Advanced Microeconomics: Game Theory

Transcription

1 Advanced Microeconomics: Game Theory P. v. Mouche Wageningen University 2018

2 Outline 1 Motivation 2 Games in strategic form 3 Games in extensive form

3 What is game theory? Traditional game theory deals with mathematical models of conflict and cooperation in the real world between at least two rational intelligent players. Player: humans, organisations, nations, animals, computers,... Situations with one player are studied by the classical optimisation theory. Traditional because of rationality assumption. Rationality and intelligence are completely different concepts.

4 Nature of game theory Applications: parlour games, military strategy, computer games, biology, economics, sociology, psychology anthropology, politocology. Game theory provides a language that is very appropriate for conceptual thinking. Many game theoretical concepts can be understood without advanced mathematics.

5 Outcomes and payoffs A game can have different outcomes. Each outcome has its own payoffs for every player. Nature of payoff: money, honour, activity, nothing at all, utility, real number,.... Interpretation of payoff: satisfaction at end of game. In general it does not make sense to speak of winners and losers.

6 Rationality Because there is more than one player, especially rationality becomes a problematic notion. For example, what would You as player 1 play in the following bi-matrix-game: ( 300; ; ; ; 500 ). (One player chooses a row, the other a column; first (second) number is payoff to row (column) player.)

7 Tic-tac-toe Notations: Player 1: X. Player 2: O. Many outcomes (more than three). Three types of outcomes: player 1 wins, draw, player 1 loses. Payoffs (example): winner obtains 13 Euro from loser. When draw, then each player cleans the shoes of the other. (In fact it is a a zero-sum game.) Example of a play of this game:

8 Tic-tac-toe (cont.) X X X X O X X O X X O O X O So: player 2 is the winner. O X X O O X O Question: Is player 1 intelligent? Is player 1 rational? Answer: We do not know.

9 Hex

10 Real-world types all players are rational players may be not rational all players are intelligent players who may be not intelligent binding agreements no binding agreements chance moves no chance moves communication no communication static game dynamic game transferable payoffs no transferable payoffs interconnected games isolated games (In red what we will assume always.) perfect information imperfect information complete information incomplete information

11 Perfect information A player has perfect information if he knows at each moment when it is his turn to move how the game was played untill that moment. A player has imperfect information if he does not have perfect information. A game is with (im)perfect information if (not) all players have perfect information. Chance moves are compatible with perfect information. Examples of games with perfect information: tic-tac-toe, chess,... Examples of games with imperfect information: poker, monopoly (because of the cards, not because of the die).

12 Complete information A player has complete information if he knows all payoff functions. A player has incomplete information if he does not have complete information. A game is with (in)complete information if (not) all players have complete information. Examples of games with complete information: tic-tac-toe, chess, poker, monopoly,... Examples of games with incomplete information: auctions, oligopoly models where firms only know the own cost functions,...

13 Common knowledge Something is common knowledge if everybody knows it and in addition that everybody knows that everybody knows it and in addition that everybody knows that everybody knows that everybody knows it and...

14 Common knowledge A group of dwarfs with red and green caps are sitting in a circle around their king who has a bell. In this group it is common knowledge that every body is intelligent. They do not communicate with each other and each dwarf can only see the color of the caps of the others, but does not know the color of the own cap. The king says: Here is at least one dwarf with a red cap.. Next he says: I will ring the bell several times. Those who know their cap color should stand up when i ring the bell.. Then the king does what he announced. The spectacular thing is that there is a moment where a dwarf stands up. Even, when there are M dwarfs with red caps that all these dwarfs simultaneously stand up when the king rings the bell for the M-th time.

15 Mathematical types Game in strategic form. Game in extensive form. Game in characteristic function form.

16 Game in strategic form Definition Game in strategic form, specified by n players: 1,..., n. for each player i a strategy set (or action set) X i. for each player i payoff function f i : X 1 X n R. X := X 1 X n : set of strategy profiles (or multi-strategies). Interpretation: players choose simultaneously a strategy. A game in strategic form is called finite if each strategy set X i is finite.

17 Some concrete games (ctd). Stone-paper-scissors 0; 0 1; 1 1; 1 1; 1 0; 0 1; 1 1; 1 1; 1 0; 0

18 Some concrete games (ctd). Cournot-duopoly: n = 2, X i = [0, m i ] or X i = R + f i (x 1, x 2 ) = p(x 1 + x 2 )x i c i (x i ). Transboundary pollution game: n arbitrary, X i = [0, m i ] f i (x 1,..., x n ) = P i (x i ) D i (T i1 x 1 + +T in x n ).

19 Some concrete games (ctd.) The Hotelling bi-matrix game depends on two parameters: integer n 1 and w ]0, 1]. Consider the n+1 points of H := {0, 1,..., n} on the real line, to be referred to as vertices n Two players simultaneously and independently choose a vertex. If player 1 (2) chooses vertex x 1 ( x 2 ), then: Case w = 1: the payoff f i (x 1, x 2 ) of player i is the number of vertices that is the closest to his choice x i ; however, a shared vertex, i.e. one that has equal distance to both players, contributes only 1/2. General case: 0 < w 1: exactly the same vertices as in the above for w = 1 contribute. Take such a vertex. If it is at distance d to x i, then it contributes w d if it is not a shared vertex, and otherwise it contributes w d /2.

20 Some concrete games (ctd.) Example n = 7 and w = 1. Action profile ( 5,2 ) : Payoffs: = = 4 Action profile ( 0,3 ) : Payoffs 1+1 = = 6

21 Some concrete games (ctd.) Example n = 7 and w = 1. Action profile ( 2,6 ) : Payoffs: = = Action profile ( 3,3 ) : Payoffs: = = 4

22 Some concrete games (ctd.) Example n = 5 and w = 1/4: Action profile ( 1,3 ) : Payoffs: = = Action profile ( 1,4 ) : Payoffs: = = 1 1 2

23 Normalisation Many games which are not defined as a game in strategic form can be represented in a natural way by normalisation as a game in strategic form. For example, chess and tic-tac-toe: n = 2, X i is set of completely elaborated plans of playing of i, f i (x 1, x 2 ) { 1, 0, 1}.

24 Fundamental notions Conditional payoff function f (z) i of player i: f i as a function of x i for fixed strategy profile z of the opponents. Best reply correspondence R i of player i: assigns to each strategy profile z of the opponents of player i the set of maximisers R i (z) of f (z) i. (Strictly) dominant strategy of a player i: (the) best strategy of player i independently of strategies of the other players. Strongly (or strictly) dominated strategy of a player: a strategy of a player for which there exists another strategy that independently of the strategies of the other players always gives a higher payoff.

25 Solution concepts Strictly dominant equilibrium: strategy profile where each player has a strictly dominant strategy. Procedure of iterative (simultaneous) elimination of strongly dominated strategies Strategy profile that survives this procedure. If there is a unique strategy profile that survives the above procedure this strategy profile is called the iteratively not strongly dominated equilibrium. Nash equilibrium: strategy profile such that no player wants to change his strategy in that profile.

26 Solution concepts (ctd.) Theorem 1 Each strictly dominant equilibrium is an iteratively not strongly dominated equilibrium. And if the game is finite: 1 An iteratively not strongly dominated equilibrium is a unique nash equilibrium. 2 Each nash equilibrium is an iteratively not strongly dominated strategy profile. (So each nash equilibrium survives the procedure.) Proof. 1. Already in first steps of procedure all strategies are removed with the exception of strictly dominant ones. 2, 3. One verifies that in each step of the procedure the set of nash equilibria remains the same. (See the text book.)

27 Nash equilibria A strategy profile e = (e 1,..., e n ) is a nash equilibrium if and only if for each player i one has e i R i (e 1,...,e i 1, e i+1,..., e n ). Sometimes (in economics even often ) can be determined by f i x i = 0 (i = 1,...,n)

28 Example Question: determine (if any), the strictly dominant equilibrium, the iteratively not strongly dominated equilibrium (if any) and the nash equilibria of the game Example 2; 4 1; 4 4; 3 3; 0 1; 1 1; 2 5; 2 6; 1 1; 2 0; 5 3; 4 7; 3 0; 6 0; 4 3; 4 1; 5. Answer: ( no strictly) dominant equilibrium. The procedure gives 2; 4 1; 4 4; 3. Thus the game does not have an 1; 1 1; 2 5; 2 iteratively not strongly dominated equilibrium. Nash equilibria: (1, 1),(1, 2),(2, 2) and (2, 3) (i.e. row 2 and column 3).

29 Example Question: determine (if any), the strictly dominant equilibrium, the iteratively not strongly dominated equilibrium and the nash equilibria of the game Example 6; 1 3; 1 1; 5 2; 4 4; 2 2; 3 5; 1 6; 1 5; 2 Answer: No player has as strictly dominant strategy, thus the game does not have a strictly dominant equilibrium. The procedure of iterative elimination of strongly dominated strategies gives the bi-matrix (5; 2). Thus the game has an iteratively not strongly dominated equilibrium: (3, 3). The game has one nash equilibrium: (3, 3).

30 Example Question: consider the Hotelling bi-matrix game in the case n = 2 and w = 1/2. Determine the nash equilibria of this game (by representing it as a 3 3-bi-matrix game with at the first row strategy 0 for player 1, at the second row strategy 1 for player 1, etc.) Answer: 7/8; 7/8 1; 3/2 5/4; 5/4 3/2; 1 1; 1 3/2; 1 5/4; 5/4 1; 3/2 7/8; 7/8 Nash equilibria: (0, 1),(1, 0),(1, 1),(1, 2),(2, 1)..

31 Mixed strategies Some games do not have a nash equilibrium. Mixed strategy of player i: probability density over his strategy set X i. With mixed strategies, payoffs have the interpretation of expected payoffs. Nash equilibrium in mixed strategies. Remark: each nash equilibrium is a nash equilibrium in mixed strategies. (See text book for formal proof.)

32 Bi-matrix-game with mixed strategies Consider a 2 2 bi-matrix-game (A; B) Strategies: (p, 1 p) for player 1 and (q, 1 q) for player B. Expected payoffs: f 1 (p, q) = (p, 1 p) A f 2 (p, q) = (p, 1 p) B ( q 1 q ( q 1 q ), ).

33 Example Example Determine the nash equilibria in mixed strategies for ( ) 0; 0 1; 1. 2; 2 1; 1 Answer: f 1 (p; q) = ( 4q + 2)p+3q 1, f 2 (p; q) = (4p 3)q + 1 2p. This leads to the nash equilibrium p = 3/4, q = 1/2.

34 Example Example Determine ( the nash ) equilibria in mixed strategies for 1; 1 1; 1. 1; 1 1; 1 Answer: p = q = 1/2.

35 Other fundamental notions If x = (x 1,...,x n ) and y = (y 1,...,y n ) are strategy profiles, then one says: z is an unanimous pareto-improvement of x if f i (y) > f i (x) (i N); y is a pareto-improvement of x if f i (y) f i (x) (i N) with at least one of these inequalities strict. A strategy profile x is called (strongly) pareto-efficiënt if there does not exist a pareto-improvement of x. weakly pareto-efficiënt if there does not exist an unanimous pareto-improvement of x. (strongly) pareto-inefficient if it is not pareto efficient. (weakly) pareto-inefficient if it is not weakly pareto efficient.

36 Other fundamental notions (cont.) Full cooperative strategy profile: a strategy profile that maximizes the total payoff. Prisoners dilemma: a game in strategic form where there is a strictly dominant weakly pareto-inefficient nash equilibrium.

37 Existence of nash equilibria Conditional payoff function of player i: f i as a function of x i, given strategies of the other players. Theorem (Nikaido-Isoda.) Each game in strategic form where 1 each strategy set is a convex compact subset of some R n, 2 each payoff function is continuous, 3 each conditional payoff function is quasi-concave, has a nash equilibrium. Proof. This is a deep theoretical result. A proof can be based on Brouwer s fixed point theorem. See text book for the proof of a simpler case (Theorem 7.2., i.e. the next theorem).

38 Theorem of Nash Theorem Each bi-matrix-game has a nash equilibrium in mixed strategies. Proof. Apply the Nikaido-Isoda result.

39 Antagonistic game Consider an antagonistic game: two players and f 1 + f 2 = 0. (Cfr. with Exercise 7.7 in the text book.) Theorem If (a 1, a 2 ) and (b 1, b 2 ) are nash equilibria, then f 1 (a 1, a 2 ) = f 1 (b 1, b 2 ) and f 2 (a 1, a 2 ) = f 2 (b 1, b 2 ). Proof. f 1 (a 1, a 2 ) f 1 (b 1, a 2 ) = f 2 (b 1, a 2 ) f 2 (b 1, b 2 ) = f 1 (b 1, b 2 ). In the same way f 1 (b 1, b 2 ) f 1 (a 1, a 2 ). Therefore f 1 (a 1, a 2 ) = f 1 (b 1, b 2 ) and thus f 2 (a 1, a 2 ) = f 2 (b 1, b 2 ).

40 Little test Are the following statements about games in strategic form true or false? a. If each player has a dominant strategy, then there exists a unique Nash equilibrium. F. b. A player has at most one strictly dominant strategy. T c. The 2 2-bi-matrix-game: ( 4; 0 2; 2 0; 1 1; 0 has a strictly dominant equilibrium. T )

41 Little test (ctd.) d. The 3 2-bi-matrix-game: 4; 0 2; 2 0; 1 1; 0 2; 1 3; 2 does not have a Nash equilibrium in mixed strategies. F e. If each strategy profile is a Nash equilibrium, then each payoff function is constant. F f. Each fully cooperative strategy profile is pareto efficient. T g. In a zero-sum game each strategy profile is pareto efficient. T h. It is possible that a pure strategy is not strongly dominated by a pure strategy, but is by a mixed strategy. T i. It is possible that a best-reply-correspondence of a player is empty-valued, i.e. that given strategies of the other players there does not exist a best reply of that player. T

42 Appetizer t-t-t chess 8 8 checkers hex value draw not known draw 1 opt. strat. known not known known not known Value: outcome of the game in the case of two rational intelligent players. Optimal strategy for a player: a strategy that guarantees this player at least the value.

43 Hex 1 Invented independently by Piet Hein and John Nash Hex can not end in a draw. ( Equivalent with Brouwer s fixed point theorem in two dimensions.) 4 If You can give a winning strategy for hex, then You solved a 1-million-dollar problem.

44 Games in extensive form Our setting is always non-cooperative with complete information (and for the moment) perfect information and no chance moves. Game tree: Nodes (or histories): end nodes, decision nodes, unique initial node. Directed branches. Payoffs at end nodes. Each non-initial node has exactly one predecessor. No path in tree connects a node with itself. Game is finite (i.e. a finite number of branches and nodes). Actual moves can be denoted by arrows.

45 Perfect information (ctd.) Theoretically: Imperfect information can be dealt with by using information sets. The information sets form a partition of the decision nodes. (Example: Figure 7.10.) Perfect information: all information sets are singletons. Solution concept: Nash equilibrium. Games in strategic form are games with imperfect information.

46 Normalisation Strategy: specification at each decision node how to move. (This may be much more than a completely elaborated plan of play.) Normalisation: make out (in natural way) of game in extensive form a game in strategic form. So normalisation destroys the perfect information. All terminology and results for games in strategic form now also applies to games in extensive forms.

47 Solving from the end to the beginning Example Consider the following game between two (rational and intelligent) players. There is a pillow with 100 matches. They alternately remove 1, 3 or 4 matches from it. (Player 1 begins.) The player who makes the last move wins. Who will win? Answer: the loosing positions are 0, 2, 7, 9, 14, 16, 21,..., i.e. the numbers that have remainder 0 or 2 when divided by 7. Because 100/7 has remainder 2, 100 is a loosing position and player 2 has a winning strategy.

48 Procedure of backward induction (explained at the blackboard) leads to a non-empty set of backward induction strategy profiles. Theorem (Kuhn.) Each backward induction strategy profile of a finite game in extensive form with perfect information is a nash equilibrium. Proof. See text book. But (as we shall see) a nash equilibrium is not necessarily a backward induction strategy profile.

49 Hex-game revisited As the game cannot end in a draw, the above theory guarantees that player 1 or player 2 has a winning strategy. Here is a proof that player 1 has such a strategy by a strategy-stealing argument:

50 Hex-game revisited (ctd.) suppose that the second player has a winning strategy, which we will call S. We can convert S into a winning strategy for the first player. The first player should make his first move at random; thereafter he should pretend to be the second player, stealing the second player s strategy S, and follow strategy S, which by hypothesis will result in a victory for him. If strategy S calls for him to move in the hexagon that he chose at random for his first move, he should choose at random again. This will not interfere with the execution of S, and this strategy is always at least as good as S since having an extra marked square on the board is never a disadvantage in hex.

51 Subgame perfection Subgame: game starts at a decision node. Subgame perfect nash equilibrium: a nash equilibrium that remains for each subgame a nash equilibrium. Theorem For every finite extensive form game with perfect information the set of backward induction strategy profiles coincides with the set of subgame perfect nash equilibria. Proof. See text book.

52 Games in extensive form: extensions Three extensions: Imperfect information. Incomplete information: the solution concept here is that of Bayesian equilibrium (7.2.3.). [Next part of course.] Randomization.

53 Imperfect information Imperfect information. Can be dealt with by using information sets. The information sets form a partition of the decision nodes. (Example: Figure 7.10.) Perfect information: all information sets are singletons. Strategy: specification at each information set how to move. The procedure of backward induction cannot be applied anymore, but the notion of subgame perfect Nash equilibria still makes sense (when subgame is properly defined). [Next part of course.] Subgame: not all decision nodes define anymore a subgame. (Example: Figure 7.20.) [Next part of course.] Nash equilibria need not always exist. (Example: Figure 7.23.) [Next part of course.]

54 Randomization Three types of strategies: pure, mixed and behavioural strategies.

55 Randomization [Next part of the course.] A pure strategy of player i is a book with instructions where there is for each decision node for i a page with the content which move to make at that node. So the set of all pure strategies of player i is a library of such books. A mixed strategy of player i is a probability density on his library. Playing a mixed strategy now comes down to choosing a book from this library by using a chance device with the prescribed probability density.

56 Randomization (ctd.) A behavioural strategy, is like a pure strategy also a book, but of a different kind. Each page in the book still refers to a decision node, but now the content is not which move to make but a probability density between the possible moves. For many games (for instance those with perfect recall) it makes no difference whatever if players employ mixed or behavioural strategies.

57 Nash John Nash ( ). Mathematician. (Economist?) Nobel price for economics in 1994, together with Harsanyi and Selten. Abel Price for mathematics in Just after having received it he was killed in a car crash. Got this price for his PhD dissertation (27 pages) in