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Optimal Payments in Dominant-Strategy Mechanisms

Optimal Payments in Dominant-Strategy Mechanisms. Victor Naroditskiy Maria Polukarov Nick Jennings University of Southampton. 7. 15. 9. 9. 2. 7. 2. mechanism. allocation function who is allocated. payment function payment to each agent. fixed. optimized. 3.

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Optimal Payments in Dominant-Strategy Mechanisms

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  1. Optimal Payments in Dominant-Strategy Mechanisms Victor Naroditskiy Maria Polukarov Nick Jennings University of Southampton

  2. 7 15 9 9 2 7 2

  3. mechanism allocation function who is allocated payment function payment to each agent fixed optimized 3

  4. truthful mechanisms with payments: groves class 9 2 7 individual rationality minimize the amount burnt [Moulin 07] [Guo&Conitzer 07] no subsidy weak BB optimize fairness [Porter et al 04]

  5. single-parameter domains: characterization of DS mechanisms if allocated when reporting x, then allocated when reporting y ≥ x if allocated when reporting x, then allocated when reporting y ≥ x if not allocated when reporting x, then not allocated when reporting y ≤ x g(v-i) - the minimum value agent i can report to be allocated v-i = (v1,...,vi-1,vi,vi+1,...,vn) 7 9 2 x h(9,2) - g(9,2) h(7,9) h(7,2) - g(7,2) determined by the allocation function g(v-i) = minx | fi(x,v-i) = 1 g - price (critical value) h - rebate h(v-i) is the only degree of freedom in the payment function optimize h(v-i)

  6. optimal payment functionconstructive characterization e.g., maximize social welfare objective e.g., no subsidy and voluntary participation IN constraints e.g., efficient allocation function OUT optimal payment (rebate) function AMD [Conitzer, Sandholm, Guo]

  7. dominant-strategy implementation no prior on the agents' values V = [0,1]n f: V  {0,1}n W = [0,1]n-1 g, h: WR

  8. example MD problemwelfare maximizing allocation [Moulin 07] [Guo&Conitzer 07] n agents m items maxh(w)r s.t. for all v in V (social welfare within r of the efficient surplus v1 + ... + vm) v1 + ... + vm + i h(v-i) - mvm+1 ≤ r(v1 + ... + vm) i h(v-i) - mvm+1 ≤ 0 (weak BB) h(v-i) ≥ 0 (IR)

  9. generic MD problem maxh(w),objValobjVal s.t. for all v in V objective(f(v), g(v-i), h(v-i)) ≥ objVal constraints(f(v), g(v-i), h(v-i)) ≥ 0 objective and constraints are linear in f(v), g(v-i), and h(v-i) optimization is over functions infinite number of constraints

  10. example 2 agents 1 free item

  11. allocation regions f(v) = (0,1) f(v) = (1,0)

  12. regions with linear constraints constant allocation and linear critical value on each triangle f(v) = (0,1) g(v2) = v2 g(v1) = v1 constraints linear in h(w) f(v) = (1,0)

  13. linear constraints on a polytope a linear constraint c1v1 + ... + cnvn ≤ cn+1 holds at all points v in P of a polytope P iff it holds at the extreme points v in extremePoints(P) 2v1 + v2 ≤ 5 v2 v1

  14. allocation of free items [Guo&Conitzer 08] V = [0,1]2 optimal solution linear f,g,h => constraints are linear in v constraints(f(v), g(v-1), g(v-2), h(v-1), h(v-2)) restricted problem LP with variables h(0), h(1), objVal - upper bound! V = {(0,0) (1,0) (0,1) (1,1)} W = {(0) (1)} the upper bound (objVal) is achieved and the constraints hold throughout V

  15. allocation with costs hb(v2), hb(v1) hb(w1) ha(w1) ha(v2), hb(v1) ha(w1) hb(w1) w1 each payment region has n extreme points

  16. overview of the approach • find consistent V and W space subdivisions • solve the restricted problem • extreme points of the value space subdivision • payments at the extreme points of W region x define a linear function hx • optimal rebate function is h(w) = {hx(w) if w in x}

  17. subdivisions • PX - subdivision (partition) of polytope X q* q' q PX = {q,q',q*}

  18. vertex consistency project points 0 k 1 w1 v-2 1,0 v-1

  19. region consistency v2· k v2· k lift regions w1· k 0 k 1 w1 v1· k

  20. triangulation each polytope in PW is a simplex

  21. characterization • if there exist PV and PW satisfying • PV refine the initial subdivision • allocation constant on q in PV • critical value linear on q in PV • vertex consistency • region consistency • PW is a triangulation • then an optimal rebate function is given by • interpolation of optimal rebate values from the restricted problem • by construction, the optimal rebates are piecewise linear

  22. upper bound restricted problem with any subset of value space

  23. lower bound(approximate solutions) not a triangulation: cannot linearly interpolate the extreme points ha(w1) ha(w1) hb(w1) hb(w1) 0 0 k* k k 1 1 w1 w1 allocate to agent 1 if v1 ≥ kv2

  24. examples V = {v in Rn | 1 ≥ v1 ≥ v2 ≥ ... ≥ vn ≥ 0} h: WR W = {w in Rn-1 | 1 ≥ w1 ≥ w2 ≥ ... ≥ wn-1 ≥ 0}

  25. efficient allocation of free items n agents with private values m free items/tasks social welfare: [Moulin 07] [Guo&Conitzer 07] throughout V agents 1..m are allocated fairness: [Porter 04] f(v) = (1,...1,0,...0) m

  26. extreme points restricted problem is a linear program with constraints for n+1 points (0...0) (10...0) (1110...0) ... (1...1)

  27. fairness: [Porter 04] results follow immediately from the restricted problem the feasible region is empty for k<m+1 => impossibility result unique linear (m+2)-fair mechanism

  28. efficient allocation of items with increasing marginal cost n agents with private values m items with increasing costs 3 4 7 14 • tragedy of the commons: • cost of the ith item measures disutility that i agents experience from sharing the resource with one more user m+1 possible efficient allocations depending on agents' values

  29. algorithmic solution input: n, cost profile output: percentage of efficient surplus optimal payment function number of regions is exponential in the number of agents/costs piecewise linear on each region

  30. hypercube triangulation • a hypercube [0,1]n can be subdivided into n! simplices with hyperplanes xi = xj comparing each pair of coordinates • each simplex corresponds to a permutation σ(1)... σ(n) of 1...n

  31. hyperrectangle triangulation side in dimension i is of length ai subdivided via hyperplanes xi/ai= xj/aj applies to initial subdivisions that can be obtained with hyperplanes of the form xi = ci where ci is a constant

  32. arbitrary initial subdivision can be approximated with a piecewise constant function we know consistent partitions for the modified problem triangulations of hyperrectangles

  33. contribution • characterized linearity of mechanism design problems • consistent partitions • piecewise linear payments are optimal • interpolate values at the extreme points • approach for finding optimal payments • unified technique for old and new problems • algorithm for finding approximate payments and an upper bound

  34. open questions

  35. consistent partitions for public good? build a bridge if v1 + ... + vm ≤ c where c is the cost

  36. ...open questions • full characterization of allocation functions that have consistent partitions • is a consistent partition necessary for the existence of (piecewise) linear optimal payments • approximations: simple payment functions that are close to optimal

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