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Mechanism design for
computationally limited agents
(previous slide deck discussed the case where valuation determination was
complex)
Tuomas Sandholm
Computer Science Department
Carnegie Mellon University
Part I
Mechanisms 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 II
Usual-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 III
Based 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
u(t, o)?
u(t, o) = 3
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
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]