Costly valuation computation/information acquisition

in auctions:

Strategy, counterspeculation, and deliberation equilibrium

Tuomas SandholmComputer Science Department

Carnegie Mellon University

Mainly based on the following papers:Larson, K. and Sandholm, T. 2001. Costly Valuation Computation in Auctions. In Proceedings of the Theoretical Aspects of Reasoning about Knowledge (TARK). Larson, K. and Sandholm, T. 2001. Computationally Limited Agents in Auctions. In Proceedings of the International Conference on Autonomous Agents, Workshop on Agent-based Approaches to B2B .

[Sandholm NOAS-91, AAAI-93]

TRACONET, 1990-91

$ 2,000

$ 1,700

Contract:Task transferred

Auction

3

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

• In many (even private-values quasilinear) 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

• Our models apply to information gathering as well

4

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)

5

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

6

E[1pay] c v1 v2d

v 2

1

v

Simple model: can pay c to find one’s own valuation => Vickrey auction no longer has a dominant strategy

E[1nopay] v1 v2d

0

1

v1 1

2 v2

[Sandholm ICMAS-96, International J. of Electronic Commerce 2000]

Thrm. In a private value Vickrey auction with uncertainty about an agent’s own valuation, a risk-neutral agent’s best strategy can depend on others.

E.g. two bidders (1 and 2) bid for a good.v1 uniform between 0 and 1; v2 deterministic, 0 ≤ v2 ≤ 0.5Agent 1 bids 0.5 and gets item at price v2:

Say agent 1 has the choice of paying c to find out v1. Then agent 1 will bid v1 and get the item iff v1 ≥ v2 (no loss possibility, but c invested)

E[1pay] E[1

nopay] v2 2c

Same model studied more recently in the literature on “information acquisition in auctions” [Compte and Jehiel 01, Rezende 02, Rasmussen 06]

v1

pdf

v2

loss gain

1

7

Domain problem solver

(anytime algorithm)

Quest for a general fully normative model

Auctioneer

Deliberation controller

(uses performance profile)

Agent

resultCompute!

Agent

resultCompute!

bid(result) bid(result)

Deliberation controller

(uses performance profile)

Domain problem solver

(anytime algorithm)

8

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

9

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

10

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

11

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

12

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]

13

Performance profile tree [Larson & Sandholm AAAI-00, AIJ-01, 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

14

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

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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

16

“Strategic computing”

• 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” [AIJ-01]

17

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

18

Costly computing in English auctions• For rational bidders, straightforward bidding is ex post eq.

• 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 & resuming 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

19

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

20

Proof continued…• Agent 1 has straightforward (ex post eq.) 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 • If Agent 2 bids, he should bid bid1 + ε• His strategy depends on the performance profiles…

21

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

low2-low1

high2-low1-3-3

Withdraw

high2

low2

Withdraw

-2

-2high2-low1-2

high2

low2

Compute on 2’s

high2

low2

Bid-1

-1

-3

-3

Withdraw

Bid high2

low2

Withdraw

Compute on 1’shigh1

low1

-2

-1

high2-low1-1

low2-low1-1

Bid

Compute on 1’s

high1

high1

low1

low1

-3-3

-2-2

-3high2-low1-3

Agent 2’s utilitylow2 < low1 < high2 < high1

22

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

low2 =3, low1 =12, high2 =22, high1 =30

23

Other variants we solved

• Agents cannot pass on computing during the auction & resume 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.

Designing mechanisms for agents whose valuation deliberation

is limited or costly

[Larson & Sandholm AAMAS-05]

27

Mechanism desiderata• Preference formation-independent

– Mechanism should not be involved in agents’ preference formation process • (otherwise revelation principle applies trivially)• I.e., agents communicate to auctioneer in terms of valuation (or expected valuations)

• 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• E.g. agent should not want to report a valuation and willingness to pay higher than his true valuation• <= truthful (equivalence in the case of direct mechanisms)

• Thm. There exists no direct or indirect mechanism (where any agent can affect the allocation regardless of others’ revelations) that satisfies all these 3 properties

32

Recent work on overcoming the impossibility• Restricted settings

– Not too much asymmetry – tends to avoid strong strategic computing

• Relaxing properties (but not Non-Misleading)– Relax Deliberation-Proof: Encourage strategic deliberation

• Incentives for the right (cheap) agents to compute & share right information?– Some agents as “experts” [Ito et al. AAMAS-03]

• Cavallo & Parkes [AAAI-08] get efficiency and no deficit in (within-period) ex post equilibrium. Agents report deliberation states and center says which agent deliberates next

– Assumptions» Only one agent can compute at a time» Valuations increase with computation» Time is discounted

– Without strategic deliberation possibility, achievable using dynamic VCG [Bergemann&Valimaki 07]– With strategic deliberation, use payments such that equilibrium utilities are exactly as they would be if an agent’s

deliberation processes about other agents’ values were in fact about its own value

– Relax Preference-Formation Independent• Mechanism guides deliberation• Revealing only some info about agents’ deliberative capabilities?• Related to “search” & sequential preference elicitation• Generalizing [Cremer et al. 03] to multi-step info gathering & to gathering info about other

agents as well• [Larson AAMAS-06] studies mechanism design for the case where agents can only

deliberate on their own valuations