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

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]

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)

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]

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

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)

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

example 2 agents 1 free item

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

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)

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

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

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

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}

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

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

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

triangulation each polytope in PW is a simplex

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

upper bound restricted problem with any subset of value space

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

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

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

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

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

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

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

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

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

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

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

open questions

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

...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

Optimal Payments in Dominant-Strategy Mechanisms