TPPE24 Ekonomisk Analys:

Transcription

1 TPPE24 Ekonomisk Analys: Besluts- och Finansiell i Metodik Lecture 5 Game theory (Spelteori) - description of games and two-person zero-sum games 1

2 Contents 1. A description of the game 2. Two-person zero-sum games 3. Solution space for a game 4. Prisoner s dilemma 5. Other aspects of a game 2

3 Battle of the Bismarck sea The battle of the Bismarck sea, is set in the South Pacific in General Imamura has been ordered to transport Japanese troop across the Bismarck Sea to New Guinea, and General Kenney wants to bomb the troop transports. Imamura must choose between a shorter northern route (2 days) or a longer southern route (3 days) to New Guinea, and Kenney must decide where to send his planes to look for the Japanese. If the Kenney sends his planes to the wrong route he can recall them but the number of days of bombing is reduced (by 1 day). - Game theory 3

4 Description of the game Player: individuals who make decisions. Each player s goal is to maximise his utility by choice of action. Action: the choice that a player can make. Payoff: the measure of preference after the outcome of the game has been realised. 4

5 Normal form analysis (U ij, V ij ) Player B B 1 B 2... Player A A 1 U 11,VV 11 U 12,VV U 1j,VV 1j B j A 2... U 21,V U 22,V U 2j,V 2j A i U i1,v i1 U i2,v i2... U ij,v ij Player A chooses among alternatives A 1, A 2,...,A i. Player B chooses among alternatives B 1, B 2,...,B j. Player A s payoff is U ij when A i and B j are implemented. Player B s payoff is V ij when A i and B j are implemented. 5

6 Bismarck sea 1943 Player: Action: General Kenney (K) and General Imamura (I) North or South Kenney Imamura (K, I) North South North 2, -2 2, -2 South 1,-11 3, -3 6

7 Game theory - interactive decision analysis Decision analysis You are self-interested and selfish Game theory So is everyone else Never assume that your opponents behavior is fixed. Predict their reaction to your behavior. 7

8 Solutions A solution concept represents our predication of what will happened in a game or what is considered fair. A solution defined by a particular solution concept is called equilibrium. The outcome from solution concept should be individual rational and collective rational in principle. 8

9 Constant-sum sum games vs non constant-sumsum games A constant-sum game is a game in which the sum of payoffs of the players is a constant whatever the strategies they choose. Zero-sum game is a special case of constant-sum games. Zero-sum game is more interesting for game theorists. Nonzero sum game is more common in the economic system. 9

10 Two-person, zero-sum game This game involves two players. One player wins whatever the other one losses. The sum of their net winnings is zero. Let U ij Player A s payoff and -V ij Player B s payoff (U ij, V ij ) Player A A 1 Player B B 1 B 2 B A A In the equilibrium, the payoff to player A is referred as the value of the game. A game has a value of zero is called a fair game. 10

11 Minimax criterion (1) (U ij, V ij ) Player A A 1 Player B B 1 B 2 B 3 A A Minimum Maxmini value Maximum Minimax value If Minimax=Maxmini, we obtain a saddle point. This strategy is an equilibrium and also called stable solution. 11

12 Minimax criterion (2) Player B (U ij,, V ij) ) B 1 B 2 B 3 Player A A A 2 A Minimum Maximum Is it possible to have multiple saddle points? 12

13 Bismarck sea 1943 Kenney Imamura (K, I) North South North 2 2 South 1 3 Minimum 2 1 Maximum 2 3 Minimax=Maxmini = 2 North-North is the outcome in

14 Minimax criterion (3) (U ij,, V ij) ) Player B B 1 B 2 B 3 Player A A A 2 A Minimum -2 Maxmini value -3-4 Maximum Minimax value If MinimaxMaxmini, there is no saddle point 14

15 Minimax criterion (4) Player B (U ij,, V ij) ) B 1 B 2 B 3 Player A A A 2 A Minimum Maximum and thus there is only unstable solution. 15

16 Graphical solution procedure We have the following payoff table Player B (U ij, V ij ) B 1 B 2 q 1-q Player A A 1 p A 2 1-p Player A: The expected payoff by choosing A 1 =50q+80(1-q)=80-30q The expected payoff by choosing A 2 =90q+20(1-q)=20+70q Player B: The expected payoff by choosing B 1 =50p+90(1-p)=90-40p The expected payoff by choosing B 2 =80q+20(1-p)=20+60p

17 Graphical solution procedure 1. Player A may choose either A 1 or A 2 u, player A s payoff 2. However, a mixed strategy could improve the payoff. 3. Since every gain of player A is the A 2 loss of player B, Minimax A 1 4. Play B will battle in the way to minimise i i A payoff ffby changing the value of q. q=0 q=1 5. q=0.6 and u*=62 17

18 Graphical solution procedure v 1. Using the same logic, we have Maxmini B 2 B 1 p=0.7 and v*=62 2. u*=v*= * * value of fthe game 3. Minimax = Maxmini p=0 p=1 18

19 Minimax theorem In two-person zero-sum game, there exists a Minimax solution u*, which is identical with the Maxmini solution v*. - von Neumann (1928) u* Maxu( p) p* Uq* v* minv( q) p* Uq* Or in every two-person zero-sum game, there exists an equilibrium in mixed strategies. 19

20 Minimax theorem (U ij, V ij ) B 1 Column... B j... B n q 1... q j... q n Row A 1 p 1 U U 1j U 1n A i p i U i1... U ij U in A m p U m m1... U mj U mn Row palyer max z u s. t. u u p T 11 j p, j Column palyer min s. t. p 0 q 0 q T w v v u q, 11 i i 20

21 Linear programming approach Linear programming approach Primal problem Dual problem max b Ax x c s t z T min c y A y b T T s t w p p 0.. x b Ax s t 0.. y c y A s t 1) (0 max p u 1) (0 min q v w p U s t u T q U v s t v p u q v 21

22 Non constant sum game Very often, we have non constant sum game Player B (U ij, V ij ) B 1 B 2 q 1-q Player A A 1 p 3, 3-3, 4 A 2 1-p 4, -3-1,-2 22

23 Payoff diagram and payoff set A 1 B 2 v Pure A 1 A 1 B 1 1. Give the payoffs for different strategy combination. Pure B 2 Payoff set (utdelningsrum) Pure B 1 u 2. Link the points with pure strategies. A 2 B 2 Pure A 2 A 2 B 1 23

24 Pareto optimal A 1 B 2 Pure B 2 v Pure A 1 A 1 B 1 Pure B 1 u At the Pareto optimal, no player can increase its own payoff without ih decreasing the other player s payoffs. Pareto optimum: Pure B 1 and Pure A 1 A 2 B 2 Pure A 2 A 2 B 1 24

25 Security level - säkerhetsnivå u Security level: l highest h expected payoff regardless of other player s strategies. (best from worst) v B 1 A 1 p q Maxmini=-2 Maxmini=-1 B 2 A 2 p=0 Play A security level p=1 q=0 Play B security level q=1 25

26 Agreement level - avtalsmängden Agreement level: l Pareto optimum which h is larger than security level A 1 B 2 v Pure A 1 A 1 B 1 Pure B 2 Pure B 1 u Agreement level B s Bs security level A 2 B 2 Pure A 2 A s As security level A 2 B 1 26

27 Threaten level - hotnivå v Threaten strategy: a strategy used to minimise i i other player s payoff. (worst from best) u B A B 2 A 2 Minimax=-2 B 1 p A 1 q Minimax=-1 p=0 p=1 q=0 q=1 Play A threaten strategy and threaten level Play B threaten strategy and threaten level 27

28 Prisoner s s dilemma a non zero sum game Player: Action: Information: Payoff: Row (R) and column (C) Deny or Confess Players make decision simultaneously Row (R, C) Deny Confess Column Deny Confess -1, ,0 0 0,-10-8,-8 28

29 Prisoner s s dilemma 1. Individually best strategies lead to an outcome that is not efficient. Efficient - Pareto optimum: there is no other combination of actions or strategies that would make at least one player better off without making another player worse off. 2. Not a game of pure conflict. There is an opportunity for communication, cooperation, or trust. t 29

30 Prisoner s s dilemma - comments This game seems perverse and unrealistic to many people. But according to some prosecutors, it is a standard crime-fighting tools. 30

31 Prisoner s s dilemma - applications Whenever you observe individuals in a conflict that hurts them all, your first thought should be of the Prisoner ss dilemma. Applications include -Arms control, build up nuclear arsenal or not - Oligopoly pricing, price war or not - International ti ltrading, no tariff or impose tariff - Open access resourcing, restrained fishing or unrestrained fishing 31

32 Other aspects in a game (1) Time-frame for decisions: Are the moves simultaneous? Sequential? Nature of the conflict and interaction: Are players interests in conflict? Cooperation? Will players interact once, or repeatedly? Rationality assumption: Players aim to maximize their payoff Players are perfect calculators Common knowledge assumption: Each player knows the rules of the game Each player knows that each player knows the rules 32

33 Other aspects in a game (2) Nature: a pseudo-player player who takes random actions at specified points in the game with specified probabilities. Information: a knowledge about other players action, states of the nature at a particular point of time line of the game. Information: Is there full information? Advantages? 33

34 Strategic games vs sequential game Strategic games are also known as static, simultaneous-move, normal form games. Components: players, actions, payoffs Sequential games are also known as dynamic, extensive form games. Components: players, actions, payoffs, nature, game tree, information sets 34

35 Game applications inside economics 1. Buyer and seller negotiating over a price 2. Employer and employee interaction (negotiation, creating incentives) 3. Firm and its competitors 4. Auctioneer and bidders... 35

36 Game applications outside economics 1. Presidential candidates 2. Congress and the President 3. Opposing generals at War 4. Contests in the animal kingdom 5. Two football teams in their match 36