dropping user i results elmination of subtree
with the total welfare yi(u)
 xi(u) = ui - [W(N,u) - W(N-i,u)] = ui - yi(u)
Theorem 4: proof
Propagate qi
and yi
(allocation and
cost shares)
calculate
W(u) for
each node
total of exactly 2 messages per link
Theorem 4: clarification
• In our model the tree must be a subtree of a
given universal tree T(P)
• Is it computationally feasible, when we can
select ANY subtree of the original
network?
• No ! The problem becomes NP-hard to
approximate within ratio .
– even if the original graph is bounded-degree
Shapley’s cost sharing method
• Reminder :
Shapley’s mechanism is M(f*) when:
|T|!(|S| - |T| - 1)!
[C(Ti) - C(T)]
– f*i(S) = TS-i |S|!
Shapley cost sharing feasibility
• Theorem 5:
Shapley’s cost sharing requires, in the worst case,
(n · p) message exchanges ((n2) when p=O(n) )
• What’s wrong with worst case of (n2) ?
– Centralized approach’s worst-case is also (n2)
– In our complexity model, centralized approach can be
applied to any (polynomial) cost sharing mechanism
– Thus, Shapley can be considered as with “maximal”
communication complexity.
– Shapley has no benefit for being distributed !
Conclusions
NPT
VP
CS
strategy-proof
Efficiency
Marginal
Cost
Exactly 2 messages
per link ( total (n) ):
FEASIBLE
Budget-Balance
shapley
cross-monotonic
(n2) msg exchanges:
FEASIBILITY PROBLEMS
Bibliography
• Moulin H. and S. Shenker (1997).
“Strategyproof Sharing of submodular costs:
Budget Balance versus Efficiency”
Economic Theory.
http://www.aciri.org/Shenker/cost.ps
• Feigenbaum J. Papadimitriou C. and
Shenker S “Sharing the cost of multicast
transmissions”
group strategyproof
• Group strategyproof
– No coalition of agents has an incentive to jointly
misreport their true ui
• Formal defnition:
– for a fixed T  N,
– for any u,u’ such that uj = u’j jT and
allocations (q,x) and (q’,x’) repectively
– if
uiq’i - x’i  uiqi - xi iT
then all the inequalities are equalities.
• Strategyproofness is when |T| = 1
group strategyproof
•
•
•
•
Let’s see why MC is not group-strategyproof
C(1)=C(2)=6
C(12)=8
u1 = u2 = 5
s*(u) = {1,2}
x*1(u) = x*2(u) = 5 - (8 - 6) = 3
• But, agent 1 can change to u’1 = 7
her allocation stays the same
x*2(u) decreases to 2 !!!