Mechanism design for computationally limited agents (last lecture 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

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.

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

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

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

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

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

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)

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

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

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

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