Regret Minimizing Equilibria of Games with

Strict Type Uncertainty

Stony Brook Conference on Game Theory

Nathanaël Hyafil and Craig Boutilier

Department of Computer ScienceUniversity of Toronto

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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

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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)

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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 , ...)

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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

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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

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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)

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MiniMax Regret

• Single agent: make decision dD with incomplete utility function u U

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Why MiniMax Regret?

x

x’

x’x

x

x’

x’

x

xx’

x

x’

u1 u2 u4 u5u3

• In this context, MaxiMin not good:

u6

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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)

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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):

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Example

• First-Price Auction– 2 agents ; – 3 actions: .25 , .5 , .75 – Ties broken randomly

(V-.75)/2V-.75V-.750.75

0(V-.5)/2V-.50.5

00(V-.25)/20.25

0.75 0.50.25

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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 ) ?

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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:

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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:

0.25

0.25 (1-.25) / 2

0.5 (1 - .5)

0.75 (1 - .75)

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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:

Regret vs. 0.25

0.25 0

0.5 (1 - .5) - (1-.25) / 2

0.75 (1 - .75) - (1-.25) / 2

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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:

Regret vs. 0.25

0.25 0 0

0.5 (1 - .50) - (1-.25) / 2 - 0.175 < 0

0.75 (1 - .75) - (1-.25) / 2 - 0.425 < 0

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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

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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

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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

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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

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Equilibrium definitions

• MiniMax Regret Best Response to -i :

iTi,

• is a MiniMax Regret Equilibrium iff i is a MiniMax Regret best resp. to -i, i

• i is a MiniMax Regret Dominant Strategy iffit is a MiniMax Regret best resp. to all -i

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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)

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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.

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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)

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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

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Application:Partial Revelation MD

• Revelation Principle Direct, truthful mechanisms: – agents directly report their full type

• But:– hard/costly valuation problem– privacy concerns– communication costs

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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

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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

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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