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Regret Minimizing Equilibria of Games with Strict Type Uncertainty

Regret Minimizing Equilibria of Games with Strict Type Uncertainty. Stony Brook Conference on Game Theory Nathana ë l Hyafil and Craig Boutilier Department of Computer Science University of Toronto. Overview. 1. Motivation / Background Automated Mechanism design Strict Uncertainty

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Regret Minimizing Equilibria of Games with Strict Type Uncertainty

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  1. Regret Minimizing Equilibria of Games with Strict Type Uncertainty Stony Brook Conference on Game Theory Nathanaël Hyafil and Craig Boutilier Department of Computer Science University of Toronto

  2. Overview • 1. Motivation / Background • Automated Mechanism design • Strict Uncertainty • Minimax Regret • 2. Games with Strict Type Uncertainty • Definition of equilibrium • Existence of equilibrium • 3. Applications / Conclusion • Partial Revelation Mechanism Design

  3. Automated MD (AMD) • VCG: always pick efficient outcome • Myerson auction: • not always optimal outcome • but maximizes expected objective (revenue) given a prior over agents’ types • AMD: • for general objectives (not just revenue) • general outcome space (not just auctions)

  4. Automated MD (AMD) • Given: • sets of types, outcomes • objective function f(,o) (SW, revenue, ...) • prior over types • Optimization problem: • find mechanism (outcome for each type vector) • maximize expected objective value • subject to Constraints: • Incentive Compatibility (BNE or DS) • ( Individual Rationality , Budget Balance , ...)

  5. Where do priors over types come from? • “Experts”? • Costly! • Can rule out inappropriate valuations • But hard to quantify probabilistically • simple distribution (unrealistic but needed) • Observation of past behavior? • Gives linear constraints on values • Not probability distributions

  6. Strict Uncertainty • No probability distribution but subset of possible types • Agents cannot maximize expected utility • use MiniMax Regret as decision criterion • Mechanism Designer: can’t use Bayes-Nash Eq., can’t maximize expected objective  Mech Designer minimizes his regret too

  7. MiniMax Regret • Different from: • regret used to converge to equilibrium in repeated games (e.g., Hart & MasColell) • regret of Regret Theory (Bell; Loomes & Sugden) • Savage’s MiniMax Regret criterion from Decision Theory • recently used for uncertainty about utilities (as opposed to outcomes)

  8. MiniMax Regret • Single agent: make decision dD with incomplete utility function u U

  9. Why MiniMax Regret? • In this context, MaxiMin not good: x’ x’ x’ x x x’ x’ x x x x x’ u1 u2 u3 u4 u5 u6

  10. 2. Games of Incomplete Information with Strict Type Uncertainty • N players, and for each: • Actions: Ai • Types: i • Utility: ui: A  i R • Each agent knows its type, not the others’, but: • Common prior: Strict: T • Strategy: i: i  (Ai)

  11. Regret definitions • Regret of strategy i for agent i of type i, given type -i and strategy -i of the others: • MaxRegret of strategy i for i of type i, given strategy -i of the others (for prior T):

  12. 0.25 0.5 0.75 0.25 (V-.25)/2 0 0 0.5 V-.5 (V-.5)/2 0 0.75 V-.75 V-.75 (V-.75)/2 Example • First-Price Auction • 2 agents ; • 3 actions: .25 , .5 , .75 • Ties broken randomly

  13. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ?

  14. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2=.2:

  15. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2 = .2:

  16. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2 = .2:

  17. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2 = .2:

  18. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2 = .2 0

  19. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • What is MR1(bid =.25|1=.4 ; 2) ? R1(bid = .25) if 2 = .2 0 R1(bid = .25) if 2 = .4 0 R1(bid = .25) if 2 = .6 0 R1(bid = .25) if 2 = .8 0

  20. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • MR1(bid =.25|1=.4 ; 2) = max { 0, 0, 0, 0} = 0

  21. Example: Agent 1’s reasoning • 2 {.2, .4, .6, .8} 2: .2  .25 {.4 , .6 }  .5 .8  .75 • MR1(bid =.25|1=.4 ; 2) = max { 0, 0, 0, 0} = 0 • so argmina MR1(a| 1=.4 ; 2) = .25 • and MMR = 0

  22. Equilibrium definitions • MiniMax Regret Best Response to -i : iTi, •  is a MiniMax Regret Equilibrium iff i is a MiniMax Regret best resp. to -i, i • i is a MiniMax Regret Dominant Strategy iff it is a MiniMax Regret best resp. to all -i

  23. Example First-Price Auction Ti = {.2 , .4 , .6 , .8} • MiniMaxRegret Equilibrium: (i,i) with i: .2  bid .25 (MMR = 0) .4  bid .25 (MMR = 0) .6  bid (.25,.5) with p=(.6,.4) (MMR = 0.03) .8  bid (.5,.75) with p=(10/11,1/11) (MMR =.0227)

  24. Existence Results • Theorem: There exists a MiniMax Regret Eq in all games with finite number of agents, actions and types • Proposition:  is a MiniMax Regret dominant strategy equ. for a Strict incomplete information game iff it is a DS for any corresponding Bayesian game • Observation:  is a MiniMax Regret Eq. with zero regret for all types of all agents iff it is an Ex-Post Eq.

  25. Non-finite Games? • Proof relies on Kakutani’s fixed point theorem • main difference with Bayesian games: expected utility is linear, Max Regret is not • so any extension (e.g., continuous games) that doesn’t require linearity should apply to MMR (e.g., Milgrom & Weber 1987)

  26. 3. Applications: • Strict Automated Mechanism Design: • designer is regret minimizer too • regret of mechanism M1 vs. M2: difference in objective value (SW, …) between M1 and M2 when an ‘adversary’ picks the types of the agents • (Hyafil & Boutilier, UAI 2004): • formulation as optimization subject to IC, IR, … • infinite number of constraints, some non-linear • algorithm to solve as sequence of linear problems

  27. Application:Partial Revelation MD • Revelation Principle  Direct, truthful mechanisms: • agents directly report their full type • But: • hard/costly valuation problem • privacy concerns • communication costs

  28. Partial Revelation MD • Instead: partial type • v  [.4 , .6] • Partial Revelation: • Type space is partitioned in finite number of sets • Report is the subset containing full type • Choose outcome despite remaining uncertainty

  29. Partial Revelation MD • For very general form of partitions, with no structure on (quasi-linear) outcome space: • “impossible” to impose truthfulness in Dominant Strategies and Bayes-Nash equilibrium • Use MiniMax Regret equilibrium concept in Partial Revelation MD

  30. Conclusion • Games with Strict Uncertainty: • definition • proposed MiniMax Regret as Rationality concept • proved Existence of MiniMaxRegret Equilibria • Applications: • Partial Revelation MD • Multi-Attribute Bargaining • Sequential Strict Automated MD

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