Introduction to Game Theory

Transcription

1 Introduction to Game Theory Presentation vs. exam You and your partner Either study for the exam or prepare the presentation (not both) Exam (50%) If you study for the exam, your (expected) grade is 92 If you don t, your grade is 80 Presentation (50%) If both you and your partner prepare the presentation, your (expected) joint grade is 100 If one of you prepares the presentation and the other does not, your joint grade is 92 If none of you prepare the presentation, your joint grade is 84 1

2 Presentation vs. exam: tabular form you decide independently of your partner, taking this knowledge into account you want to maximize your grade Your partner Presentation Exam You Presentation 90, 90 86, 92 Exam 92, 86 88, 88 what do you do? Game theory: set-up Finite set of players Set of strategies available to each player Payoff for each player as a function of the strategies selected by all players Everything a player cares about is reflected in her payoff, which she strives to maximize (rationality) Each player knows everything about the structure of the game (common knowledge) 2

3 Back to presentation and exam Your partner Presentation Exam You Presentation 90, 90 86, 92 Exam 92, 86 88, 88 you should study for the exam Your partner Presentation Exam You Presentation 90, 90 86, 92 Exam 92, 86 88, 88 you should study for the exam Presentation and exam: outcome Your partner Presentation Exam You Presentation 90, 90 86, 92 Exam 92, 86 88, 88 You both get a grade of 88, whereas you could both get a grade of 90, if you somehow cooperated 3

4 Best response and dominant strategies S 1 is a best response of player 1 to strategy S 2 of player two if 8 S1 P 1 (S 1; S 2 ) P 1 (S 1 ; S 2 ) S 1 is a strict best response of player 1 to strategy S 2 of player two if 8 S1 6=S 1 P 1 (S 1; S 2 ) > P 1 (S 1 ; S 2 ) S 1 is a dominant strategy for player 1 if it is a best response to every strategy of player 2 8 S1 ;S 2 P 1 (S 1; S 2 ) P 1 (S 1 ; S 2 ) S 1 is a strictly dominant strategy for player 1 if it is a strict best response to every strategy of player 2 8 S1 6=S 1 ;S 2 P 1 (S 1; S 2 ) > P 1 (S 1 ; S 2 ) Other considerations Finite games finite set of players and finite set of strategies ordinal payoffs S 2 1 S 2 2 S 2 1 S 2 2 S , 90 86, 92 S , 86 88, 88 same reasoning as S 1 1 2, 2 0, 3 S 1 2 3, 0 1, 1 Simultaneity each play makes her decision without knowing the decisions of other players 4

5 Low-priced and upscale Two firms Either produce a low-priced or upscale version of a product (not both) Distribution of the population 60% of the population buy low-priced 40% of the population buy upscale Popularity of the firms If the two firms directly compete in a market segment, firm 1 gets 80% of the sales and firm 2 gets 20% of the sales If the two firms compete on different market segments, each gets 100% of the sales Low-priced vs. upscale: tabular form Firm 2 Low-priced Upscale Firm 1 Low-priced 0.48, , 0.40 Upscale 0.40, , 0.08 Firm 1 has the strictly dominant strategy of producing low-priced items Firm 2 has no strictly dominant strategy Firm 1 will choose to produce low-priced items, then Firm 2 s best response to that will be to produce upscale items 5

6 Three-client game Two firms each can hope to do business with just one of three clients, B, or C Clients The business of client is 8 The business of clients B and C is 2 each Client will either do business with the two firms simultaneously, or with none Business approaches If the two firms approach the same client, each gets half the business of that client If firm 1 approaches a client alone it gets nothing If firm 2 approaches client B or C alone, it gets it full business Three-client game: firm 1 Firm 2 B C 4, 4 0, 2 0, 2 Firm 1 B 0, 0 1, 1 0, 2 C 0, 0 0, 2 1, 1 no dominant strategy for Firm 1 6

7 Three-client game: firm 2 Firm 2 B C 4, 4 0, 2 0, 2 Firm 1 B 0, 0 1, 1 0, 2 C 0, 0 0, 2 1, 1 no dominant strategy for Firm 2 either Nash equilibrium Sometimes, there is no dominant strategy at all (S 1; S 2) is a Nash equilibrium if S 1 is a best response to S 2 and S 2 is a best response to S 1 8 S1 ;S 2 P 1 (S 1; S 2) P 1 (S 1 ; S 2) ^ P 1 (S 1; S 2) P 1 (S 1; S 2 ) If one player has a dominant strategy, then that strategy together with the other player s best response to it is a Nash equilibrium 7

8 Three-client game: Nash equilibrium Firm 2 B C 4, 4 0, 2 0, 2 Firm 1 B 0, 0 1, 1 0, 2 C 0, 0 0, 2 1, 1 (,) is the unique Nash equilibrium Looking for Nash equilibria Checking Nash equilibrium individual strategies are best responses to each other Finding Nash equilibria compute each player s best responses to each strategy of the other player find the strategies that are mutual best responses 8

9 Coordination game You and your partner Either prepare presentation in PowerPoint or in Keynote (not both) Coordination If both you and your partner choose the same application, each gets a payoff of 1 Otherwise, each gets a payoff of 0 Coordination game: Nash equilibria Your partner PowerPoint Keynote You PowerPoint 1, 1 0, 0 Keynote 0, 0 1, 1 Two Nash equilibria Focal point external to the game 9

10 Unbalanced coordination game Your partner PowerPoint Keynote You PowerPoint 2, 2 0, 0 Keynote 0, 0 1, 1 Two Nash equilibria all the same Focal point intrinsic to the game Battle of sexes Your partner PowerPoint Keynote You PowerPoint 1, 2 0, 0 Keynote 0, 0 2, 1 Two Nash equilibria once more 10

11 Stag and hare Hunter 2 Stag Hare Hunter 1 Stag 4, 4 0, 3 Hare 3, 0 3, 3 Yet, two Nash equilibria Hawk-dove Two animals foraging food Each animal can take either an aggressive (hawk) or passive (dove) stance Payoffs If the two animals behave passively, each gets a payoff of 3 If one behaves aggressively and the other passively, the aggressor gets a payoff of 5, while the passive gets a payoff of 1 If both behave aggressively, each gets a payoff of 0 11

12 Hawk-dove: Nash equilibria nimal 2 Dove Hawk nimal 1 Dove 3, 3 1, 5 Hawk 5, 1 0, 0 Two Nash equilibria You and your friend Matching cents Either can choose heads or tails of its own onecent coin Payoffs If coins match, you loose your cent to your friend If coins do not match, your friend looses her cent to you 12

13 Matching cents: no Nash equilibria Your friend Heads Tails You Heads -1, +1 +1, -1 Tails +1, -1-1, +1 No Nash equilibria zero-sum game: payoffs of the players sum to zero on every outcome Pareto-optimality (S 1; S 2) is a Pareto-optimal if 8 S1 ;S 2 P 1 (S 1; S 2) > P 1 (S 1 ; S 2 ) _ P 2 (S 1; S 2) > P 2 (S 1 ; S 2 ) _ P 1 (S 1; S 2) = P 1 (S 1 ; S 2 ) ^ P 2 (S 1; S 2) = P 2 (S 1 ; S 2 ) (S 1; S 2) is a Pareto-optimal if :9 S1 ;S 2 P 1 (S 1; S 2) < P 1 (S 1 ; S 2 ) ^ P 2 (S 1; S 2) P 2 (S 1 ; S 2 ) _ P 1 (S 1; S 2) P 1 (S 1 ; S 2 ) ^ P 2 (S 1; S 2) < P 2 (S 1 ; S 2 ) 13

14 Presentation vs. exam again Your partner Presentation Exam You Presentation 90, 90 86, 92 Exam 92, 86 88, 88 (Exam, Exam) is a Nash equilibrium that is not a Pareto-optimal outcome (Presentation, Presentation), (Presentation, Exam), and (Exam, Presentation) are Pareto-optimal outcomes, none of which is a Nash equilibrium Social optimality (S 1; S 2) is a social optimum if 8 S1 ;S 2 P 1 (S 1; S 2) + P 2 (S 1; S 2) P 1 (S 1 ; S 2 ) + P 2 (S 1 ; S 2 ) (Presentation, Presentation) is the only social optimum in the Presentation vs. Exam game social optimum is a Pareto-optimum, but not conversely 14

15 Facility location B C D E F Two firms Firm 1 can open store in, C, or E Firm 2 can open store in B, D, or F Payoffs Customers go to the nearest store Same number of customers per town Payoffs proportional to number of customers Facility location: tabular form Firm 2 B D F 1, 5 2, 4 3, 3 Firm 1 C 4, 2 3, 3 4, 2 E 3, 3 2, 4 5, 1 if firm 2 chooses B, then firm 1 s best response is C if firm 2 chooses D, then firm 1 s best response is C if firm 2 chooses F, then firm 1 s best response is E There is no strictly dominant strategy by firm 1, but there is a strictly dominated strategy by firm 1 (strategy ) There is no strictly dominant strategy by firm 2, but there is a strictly dominated strategy by firm 2 (strategy F) 15

16 Strictly dominated strategies S 1 is a strictly dominated strategy for player 1 if it is not a best response to any strategy of player 2 8 S2 9 S1 6=S 1 P 1 (S 1; S 2 ) < P 1 (S 1 ; S 2 ) Iterated deletion of strictly dominated strategies find all strictly dominated strategies and delete them consider the reduced game without the strictly dominated strategies repeat the process Facility location: reduced form Firm 1 Firm 2 B C D E F Firm 2 B D Firm 1 C 4, 2 3, 3 E 3, 3 2, 4 C is a strictly dominant strategy for firm 1 (E is strictly dominated) and D is a strictly dominant strategy for firm 2 (B is strictly dominated) 16

17 Nash equilibrium and deletion of strictly dominated strategies - I Nash equilibrium in the original game is a Nash equilibrium in the reduced game Proof (Contradiction) Suppose that (S 1; S 2) is a Nash equilibrium and S 1 is a strictly dominated strategy Because (S 1; S 2) is a Nash equilibrium, for all S 1, P 1 (S 1 ; S 2 ) P 1(S 1 ; S 2 ) Because S 1 is a strictly dominated strategy, there is S 1 such that P 1 (S 1 ; S 2 ) < P 1(S 1 ; S 2 ) Nash equilibrium and deletion of strictly dominated strategies - II Nash equilibrium in the reduced game is a Nash equilibrium in the original game Proof Suppose that (S 1; S 2) is a Nash equilibrium in the reduced game and S 0 1 is a strictly dominated strategy in the original game Because (S 1; S 2) is a Nash equilibrium, for all S 1 6= S 0 1, P 1 (S 1 ; S 2 ) P 1(S 1 ; S 2 ) Because S 0 1 is a strictly dominated strategy, there is S 1 6= S 0 1 such that P 1 (S 0 1 ; S 2 ) < P 1(S 1 ; S 2 ) Therefore, there is S 1 6= S 0 1 such that P 1 (S 0 1 ; S 2 ) < P 1(S 1 ; S 2 ) P 1(S 1 ; S 2 ) 17

18 Traffic at equilibrium x/100 C 45 B 45 D x/100 4,000 drivers Each driver chooses either path CB or DB Travel time in links D e CB is 45 minutes Travel time in links C and DB is x/100 minutes, where x is the number of drivers using the link Payoff is the negative of the driver s travel time Nash equilibria No driver has a dominant strategy CB is the best response of a driver if all others opt for path DB DB is the best response of a driver if all others opt for path CB Nash equilibria any set of 2,000 drivers that choose path CB with the other 2,000 drivers choosing path DB the travel time of each driver is 65 minutes 18

19 Braess s paradox x/100 C 45 0 B 45 D x/100 Each of the 4,000 drivers has three choices path CB with travel time 45 + x/100 minutes path DB with travel time 45 + x/100 minutes path CDB with travel time 2x/100 Nash equilibrium Choosing path CDB is a strictly dominant strategy for each player 2x/100 < 45 + x/100 for all 0 x 4,000 Unique Nash equilibrium each player chooses path CDB travel time is 80 minutes; worse than travel time without link CD! 19

20 General case Network Travel time along link e T e (x) = a e x + b e, where x is the number of drivers traversing the link n drivers each with an origin and a destination each with an arbitrary choice of paths from the origin to the destination Questions Is there a Nash equilibrium? How much worse is the social cost of the equilibrium in relation to the optimal social cost? Traffic pattern and social cost Tra±c pattern: P = fp 1 ; : : : ; P n g Tra±c intensity on link e: x e = jfe 2 P j j 1 j ngj Travel time over link e: T e (x e ) Travel time of driver i: T i (P) = P e2p i T e (x e ) Social cost: S(P) = = nx T i (P) i=1 nx X i=1 e2p i T e (x e ) = X x e T e (x e ) e 20

21 Best response dynamics while there is a driver whose best response to the paths chosen by other drivers is an alternate path to the one it is taking, switch to the alternate path Tra±c pattern without driver i: P i = fp 1 ; : : : ; P n g fp i g Tra±c pattern with Pi 0 substituted for P i: P 0 = P i [ fpi 0g ( x i x e 1 if e 2 P i e = otherwise x 0 e = x e ( x i e + 1 if e 2 Pi 0 otherwise x i e Energy Energy: E(P) = X e = X e Xx e y=1 x i T e (y) ex T e (y) + X T e (x e ) e2p i y=1 = E(P i ) + T i (P) Best-response dynamics: T i (P 0 ) < T i (P) Energy decreases with best-response dynamics: E(P 0 ) < E(P) There is a traffic equilibrium 21

22 Energy and social cost - I e energy s social cost e = s = e = C s = x 5 e = s = C x 5 e = 5 s = 5 0 B 0 B e = s = e = 26 s = 28 5 x D e = s = e = s = e = 24 s = 28 5 x D e = s = Energy and social cost - II e energy s social cost e = s = C x 5 e = s = C x 5 0 B 0 B e = s = x D e = s = e = 5 s = 5 5 x D e = s = e = 23 s = 30 e = 21 s = 30 22

23 Energy and social cost - III e energy s social cost e = s = C x 5 0 B 5 x D e = 20 s = 32 e = s = Energy and social cost Travel time over link e: T e (y) = a e y + b e Xx e x e T e (x e ) a e y + b e y=1 (x e + 1)x e = a e x et e (x e ) + x e b e S(P) E(P) 1 2 S(P) 23

24 Social cost of equilibrium Tra±c pattern at optimal cost: P Tra±c pattern at equilibrium: P S(P ) E(P ) E(P) 1 2 S(P) social cost of equilibrium is at most twice the socially optimal cost Sequential games Players take an order in choosing their strategies Finite and deterministic strategies Non-cooperative Players seek to maximize their payoff at every stage of the game (sub-game perfection) Perfect information 24

25 Choosing regions for advertising Two firms trying to decide whether to focus their advertising on two possible regions and B Firm 1 chooses first If firm 2 follows firm 1 into the same region then firm 1 gets 2/3 of the market in that region firm 2 gets 1/3 of the market in that region If firm 2 chooses the other region then each firm gets the full market in that region The market for region is 12 and for region B is 6 Extensive form of the game Firm 2 B 4, 2 Firm 1 B 6, 12 B 12, 6 Firm 2 8, 4 25

26 Rollback equilibrium Firm 2 B 4, 2 Firm 2 B 4, 2 Firm 1 B 6, 12 Firm 1 B 6, 12 B 12, 6 B 12, 6 Firm 2 8, 4 Firm 2 8, 4 To look forward, reason backward Rollback equilibrium: firm 1 advertises in and, then, firm 2 advertises in B Normal form of the game Firm 2 if, if B if, B if B B if, if B B if, B if B Firm 1 8, 4 8, 4 12, 6 12, 6 B 6, 12 4, 2 6, 12 4, 2 dvertising in is a stritcly dominant strategy for firm 1 Firm 1 advertises in Firm 2 plays B if, if B or B if, B if B ; it advertises in B rollback equilibrium is a Nash equilibrium 26

27 Market entry game Entry of firm 1 in a market where firm 2 is already established If firm 1 decides not to enter the market, then firm 1 has a payoff of 0 and firm 2 has a payoff of 2 If firm 1 decides to enter the market, then if firm 2 cooperates, then both firms have a payoff of 1 if firm 1 retaliates, then both firms have a payoff of -1 Extensive form of the game Cooperate Firm 1 1, 1 Firm 2 Enter Retaliate -1, -1 Stay out 0, 2 Rollback equilibrium: firm 2 enters and, then, firm 2 cooperates 27

28 Normal form of the game Firm 2 Retaliate if enter Cooperate if enter Firm 1 Stay out 0, 2 0, 2 Enter -1, -1 1, 1 The rollback equilibrium (Enter, Cooperate if enter) is a Nash equilibrium (Stay out, Retaliate if enter) is also a Nash equilibrium irbus, Boeing, EU, and US irbus and Boeing it costs irbus 1000 ME to enter the markets Boeing is already in the markets monopoly in a market yields profits of 900 ME to the airline competition in a market yields a profit of 300 ME to the airline EU and US EU and the US profits whatever the respective airlines profit plus a 700 ME bonus if there is competition in the respective market Ordering of moves EU decides whether or not to pass protective legislation (PL) US then decides whether or not it passes PL finally, airbus decides whether or not to enter the markets 28

29 Computing payoffs: PL in both markets, irbus builds irbus Boeing EU US EU market US market entry competition Payoffs Computing payoffs: PL in EU only, irbus builds irbus Boeing EU US EU market US market entry competition Payoffs

30 Computing payoffs: No PL, irbus builds irbus Boeing EU US EU market US market entry competition Payoffs Computing payoffs: irbus does not build irbus Boeing EU US EU market US market entry competition Payoffs

31 Extensive form of the game EU PL US PL no PL irbus build not build build not build 900, 900, , 1800, , 1000, 200 0, 1800, 0 no PL irbus build 1300, 1300, -400 not build 0, 1800, 0 Rollback equilibrium EU PL US PL no PL irbus build not build build not build 900, 900, , 1800, , 1000, 200 0, 1800, 0 no PL irbus build 1300, 1300, -400 not build 0, 1800, 0 31

32 irbus moves before US EU PL build irbus not build US PL no PL PL no PL 900, 900, , 1000, 200 0, 1800, 0 0, 1800, 0 no PL irbus build 1300, 1300, -400 not build 0, 1800, 0 irbus moves first of all EU PL US PL no PL 900, 900, , 1000, 200 irbus build no PL 1300, 1300, -400 not build 0, 1800, 0 No first mover advantage 32

33 Mixed strategies Strategies are randomized over the pure strategies Payoffs become average payoffs Concept of best response still holds; best set of probabilities chosen by a player given the sets of probabilities chosen by the other players It can be shown that a Nash equilibrium always exists Matching cents mixed strategies You and your friend If coins match, you loose your cent to your friend If coins do not match, your friend looses her cent to you You choose Heads with probability p Your friend chooses Heads with probability q Payoffs If you play Heads, your average payoff is (1 2q) If you play Tails, your average payoff is ( q) nalogous reasoning for your friend 33

34 Matching cents: mixed Nash equilibrium There are no pure strategies that yield a Nash equilibrium What is your best response to strategy q by your friend? If (1 2q) > (-1 + 2q) you should play Heads If (1 2q) < (-1 + 2q) you should play Tails However, pure strategies do not have a Nash equilibrium For a Nash equilibrium we must have (1 2q)=(-1 + 2q), so that q = ½ (also p = ½) Mixed Nash equilibrium Equilibrium probability q for player 2 qp 1 (S 1 ; S 2 ) + (1 q)p 1 (S 1 ; ¹ S 2 ) = qp 1 ( ¹ S 1 ; S 2 ) + (1 q)p 1 ( ¹ S 1 ; ¹ S 2 ) Expected payo of player 1 is P E 1 (p; q) = p(qp 1 (S 1 ; S 2 ) + (1 q)p 1 (S 1 ; ¹ S 2 )) + (1 p)(qp 1 ( ¹ S 1 ; S 2 ) + (1 q)p 1 ( ¹ S 1 ; ¹ S 2 )) 34

35 Penalty-Kick Kicker and goalie Kicker can shoot left or right Goalie can dive left or right Payoffs fraction of scores when kicker shoots left and goalie dives left is 0.58 fraction of scores when kicker shoots left and goalie dives right is 0.95 fraction of scores when kicker shoots right and goalie dives left is 0.90 fraction of scores when kicker shoots right and goalie dives right is 0.70 Penalty-kick Kicker L (p) R (1-p) L (q) Goalie R (1-q) 0.58, , , , p is the probability of the kicker shooting left and q the probability goalie diving left No pure Nash equilibrium Mixed Nash equilibrium for q = 0.42 and p =

36 Unbalanced coordination game You PowerPoint (q) Your partner Keynote (1-q) PowerPoint (p) 1, 1 0, 0 Keynote (1-p) 0, 0 2, 2 Two pure Nash equilibria all the same One mixed Nash equilibrium at q = p = 2/3 36