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Heuristic Mechanism Design. David C. Parkes Harvard University. Joint with Florin Constantin, Ben Lubin and Quang Duong Cornell CS/Econ Workshop September 4, 2009. Embracing messiness. Early development of MD theory focused on an “in principle” mathematical approach.

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heuristic mechanism design

Heuristic Mechanism Design

David C. Parkes

Harvard University

Joint with Florin Constantin, Ben Lubin and Quang Duong

Cornell CS/Econ WorkshopSeptember 4, 2009

embracing messiness
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, ...
slide3

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
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.
observations
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.”
one starting point
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
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
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.)
online stochastic optimization
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.
online stochastic optimization1
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
obtaining sp
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.
results knapsack auction
Results: Knapsack auction
  • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency:

Regular

results knapsack auction1
Results: Knapsack auction
  • 10 items; 5 periods; 2 arrivals/period; U[1,5] demand; U[1,5] patience. Efficiency:

Regular

results knapsack auction2
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)

results knapsack auction3
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)

cas and ces
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?
possible answers
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)
possible answers1
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)
example threshold rule

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

example threshold rule1

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

a surprise
A Surprise!
  • Single-minded. Computing approx. BNE
    • c.f., Reeves & Wellman’04, Vorobeychik & Wellman ‘08, Rabinovich et al, ‘09
small rule
Small Rule
  • max Count(¢i = 0)
  • maximizes # of agents with nothing to gain
  • ... also maximizes worst-case regret!

1

2

3

4

a bayesian viewpoint
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.
a bayesian viewpoint1
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!
distributional analysis
Distributional analysis
  • z = (p1, b1,... bn) instance
  • H*(z), Hm(z): if ||H*(z), Hm(z)|| small then payments in m almost always = reference.
distributional analysis1
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.

slide28

average ||H*, Hm ||

over all environments

slide29

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

slide30

discount in VCG

discount in VCG

discount in m

discount in m

Threshold

Small

summary heuristic approach
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