Mechanism design for computationally limited agents

(previous slide deck discussed the case where valuation determination was complex)

Tuomas SandholmComputer Science Department

Carnegie Mellon University

Part IMechanisms that are computationally

(worst-case) hard to manipulate

Voting mechanisms that are worst-case hard to manipulate

• Bartholdi, Tovey, and Trick. 1989. The computational difficulty of manipulating an election, Social Choice and Welfare, 1989.

• Bartholdi and Orlin. Single Transferable Vote Resists Strategic Voting, Social Choice and Welfare, 1991.

• Conitzer, V., Sandholm, T., Lange, J. 2007. When are elections with few candidates hard to manipulate? JACM.

• Conitzer, V. and Sandholm, T. 2003. Universal Voting Protocol Tweaks to Make Manipulation Hard. International Joint Conference on Artificial Intelligence (IJCAI).

• Elkin & Lipmaa …

2nd-chance mechanism [in paper “Computationally Feasible VCG Mechanisms” by Nisan & Ronen, EC-00]

• (Interesting unrelated fact: Any VCG mechanism that is maximal in range is incentive compatible)

• Observation: only way an agent can improve its utility in a VCG mechanism where an approximation algorithm is used is by helping the algorithm find a higher-welfare allocation

• Second-chance mechanism: let each agent i submit a valuation fn vi and an appeal fn li: V->V. Mechanism (using alg k) computes k(v), k(li(v)), k(l2(v)), … and picks the among those the allocation that maximizes welfare. Pricing based on unappealed v.

Other mechanisms that are worst-case hard to manipulate

• O’Connell and Stearns. 2000. Polynomial Time Mechanisms for Collective Decision Making, SUNYA-CS-00-1

• …

Part IIUsual-case hardness of manipulation

Impossibility of usual-case hardness• For voting:

– Procaccia & Rosenschein JAIR-97• Assumes constant number of candidates• Impossibility of avg-case hardness for Junta

distributions (that seem hard)– Conizer & Sandholm AAAI-06

• Any voting rule, any number of candidates, weighted voters, coalitional manipulation

• Thm. <voting rule, instance distribution> cannot be usually hard to manipulate if

– It is weakly monotone (either c2 does not win, or if everyone ranks c2 first and c1 last then c1 does not win), and

– If there exists a manipulation by the manipulators, then with high probability the manipulators can only decide between two candidates

– Elections can be Manipulated Often by E. Friedgut, G. Kalai, and N. Nisan. Draft, 2007.

• For 3 candidates• Shows that randomly selected manipulations work

with non-vanishing probability– Still open directions available

• Multi-stage voting protocols• Combining randomization and manipulation

hardness…

• Open for other settings

Problems with mechanisms that are worst-case hard to manipulate

• Worst-case hardness does not imply hardness in practice

• If agents cannot find a manipulation, they might still not tell the truth– Solution: mechanisms like the one in Part III of

this slide deck.

Part IIIBased on “Computational Criticisms

of the Revelation Principle” by Conitzer & Sandholm

Criticizing truthful mechanisms

• Theorem. There are settings where:– Executing the optimal truthful (in terms of social welfare)

mechanism is NP-complete– There exists an insincere mechanism, where

• The center only carries out polynomial computation• Finding a beneficial insincere revelation is NP-complete for the agents• If the agents manage to find the beneficial insincere revelation, the

insincere mechanism is just as good as the optimal truthful one• Otherwise, the insincere mechanism is strictly better (in terms of s.w.)

• Holds both for dominant strategies and Bayes-Nash implementation

Proof (in story form)• k of the n employees are needed for a project• Head of organization must decide, taking into account

preferences of two additional parties:– Head of recruiting– Job manager for the project

• Some employees are “old friends”:

• Head of recruiting prefers at least one pair of old friends on team (utility 2)

• Job manager prefers no old friends on team (utility 1)• Job manager sometimes (not always) has private

information on exactly which k would make good team (utility 3)– (n choose k) + 1 types for job manager (uniform distribution)

Proof (in story form)…Recruiting: +2 utility for pair of friends

Job manager: +1 utility for no pair of friends, +3 for the exactly right team (if exists)

• Claim: if job manager reports specific team preference, must give that team in optimal truthful mechanism

• Claim: if job manager reports no team preference, optimal truthful mechanism must give team without old friends to the job manager (if possible)– Otherwise job manager would be better off reporting type

corresponding to such a team

• Thus, mechanism must find independent set of k employees, which is NP-complete

Proof (in story form)…Recruiting: +2 utility for pair of friends

Job manager: +1 utility for no pair of friends, +3 for the exactly right team (if exists)

• Alternative (insincere!) mechanism:– If job manager reports specific team preference, give that team– Otherwise, give team with at least one pair of friends

• Easy to execute• To manipulate, job manager needs to solve (NP-complete)

independent set problem– If job manager succeeds (or no manipulation exists), get same

outcome as best truthful mechanism– Otherwise, get strictly better outcome

Criticizing truthful mechanisms…• Suppose utilities can only be computed by

(sometimes costly) queries to oracle

• Then get similar theorem:– Using insincere mechanism, can shift burden of

exponential number of costly queries to agent– If agent fails to make all those queries, outcome can

only get better

oracle

u(t, o)?

u(t, o) = 3

Is there a systematic approach?

• Previous result is for very specific setting• How do we take such computational issues into account in general

in mechanism design?• What is the correct tradeoff?

– Cautious: make sure that computationally unbounded agents would not make mechanism worse than best truthful mechanism (like previous result)

– Aggressive: take a risk and assume agents are probably somewhat bounded

• Recent results on these manipulation-optimal mechanisms in [Othman & Sandholm SAGT-09]