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

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

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

single-parameter domains: characterization of DS mechanisms 27 if allocated when reporting x, then allocated when reporting y ≥ x if not allocated when reporting x, then not allocated when reporting y ≤ x h(7,9) h(7,2) - g(7,2) h(v -i ) is the only degree of freedom in the payment function optimize h(v -i ) 5 9 if allocated when reporting x, then allocated when reporting y ≥ x h(9,2) - g(9,2) g(v -i ) - the minimum value agent i can report to be allocated v -i = (v 1,...,v i-1,v i,v i+1,...,v n ) x determined by the allocation function g(v -i ) = min x | f i (x,v -i ) = 1 g - price (critical value) h - rebate

optimal payment function constructive characterization optimal payment (rebate) function IN OUT objective e.g., maximize social welfare constraints e.g., no subsidy and voluntary participation allocation function e.g., efficient 6 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

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

generic MD problem max h(w),objVal objVal 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 ) 9 optimization is over functions infinite number of constraints

example 2 agents 1 free item 10

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

12 f(v) = (0,1) f(v) = (1,0) g(v 1 ) = v 1 g(v 2 ) = v 2 regions with linear constraints constant allocation and linear critical value on each triangle constraints linear in h(w)

linear constraints on a polytope a linear constraint c 1 v c n v n ≤ c n+1 holds at all points v in P of a polytope P iff it holds at the extreme points v in extremePoints(P) 13 v2v2 v1v1 2v 1 + v 2 ≤ 5

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

h a (v 2 ), h b (v 1 ) h b (v 2 ), h b (v 1 ) 15 h a (w 1 )h b (w 1 ) w1w1 h a (w 1 ) h b (w 1 ) allocation with costs 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 h x optimal rebate function is h(w) = {h x (w) if w in x} 16

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

vertex consistency w1w1 01k 1,0 v -1 v -2 project points 18

region consistency w1w1 01k w 1 · k lift regions v 2 · k v 1 · k v 2 · k 19

triangulation 20 each polytope in P W is a simplex

characterization if there exist P V and P W satisfying – P V refine the initial subdivision allocation constant on q in P V critical value linear on q in P V – vertex consistency – region consistency – P W 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 21

upper bound 22 restricted problem with any subset of value space

lower bound (approximate solutions) 23 not a triangulation: cannot linearly interpolate the extreme points allocate to agent 1 if v 1 ≥ kv 2 h a (w 1 )h b (w 1 ) w1w1 k*k10 h a (w 1 )h b (w 1 ) w1w1 k10

24 examples V = {v in R n | 1 ≥ v 1 ≥ v 2 ≥... ≥ v n ≥ 0} h: W  R W = {w in R n-1 | 1 ≥ w 1 ≥ w 2 ≥... ≥ w n-1 ≥ 0}

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

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

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 27

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

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

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

hyperrectangle triangulation 31 applies to initial subdivisions that can be obtained with hyperplanes of the form x i = c i where c i is a constant side in dimension i is of length a i subdivided via hyperplanes x i /a i = x j /a j

arbitrary initial subdivision can be approximated with a piecewise constant function 32 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 33

open questions 34

consistent partitions for public good? 35 build a bridge if v v m ≤ 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 36

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