Perloff (2014, 3e, GE), Chap. 13; Nicholson & Snyder (2012, 11e), Chap

Transcription

1 3. Part 5B. Game Theory EQM Concepts for Static Games 靜態賽局的均衡概念 Dominant Strategies Iterated Elimination of Strictly Dominated Strategies Nash Equilibrium Mixed Strategies Nash Equilibrium for Game with Continuum of Actions Perloff (2014, 3e, GE), Chap. 13; Nicholson & Snyder (2012, 11e), Chap

2 Static Games In a static game each player acts simultaneously, only once and has complete information about the payoff functions but imperfect information about rivals moves. e.g., employer negotiations with a potential new employee teenagers playing chicken in cars street vendors choice of locations and prices 2

3 Predicting a Game s Outcome Rational players will avoid strategies that are dominated by other strategies. In fact, we can precisely predict the outcome of any game in which every player has a dominant strategy. 3

4 Dominant Strategies Dominant Strategy ( 優勢策略 ) A strategy that yields a higher payoff no matter what the other players in a game choose. 不管對手採取何種策略, 當參與者採取某一策略時, 其報償皆大於所有其他策略, 則此策略謂之該參與者的優勢策略優勢策略 4

5 Definition: Strictly Dominant Strategies ( 絕對優勢策略 ) A strategy, ŝ i, for player i is strictly dominant if u ( ˆ i si, s i) u1( si, s i) s ˆ i si,( si, s i) S i.e., ŝ i is a strictly dominant strategy in S for player i. Definition: Dominant Strategies ( 優勢策略 ) A strategy, ŝ i, for player i is strictly dominant if u ( ˆ i si, s i) u1( si, s i) ( si, s i) S i.e., ŝ i is a dominant strategy in S for player i. 5

6 Example: 汽車公司的促銷策略 Both firms have dominant strategies. (0 利率, 免頭期款 ) is the DS EQM. 6

7 Table: Quantity-Setting Game American Airlines q U \ q A United , , Airline , , 4.6 Both firms have dominant strategies. (q U =64 64, q A = 64) is the DS EQM. Players choose strategies that don t maximize joint profits. Called a prisoners dilemma game; all players have dominant strategies that lead to a profit that t is less than if they cooperated. Perloff (2014, 3e, GE), Table 13.1, p

8 Example: Cournot Model Firm 1 chooses q 1 Firm 2 chooses q , 18 15, , 15 16, 16 No mater what firm 2 chooses, firm 1 will choose q 1 = 4, because When firm 2 choose q 2 =3 3, u 1 (4,3) = 20 > u 1 (3,3) 3) = 18 When firm 2 choose q 2 = 4, u 1 (4,4) = 16 > u 1 (3,4) = 15 Likewise, i firm 2 will choose q 2 = 4. Both firms have a dominant strategy, q = 4. Therefore, (q 1, q 2 ) = (4, 4) is the DS EQM. 8

9 Example: Battle of the Sexes ( 兩性戰爭 ) A wife and husband may either go to the ballet or to a boxing match: Both prefer spending time together The wife prefers ballet and the husband prefers boxing Player 1 (Wife) Player 2 (Husband) Ballet Boxing Ballet 2, 1 0, 0 Boxing 0, 0 1, 2 There is no dominant strategy for both. No DS EQM. Nicholson & Snyder (2012, 11e), Figure 8.3, p

10 Comments Dominant Strategies EQM The only assumption under DS EQM is the Optimization Behavior. DS EQM may not be the maximum joint profit outcome. market failure A firm with a strictly dominant strategy does not need to know the choices and payoffs of other firms. The order of moves does not matter. More importantly, not every game has a DS EQM. ( 並不是所有賽局都存在優勢策略均衡解 ) 10

11 Iterated Elimination of Strictly Dominated Strategies 逐步消去絕對劣勢策略 Dominated strategy ( 劣勢策略 ) Any other strategy available to a player who has a dominant strategy. 對參賽者來說, 除了優勢策略之外的其他策略 11

12 Definition: Strictly Dominated Strategies ( 絕對劣勢策略 ) A strategy, s i, for player i is strictly dominated if s s, ( s, s ) S i i i i u ( s, s ) u ( s, s ) i i i 1 i i 12

13 Example: Cournot Model Firm 1 chooses q 1 Firm 2 chooses q , 0 0, 20 0, , 0 900, , , 0 675, 18 0, 0 No dominant strategy for both firms but there exist dominated strategies. Continuously eliminate dominated strategies gives the IEDS EQM, (q 1, q 2 ) = (30, 30). 13

14 Example: The illustration of the case that order matters in IEDS Start with player 1 Player 1 The IEDS EQM is (U, M). Start with player 2 Player 1 Player 2 L M R U 10, 0 5, 1 4, -100 D 10, 100 5, 0 0, -50 Player 2 L M R U 10,0 0 5,1 4, -100 D 10, 100 5, 0 0, -50 The most likely l IEDS EQMs is (D, L) by the assumption of complete information. 14

15 Comments on IDES EQM: The assumptions under IEDS EQM are the common knowledge of the structure of the game rationality However, the order that who play first may lead to different EQM outcome. the order matters in IEDS 15

16 Nash Equilibrium John Forbes Nash, Jr. (born June 13, 1928) An American mathematician 1958 Schizophrenia ( 精神分裂 ) 1994 Nobel Prize in Economics 16

17 Addition Assumption on Game Structure Common knowledge of the beliefs of what his rivals will be playing should be correct. Definition: Nash Equilibrium (NE, 納許均衡 ) A strategy, s *, is a NE if i & s i S i u s s u s s * i ( i, i ) i ( i, i ) NE is the best choice (response) for each player given the other players equilibrium strategies. NE once achieved, there is no incentives for players to alter their behavior. 17

18 Definition: Nash Equilibrium (NE, 納許均衡 ) A NE is a strategy profile s * such that s * i is a best response to other players equilibrium i strategies, t s * -i s * i BR i (s * -i) where s * i is the best response for player i to rivals strategies s -i, denoted s i BR i (s -i ) if s i S i u i (s * i, s -i ) u i (s i, s -i ) Example: A 2-Player Game (s * * 1, s 2)isaNEif 2 if u 1 (s * 1, s * 2) u 1 (s 1, s * 2) for all s 1 S 1 u 2 (s * 2, s * 1) u 2 (s 2, s * 1) for all s 2 S 2 18

19 Example: Prisoners Dilemma Suspect 1 Suspect 2 Fink Silent Fink 1, 1 3, 0 Silent 0, 3 2, 2 Finking is player 1 s best response to player 2 s finking, the same logic applies for player 2. Therefore, the NE is (Fink, Fink) Nicholson & Snyder (2012, 11e), Figure 8.1, p

20 Dominant Strategies & NE A strategy that is a best response to any strategy the other players might choose Finking is a dominant strategy for both players s * i BR i (s -i ) s -i When a dominant strategy exists, it is the unique NE. 20

21 Table: Dominant Strategy Equilibrium & NE Every dominant strategy equilibrium is a Nash equilibrium. Perloff (2014, 3e, GE), Table 13.1, p

22 Table: Best Responses & NE In a game without dominant strategies, calculate best responses to determine Nash equilibrium. Perloff (2014, 3e, GE), Table 13.2, p

23 Table: Failure to Maximize Joint Payoffs & Cooperation (a) Firms don t cooperate in this game and the sum of firms profits is not maximized in the NE. (b) If advertising by either firm attracts new customers to the market, then NE does maximize joint profit. Perloff (2014, 3e, GE), Table 13.3, p

24 Table: Multiple Nash Equilibria Many oligopoly games have more than one Nash equilibrium. Perloff (2014, 3e, GE), Table 13.4, p

25 Example: Battle of the Sexes ( 兩性戰爭 ) Player 1 (Wife) Player 2 (Husband) Ballet Boxing Ballet 2, 1 0, 0 Boxing 0, 0 1, 2 There are 2 NEs. (Ballet, Ballet) (Boxing, Boxing) There is no dominant strategy. No DS EQM Nicholson & Snyder (2012, 11e), Figure 8.3, p

26 Example: Rock, Paper, Scissors Two players simultaneously display one of three hand signals: rock breaks scissors scissors cut paper paper covers rock Player 2 Rock Paper Scissors Player 1 Rock 0, 0-1, 1 1, -1 Paper 1, -1 0, 0-1, 1 Scissors -1, 1 1, -1 0, 0 None of the strategies is a NE. Nicholson & Snyder (2012, 11e), Figure 8.6, p

27 Comments on NE A NE is not always present in two-player (pure- strategy) games. It is unclear how a player can choose a best- response strategy before knowing how rivals will play. ANE may not tbe optimum from the viewpoint i of society. There may be multiple NEs. There may have no (pure strategy) NE. 27

28 Mixed Strategies 混和策略 Pure Strategy: Players choose one action with certainty. Mixed Strategy: Players randomly select from several possible actions. Reasons for Studying Mixed Strategies Some games have no NE in pure strategies but will have one in mixed strategies Strategies involving randomization are familiar and natural in certain settings. It is possible to purify mixed strategies. 28

29 Mixed Strategies Suppose that player i has a set of M possible actions, A = 1 i, a m i,,a M i {a i i i} i } A mixed strategy is a probability distribution over the M actions, s =( i 1 i,, m i,, M i) ) where 0 m i 1 and 1 i+ + m i+ + M i = 1 A pure strategy is a special case of a mixed strategy; i.e., only one action is played with positive probability. Mixed strategies that involve two or more actions being gplayed with positive probability are called strictly mixed strategies. 29

30 EQM Concept for Mixed-Strategy Games A player randomizes over only those actions among which he or she is indifferent. One player s indifference condition pins down the other player s mixed strategy. 30

31 Example: Battle of the Sexes ( 兩性戰爭 ) Player 1 (Wife) Player 2 (Husband) Ballet (q) Boxing (1-q) Ballet (p) 2, 1 0, 0 Boxing (1-p) 0, 0 1, 2 The wife plays ballet with probability p and the husband with probability q. Wife maximizes her expected utilities by choosing the possibility, p. Husband maximizes his expected utilities by choosing the possibility, q. Nicholson & Snyder (2012, 11e), Figure 8.3, p

32 Wife s Choice max EU ( p, q) 2 p q 0 p (1 q) { p} W 0 (1 p ) q 1 (1 p ) (1 q ) 3 pq 1 p q EU W 3q 1 p p 1 EU * q W 0 p 1 WifeplaysBallet Ballet. 3 p 1 EU W * q 0 p 0 Wife plays Boxing. 3 p 1 EU q W 0 No difference. 3 p 32

33 p Husband s Choice max EU ( p, q) 1 p q 0 p (1 q) { q} H 0(1 p ) q 2(1 p )(1 q ) 3 pq 2 2 p 2 q EU H 3p 2 q q 2 EU * H 0 q 1 Husband plays Ballet. 3 q 2 EU H * p 0 q 0 Husband plays Boxing. 3 q 2 EU p H 0 No difference. 3 q 33

34 Figure: Expected Payoffs in the Battle of the Sexes q 1 2/3 Husband s best response (BR ) There are 2 pure strategy NEs. (BR 2 ) (Ballet,, Ballet) ) = (p = 1, q = 1) (Boxing, Boxing) = (p = 0, q = 0) There is 1 mixed strategy NE. (p = 2/3, q = 1/3) Mixed-Strategy NE 1/3 Wife s best response (BR 1 ) 1/3 2/3 1 p Nicholson & Snyder (2012, 11e), Figure 8.7, p

35 Example: Battle of the Sexes ( 兩性戰爭 ) Player 1 (Wife) Player 2 (Husband) Ballet (q) Boxing (1-q) Ballet (p) 2, 1 0, 0 Boxing (1-p) 0, 0 1, 2 The wife plays ballet with probability p and the husband with probability q. There are 2 pure strategy NEs. (Ballet, Ballet) = (p = 1, q = 1) (Boxing, Boxing) = (p =0 0, q =0) There is one mixed strategy NE. (p = 2/3, q = 1/3) Nicholson & Snyder (2012, 11e), Figure 8.3, p

36 The Existence Theorem (Nash) In all finite games, there is at least one NE. The theorem does not guarantee the existence of a pure-strategy NE. It does guarantee that, t if a pure-strategy t NE does not exist, a mixed-strategy NE does. 36

37 Nash Equilibrium for Game with Continuum of Actions Continuum of Actions Some settings are more realistically modeled via a continuous range of actions. e.g., quantity or price. Using calculus to solve for NEs makes it possible to analyze how the equilibrium actions vary with changes in underlying parameters. 37

38 Application: Tragedy of the Commons The Tragedy of the Commons describes the overuse that arises when scarce resources are treated as common property. 2 herders decide how many sheep to graze on the village commons. The commons is quite small and can rapidly succumb to overgrazing. Nicholson & Snyder (2012, 11e), Example 8.5, pp

39 Let q i = the number of sheep chosen by herder i Suppose that the per-sheep value of grazing on the commons is v(q 1, q 2 ) = 120 (q 1 + q 2 ) The normal form is a listing of the herders payoff functions u 1 (q 1, q 2 ) = q 1 v(q 1, q 2 ) = q 1 (120 q 1 q 2 ) u 2 (q 1, q 2 ) = q 2 v(q 1, q 2 ) = q 2 (120 q 1 q 2 ) 39

40 To solve for the NE, we solve herder 1 s maximization problem and get his best-response function. q2 q 1 60 BR1 q2 2 q1 Similarly, q 2 60 BR2 q1 2 The NE will satisfy both best-response functions simultaneously. 1 q1 q * * q 40, q u * * 1 u2 40 ( ) 1,

41 Figure: Best-Response Functions in the Tragedy of the Commons S B 120 A s best-response function NE B sbest-response Bs function S A Nicholson & Snyder (2012, 11e), Figure 8.8, p

42 Suppose the per-sheep value of grazing rises for herder 1 It would result in more sheep for herder 1 and fewer for herder 2. The NE is not the best use of the commons. If both herders grazed 30 sheep each, their payoffs would rise. u * * 1 u2 30 ( ) 1,800 Solving a joint-maximization problem will lead to the higher h payoffs. 42