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

CS 758/858: Algorithms http://www.cs.unh.edu/~ruml/cs758 Game Theory 1 handout: slides Wheeler Ruml (UNH) Class 27, CS 758 – 1 / 16

Game Theory ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games ■ Terminology ■ Uncertainty Game Theory ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 2 / 16

Decision Science note that people are autonomous: act in their own interest. Game Theory internet, economics, game theory, multi-agent systems ■ Decision Science ■ Alice ■ Not a Game what happens in such situations? game dynamics, equilibria and ■ A Dilemma other ‘solution concepts’ ■ Other Games ■ More Games ■ Terminology coordination, competition, coalition, allocation ■ Uncertainty ■ Break how to design the interaction? mechanism design ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 3 / 16

What should Alice do? Alice has three options: going to the club (c), going to a movie Game Theory (m), or watching a video at home (h). If she is on her own, Alice ■ Decision Science ■ Alice has a utility of 100 for c, 50 for m, and 50 for h. However, Alice ■ Not a Game ■ A Dilemma is also interested in the activities of two other agents, Bob and ■ Other Games Carol, who frequent both the club and the movie theater. Bob is ■ More Games ■ Terminology Alices nemesis; he is downright painful to be around. If Alice ■ Uncertainty runs into Bob at the movies, she can try to ignore him and only ■ Break ■ Computing Nash suffers a disutility of 40; however, if she sees him at the club he ■ Extensive Form ■ Congestion Games will pester her endlessly, yielding her a disutility of 90. ■ Other Topics Unfortunately, Bob prefers the club: he is there 60% of the time, ■ EOLQs spending the rest of his time at the movie theater. Carol, on the other 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. Wheeler Ruml (UNH) Class 27, CS 758 – 4 / 16

Not a Game B=c B=m Game Theory ■ Decision Science For A=c: C=c 15 150 ■ Alice C=m 10 100 ■ Not a Game ■ A Dilemma ■ Other Games ■ More Games B=c B=m ■ Terminology For A=m: C=c 50 10 ■ Uncertainty ■ Break C=m 75 15 ■ Computing Nash ■ Extensive Form ■ Congestion Games For A=h: 50 ■ Other Topics ■ EOLQs Note: not a game, but an example of the usefulness of expected utility. Wheeler Ruml (UNH) Class 27, CS 758 – 5 / 16

The Prisoner’s Dilemma in ‘normal form’: Game Theory player 2 ■ Decision Science ■ Alice C D ■ Not a Game player 1 C -1,-1 -4,0 ■ A Dilemma ■ Other Games D 0,-4 -3,-3 ■ More Games ■ Terminology ■ Uncertainty any rational player chooses D (check utilities!) ■ Break ■ Computing Nash communication beforehand doesn’t matter. finite number of ■ Extensive Form ■ Congestion Games repetitions doesn’t matter. however, infinite or unknown ■ Other Topics ■ EOLQs repetitions can change strategy. Wheeler Ruml (UNH) Class 27, CS 758 – 6 / 16

Other Games coordination game (‘chicken’): Game Theory L R ■ Decision Science ■ Alice L 1,1 0,0 ■ Not a Game ■ A Dilemma R 0,0 1,1 ■ Other Games ■ More Games ‘common payoff’ ■ Terminology ■ Uncertainty ■ Break competition game (‘matching pennies’): ■ Computing Nash ■ Extensive Form H T ■ Congestion Games H 1,-1 -1,1 ■ Other Topics ■ EOLQs T -1,1 1,-1 ‘zero-sum’. mixed strategy with p = 1 2 is Nash Wheeler Ruml (UNH) Class 27, CS 758 – 7 / 16

More Games Rock, Paper, Scissors (‘Rochambeau’): Game Theory rock paper scissors ■ Decision Science ■ Alice rock 0,0 -1,1 1,-1 ■ Not a Game ■ A Dilemma paper 1,-1 0,0 -1,1 ■ Other Games scissors -1,1 1,-1 0,0 ■ More Games ■ Terminology ■ Uncertainty Battle of Sexes: ■ Break ■ Computing Nash husband ■ Extensive Form A B ■ Congestion Games ■ Other Topics wife A 2,1 0,0 ■ EOLQs 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. Wheeler Ruml (UNH) Class 27, CS 758 – 8 / 16

Terminology pure strategy: play one action Game Theory mixed strategy: probability distribution over actions ■ Decision Science ■ Alice strategy profile: strategy for each player ■ Not a Game ■ A Dilemma Pareto optimal: not Pareto dominated ■ Other Games Nash equilibrium: each agents’ strategy is best response to ■ More Games ■ Terminology others’. always exists (Nash, 1951) ■ Uncertainty maxmin strategy: maximize worst-case. same as Nash in ■ Break ■ Computing Nash finite two-player zero-sum game (von Neumann, 1928) ■ Extensive Form max regret: maximum difference from best response ■ Congestion Games ■ Other Topics minimax regret: minimize max possible regret ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 9 / 16

Uncertainty we don’t know other player’s payoffs: Game Theory L R ■ Decision Science ■ Alice T 100, a 1 − ǫ, b ■ Not a Game ■ A Dilemma B 2, c 1, d ■ Other Games ■ More Games ■ Terminology B is safe (maxmin, 1 > 1 − ǫ ) ■ Uncertainty T minimizes regret (Minimax regret, 98 > ǫ ). ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 10 / 16

Break Final Exam: Wed Dec 17, 1-3pm, Kingsbury N113 ■ Game Theory no books, notes, gadgets, ... ■ Decision Science ■ ■ Alice bring questions from practice final to (last!) recitation ■ ■ 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

Computing Nash equilibria two-player zero-sum: LP ■ Game Theory two-player: find a NE is PPAD-complete (2006). PPAD not ■ Decision Science ■ ■ Alice believed to equal P. ■ Not a Game ■ A Dilemma n -player zero-sum: ■ ■ Other Games uniqueness, guaranteed payoff, and action inclusion/exclusion ■ ■ More Games ■ Terminology are NP-hard ■ Uncertainty computing all equilibria of two-player game requires ■ Break ■ ■ Computing Nash exponential time in the number of players ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 12 / 16

Extensive Form game tree Game Theory ■ Decision Science every finite perfect-information game in extensive form has a ■ Alice ■ Not a Game pure-strategy Nash equilibrium (Zermelo, 1913). no need for ■ A Dilemma randomness because everyone sees others’ play. ■ 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

Congestion Games each player selects subset of resources, price of each depends on Game Theory how many select it. ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma any congestion game has a pure-strategy Nash equilibrium. This ■ Other Games equilibrium can be found by iterating myopic best response (hill ■ More Games ■ Terminology climbing): ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form while some agent’s action is not a best response ■ Congestion Games ■ Other Topics set it to one ■ EOLQs Things change when game is non-atomic (multiple steps)! Wheeler Ruml (UNH) Class 27, CS 758 – 14 / 16

Other Topics taxes ■ Game Theory auctions ■ ■ Decision Science ■ Alice social choice ■ ■ Not a Game coalitions ■ A Dilemma ■ ■ Other Games logics of belief ■ ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 15 / 16

EOLQs Game Theory Nope. ■ Decision Science ■ Alice ■ Not a Game ■ A Dilemma Feel free to collaborate on your studying, ■ Other Games and good luck on the final exam! ■ More Games ■ Terminology ■ Uncertainty ■ Break ■ Computing Nash ■ Extensive Form ■ Congestion Games ■ Other Topics ■ EOLQs Wheeler Ruml (UNH) Class 27, CS 758 – 16 / 16