Heuristic Mechanism Design

1. Heuristic Mechanism Design David C. Parkes Harvard University Joint with Florin Constantin, Ben Lubin and Quang Duong Cornell CS/Econ WorkshopSeptember 4, 2009

2. Embracing messiness • Early development of MD theory focused on an “in principle” mathematical approach. • Today, we see great demand for mechanisms and markets that need to manage (messy) real world details • e.g., dynamics, complex preferences, scalability, transparency, stability, ...

3. Krugman, 9/2/09 New York Times • How did Economists Get it So Wrong? “...economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth. ... economists will have to learn to live with messiness.”

4. Examples of “Messy” systems • Sponsored search • dynamic system, massive # of goods, use of machine learning, bidder tools, asynch. updating • Medical matching (Roth & Peranson) • couples have preferences, hospitals may specify revisions; may be no stable match, empirically “set of stable matchings is very stable empty” • HBS draft mechanism (Cantillon & Budish’09) • Non-SP. But ex ante welfare higher than under RSD (only anon., SP and andex post efficient.) • UK wireless spectrum auction (Cramton) • use a bidder-optimal core for final stage; not SP but can avoid other instability of VCG.

5. Observations • Most deployed mechanisms and markets are not strategyproof • need to develop solutions that are “truthful and stable enough” given complexities of environ. • balance SP with other considerations • Real-world problems are multi-dimensional, and dynamic. Not isolated events. • need theory and engineering knowledge to guide practical design • Al Roth, 1999 “...if we fail to develop... an "engineering" literature, we will fail to profit from design experience in a cumulative way.”

6. c.f. Yoav’s comment

7. One starting point • Adopt computational approach that would be desirable without incentive/stability concerns • Modify decisions, and/or design payments to make the method truthful and stable enough. • How to evaluate? • comparative study of initial algorithm and modified algorithm • analysis of strategic properties (through comput. and/or theoretical approaches) • identify good and bad cases

8. Two examples • Dynamic knapsack auctions • would use an online stochastic algorithm to solve • with incentive concerns, adopt a “self-correction” approach to obtain SP • CAs and CEs • would use a branch-and-bound, cutting-plane approach to solve • with incentive concerns, adopt a “reference mechanism” approach to obtain approx-SP

9. Dynamic knapsack auction • Input: { (a1, d1, v1, q1), ... (an, dn, vn, qn) } • Capacity C to sell. Probabilistic arrival model. • Patience ) no simple characterization of optimal policies available • c.f., threshold policies; optimal mechanisms (Kleywegt & Papastavrou’01, Pai & Vohra’08, Dizdar et al.’09.)

10. Online Stochastic Optimization (Van Hentenryck and Bent; Mercier; Shapiro’06) • Multistage stochastic integer program Q = maxx1 E[ maxx2 E[... maxxT v(x, ») ]] • where » = (»1, ..., »T) is a stochastic process, »t observation at time t, (x1..t-1, »1..t) state at time t.

11. Online Stochastic Optimization (Van Hentenryck and Bent; Mercier; Shapiro’06) • Multistage stochastic integer program Q = maxx1 E[ maxx2 E[... maxxT v(x, ») ]] • where » = (»1, ..., »T) is a stochastic process, »t observation at time t, (x1..t-1, »1..t) state at time t. • Solve anticipatory relaxation: maxxt E» [Opt(st, xt, »>t)] • i.e., construct scenarios »1,..., »w. For each xt, compute g(xt) = 1/w i Opt(st, xt, »i). Pick best. • need exogenous uncertainty

12. Obtaining SP • Self-correction (P. & Duong’07, Constantin & P.’09): • check a proposed allocation is consistent with a monotonic policy, cancel allocation otherwise. • A local check: • Fixing reports of other agents, just verify that the agent is still allocated for higher types. • i.e., no need to check for other “-i” type profiles • Combine (a,v,q)-ironing + departure-mon, obtain SP. • Make sensitivity analysis tractable.

13. Results: Knapsack auction • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency: Regular

14. Results: Knapsack auction • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency: Regular

15. Results: Knapsack auction • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency: Regular consensus algorithm strategyproof (via output-ironing)

16. Results: Knapsack auction • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency: • Typical: IgnoDep <5% NowWait <5% OSCO • w/ VVs; as much as 35% rev. improvement Regular consensus algorithm strategyproof (via output-ironing)

17. CAs and CEs • NP-hard WD, but generally solvable on (current!) problems of practical importance. • But, VCG mechanism not desirable: • outside core for CAs • budget deficit for CEs ) can’t obtain SP together with desired computational approach. • How should we set prices, to achieve “almost SP” and stability?

18. Possible Answers • Example: (A,$10), (B,$10), (AB,$15) • “It doesn’t matter” • Just use first price, and in any NE agents will have an efficient outcome with core payoffs • E.g., outcome (A,$6) (B,$9) (AB,$15)

19. Possible Answers • Example: (A,$10), (B,$10), (AB,$15) • “It doesn’t matter” • Just use first price, and in any NE agents will have an efficient outcome with core payoffs • E.g., outcome (A,$6) (B,$9) (AB,$15) • “At least minimize distance to VCG” • respecting no-deficit in CEs (Parkes et al.’01) • respecting core in CAs (Day & Milgrom ‘07) • E.g., outcome (A,$7.50) (B,$7.50)

20. pvcg,i= bid - ¢vcg,i Example: Threshold rule (Parkes et al.’01) • Agent can always extract profit ¢vcg,i • Regret = ¢vcg,i - ¢i ¢vcg,1 ... ¢1 ... ¢vcg,4 = b4 – pvcg,4 1 2 3 4

21. pvcg,i= bid - ¢vcg,i Example: Threshold rule (Parkes et al.’01) • Agent can always extract profit ¢vcg,i • Regret = ¢vcg,i - ¢i • Threshold rule minimizes max regret • “²-SP” for minimal ². “Truthful most often,” for costly manipulation C. ¢i ¢vcg,1 ... ... ¢vcg,4 = b4 – pvcg,4 1 2 3 4

22. A Surprise! • Single-minded. Computing approx. BNE • c.f., Reeves & Wellman’04, Vorobeychik & Wellman ‘08, Rabinovich et al, ‘09

23. Small Rule • max Count(¢i = 0) • maximizes # of agents with nothing to gain • ... also maximizes worst-case regret! 1 2 3 4

24. A Bayesian viewpoint • Agent with type vi responds to distribution on strategic environments induced by F(b-i) • Maximize expected utility: faces uncertainty • e.g., consider Eb-i [ |¼i (vi, b-i)/vi| ] • set prices to minimize expected marginal gain (c.f., Erdil & Klemperer’09) • Sensitivity of bid price in distr., not regret.

25. A Bayesian viewpoint • Agent with type vi responds to distribution on strategic environments induced by F(b-i) • Maximize expected utility: faces uncertainty • e.g., consider Eb-i [ |¼i (vi, b-i)/vi| ] • set prices to minimize expected marginal gain (c.f., Erdil & Klemperer’09) • Sensitivity of bid price in distr., not regret. • SP mechanisms provide a reference) try to “best conform” to payments in distribution!

26. Distributional analysis • z = (p1, b1,... bn) instance • H*(z), Hm(z): if ||H*(z), Hm(z)|| small then payments in m almost always = reference.

27. Distributional analysis • z = (p1, b1,... bn) instance • H*(z), Hm(z): if ||H*(z), Hm(z)|| small then payments in m almost always = reference. • Simplify, obtain univariate distribution: (p1, v1, ..., vn) (¼1, v1, ..., vn) (¼i, V) (¼i / V) where V is total value for allocation.

28. average ||H*, Hm || over all environments

29. 3 equilibrium x 3 environments x 6 rules • 54 data points {(eff,metric), (shaving,metric)} • corr(KL,eff) = -0.381; corr(KL,shave)=+0.379, both at 0.05 signif. level. average ||H*, Hm || over all environments

30. discount in VCG discount in VCG discount in m discount in m Threshold Small

32. Summary: Heuristic approach • Start with good, cooperative algorithm • Modify it, or associate payments with it, to achieve “good enough” SP, stability. • output ironing; reference mechanism fitting. • Still need theory  • in which problems can “local correction” work in achieving SP? • a theory for “approximate SP”, “approximate stability” and so forth, • alt. models of behavior

33. Thank you!