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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 - PowerPoint PPT Presentation
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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
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)
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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
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
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
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:
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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
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:
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
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 :
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
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