Game Theory Week 1. Game Theory Course: Jackson, Leyton-Brown & Shoham. Game Theory Course: Jackson, Leyton-Brown & Shoham Game Theory Week 1

Transcription

1 Game Theory Week 1 Game Theory Course: Jackson, Leyton-Brown & Shoham

2 A Flipped Classroom Course Before Tuesday class: Watch the week s videos, on Coursera or locally at UBC Hand in the previous week s assignment electronically Tuesday class: A lecture with high-level review of concepts from the week s videos Enrichment lectures about concepts not covered online Discussion, interactive activities Thursday class: A lab focusing on group work We ll review the solutions to the previous week s assignment Then we ll give you the next assignment (usually 1 or 2 questions) and you ll work in groups Kevin and Dave/James will be there to offer help, hints, and advice about how to improve answers

3 This begins now! Before Tuesday s class, watch the first week of videos:

4 Auction Results Game Theory Course: Jackson, Leyton-Brown & Shoham Game Theory Week 1

5 Auction Results

6 TCP Backoff Game Should you send your packets using correctly-implemented TCP (which has a backoff mechanism) or using a defective implementation (which doesn t)? both use a correct implementation: both get 1 ms delay one correct, one defective: 4 ms for correct, 0 ms for defective both defective: both get a 3 ms delay Some questions to discuss after playing: What action should a player of the game take? Would all users behave the same in this scenario? What global behavior patterns should a system designer expect? For what changes to the numbers would behavior be the same? What effect would communication have? Repetitions? (finite? infinite?) Does it matter if I believe that my opponent is rational?

7 Defining Games - The Normal Form Finite, n-person normal form game: N, A, u : Players: N = {1,, n} is a finite set of n, indexed by i Action set for player i A i a = (a 1,, a n ) A = A 1 A n is an action profile Utility function or Payoff function for player i: u i : A R u = (u 1,, u n ), is a profile of utility functions Writing a 2-player game as a matrix: row player is player 1, column player is player 2 rows correspond to actions a 1 A 1, columns correspond to actions a 2 A 2 cells listing utility or payoff values for each player: the row player first, then the column

8 More General Form 3 Competition and Coordination: Normal form games Prisoner s dilemma is any game C D C a,a b,c D c,b d,d Figure 33 with c > a > d > b Any c > a > d > b define an instance of Prisoner s Dilemma To fully understand the role of the payoff numbers we would need to enter into a discussion of utility theory Here, let us just mention that for most purposes, the analysis of any game is unchanged if the payoff numbers undergo any positive affine transformation; this simply means that each payoff x is replaced by a payoff ax + b,

9 whose own payoffs are chosen to make the payoffs in each outcome sum to zero A classical example of a zero-sum game is the game of Matching Pennies In this game, each of the two players has a penny and independently chooses to display either heads or tails The two players then compare their pennies If they are the same then player 1 pockets both, and otherwise player 2 pockets them The payoff matrix Oneisplayer shownwants in Figure to 36 match; the other wants to mismatch Matching Pennies Heads Tails Heads 1, 1 1, 1 Tails 1, 1 1, 1 Figure 36: Matching Pennies game The popular children s game of Rock, Paper, Scissors, also known as Rochambeau, provides a three-strategy generalization of the matching-pennies game The payoff matrix of this zero-sum game is shown in Figure 37 In this game, each of Game thetheory twocourse: players Jackson, canleyton-brown choose either & Shohamrock, paper, or scissors If both Game players Theory Week choose 1

10 Game AtTheory the other Course: end Jackson, of Leyton-Brown the spectrum & Shoham from pure coordination games liegame zero-sum Theory Week games, 1 coordinate on an action that is maximally beneficial to all As an example, imagine two drivers driving towards each other in a country having no traffic rules, and who must independently decide whether to drive on the left or on the right If the drivers choose the same side (left or right) they have some high utility, and otherwise they have a low utility The game matrix is shown inwhich Figure 35 side of the road should you drive on? Coordination Game Left Right Left 1,1 0,0 Right 0, 0 1, 1 Figure 35: Coordination game Zero-sum games

11 General Games: Battle of the Sexes Paper Scissors The most interesting games combine elements of cooperation and competition Figure 36 Rock, Paper, Scissors game B F B 2,1 0,0 F 0,0 1,2 Figure 37 Battle of the Sexes game 2 Strategies in normal-form games We have so far defined the actions available to each player in a game, but not yet his

12 Keynes Beauty Contest Game: The Stylized Version Each player names an integer between 1 and 100 The player who names the integer closest to two thirds of the average integer wins a prize, the other players get nothing Ties are broken uniformly at random

13 Best Response If you knew what everyone else was going to do, it would be easy to pick your own action Let a i = a 1,, a i 1, a i+1,, a n now a = (a i, a i ) Definition (Best response) ạ i BR(a i ) iff a i A i, u i (a i, a i ) u i (a i, a i )

14 Nash Equilibrium Really, no agent knows what the others will do What can we say about which actions will occur? Idea: look for stable action profiles Definition (Nash Equilibrium) a = a 1,, a n is a ( pure strategy ) Nash equilibrium iff i, a i BR(a i )

15 r, minus your delay), and the second number represents your colleague s Nash Equilibria of Example Games C D C 1, 1 4,0 D 0, 4 3, 3 Figure 31 The TCP user s (aka the Prisoner s) Dilemma options what should you adopt, C or D? Does it depend on what you league will do? Furthermore, from the perspective of the network operaof behavior can he expect from the two users? Will any two users behave n presented with this scenario? Will the behavior change if the network s the users to communicate with each other before making a decision? hanges to the delays would the users decisions still be the same? How rs behave if they have the opportunity to face this same decision with the art multiple times? Do answers to the above questions depend on how ents are and how they view each other s rationality?

16 r, minus your delay), and the second number represents your colleague s in Figure 35 Nash Equilibria of Example Games C D Left Right C 1, 1 4,0 D 0, 4 3, 3 Left 1,1 0,0 Right 0, 0 1, 1 Figure 31 The TCP user s (aka the Prisoner s) Figure Dilemma 35: Coordination game options what should you adopt, C or D? Does it depend on what you league will do? Furthermore, from the perspective of the network operaof behavior can he expect Zero-sum from games the two users? Will any two users behave nzero-sum presented game with this Atscenario? the otherwill end of the the behavior spectrum change fromifpure the network coordination games lie zero-sum games, s the users to communicate which (bearing with each in mind other the before comment makingwe a decision? made earlier about positive affine trans- would the users are more decisions properly still called be theconstant-sum same? How games Unlike common-payoff hanges constant-sum to the delaysformations) rs game behave if they have games, the opportunity constant-sum to face games this are same meaningful decision with primarily the in the context of two-player art multiple times? Do answers to the above questions depend on how (though not necessarily two-strategy) games ents are and how they view each other s rationality?

17 r, minus your delay), and the second number represents your colleague s in Figure 35 Nash Equilibria of Example Games Rock C D Left Right Paper C 1, 1 4,0 Scissors D 0, 4 3, 3 Left 1,1 0,0 Right 0, 0 1, 1 Figure 36 Rock, Paper, Scissors game Figure 31 The TCP user s (aka the Prisoner s) Figure Dilemma 35: Coordination game B F options what should you adopt, C or D? Does it depend on what you league will do? Furthermore, from the perspective of the network operaof behavior can he expect B Zero-sum 2,1 from games 0,0 the two users? Will any two users behave nzero-sum presented game with this Atscenario? the otherwill end of the the behavior spectrum change fromifpure the network coordination games lie zero-sum games, s the users to communicate which F 0,0 (bearing with 1,2 each in mind other the before comment makingwe a decision? made earlier about positive affine trans- would the users are more decisions properly still called be theconstant-sum same? How games Unlike common-payoff hanges constant-sum to the delaysformations) rs game behave if they have the opportunity to face this same decision with the Figure 37 games, Battle constant-sum of the Sexes game games are meaningful primarily in the context of two-player art multiple times? Do answers to the above questions depend on how (though not necessarily two-strategy) games ents are and how they view each other s rationality?

18 situations of pure competition; one player s gain must come at the expense of the r, minus your delay), and the second number represents your colleague s Nash Equilibria other in Figure player 35 of Example This property requires Games that there be exactly two agents Indeed, if Rock you 0 allow more 1 agents, any 1 game can be turned into a zero-sum game by adding a dummy C player Dwhose actions do notleft impact the Right payoffs to the other agents, and whose own payoffs are chosen to make the payoffs in each outcome sum to zero Paper Matching C 1, A classical 1 example 4,0 of a zero-sum Left game 1,1 is the0,0 game of Matching Pennies In this Pennies game game, each of the two players has a penny and independently chooses to display Scissors either 1 heads or 1 tails The0two players then compare their pennies If they are the D same 0, 4 then player 3, 3 1 pockets both, Rightand otherwise 0, 0 player 1, 1 2 pockets them The payoff Figure 36 matrix Rock, is Paper, shown Scissors in Figure game 36 Figure 31 The TCP user s (aka the Prisoner s) Figure Dilemma 35: Coordination game B F Heads Tails options what should you adopt, C or D? Does it depend on what you league will do? Furthermore, from the perspective of the network operaof behavior can he expect B Zero-sum 2,1 from games 0,0 Heads 1, 1 1, 1 the two users? Will any two users behave nzero-sum presented game with this Atscenario? the otherwill end of the the behavior spectrum change fromifpure the network coordination games lie zero-sum games, s the users to communicate which F 0,0 (bearing with 1,2 each in mind other the before Tails comment making 1, we a1decision? made1, earlier 1 about positive affine trans- would the users are more decisions properly still called be theconstant-sum same? How games Unlike common-payoff hanges constant-sum to the delaysformations) rs game behave if they have games, the opportunity constant-sum to face games this are same meaningful decision with primarily the Figure 37 Battle of the Sexes game Figure 36: Matching Pennies game in the context of two-player art multiple times? Do answers to the above questions depend on how (though not necessarily two-strategy) games ents are and how they view each other s rationality?

19 Domination Let s i and s i be two strategies for player i, and let S i be is the set of all possible strategy profiles for the other players What s a strategy? For now, just choosing an action ( pure strategy ) Definition ṣ i strictly dominates s i if s i S i, u i (s i, s i ) > u i (s i, s i ) Definition ṣ i very weakly dominates s i if s i S i, u i (s i, s i ) u i (s i, s i )

20 Pareto Optimality When one outcome o is at least as good for every agent as another outcome o, and there is some agent who strictly prefers o to o : it seems reasonable to say that o is better than o we say that o Pareto-dominates o Definition (Pareto Optimality) An outcome o is Pareto-optimal if there is no other outcome that Pareto-dominates it

21 Pareto Optimality When one outcome o is at least as good for every agent as another outcome o, and there is some agent who strictly prefers o to o : it seems reasonable to say that o is better than o we say that o Pareto-dominates o Definition (Pareto Optimality) An outcome o is Pareto-optimal if there is no other outcome that Pareto-dominates it can a game have more than one Pareto-optimal outcome?

22 Pareto Optimality When one outcome o is at least as good for every agent as another outcome o, and there is some agent who strictly prefers o to o : it seems reasonable to say that o is better than o we say that o Pareto-dominates o Definition (Pareto Optimality) An outcome o is Pareto-optimal if there is no other outcome that Pareto-dominates it can a game have more than one Pareto-optimal outcome? does every game have at least one Pareto-optimal outcome?

23 35 Pareto Optimal Outcomes in Example Games Left Right Left 1,1 0,0 Right 0, 0 1, 1 Figure 35: Coordination game games er end of the spectrum from pure coordination games lie zero-sum games, aring in mind the comment we made earlier about positive affine transs) are more properly called constant-sum games Unlike common-payoff nstant-sum games are meaningful primarily in the context of two-player t necessarily two-strategy) games

24 35 Figure 36 Pareto Optimal Outcomes in Example Games Rock, Paper, Scissors game Left Right B F Left 1,1 0,0 Right 0, 0 1, 1 B 2,1 0,0 F 0,0 1,2 Figure 35: Coordination game Figure 37 Battle of the Sexes game 322 Strategies in normal-form games games We have so far defined the actions available to each player in a game, but not yet his er end of the spectrumset fromof pure strategies, coordination or his available games liechoices zero-sumcertainly games, one kind of strategy is to select aringpure strategy mind the comment a single weaction madeand earlier playabout it; wepositive call suchaffine a strategy trans-s) are more properly called the notation constant-sum we havegames alreadyunlike developed common-payoff for actions to represent it There is, however, pure strategy, and we will use nstant-sum games are another, meaningful less obvious primarilytype in the of strategy; context aof two-player can choose to randomize over the set of t necessarily two-strategy) available games actions according to some probability distribution; such a strategy is called mixed strategy a mixed strategy Although it may not be immediately obvious why a player should

25 of pure competition; one player s gain must come at the expense of the 35 er This property requires that there be exactly Figure two36agents Rock, Indeed, Paper, Scissors if game Pareto Optimal Outcomes in Example Games more agents, any game can be turned into a zero-sum game by adding player whose actions do notleft impact the Right payoffs to the other agents, B Fand n payoffs are chosen to make the payoffs in each outcome sum to zero cal example of a zero-sum Left game 1,1 is the0,0 game of Matching B Pennies 2,1 In0,0 this h of the two players has a penny and independently chooses to display ds or tails The two players then compare their pennies If they are the Right 0, 0 1, 1 F 0,0 1,2 player 1 pockets both, and otherwise player 2 pockets them The payoff hown in Figure 36 Figure 35: Coordination game Figure 37 Battle of the Sexes game Heads Tails 322 Strategies in normal-form games games Heads 1, 1 1, 1 We have so far defined the actions available to each player in a game, but not yet his er end of the spectrumset fromof pure strategies, coordination or his available games liechoices zero-sumcertainly games, one kind of strategy is to select aringpure strategy mind thetails comment a single 1, weaction 1madeand 1, earlier 1 playabout it; wepositive call suchaffine a strategy trans-s) are more properly called the notation constant-sum we havegames alreadyunlike developed common-payoff for actions to represent it There is, however, pure strategy, and we will use nstant-sum games Figureare 36: another, meaningful Matching less obvious primarily Penniestype game in the of strategy; context aof two-player can choose to randomize over the set of t necessarily two-strategy) available games actions according to some probability distribution; such a strategy is called mixed strategy a mixed strategy Although it may not be immediately obvious why a player should

26 of pure competition; one player s gain must come at the expense of the 35 er This property requires when that congestion there be occurs exactly Figure You two36 have agents Rock, two possible Indeed, Paper, Scissors if strategies: game C (for using a Correct Pareto Optimal Outcomes in Example Games more agents, any game implementation) can be turned and into D a (for zero-sum using a game Defective by adding one) If both you and your colleague player whose actions adopt do not CLeft impact then your the Right average payoffs packet to the delay other is agents, B 1ms (millisecond) Fand If you both adopt D the delay is 3ms, because of additional overhead at the network router Finally, if one of n payoffs are chosen to make the payoffs in each outcome sum to zero you adopts D and the other adopts C then the D adopter will experience no delay at all, cal example of a zero-sum Left game 1,1 is the0,0 game of Matching B Pennies 2,1 In0,0 this but the C adopter will experience a delay of 4ms h of the two players has a penny and independently chooses to display These consequences are shown in Figure 31 Your options are the two rows, and ds or tails The two players then compare their pennies If they are the Right your colleague s 0, 0 1, options 1 are the Fcolumns 0,0 In each 1,2 cell, the first number represents player 1 pockets both, your and payoff otherwise (or, minus player your 2 pockets delay), them and the The second payoff number represents your colleague s hown TCP inuser s Figure 36 payoff 1 game Figure 35: Coordination game Figure 37 Battle of the Sexes game Heads Tails C D Prisoner s dilemma game 322 Strategies in normal-form games games Heads 1, 1 1, 1 C 1, 1 4,0 We have so far defined the actions available to each player in a game, but not yet his er end of the spectrumset fromof pure strategies, coordination or his available games liechoices zero-sumcertainly games, one kind of strategy is to select aringpure strategy mind thetails comment a single 1, weaction 1madeand 1, earlier 1 playabout it; we Dpositive call0, such 4 affine a strategy 3, transs) are more properly called the notation constant-sum we havegames alreadyunlike developed common-payoff for actions to represent it There is, however, 3 a pure strategy, and we will use nstant-sum games Figureare 36: another, meaningful Matching less obvious primarily Pennies Figure type game 31 in the of The strategy; context TCP user s aof two-player (aka can the Prisoner s) choose to Dilemma randomize over the set of t necessarily two-strategy) available games actions according to some probability distribution; such a strategy is called mixed strategy a mixed strategy Although it may not be immediately obvious why a player should

27 of pure competition; one player s gain must come at the expense of the 35 er This property requires when that congestion there be occurs exactly Figure You two36 have agents Rock, two possible Indeed, Paper, Scissors if strategies: game C (for using a Correct Pareto Optimal Outcomes in Example Games more agents, any game implementation) can be turned and into D a (for zero-sum using a game Defective by adding one) If both you and your colleague player whose actions adopt do not CLeft impact then your the Right average payoffs packet to the delay other is agents, B 1ms (millisecond) Fand If you both adopt D the delay is 3ms, because of additional overhead at the network router Finally, if one of n payoffs are chosen to make the payoffs in each outcome sum to zero you adopts D and the other adopts C then the D adopter will experience no delay at all, cal example of a zero-sum Left game 1,1 is the0,0 game of Matching B Pennies 2,1 In0,0 this but the C adopter will experience a delay of 4ms h of the two players has a penny and independently chooses to display These consequences are shown in Figure 31 Your options are the two rows, and ds or tails The two players then compare their pennies If they are the Right your colleague s 0, 0 1, options 1 are the Fcolumns 0,0 In each 1,2 cell, the first number represents player 1 pockets both, your and payoff otherwise (or, minus player your 2 pockets delay), them and the The second payoff number represents your colleague s hown TCP inuser s Figure 36 payoff 1 game Figure 35: Coordination game Figure 37 Battle of the Sexes game Heads Tails C D Prisoner s dilemma game 322 Strategies in normal-form games games Heads 1, 1 1, 1 C 1, 1 4,0 We have so far defined the actions available to each player in a game, but not yet his er end of the spectrumset fromof pure strategies, coordination or his available games liechoices zero-sumcertainly games, one kind of strategy is to select aringpure strategy mind thetails comment a single 1, weaction 1madeand 1, earlier 1 playabout it; we Dpositive call0, such 4 affine a strategy 3, transs) are more properly called the notation constant-sum we havegames alreadyunlike developed common-payoff for actions to represent it There is, however, 3 a pure strategy, and we will use nstant-sum games Figureare 36: another, meaningful Matching The less paradox obvious primarily Pennies Figure oftype game Prisoner s 31 in the of The strategy; context TCP dilemma: user s aof two-player (aka can the Prisoner s) choose to Dilemma randomize over the set of t necessarily the (DS) two-strategy) Nash available equilibrium games actionsisaccording the onlyto non-pareto-optimal some probability distribution; outcome! such a strategy is called mixed strategy a mixed strategy Although it may not be immediately obvious why a player should