Strategyproof Sharing of submodular costs: Budget Balance Vs. Efficiency

Strategyproof Sharing of submodular costs:Budget Balance Vs. Efficiency Liad Blumrosen May 2001

Welfare: ui - Cost({1,2}) = 2 + 2 - 3 = 1 1 2 3   X Budget Balance: xi - Cost({1,2 }) = 2 + 1 - 3 = 0 2$ 1$ 0$ Motivation U1 = 2 U2 = 2 U3 = 3 • M • Knows costs Cost( {1,2} ) = 3

} Game theory } cs Lecture outline • Introduction • Budget Balance Vs. Efficiency • Suggested mechanisms • Marginal Cost • Shapley • Multicast networks • Feasibilty of mechanisms in multicast networks • conclusions

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)

T S Submodular cost function • We will deal with submodular cost functions: • C is submodularif  S,T  N C(T) - C(ST)  C(ST) - C(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.) The social welfare is not the sum of the agent surpluses, and doesn’t 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N[uT -C(T)] (where uT = iRuj) • Remark: In our model the utilities are quasi-linear (uiqi - xi)

NPT VP CS strategy-proof Budget-Balance Efficiency Budget-balance and Efficiency are mutual exclusive !!! Model’s desired properties

Model’s desired properties NPT VP CS strategy-proof Efficiency Budget-Balance shapley Marginal 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: 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 • 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) = 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)! • f*i(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*) < (f) ff* [C(Ti) - C(T)]

Model’s desired properties NPT VP CS strategy-proof Efficiency Budget-Balance shapley Marginal Cost cross-monotonic

Marginal Cost Mechanism surplusi = uiq*i - x*i = ( w(N,u) - w(n - i,u) ) • The welfare of coalition S isw(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

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: us - C(S)  us-{i} - C(S-{i})  xi(u) = uiqi(u) - [ W(N,u) - 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*

M and MC are welfare equivalent Marginal Cost Mechanism:proof S* • 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: ui*qi - xi = 0 • In MC: qi(u)= 1, xi(u) = ui  Agent i has welfare of: ui*qi - xi = 0 S

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) 

} Game theory } cs Lecture outline • Introduction • Budget Balance Vs. Efficiency • Suggested mechanisms • Marginal Cost • Shapley • Multicast networks • Feasibilty of mechanisms in multicast networks • conclusions

Multicast transmission 7 5 • Pick set of receivers 4 2 2 1 3 3 source

multicast the movie on the tree. Multicast transmission • Pick set of receivers • create a tree connecting the receivers 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)

We will ignore these properties 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

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) = maxTN[ uT - C( R(T) ) ] • How can we compute W(N-i,u) ?

Theorem 4: proof  node on the path from i to the root yi(u) : minw(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)

Propagate qi and yi (allocation and cost shares) calculate W(u) for each node Theorem 4: proof 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) = TS-i |S|! [C(Ti) - 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 VP CS strategy-proof Efficiency Budget-Balance shapley Marginal Cost cross-monotonic 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 jT and allocations (q,x) and (q’,x’) repectively • if uiq’i - x’i  uiqi - xi iTthen 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 !!!

Strategyproof Sharing of submodular costs: Budget Balance Vs. Efficiency