Lecture 3: Nash Equilibrium

Transcription

1 Microeconomics I: Game Theory Lecture 3: Nash Equilibrium (see Osborne, 2009, Sect ) Dr. Michael Trost Department of Applied Microeconomics November 8, 2013 Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 1 / 38

2 Strategic games and Nash equilibrium Our study of games starts with the study of the simplest class of games. The so called class of strategic games. After introducing the class of strategic games we present the most prominent solution concept that is applied on such games. This solution concept has been devised by John F. Nash and is named after its inventor, the Nash equilibrium concept. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 2 / 38

3 Strategic game A strategic game is a games in which each player chooses her action once and for all, the players choose their actions simultaneously in the sense that no player is informed of the actions chosen by the other players, when she chooses her action. Strategic games are also referred to as simultaneous-move games. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 3 / 38

4 Strategic game Definition 3.1 (Strategic game) A strategic game Γ := (I, (A i ) i I, ( i ) i I ) consists of a finite set of players I, for each player i I, a set of actions A i for each player i I, a preference relation i on the set of action profiles A := i I A i which is representable by some utility function U i Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 4 / 38

5 Action and action profile Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action a i of player i is an element of player i s action set A i. An action profile a:=(a i)i I is a list specifying the actions of every player. It is an element of the Cartesian product A := i I A i of the players action sets. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 5 / 38

6 Preferences on action profiles NOTE: The preferences of the decision makers in a strategic game are defined on the set of all action profiles. This captures the interaction between the decision makers. Each player is affected by the actions of all players, not only by her own action. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 6 / 38

7 Other players action Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). Let a A be an action profile and a i A i an action of player i. List a i := (a j)j I\{i} denotes the actions of all players except i. Cartesian product A i := lists of action. j I\{i}A j is the set of all such Action profile (a i, a i) denotes the action profile in which player i chooses action a i and each player j different to i chooses action a j. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 7 / 38

8 Subclasses of strategic games A strategic game Γ := (I, (A i ) i I, ( i ) i I ) is called finite if the action set A i is finite for every player i I. a two-player strategic game if only two players participate in the game (i.e., #I = 2). Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 8 / 38

9 Symmetric two-player strategic games A two-player strategic game Γ := ( ) {k, l}, (A i ) i {k,l}, ( i ) i {k,l} is called symmetric if the actions sets and the preferences of the players are identical, i.e., A k = A l holds and (a k, a l ) k (b k, b l ) if and only if (a k, a l ) l (b k, b l ) is satisfied for every action profiles (a k, a l ), (b k, b l ) A k A l. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 9 / 38

10 Competitive two-player strategic games A two-player strategic game Γ := ( ) {k, l}, (A i ) i {k,l}, ( i ) i {k,l} is called competitive (or alternatively, zero-sum game) if the preferences of the players are opposite, i.e., (a k, a l ) k (b k, b l ) if and only if (b k, b l ) l (a k, a l ) is satisfied for every action profiles (a k, a l ), (b k, b l ) A k A l. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 10 / 38

11 Example: Prisoner s Dilemma The two suspects A and B in a major crime are questioned separately. 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 informer against the other ( finks ). If both stay quiet, each must spend 1 year in prison. If one and only one finks, then the one who finks is freed and the other one spends 4 years in prison. If both fink, each of them will spend 3 years in prison. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 11 / 38

12 The strategic game of Prisoner s Dilemma EXERCISE: Describe the strategic game Γ PD := (I, (A i ) i I, ( i ) i I ) behind the PRISONER S DILEMMA story! The players are The players action sets are The players preferences are Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 12 / 38

13 Bimatrix of Prisoner s Dilemma EXERCISE: Describe the PRISONER S DILEMMA by a bimatrix! Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 13 / 38

14 Example: Guessing Game Each person of group I of participants must announce a real number from 0 to 100. The person whose guessed number is the closest to the two-third of the average of all guessed numbers is the winner and receives a monetary prize of e 10. If two or more persons announce numbers which turn out to be the closest to the two-third of the average of all guessed numbers, then the monetary prize is split equally between these persons. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 14 / 38

15 The strategic game of the Guessing Game EXERCISE: Describe the strategic game Γ GG := (I, (A i ) i I, ( i ) i I ) behind the GUESSING GAME story! The players are The players action sets are The players preferences are Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 15 / 38

16 The two issues of game theory Having specified the rules of the game, game theory aims at tackling the positive issue: Which of the available actions the players will choose? or the normative issue: Which of the available actions the players should choose? Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 16 / 38

17 Solution concept A theory that provides answers to the positive or normative issue of game theory is called a solution theory or, more common, solution concept. The most prominent solution concept for strategic games is the Nash equilibrium concept. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 17 / 38

18 Nash equilibrium as a steady state The Nash equilibrium describes a steady state of the strategic game: An action profile is a Nash equilibrium whenever no player has an incentive to deviate from her action given the other players adhere to their actions. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 18 / 38

19 Nash equilibrium Definition 3.2 (Nash equilibrium) Consider a strategic game Γ := (I, (A i ) i I, ( i ) i I ). An action profile a := (ai ) i I A is a Nash equilibrium of Γ if, for every player i I, U i (ai, a i) U i (a i, a i) holds for every action a i A i. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 19 / 38

20 Nash equilibrium of Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 QUESTION: Is action profile (quiet,quiet) a Nash equilibrium? ANSWER: No, because if suspect B chooses to be quiet, then A is better off if she finks. (Moreover, if suspect A chooses to be quiet, then B is better off if he finks, too). Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 20 / 38

21 Nash equilibrium of Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 QUESTION: Is action profile (fink,quiet) a Nash equilibrium? ANSWER: No, because if suspect A chooses to fink, then B is better off if he finks, too. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 21 / 38

22 Nash equilibrium of Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 QUESTION: Is action profile (quiet,fink) a Nash equilibrium? ANSWER: No, because if suspect B chooses to fink, then A is better off if he finks, too. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 22 / 38

23 Nash equilibrium of Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 QUESTION: Is action profile (fink,fink) a Nash equilibrium? ANSWER: Yes! To prove this answer, you must show that, for every player, it is not profitable to be quiet given the other suspect finks. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 23 / 38

24 Nash equilibrium of Prisoner s Dilemma Suspect B Suspect A quiet fink quiet 2,2 0,3 fink 3,0 1,1 The action profile (fink,fink) is a Nash equilibrium, because given suspect B finks, then suspect A would not experience a higher utility if she chooses to be quiet. given suspect A finks, then suspect B would not experience a higher utility if she chooses to be quiet. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 24 / 38

25 The Prisoner s Dilemma The salient feature of PRISONER S DILEMMA is that the Nash equilibrium is an inefficient outcome of the game in the sense that there are other outcomes that are preferred to the Nash outcome by both players. Indeed, both players would be better off in outcome (quiet,quiet) than in Nash outcome (fink,fink) Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 25 / 38

26 The Prisoner s Dilemma The reason why an inefficient outcome is reached is that the efficient outcome is instable. In the efficient situation, players have an incentive to deviate from it. This behavioral instability of efficient outcomes is presumably inherent in numerous real-life situations (e.g., in the competition of firms for profits and of states on international power) Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 26 / 38

27 Example: Price-competing firms For all firms, the best outcome would be to charge a high price for their goods. However, to receive a higher profit every firm has an incentives to undercut the prices of its competitors. The process of undercutting prices ends up in an inefficient outcome for the firms. Finally, all firms charge a low price for their goods. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 27 / 38

28 Example: Arms-racing states For states, the best outcome would be to spend its revenues on social and economic welfare. However, each state has an incentive to gain international power by building an arsenal of weapons. This incentive ends up in an arms race between states and thus in an inefficient outcome. Finally, because of the mutual military buildup, no state has gained additional international power, but less financial capabilities for social and economic initiatives. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 28 / 38

29 Nash equilibrium of Matching Pennies Individual B Individual A Head Tail Head 1,-1-1,1 Tail -1,1 1,-1 EXERCISE: Figure out the Nash equilibria of MATCHING PENNIES! Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 29 / 38

30 Nash equilibrium of Battle of Sexes Individual B Individual A Bach Stravinsky Bach 2,1 0,0 Stravinsky 0,0 1,2 EXERCISE: Figure out the Nash equilibria of BATTLE OF SEXES! Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 30 / 38

31 The issue of existence and uniqueness The last two example of strategic games reveal that there are games in which there is no Nash equilibrium solution (see MATCHING PENNIES). there are more than one Nash equilibrium solution (see BATTLE OF SEXES). The first observation is referred to as the problem of existence, and the second one is referred to as the problem of uniqueness. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 31 / 38

32 Nash equilibrium of the Guessing Game QUESTION: What are the Nash equilibria of the GUESSING GAME with two players called henceforth player 1 and player 2. We demonstrate that the action profile (0, 0) (i.e., both players guess number 0) is the only Nash equilibrium of this game. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 32 / 38

33 Nash equilibrium of the Guessing Game CLAIM: The action profile (0, 0) is a Nash equilibrium of the two player GUESSING GAME. PROOF: If both players guess number 0, there is a draw and each player receives a monetary prize of e 5. Given player 2 guessed number 0, then player 1 would be worse off and gain e 0 if she guesses a number being different to 0. Thus, there is no incentive for her to deviate from the guessed number. The same arguments holds for player 2 Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 33 / 38

34 Nash equilibrium of the Guessing Game CLAIM: Any action profile (a 1, a 2 ) different to (0, 0) is not a Nash equilibrium of the two player GUESSING GAME. PROOF: We show that in each of the three possible cases 1 a 1 = a 2 > 0 2 a 1 > a 2 3 a 2 > a 1 the profile (a 1, a 2 ) of guesses does not constitute a Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 34 / 38

35 Nash equilibrium of the Guessing Game CLAIM: Any action profile (a 1, a 2 ) different to (0, 0) is not a Nash equilibrium of the two player GUESSING GAME. CASE 1 a 1 = a 2 > 0 In this case, both players are winners and both players receive a prize of e 5. However, both players have an incentive to revise their guesses. Given player 2 announces guess a 2, player 1 could win e 10 if she announces a guess that is below a 2. A similar argument holds for player 2. Therefore in this case the profile of guesses is not a Nash equilibrium. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 35 / 38

36 Nash equilibrium of the Guessing Game CLAIM: Any action profile (a 1, a 2 ) different to (0, 0) is not a Nash equilibrium of the two player GUESSING GAME. CASE 2 a 1 > a 2 In this case, player 1 receives nothing because a 1 2 a 1 + a 2 > 2 a 1 + a 2 a holds. Given player 2 adheres to a 2, then player 1 could be a winner (and receives at least e 5) if she chooses a number which is smaller or equal than a 2. Hence, the profile of guesses is not a Nash equilibrium in this case. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 36 / 38

37 Nash equilibrium of the Guessing Game CLAIM: Any action profile (a 1, a 2 ) different to (0, 0) is not a Nash equilibrium of the two player GUESSING GAME. CASE 3 a 2 > a 1 To show that, in this case, the profile of guesses is not a Nash equilibrium one has to adopt only the arguments of case 2 and swap the roles of player 1 and player 2. Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 37 / 38

38 Nash equilibrium of the Guessing Game EXERCISE: Show that the only Nash equilibrium in GUESSING GAMES with n 2 players is the profile of guesses in which every player announces number 0. HINT: Proceed as follows. Show that the profile of guesses in which every player chooses 0 is a Nash equilibrium. Thereafter demonstrate that in every other profile of guesses a player announcing the highest number could be better off by guessing a different number (which one?). Dr. Michael Trost Microeconomics I: Game Theory Lecture 3 38 / 38