CS 758/858: Algorithms http://www.cs.unh.edu/~ruml/cs758 Game - - PowerPoint PPT Presentation

CS 758/858: Algorithms

Game Theory

Wheeler Ruml (UNH) Class 27, CS 758 – 1 / 16

http://www.cs.unh.edu/~ruml/cs758 1 handout: slides

Game Theory

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 2 / 16

Decision Science

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 3 / 16

note that people are autonomous: act in their own interest. internet, economics, game theory, multi-agent systems what happens in such situations? game dynamics, equilibria and

• ther ‘solution concepts’

coordination, competition, coalition, allocation how to design the interaction? mechanism design

What should Alice do?

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 4 / 16

Alice has three options: going to the club (c), going to a movie (m), or watching a video at home (h). If she is on her own, Alice has a utility of 100 for c, 50 for m, and 50 for h. However, Alice is also interested in the activities of two other agents, Bob and Carol, who frequent both the club and the movie theater. Bob is Alices nemesis; he is downright painful to be around. If Alice runs into Bob at the movies, she can try to ignore him and only suffers a disutility of 40; however, if she sees him at the club he will pester her endlessly, yielding her a disutility of 90. Unfortunately, Bob prefers the club: he is there 60% of the time, spending the rest of his time at the movie theater. Carol, on the

• ther hand, is Alices friend. She makes everything more fun.

Specifically, Carol increases Alices utility for either activity by a factor of 1.5 (after taking into account the possible disutility of running into Bob). Carol can be found at the club 25% of the time, and the movie theater 75% of the time.

Not a Game

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 5 / 16

For A=c: B=c B=m C=c 15 150 C=m 10 100 For A=m: B=c B=m C=c 50 10 C=m 75 15 For A=h: 50 Note: not a game, but an example of the usefulness of expected utility.

The Prisoner’s Dilemma

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 6 / 16

in ‘normal form’: player 2 C D player 1 C

• 1,-1
• 4,0

D 0,-4

• 3,-3

any rational player chooses D (check utilities!) communication beforehand doesn’t matter. finite number of repetitions doesn’t matter. however, infinite or unknown repetitions can change strategy.

Other Games

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 7 / 16

coordination game (‘chicken’): L R L 1,1 0,0 R 0,0 1,1 ‘common payoff’ competition game (‘matching pennies’): H T H 1,-1

• 1,1

T

• 1,1

1,-1 ‘zero-sum’. mixed strategy with p = 1

2 is Nash

More Games

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 8 / 16

Rock, Paper, Scissors (‘Rochambeau’): rock paper scissors rock 0,0

• 1,1

1,-1 paper 1,-1 0,0

• 1,1

scissors

• 1,1

1,-1 0,0 Battle of Sexes: husband A B wife A 2,1 0,0 B 0,0 1,2 two pure-strategy Nash equilibria also mixed-strategy Nash equilibrium: each each player plays preferred with p=2

• 3. Pareto dominated by other two equilibria.

Terminology

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 9 / 16

pure strategy: play one action mixed strategy: probability distribution over actions strategy profile: strategy for each player Pareto optimal: not Pareto dominated Nash equilibrium: each agents’ strategy is best response to

• thers’. always exists (Nash, 1951)

maxmin strategy: maximize worst-case. same as Nash in finite two-player zero-sum game (von Neumann, 1928) max regret: maximum difference from best response minimax regret: minimize max possible regret

Uncertainty

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 10 / 16

we don’t know other player’s payoffs: L R T 100,a 1 − ǫ, b B 2,c 1,d B is safe (maxmin, 1 > 1 − ǫ) T minimizes regret (Minimax regret, 98 > ǫ).

Break

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 11 / 16

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Final Exam: Wed Dec 17, 1-3pm, Kingsbury N113

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no books, notes, gadgets, ...

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bring questions from practice final to (last!) recitation

Computing Nash equilibria

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 12 / 16

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two-player zero-sum: LP

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two-player: find a NE is PPAD-complete (2006). PPAD not believed to equal P.

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n-player zero-sum:

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uniqueness, guaranteed payoff, and action inclusion/exclusion are NP-hard

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computing all equilibria of two-player game requires exponential time in the number of players

Extensive Form

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 13 / 16

game tree every finite perfect-information game in extensive form has a pure-strategy Nash equilibrium (Zermelo, 1913). no need for randomness because everyone sees others’ play.

Congestion Games

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 14 / 16

each player selects subset of resources, price of each depends on how many select it. any congestion game has a pure-strategy Nash equilibrium. This equilibrium can be found by iterating myopic best response (hill climbing): while some agent’s action is not a best response set it to one Things change when game is non-atomic (multiple steps)!

Other Topics

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 15 / 16

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taxes

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auctions

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social choice

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coalitions

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logics of belief

EOLQs

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs

Wheeler Ruml (UNH) Class 27, CS 758 – 16 / 16

Nope. Feel free to collaborate on your studying, and good luck on the final exam!