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

2. Motivation U 1 = 2U 2 = 2U 3 = 3 M Knows costs  X 2$1$0$ 132 Cost( {1,2} ) = 3 Welfare:  u i - Cost({1,2}) = 2 + 2 - 3 = 1 Budget Balance:  x i - Cost({1,2 }) = 2 + 1 - 3 = 0

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

4. The Model N agents –Agent can either receive service or not (binary) u i - 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: q i - does agent i receive the service? –if q i = 1 she receives. if q i = 0, she doesn’t x i - 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) TS

6. Mechanism’s desired properties No Positive Transfers (NPT) –Cost shares (payments) are nonnegative:  i x i  0 Voluntary Participation (VP) –Welfare level (u - x) of no service at no cost (q i =0,x i =0) is guaranteed for truthful agents Consumer Sovereignty (CS) –Each agent has u i guaranteeing getting the service (regardless of the other reported values u -i )

7. Mechanism’s desired properties: Incentive Compatibility Strategyproof mecahnsim –Telling the true u i is a dominant straegy for any agent Group-strategyproof mechanism –No coalition of agents has an incentive to jointly misreport their true u i –Stronger form of Incentive Compatibility.

8. Model’s desired properties (cont.) Budget Balance –  x i = C(R) (when R is the receivers set) Efficiency –For any u, the mechanism should maximize the social welfare: W(N,u) = max T  N [u T -C(T)] (where u T =  i  R u j ) Remark: In our model the utilities are quasi- linear (u i q i - x i ) The social welfare is not the sum of the agent surpluses, and doesn’t depend on payments (x i )

9. Model’s desired properties NPTCSVP strategy-proof Budget-Balance Efficiency Budget-balance and Efficiency are mutual exclusive !!!

10. Model’s desired properties NPTCSVP strategy-proof Budget-BalanceEfficiency shapley Marginal Cost

11. Cost Sharing Methods A Cost Sharing Method f allocates C(S) among the agents in S –f i (S) - is the payment of agent i when the receivers set is S –  f i (S) = C(S) (budget-balance) Cost Sharing Function is cross-monotonic if: S  T, i  S  f i (S)  f i (T) –Agent can’t 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) S 0 = N S t+1 = { i | u i  f i ( S t ) } (proceed untill S t 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) = sup u [ bestWelfare(u) - welfare M(f) (u) ] –where: bestWelfare(u) = max T  N (u T - C(T)) welfare M(f) (u) = (U s*(f,u) - C(s*(f,u))

15. 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)]

16. Model’s desired properties NPTCSVP strategy-proof Budget-BalanceEfficiency cross-monotonic shapley Marginal Cost

17. Marginal Cost Mechanism The welfare of coalition S is w(S,u) = max T  S ( U T - C(T) ) Coalition S is called efficent if u s - 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 = u i q* i - ( w(N,u) - w(n - i,u) ) marginal welfare of agent i surplus i = uiq* i - x* i = ( w(N,u) - w(n - i,u) )

18. 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 u i q i (u) - x i (u) = u i q* i (u) - x* i (u)

19. 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: x i (u) = u i q i (u) - [ W(N,u) - h i (u -i ) ] I’ll prove the following: –h i (u -i ) = W(N-i,u) (as in the MC mechanism) –if efficient set is not maximal, welfare equivalence maintains

20. Marginal Cost Mechanism:proof We know x i (u) = u i q i (u) - [ W(N,u) - h i (u -i ) ] I’ll show h i (u -i ) = W(N-i,u) Consider arbitrary u -i u 0 - the completion of u -i by u 0 i = 0 –NPT, VP  x i (u 0 ) = 0 x i (u 0 ) = u i q i (u 0 ) - [ W(N,u 0 ) - h i (u -i ) ]  h i (u -i ) = W(N,u 0 ) = W(N - i,u 0 ) = W(N - i,u)  x i (u) = u i q i (u) - [ W(N,u) - W(N - i,u)] if S efficient, S-{i} also efficient: u s - C(S)  u s-{i} - C(S-{i})

21. 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 u s - C(S) = W(N,u) ( = max T  N (u T - C(T) ) –consequence of submodularity of C  if S efficient, and S* is largest-efficient thenS  S*

22. Marginal Cost Mechanism:proof If i  S*, in both M, MC: –q i (u) = 0, x i (u) = 0 If i  S*  S, in both M, MC: –q i (u) = 1, x i (u) = u i q i (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: q i (u) = 0, x i (u) = 0  Agent i has welfare of: u i *q i - x i = 0 –In MC: q i (u) = 1, x i (u) = u i  Agent i has welfare of: u i *q i - x i = 0 S* S M and MC are welfare equivalent

23. 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 

24. Marginal Cost Mechanism:proof Strategypoofness and efficiency are known properties of the VCG mechanism. NPT: W(N,u) = u s* - C(S*)  u i + u s* - i - C(S* - i)  u i + W(N-i, u)  x* i (u) = u i q i (u) - [ W(N,u) - W(N - i,u)]  u i - [ W(N,u) - W(N - i,u)]  0 VP: welfare i = u i q i (u) - x i (u) = = u i q i (u) - u i q i (u) - [ W(N,u) - W(N - i,u)]  0 = = welfare i (q i =0, x i = 0 )

25. Marginal Cost Mechanism:proof CS: lemma: If u i  C( {i} ) then u s  {i} - C( s  {i} )  u s - 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) u s -C(S  {i}) - C({i})  u s -C(S) u s  {i} - C( s  {i} ) = u s + u i - C( s  {i} )  u s + C({i}) - C( s  {i} )  u s -C(S)   for big enough u i (  C(i) ), any largest efficient set will contain i

26. Marginal Cost Mechanism shapleymarginal cost NPT  VP  CS  (not needed)  Incentive Compatibilitygroupsingelton Budget Balance  X (never surplus) EfficiencyX (minmax loss) 

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

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

29. Multicast transmission source Pick set of receivers create a tree connecting the receivers multicast the movie on the tree.

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  {u i } 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

32. 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.

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 CC

34. Theorem 4: proof Following is an algorithm for the implementation of MC in multicast network The allocation (q  {0,1} |P| ): q i (u) = 1 if W  (u)  0 for all nodes  on the path from user i to the root Else, q i (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 q i (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: x i (u) = u i q i (u) - [ W(N,u) - W(N-i,u) ] –Recall that W(N,u) = max T  N [ u T - C( R(T) ) ] How can we compute W(N-i,u) ?

37. Theorem 4: proof y i (u) : min w  (u) Case 1: If u i  y i (u) –Receivers set stays the same when dropping i. Thus, W(N,u) - W(N-i,u) = u i  x i (u) = u i - [W(N,u) - W(N-i,u)] = 0 Case 2: If u i > y i (u) –Dropping user i results elmination of subtree with the total welfare y i (u)  x i (u) = u i - [W(N,u) - W(N-i,u)] = u i - y i (u)  node on the path from i to the root

38. Theorem 4: proof calculate W  (u) for each node Propagate q i and y i (allocation and cost shares) 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. 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)]

41. Shapley cost sharing feasibility Theorem 5: Shapley’s cost sharing requires, in the worst case,  (n · p) message exchanges (  (n 2 ) when p=O(n) ) What’s wrong with worst case of  (n 2 ) ? –Centralized approach’s worst-case is also  (n 2 ) –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 NPTCSVP strategy-proof Budget-BalanceEfficiency cross-monotonic shapley Marginal Cost Exactly 2 messages per link ( total  (n) ): FEASIBLE  (n 2 ) 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 u i Formal defnition: –for a fixed T  N, –for any u,u’ such that u j = u’ j  j  T and allocations (q,x) and (q’,x’) repectively –if u i q’ i - x’ i  u i q i - x i  i  T then all the inequalities are equalities. Strategyproofness is when |T| = 1

45. group strategyproof Let’s see why MC is not group- strategyproof C(1)=C(2)=6 C(12)=8 u 1 = u 2 = 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 !!!