Introduction to Game Theory

Transcription

1 Introduction to Game Theory Managing with Game Theory Hongying FEI

2 Poker Game ( 2 players) Each player is dealt randomly 3 cards Both of them order their cards as they want Cards at the corresponding position will be paired, each card with larger number than the counterpart will earn 1 point The player with more points wins.

3 3 Player 1 Player 2

4 3 Player 1 Player 2 Round 1 Round 2 Round 3 Winner Player Player

5 3 Player 1 Player 2 Round 1 Round 2 Round 3 Winner Player Player

6 Some Definitions Strategic game Situation in which players make strategic decisions that take into account each other s actions and responses; A strategic game consists of A set of players For each player, a set of actions (strategies) For each player, preferences over the set of action profiles (ordinal expression: payoff functions) The players choose their actions simultaneously once and for all!

7 What is Game Theory Game theory is a formal way to analyze interaction among a group of rational agents who behave strategically. More formally, it is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Roger B. Myerson (1991). Game Theory: Analysis of conflict, Harvard University Press, P. 1. Chapter-Preview links, pp. vii-xi

8 How will the suspects act? Prisoners Dilemma Two suspects in a major crime are held in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they both stay quiet, each will be convicted of the minor offense and spend one year in prison. If one and only one of them finks, she will be freed and used as a witness against the other, who will spend four years in prison. If they both fink, each will spend three years in prison.

9 Suspect 1 Actions Fink? Quiet? Prisoners Dilemma Fink? Players Quiet? Fink Quiet Suspect 2 Fink Quiet -3, -3 0, -4-4, 0-1,-1 Preferences of suspect 1: u 1 Fink, Quiet = 0 >u 1 Quiet, Quiet = 1 > u 1 Fink, Fink = 3 > u 1 Fink, Fink = -4 Preferences of suspect 2: u 2 Fink, Quiet = 0 >u 2 Quiet, Quiet = 1 > u 2 Fink, Fink = 3 > u 2 Fink, Fink = -4

10 Prisoners Dilemma Fink Suspect 2 Quiet Suspect 1 Fink Quiet For suspect A, the strategy stay quiet is strictly dominated by the strategy betray! -3,-3 0,-4-4,0-1,-1

11 Definition (strict domination) In a strategic game with ordinal preferences, player i s action a i strictly dominates her action a Prisoners Dilemma i if u i (a i ", a i ) > u i (a i, a i ) for every list a i of the other players action, where u i is a payoff function that represents player i s preferences, and a i is strictly dominated by action a i ". Fink Suspect 2 Quiet Lesson 1: Do Fink not play a strictly -3,-3 dominated 0,-4 strategy. Suspect 1 Quiet -4,0-1,-1 For suspect A, the strategy stay quiet is strictly dominated by the strategy betray!

12 Lesson 2: Rational choices can lead to bad outcomes. Prisoners Dilemma Fink Suspect 2 Quiet Suspect 1 Fink Quiet -3,-3 0,-4-4,0-1,-1

13 Symmetric games A two-player strategic game with ordinal preferences is symmetric if the players set of actions are the same and the players preferences are represented by payoff functions u 1 and u 2 for which u 1 a 1, a 2 = u 2 (a 2, a 1 ) for every action pair(a 1, a 2 ).

14 Prisoners Dilemma Best available choice of Suspect 1 Best available choice of Suspect 2 Fink Suspect 2 Quiet Suspect 1 Fink Quiet -3,-3 0,-4-4,0-1,-1 a = a 1, a 2 = ( 3, 3)

15 A Nash equilibrium is an action profile a with the property that no player i can do better by choosing an action different from a i, given that every other Prisoners Dilemma player j adhere to a j. Best available choice of Suspect 1 Best available choice of Suspect 2 Fink Suspect 2 Quiet Suspect 1 Fink Quiet -3,-3 0,-4-4,0-1,-1 a = a 1, a 2 = ( 3, 3)

16 Symmetric Nash Equilibrium An action profile a in a strategic game with ordinal preferences in which each player has the same set of actions is a symmetric Nash equilibrium if it is a Nash equilibrium and a i is the same for every player i.

17 Definition (Nash equilibrium of strategic game with ordinal preferences) The action profile a in s strategic game with ordinal preferences is a Nash equilibrium if, for every player i and every action a i of player i, a is at least as good according to player i s preferences as the action profile (a i, a i ) in which player i choose a i while every other player j choose a j. Equivalently, for every player i, u i a u i (a i, a i ) for every action a i of player i, Where u i is a payoff function that represents player i s preferences.

18 Nash Equilibrium Idea is to identify the more likely outcomes of a game Assumes players play rationally (i.e., as they should) Know the game (agents, actions, payoffs) Can be significantly generalized (e.g., agents know none of these) Nash equilibrium: every player chooses the best response Finding the Nash Equilibria Pick a player Find his/her best response to each choice of all the others Move to the next player, and the next Any box in which all players are at the best response, i.e. intersect of best responses, is Nash Equilibrium

19 Exercise 1: Roommate game Roommate 2 3 hours 6 hours 9 hours 3 hours 1, 1 3, - 4 2, - 8 Roommate 1 6 hours - 4, 3 4, 4 6, hours - 8, 2-2, 6 3, 3

20 Roommate Game: How much will we clean? Two Nash Equilibria Roommate B 3 hours 6 hours 9 hours 3 hours 1, 1 3, - 4 2, - 8 Roommate A 6 hours - 4, 3 4, 4 6, hours - 8, 2-2, 6 3, 3

21 Game of Boxed Pigs First Arrival Little Pig s gain Big Pig s gain Little Pig 4 6 Big pig 1 9 simultaneously 3 7 Energy spent for pressing the button = 2 units When button is pressed, food given = 10 units

22 Lesson 3: Put yourself in other people's shoes. Decisions, decisions... First Arrival Little Pig s gain Big Pig s gain Little Pig 4 6 Big pig 1 9 simultaneously 3 7

23 Exercise : Hannibal s Selection In 218 BC., Hannibal the general set out, with a relatively small army of select troops, to invade Italy. Here are two routes towards Rome, the normal route (the easy pass) and a little-known route over the Alps (the hard pass). If the invader chooses the hard pass, he will lose one battalion of his army simply in getting over the mountains. If he meets the Italian defender, whichever pass he chooses, he ll lose another battalion. Question: For the Italian defender, which pass to defend?

24 Suspect 1 Lesson 4: You can't get what you want, till you know what you want. Rethink the prisoners dilemma Indignant angel Fink Quiet Suspect 2 Fink Quiet -2, -6-3, -3 0, -4-4, 0-1,-1-6, -2 No Nash equilibrium!

25 Normative admonition: Managers should choose best responses to the expected actions of their employees This has strong implications! Everyone s actions depend upon what they think others will do

26 Suspect 1 Prisoners Dilemma with indignant angels Suspect 2 Fink Quiet Fink -3,-3-2,-6 Quiet -6,-2-1,-1 r 1-r Randomize your strategies A probability Specifies that an action be chosen randomly from the set of pure strategies with some specific probabilities. A mixed strategy of a player is a probability distribution over the player s (pure) strategies. q 1-q

27 Suspect 1 Prisoners Dilemma with indignant angels Expected payoffs 2 q 1 5q Fink Suspect 2 Quiet Fink -3,-3-2,-6 Quiet -6,-2-1,-1 Suspect 1 s expected payoffs q 1-q If suspect 1 chooses Fink : 3q 2 1 q = 2 q r 1-r If suspect 1 chooses Quiet : 6q 1 q = 1 5q

28 Suspect 1 Prisoners Dilemma with indignant angels Expected payoffs 2 q 1 5q Suspect 1 s best response B 1 q : For q < 0.25, Quiet (r = 0) For q > 0.25, Fink (r = 1) Fink Suspect 2 Quiet Fink -3,-3-2,-6 Quiet -6,-2-1,-1 For q = 0.25, indifferent (0 r 1) q r 1 1-q r 1-r 1/4 1 q

29 Suspect 1 Prisoners Dilemma with indignant angels Expected payoffs Fink Suspect 2 Quiet Fink -3,-3-2,-6 Quiet -6,-2-1,-1 q 2 r 1-q 1 5r r 1-r Suspect 2 s expected payoffs If suspect 2 chooses Fink : 3r 2 1 r = 2 r If suspect 2 chooses Quiet : 6r 1 r = 1 5r

30 Suspect 1 r 1 1/4 Prisoners Dilemma with indignant angels 1 q Fink Suspect 2 Quiet Fink -3,-3-2,-6 Quiet -6,-2-1,-1 Expected payoffs q 2 r 1-q 1 5r r 1-r Suspect 2 s best response B 2 r : For r < 0.25, Fink (q = 1) For r> 0.25, Quiet (q = 0) For r = 0.25, indifferent (0 q 1)

31 Prisoners Dilemma with indignant angels r 1 Mixed strategy Nash equilibrium 1/4 1/2 1 q

32 Expected payoffs: 2 players each with two pure strategies

33 Expected payoffs: 2 players each with two pure strategies

34 Exercise 2: Calculate the expected payoffs Play 2 H(0.3) T(0.7) H(0.4) -1,1 1,-1 Player 1 T(0.6) 1,-1-1, 1

35 Exercise 3: Calculate the expected payoffs Play 2 Player 1

36 Mixed Strategy (Nash) Equilibrium Mixed strategy equilibrium A probability distribution for each player The distributions are mutual best response to one another in the sense of expected payoffs It is a stochastic state

37 Mixed strategy equilibrium : 2-player each with two pure strategies

38 Mixed strategy equilibrium : 2-player each with two pure strategies Steps to find mixed strategy equilibrium in 2- player each with two pure strategies: Find the best response correspondence for player 1, given player 2 s mixed strategy; Find the best response correspondence for player 2, given player 2 s mixed strategy; Use the best response correspondences to determine mixed strategy Nash equilibria

39 John Forbes Nash John Forbes Nash, Jr. (June 13, 1928 May ) American mathematician who made fundamental contributions to GAME THEORY. Shared the 1994 Nobel Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

40 Theorem (Nash, 1950) Existence of Nash equilibrium: In the n-player normal-formal game G = {s 1, s n ; u 1,, u n },if n is finite and s i is finite for every i, then there exists at least one Nash equilibrium, possibly involving mixed strategies. Nash, John F Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences, 36(1):

41 Recap Lesson 1 Do not play a strictly dominated strategy. Lesson 2 Rational choices can lead to bad outcomes. Lesson 3 Put yourself in other people's shoes. Lesson 4 You can't get what you want, till you know what you want.

42 Question & Answer Thanks! Hongying FEI