Strategyproof Sharing of submodular costs:

Budget Balance Vs. Efficiency

Liad Blumrosen

May 2001

MotivationU1 = 2 U2 = 2 U3 = 3

M

• Knows costs

X

2$ 1$ 0$

1 32

Cost( {1,2} ) = 3

Welfare:

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 NC(T) - C(ST) C(ST) - C(S)

• (In our model C is also non-decreasing and C() = 0)

T S

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.)• Budget Balance

xi = C(R) (when R is the receivers set)

• Efficiency– For any u, the mechanism should maximize the

social welfare: W(N,u) = maxTN[uT -C(T)] (where uT = iRuj)

• Remark: In our model the utilities are quasi-linear (uiqi - xi)

The social welfare is not the sum of the agent surpluses, and doesn’t depend on payments (xi)

Model’s desired properties

NPT CSVP

strategy-proof

Budget-Balance

Efficiency

Budget-balance and Efficiency are mutual exclusive !!!

Model’s desired properties

NPT CSVP

strategy-proof

Budget-BalanceEfficiency

shapleyMarginal Cost

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:ST, 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

• The mechanism that uses f with allocation S*(f,u) is denoted by M(f)

• S0 = N• St+1 = { i | ui fi( St ) }(proceed untill St is unchanged)

S*(f,u) is the final allocation

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) = maxTN(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)!– f*i(S) = TS-i |S|!

• Theorem 2: (without proof)

Among all M(f) derived from cross-monotonic functions, M(f*) has the uniquely smallest maximal welfare loss (f*) < (f) ff*

[C(Ti) - C(T)]

Model’s desired properties

NPT CSVP

strategy-proof

Budget-BalanceEfficiency

cross-monotonic

shapleyMarginal Cost

Marginal Cost Mechanism• The welfare of coalition S is

w(S,u) = maxT S ( UT - C(T) )

• Coalition S is called efficent if 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

surplusi = uiq*i - x*i = ( w(N,u) - w(n - i,u) )

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) xi(u) = uiqi(u) - [ W(N,u) - W(N - i,u)]

if S efficient, S-{i} also efficient: us - C(S) us-{i} - C(S-{i})

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 ST – S is efficient if

us - C(S) = W(N,u) ( = maxTN(uT - C(T) )

– consequence of submodularity of C

if S efficient, and S* is largest-efficientthen S S*

Marginal Cost Mechanism:proof• If iS*, in both M, MC:

– qi(u) = 0 , xi(u) = 0

• If iS*S, in both M, MC: – qi(u) = 1,

xi(u) = uiqi(u) - [ W(N,u) - W(N - i,u)]

• If iS* - 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

S*

S

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}) (iS, 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

source

• Pick set of receivers

3

1

2

4

2

3

57

Multicast transmission

source

• Pick set of receivers

• create a tree connecting the receivers

• multicast the movie on the tree.

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 lL 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) ) = lT(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)}lL{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() 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

child() | W

(u) 0

p()

root

C

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 rootElse, 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) = maxTN[ uT - C( R(T) ) ]

• How can we compute W(N-i,u) ?

Theorem 4: proofyi(u) : min w(u)

• Case 1: 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)

node on the path from i to the root

Theorem 4: proof

calculate W(u) for each node

Propagate qi and yi (allocation and cost shares)

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)!– f*i(S) = TS-i |S|! [C(Ti) - C(T)]

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 CSVP

strategy-proof

Budget-BalanceEfficiency

cross-monotonic

shapleyMarginal

Cost

Exactly 2 messages per link ( total (n) ): FEASIBLE

(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 jT and allocations (q,x) and (q’,x’) repectively

– if uiq’i - x’i uiqi - xi iT

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 = 7her allocation stays the samex*2(u) decreases to 2 !!!