Strategyproof Sharing of submodular costs: Budget Balance Vs. Efficiency

1
Strategyproof Sharing of submodular costsBudget
Balance Vs. Efficiency

• Liad Blumrosen
• May 2001

2
Motivation
U1 2
U2 2
U3 3

• M
• Knows costs

Cost( 1,2 ) 3
3
Lecture outline

• Introduction
• Budget Balance Vs. Efficiency
• Suggested mechanisms
• Marginal Cost
• Shapley
• Multicast networks
• Feasibilty of mechanisms in multicast networks
• conclusions

4
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 mechanisms output
• qi - does agent i receive the service?
• if qi 1 she receives. if qi 0, she doesnt
• xi - the payment of agent i (cost shares)

5
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)
• (In our model C is also non-decreasing and C(?)
  0)

6
Mechanisms 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
  (qi0,xi0) is guaranteed for truthful agents
• Consumer Sovereignty (CS)
• Each agent has ui guaranteeing getting the
  service (regardless of the other reported values
  u-i)

7
Mechanisms 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.

8
Models desired properties (cont.)
The social welfare is not the sum of the agent
surpluses, and doesnt depend on payments (xi)

• 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) maxT?NuT -C(T)
  (where uT ?i?Ruj)
• Remark In our model the utilities are
  quasi-linear (uiqi - xi)

9
Models desired properties
10
Models desired properties
NPT
CS
VP
strategy-proof
Budget-Balance
Efficiency
shapley
Marginal Cost
11
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 cant pay more when receivers set expands

12
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
• St1 i ui ? fi( St )
• (proceed untill St is unchanged)
• S(f,u) is the final allocation

13
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

14
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))

15
Shapleys cost sharing method

• Consider the following cost sharing function,
  based on Shapley Value
• T!(S - T - 1)!
• fi(S) ?T?S-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) lt ?(f) ?f?f

C(T?i) - C(T)
16
Models desired properties
NPT
CS
VP
strategy-proof
Budget-Balance
Efficiency
shapley
Marginal Cost
cross-monotonic
17
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)

surplusi uiqi - xi ( w(N,u) - w(n -
i,u) )

• Marginal cost pricing mechanism
• The reciever set (q) is the largest efficent
  coalition
• The cost shares (payments) given by VCG xi
  uiqi - ( w(N,u) - w(n - i,u) )

marginal welfare of agent i
18
Marginal Cost Mechanism

• Theorem 3If 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 (us)
• Welfare equivalent means that?u ?i uiqi(u) -
  xi(u) uiqi(u) - xi(u)

19
Marginal Cost Mechanism proof

• Let M be any strategyproof and efficient
  mechanism (also meets NPT,VP)
• Ill show that M is welfare equivalent to MC
• strategyproofness efficiency ? x(u) is
  xi(u) uiqi(u) - W(N,u) - hi(u-i)
• Ill prove the following
• hi(u-i) W(N-i,u) (as in the MC mechanism)
• if efficient set is not maximal, welfare
  equivalence maintains

20
Marginal Cost Mechanismproof

• We know xi(u) uiqi(u) - W(N,u) - hi(u-i)
  Ill 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 us -
C(S) ? us-i - C(S-i)
? xi(u) uiqi(u) - W(N,u) - W(N - i,u)
21
Marginal Cost Mechanismproof

• 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 the
  n S ? S

22
Marginal Cost Mechanismproof

• If i?S, in both M, MC
• 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)
• 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 uiqi - xi 0
• In MC qi(u) 1, xi(u) ui ? Agent i has
  welfare of uiqi - xi 0

S
S
23
Marginal Cost Mechanism

• Theorem 3If 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

?
24
Marginal Cost Mechanismproof

• Strategypoofness and efficiency are known
  properties of the VCG mechanism.
• NPTW(N,u) us - C(S) ? ui us - i - C(S -
  i) ? ui W(N-i, u) ? xi(u) uiqi(u) -
  W(N,u) - W(N - i,u) ? ui -
  W(N,u) - W(N - i,u) ? 0
• VPwelfarei uiqi(u) - xi (u)
  uiqi(u) - uiqi(u) - W(N,u) - W(N - i,u) ? 0
  welfarei(qi0, xi 0 )

25
Marginal Cost Mechanismproof

• 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

26
Marginal Cost Mechanism
shapley
marginal cost
NPT
?
?
VP
?
?
CS
?
(not needed) ?
Incentive Compatibility
group
singelton
Budget Balance
?
X (never surplus)
Efficiency
X (minmax loss)
?
27
Lecture outline

• Introduction
• Budget Balance Vs. Efficiency
• Suggested mechanisms
• Marginal Cost
• Shapley
• Multicast networks
• Feasibilty of mechanisms in multicast networks
• conclusions

28
Multicast transmission
7
5

• Pick set of receivers

4
2
2
1
3
3
source
29
Multicast transmission

• Pick set of receivers
• create a tree connecting the receivers

source
30
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)

31
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?uii ?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

32
MC cost sharing feasibility

• Theorem 4 MC cost sharing requires exactly two
  messages per link. Proof ideaThere is an
  algorithm that computes the cost shares by
  performing one bottom-up traversal on tree,
  followed by one top-down traversal.

33
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?
?
34
Theorem 4 proof

• Following is an algorithm for the implementation
  of MC in multicast network
• The allocation (q ? 0,1P )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 !

35
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

36
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) ?

37
Theorem 4 proof

• yi(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 gt 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
38
Theorem 4 proof
total of exactly 2 messages per link
39
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

40
Shapleys cost sharing method

• Reminder Shapleys mechanism is M(f) when
• T!(S - T - 1)!
• fi(S) ?T?S-i S!

C(T?i) - C(T)
41
Shapley cost sharing feasibility

• Theorem 5Shapleys cost sharing requires, in
  the worst case, ?(n p) message exchanges (?(n2)
  when pO(n) )
• Whats wrong with worst case of ?(n2) ?
• Centralized approachs 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 !

42
Conclusions
NPT
CS
VP
strategy-proof
Budget-Balance
Efficiency
shapley
Marginal Cost
cross-monotonic
Exactly 2 messages per link ( total ?(n) )
FEASIBLE
?(n2) msg exchanges FEASIBILITY PROBLEMS
43
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

44
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 uj ?j?T and
  allocations (q,x) and (q,x) repectively
• if uiqi - xi ? uiqi - xi ?i?Tthen all the
  inequalities are equalities.
• Strategyproofness is when T 1

45
group strategyproof

• Lets see why MC is not group-strategyproof
• C(1)C(2)6 C(12)8
• u1 u2 5
• s(u) 1,2x1(u) x2(u) 5 - (8 - 6) 3
• But, agent 1 can change to u1 7her allocation
  stays the samex2(u) decreases to 2 !!!