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

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,...

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.”

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 and ex post efficient.) UK wireless spectrum auction ( Cramton ) –use a bidder-optimal core for final stage; not SP but can avoid other instability of VCG.

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.”

c.f. Yoav’s comment

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

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

Dynamic knapsack auction Input: { (a 1, d 1, v 1, q 1 ),... (a n, d n, v n, q n ) } 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.)

Online Stochastic Optimization Multistage stochastic integer program Q = max x1 E[ max x2 E[... max xT v(x, » ) ]] –where » = ( » 1,..., » T ) is a stochastic process, » t observation at time t, (x 1..t-1, » 1.. t ) state at time t. (Van Hentenryck and Bent; Mercier; Shapiro’06)

Online Stochastic Optimization Multistage stochastic integer program Q = max x1 E[ max x2 E[... max xT v(x, » ) ]] –where » = ( » 1,..., » T ) is a stochastic process, » t observation at time t, (x 1..t-1, » 1.. t ) state at time t. Solve anticipatory relaxation: max xt E » [Opt(s t, x t, » > t )] –i.e., construct scenarios » 1,..., » w. For each x t, compute g(x t ) = 1/w  i Opt(s t, x t, » i ). Pick best. –need exogenous uncertainty (Van Hentenryck and Bent; Mercier; Shapiro’06)

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. Obtaining SP

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

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

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

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 strategyproof (via output-ironing) consensus algorithm

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?

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) 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) Possible Answers

Agent can always extract profit ¢ vcg,i Regret = ¢ vcg,i - ¢ i Example: Threshold rule 1234 ¢ vcg,1 ¢ vcg,4 = b 4 – p vcg,4... p vcg,i = bid - ¢ vcg,i (Parkes et al.’01) ¢1¢1

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. Example: Threshold rule 1234 ¢i¢i ¢ vcg,1 ¢ vcg,4 = b 4 – p vcg,4... p vcg,i = bid - ¢ vcg,i (Parkes et al.’01)

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

max Count( ¢ i = 0) maximizes # of agents with nothing to gain... also maximizes worst-case regret! Small Rule 1234

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

Agent with type v i responds to distribution on strategic environments induced by F(b -i ) Maximize expected utility: faces uncertainty –e.g., consider E b -i [ |  ¼ i (v i, b -i )/  v i | ] –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! A Bayesian viewpoint

z = (p 1, b 1,... b n ) instance H * (z), H m (z): if ||H * (z), H m (z)|| small then payments in m almost always = reference. Distributional analysis

z = (p 1, b 1,... b n ) instance H * (z), H m (z): if ||H * (z), H m (z)|| small then payments in m almost always = reference. Simplify, obtain univariate distribution: (p 1, v 1,..., v n ) ( ¼ 1, v 1,..., v n ) ( ¼ i, V) ( ¼ i / V) Distributional analysis where V is total value for allocation.

average ||H *, H m || over all environments

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

discount in m discount in VCG discount in m ThresholdSmall

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

Thank you!