1. Costly valuation computation/information acquisition in auctions: Strategy, counterspeculation, and deliberation equilibrium Tuomas Sandholm Computer 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).Costly Valuation Computation in Auctions. 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.Computationally Limited Agents in Auctions.

2. [Sandholm NOAS-91, AAAI-93] TRACONET, 1990-91 $ 2,000 $ 1,700 Contract: Task transferred Auction

3. 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. 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. 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. 6 Simple model: can pay c to find one’s own valuation => Vickrey auction no longer has a dominant strategy [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. v 1 uniform between 0 and 1; v 2 deterministic, 0 ≤ v 2 ≤ 0.5 Agent 1 bids 0.5 and gets item at price v 2 : Say agent 1 has the choice of paying c to find out v 1. Then agent 1 will bid v 1 and get the item iff v 1 ≥ v 2 (no loss possibility, but c invested) Same model studied more recently in the literature on “information acquisition in auctions” [Compte and Jehiel 01, Rezende 02, Rasmussen 06]

7. 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) Deliberation controller (uses performance profile) Domain problem solver (anytime algorithm)

8. 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. 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. 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. 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. 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 Computing time Solution quality [Zilberstein & Russell IJCAI-91, AIJ-96] Conditioning on solution quality so far [Hansen & Zilberstein AAAI-96]

13. 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 C P(C|A) Solution quality

14. 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 4 5 10 3 15 20 p(0) p(1) Random node Value node Our results hold in this augmented setting

15. 15 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. 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. 17 Theorems on strategic computing yes no Generalized Vickrey On which pair to allocate next computation step ? Multiple items for sale noEnglish (1 st price ascending) yes no Vickrey (2 nd price sealed bid) yes Dutch (1 st price descending) yes First 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. 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. 19 Performance profiles of the proof Agent 1’s problemAgent 2’s problem p(high 1 ) 1-p(high 1 ) p(high 2 ) 1-p(high 2 ) high 1 low 1 high 2 low 2 low 2 < low 1 < high 2 < high 1 000 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. 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: bid 1 > low 1 Agent 2 knows that agent 1 has valuation high 1 Agent 2 cannot win, and thus has no incentive to compute or bid –CASE 2: bid 1 < low 2 Agent 2 continues to increment its own bid No need to compute since it knows that its valuation is at least low 2 –CASE 3: low 1  bid 1  low 2 If Agent 2 bids, he should bid bid 1 + ε His strategy depends on the performance profiles…

21. 21 Decision problem of agent 2 in CASE 3 Withdraw Bid Compute on 2’s problem Compute on 1’s problem high 1 low 1 high 1 low 1 high 2 Compute on 2’s low 2 Decision node for agent 2 Chance node for agent 1’s performance profile Chance node for agent 2’s performance profile Bid 0 0 high 1 low 1 high 2 -low 1 low 2 -low 1 high 2 -low 1 -3 -3 Withdraw high 2 low 2 Withdraw -2 high 2 -low 1 -2 high 2 low 2 Compute on 2’s high 2 low 2 Bid -3 Withdraw Bid high 2 low 2 Withdraw Compute on 1’s high 1 low 1 -2 high 2 -low 1 -1 low 2 -low 1 -1 Bid Compute on 1’s high 1 low 1 -3 -2 -3 high 2 -low 1 -3 Agent 2’s utility low 2 < low 1 < high 2 < high 1

22. 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 low 2 =3, low 1 =12, high 2 =22, high 1 =30

23. 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.Bargaining with Limited Computation: Deliberation Equilibrium. –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.An Alternating Offers Bargaining Model for Computationally Limited Agents.

24. 24 Conclusions on this part Agents (human or software) 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 –Performance profile tree –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 Results apply to info acquisition as well

25. 25 Current & future research Using our deliberation control method in systems –Manufacturing planning, trucking [Larson & Sandholm AAMAS-04, AAAI] –Networks, … Miscomputing ratio [Larson & Sandholm AAMAS-03] New (auction) mechanisms –Game-theoretically engineered to work well under models of bounded rationality –Our normative deliberation control method = basis for new design principles ? –Our results show that even the most common mechanism design principles (e.g., revelation principle) cease to hold

26. Designing mechanisms for agents whose valuation deliberation is limited or costly [Larson & Sandholm AAMAS-05]

27. 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

28. 28 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 This holds even with 2 agent with 2 possible valuations each Proof sketch. - Direct: non-misleading => truthful - Each agent’s valuation can be resolved with 1 computation step - Agent 1 can compute his valuation cheaply => has a dominant strategy to compute it - Agent 2’s best response depends on Agent 1’s valuation => not deliberation-proof

29. 29 Indirect/multi-step mechanisms provide information to agents –Example: Ascending auction Bidders Information At price p there are k bidders remaining in the auction

30. 30 Is it possible to satisfy the 3 desiderata via a multi-stage mechanism? Thm. There does not exist any strategy-dependent mechanism that is –preference-formation independent, –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)

31. 31 Is it possible to satisfy the 3 desiderata via a multi-stage mechanism? Thm. There does not exist any strategy-dependent mechanism that is –preference-formation independent, –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)

32. 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