Mechanism design for computationally limited agents

Tuomas SandholmComputer Science Department

Carnegie Mellon University

Outline

• Part I: Limited deliberation to determine valuations: A study of common auction mechanisms

• Part II: Limited deliberation to determine valuations: Designing new mechanisms

• Part III: Other ideas for mechanism design for computationally limited agents

Part I: Limited deliberation to determine valuations: A study of

common auction mechanisms

Bidders may need to compute their valuations for (bundles of) goods

• In many applications, e.g.

– Vehicle routing problem in transportation exchanges

– Manufacturing scheduling problem in procurement

• Value of a bundle of items (tasks, resources, etc) =

value of solution with those items - value of solution without them

• Information gathering fits the model as well

Software agents for auctions

• Software agents exist that bid on behalf of user• We want to enable agents to not only bid in auctions,

but also determine the valuations of the items• Agents use computational resources to compute

valuations• Valuation determination can involve computing on NP-

complete problems (scheduling, vehicle routing, etc.)

• Optimal solutions may not be possible to determine due to limitations in agents’ computational abilities (i.e. agents have bounded rationality)

Recall

• A bidder in an auction can pay cost c to find out his own valuation => Vickrey auction ceases to have a dominant strategy [Sandholm ICMAS-96, International J. of Electronic Commerce 2000]– Same model studied in “information acquisition

in auctions” [Compte and Jehiel 01, Rezende 02, Rasmussen 03]

Bounded rationality• Work in economics has largely focused on descriptive models• Some models based on limited memory in repeated games

[Papadimitriou, Rubinstein, …]

• Some AI work has focused on models that prescribe how computationally limited agents should behave [Horvitz; Russell & Wefald; Zilberstein & Russell; Sandholm & Lesser; Hansen & Zilberstein, …]– Simplifying assumptions

• Myopic deliberation control• Asymptotic notions of bounded optimality• Conditioning on performance but not path of an algorithm

• Simplifications can work well in single agent settings, but any deviation from full normativity can be catastrophic in multiagent settings

Incorporate deliberation (computing) actions into agents’ strategies => deliberation equilibrium

Normative control of deliberation

• In our setting agents have – Limited computing, or– Costly computing

• Agents must decide how to use their limited resources in an efficient manner

• Agents have anytime algorithms and use performance profiles to control their deliberation

Anytime algorithms can be used to approximate valuations

• Solution improves over time

• Can usually “solve” much larger problem instances than complete algorithms can

• Allow trading off computing time against quality– Decision is not just which bundles to evaluate, but how carefully

• Examples

– Iterative refinement algorithms: Local search, simulated annealing

– Search algorithms: Depth first search, branch and bound

Performance profiles of anytime algorithms

• Statistical performance profiles characterize the quality of an algorithm’s output as a function of computing time

• There are different ways of representing performance profiles– Earlier methods were not normative: they do not capture all the

possible ways an agent can control its deliberation• Can be satisfactory in single agent settings, but catastrophic in multiagent

systems

Performance profiles

Computing time

Solution quality

Deterministic performance profile

Solution quality

Variance introduced by different problem instances

Computing time

[Horvitz 87, 89, Dean & Boddy 89]

Optimum

Ignores conditioning on the path

Table-based representation of uncertainty in performance profiles

.08 .19 .24

.15 .30 .17 .39

.16 .10 .16 .25 .30 .22

.08 .04 .17 .20 .22 .30 .24 .19 .15

.09 .10 .20 .22 .23 .37 .31 .13 .15

.11 .14 .33 .18 .21 .18 .08

.22 .17 .25 .24 .15 .13

.40 .31 .15 .19 .05

.15 .20 .03

.03

Computing time

Solutionquality

[Zilberstein & Russell IJCAI-91, AIJ-96]

Conditioning on solution quality so far [Hansen & Zilberstein AAAI-96]

Performance profile tree [Larson & Sandholm TARK-01]

• Normative– Allows conditioning on path of solution quality

– Allows conditioning on path of other solution features

– Allows conditioning on problem instance features (different trees to be used for different classes)

• Constructed from statistics on earlier runs

0

4

2

6

4

5

10

3

15

20

A

P(B|A) B

5

CP(C|A)Solution quality

Performance profile tree…• Can be augmented to model

– Randomized algorithms

– Agent not knowing which algorithms others are using

– Agent having uncertainty about others’ problem instances

• Agent can emulate different scenarios of others

0

4

2

6

45

10

3

15

20

p(0)

p(1)

Random node

Value node

Our results hold in this augmented setting

Roles of computing

• Computing by an agent– Improves the solution to the agent’s own problem(s)– Reduces uncertainty as to what future computing steps will

yield– Improves the agent’s knowledge about others’ valuations– Improves the agent’s knowledge about what problems

others may have computed on and what solutions others may have obtained

• Our results apply to different settings– Computing increases the valuation (reduces cost)– Computing refines the valuation estimate

Strategic computing [Larson & Sandholm]

• Good estimates of the other bidders’ valuations can allow an agent to tailor its bids to achieve higher utility

• Definition. Strong strategic computing: Agent uses some of its deliberation resources to compute on others’ problems

• Definition. Weak strategic computing: Agent uses information from others’ performance profiles

• How an agent should allocate its computation (based on results it has obtained so far) can depend on how others allocate their computation– Deliberation equilibrium

Theorems on strategic computing

yesyesno

Generalized Vickrey

On which <bidder, bundle> pair to allocate next computation step ?

Multiple items for

sale

noEnglish (1st price ascending) yes

yes

no

nonoVickrey (2nd price sealed bid)

yesyesyesDutch (1st price descending)

yesyesyesFirst price sealed-bidSingle item for

sale

Costly computing

Limited computing

Strategic computing ?Counter-speculation by rational

agents ?

Auction

mechanism

If performance profiles are deterministic, only weak strategic computing can occur

New normative deliberation control method uncovered a new phenomenon

Costly computing in English auctions• Dominant strategy mechanism for rational bidders

• Thrm: If at most one performance profile is stochastic, no strong strategic computing occurs in equilibrium

• Thrm: If at least two performance profiles are stochastic, strong strategic computing can occur in equilibrium

– Despite the fact that agents learn about others’ valuations by waiting and observing others’ bids

– Passing & restarting computation during the auction is allowed

– Proof. Consider an auction with two bidders:

• Agent 1 can compute for free

• Agent 2 incurs cost 1 for each computing step

Performance profiles of the proof

Agent 1’s problem Agent 2’s problem

p(high1)

1-p(high1)

p(high2)

1-p(high2)

high1

low1

high2

low2

low2 < low1 < high2 < high1

000

Since computing one step on 2’s problem does not yield any information, we can treat computing for two steps on 2’s problem atomically

Proof continued…• Agent 1 has a dominant strategy:

– Compute only on own problem & increment bid whenever • Agent 1 does not have the highest bid and• Highest bid is lower than agent 1’s valuation

• Agent 2’s strategy:– CASE 1: bid1 > low1

• Agent 2 knows that agent 1 has valuation high1

• Agent 2 cannot win, and thus has no incentive to compute or bid

– CASE 2: bid1< low2 • Agent 2 continues to increment its own bid• No need to compute since it knows that its valuation is at least low2

– CASE 3: low1 bid1 low2 • Agent 2’s strategy depends on the performance profiles

Decision problem of agent 2 in CASE 3

Withdraw

Bid

Compute on 2’s problem

Compute on 1’s problem

high1

low1

high1

low1

high2

high2

Compute on 2’s

low2

low2

Decision node for agent 2

Chance node for agent 1’sperformance profile

Chance node for agent 2’sperformance profile

Bid

0

-1

0

high1

low1

high2-low1-2

low2-low1-2

high2-low1-3-3

Withdraw

high2

low2

Withdraw

-2

-2high2-low1-2

high2

low2

Compute on 2’s

high2

low2

Bidhigh2-high1-3

low2-high1-3

high2-low1-3

-3

Withdraw

Bid high2

low2

Withdraw

Compute on 1’shigh1

low1

-2

-1

high2-low1-3

low2-low1-3

Bid

Compute on 1’s

high1

high1

low1

low1

-3-3

-2-2

-3high2-low1-3

Agent 2’s utilitylow2 < low1 < high2 < high1

Under what conditions does strong strategic computing occur?

Probability that agent 1 will have its high valuation

Probability that agent 2 will have its high valuation

0 0.2 0.4 0.6 0.8 1

1

0.8

0.6

0.4

0.2

0

Other variants we have solved

• Agents cannot pass on computing during the auction & continue computing later during the auction– Can make a difference in English auctions with costly computing, but

strong strategic computing is still possible in equilibrium• Agents can/cannot compute after the auction• 2-agent bargaining (again with performance profile trees)

– Larson, K. and Sandholm, T. 2001. Bargaining with Limited Computation: Deliberation Equilibrium. Artificial Intelligence, 132(2), 183-217.

– Larson, K. and Sandholm, T. 2002. An Alternating Offers Bargaining Model for Computationally Limited Agents. In Proceedings of the First International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS), Bologna, Italy, July.

Conclusions• Software agents participating in auctions may need to compute

valuations under computational limitations– This adds other possibilities to the agents’ strategies

• Modeled computing normatively as part of each agent’s strategy– Deliberation equilibrium– Showed under which auction mechanisms and which models of

bounded rationality strategic computing can/cannot occur

• Deliberation resources may be used strategically– Strong strategic vs. weak strategic computing– Deep interaction between incentives and computing

• Dominant strategy mechanisms can become strategy-prone• Even English auction with costly computing

The future ?• In many B2B settings, automated bidders can

compute valuations dynamically faster than humans

• Some future research directions– Using our deliberation control method in systems

• Manufacturing planning, networks, …

– New (market) mechanisms• Game-theoretically engineered to work well under (different)

models of bounded rationality• Our results show that even the most common mechanism

design principles (e.g., revelation principle) cease to hold• Our normative deliberation control method = basis for new

design principles ?

Part II: Limited deliberation to determine valuations:

Designing new mechanisms

[Larson & Sandholm AAMAS-05]

Mechanism desiderata• Preference formation-independent

– Mechanism should not be involved in agents’ preference formation process (otherwise revelation principle applies trivially)

• Deliberation-proof– In equilibrium, no agent should have incentive to strategically

deliberate

• Non-misleading– In equilibrium, no agent should follow a strategy that causes others

to believe that its true preferences are impossible

Proof sketch. Given any outcome function it is always possible to construct an example where agents are best off knowing the valuation of another agent

Indirect/multi-step mechanisms provide information to agents

– Example: Ascending auction

Bidders

InformationAt price p there are k bidders remaining in

the auction

Is it possible to satisfy the three desiderata via a multi-stage

mechanism? • Thm: There does not exist any strategy-dependent,

preference-formation independent mechanism that is both– deliberation proof, and– non-misleading

• Proof sketch. Look at information sets in the game induced by the indirect mechanism– Case 1: Game does not provide enough information to stop

strategic-deliberation (ascending auction)– Case 2: Game does provide enough information BUT agents’ play

a signaling game• Pooling equilibria (misleading)

Future work

• Overcoming the impossibility result by relaxing the properties– Encourage strategic deliberation

• Incentives for agents to share information?

– Relax preference-formation independent property

• Mechanism guides deliberation

Part III: Other ideas for mechanism design for computationally limited

agents

Recall from last lecture

• With computationally limited agents, a non-truthful mechanism can be better than a truth-promoting one – [Conitzer & Sandholm: “Computational

Criticisms of the Revelation Principle”, 2003]

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.

Work based on the assumption that agents can only solve problems that

are worst-case polynomial time

• 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.

• Nisan and Ronen. 2000. Computationally Feasible VCG Mechanisms, EC-00.• O’Connell and Stearns. 2000. Polynomial Time Mechanisms for Collective

Decision Making, SUNYA-CS-00-1• Conitzer, V. and Sandholm, T. 2002.

Complexity of Manipulating Elections with Few Candidates. National Conference on Artificial Intelligence (AAAI).

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

• Conitzer, V. and Sandholm, T. 2003. How Many Candidates Are Needed to Make Elections Hard to Manipulate? Conference on Theoretical Aspects of Rationality and Knowledge (TARK).

• …