1. Mechanism design for computationally limited agents (last lecture discussed the case where valuation determination was complex)
Tuomas Sandholm
Computer Science Department
Carnegie Mellon University

2. Part I Mechanisms that are computationally (worst-case) hard to manipulate

3. Voting mechanisms that are worst-case hard to manipulate
Bartholdi, Tovey, and Trick 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 When are elections with few candidates hard to manipulate? JACM.
Conitzer, V. and Sandholm, T Universal Voting Protocol Tweaks to Make Manipulation Hard. International Joint Conference on Artificial Intelligence (IJCAI).
Elkin & Lipmaa …

4. 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.

5. Other mechanisms that are worst-case hard to manipulate
O’Connell and Stearns Polynomial Time Mechanisms for Collective Decision Making, SUNYA-CS-00-1 …

6. Part II Usual-case hardness of manipulation

7. 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.

8. 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

9. 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.

10. Part III Based on “Computational Criticisms of the Revelation Principle” by Conitzer & Sandholm

11. Mechanism design Agents report strategically, rather than truthfully
One outcome must be selected
Select based on agents’ reported preferences (or types)
mechanism
outcome
types
Agents report strategically, rather than truthfully
Mechanism is called truthful if truthful = strategic

12. The revelation principle
Key tool in mechanism design
If there exists mechanism that performs well when agents act strategically, then there exists a truthful mechanism that performs just as well
new mechanism
types P1 P2 P3
mechanism
outcome
types

13. Computational criticisms of the revelation principle
Revelation principle says nothing about computational implications of using direct, truthful mechanisms
Does restricting oneself to such mechanisms lead to computational hassles? YES
If the participating agents have computational limits, does restricting oneself to such mechanisms lead to loss in objective (e.g. social welfare)? YES

14. Criticizing one-step mechanisms
Theorem. There are settings where:
Executing the optimal single-step mechanism requires an exponential amount of communication and computation
There exists an entirely equivalent two-step mechanism that only requires a linear amount of communication and computation
Holds both for dominant strategies and Bayes-Nash implementation

15. 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

16. 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)

17. Recruiting: +2 utility for pair of friends
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

18. Recruiting: +2 utility for pair of friends
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

19. Criticizing truthful mechanisms…
Suppose utilities can only be computed by (sometimes costly) queries to oracle
u(t, o)?
oracle
u(t, o) = 3
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

20. 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