Advanced Microeconomics: Game Theory P . v. Mouche Wageningen - - PowerPoint PPT Presentation

Motivation Games in strategic form Games in extensive form

Advanced Microeconomics: Game Theory

P . v. Mouche

Wageningen University

2017

Motivation Games in strategic form Games in extensive form

Outline

1

Motivation

2

Games in strategic form

3

Games in extensive form

Motivation Games in strategic form Games in extensive form

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

• ptimisation theory.

‘Traditional’ because of rationality assumption. ‘Rationality’ and ’intelligence’ are completely different concepts.

Motivation Games in strategic form Games in extensive form

Nature of game theory

Applications: parlor games, military strategy, computer games, biology, economics, sociology, psychology antropology, politicology. Game theory provides a language that is very appropriate for conceptual thinking. Many game theoretical concepts can be understood without advanced mathematics.

Motivation Games in strategic form Games in extensive form

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’.

Motivation Games in strategic form Games in extensive form

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; 400 600; 250 200; 600 450; 500

• .

(One player chooses a row, the other a column; first (second) number is payoff to row (column) player.)

Motivation Games in strategic form Games in extensive form

Tic-tac-toe

Notations: 1 2 3 4 5 6 7 8 9 Player 1: X. Player 2: O. Many outcomes (more than three). Three types of

• utcomes: 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

• ther. (In fact it is a a zero-sum game.)

Example of a play of this game:

Motivation Games in strategic form Games in extensive form

Tic-tac-toe (cont.)

X X O X X O X X O O X X O X O X X O O X O So: player 2 is the winner. Question: Is player 1 intelligent? Is player 1 rational? Answer: We do not know.

Motivation Games in strategic form Games in extensive form

Hex

http://www.lutanho.net/play/hex.html.

Motivation Games in strategic form Games in extensive form

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

Motivation Games in strategic form Games in extensive form

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).

Motivation Games in strategic form Games in extensive form

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,

• ligopoly models where firms only know the own cost

functions, ...

Motivation Games in strategic form Games in extensive form

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 ...

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

Mathematical types

Game in strategic form. Game in extensive form. Game in characteristic function form.

Motivation Games in strategic form Games in extensive form

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) Xi. for each player i payoff function fi : X1 × · · · × Xn → R. X := X1 × · · · × Xn: 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 Xi is finite.

Motivation Games in strategic form Games in extensive form

Some concrete games in strategic form

Cournot-duopoly: n = 2, Xi = [0, mi] or Xi = R+ fi(x1, x2) = p(x1 + x2)xi − ci(xi). Transboundary pollution game: n arbitrary, Xi = [0, mi] fi(x1, . . . , xn) = Pi(xi) − Di(Ti1x1 + · · · + Tinxn).

Motivation Games in strategic form Games in extensive form

Some concrete games in strategic form (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. 1 2 3 4 5 · · · n Two players simultaneously and independently choose a

• vertex. If player 1 (2) chooses vertex x1 ( x2), then:

Case w = 1: the payoff fi(x1, x2) of player i is the number

• f vertices that is the closest to his choice xi; 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 xi, then it contributes wd if it is not a shared vertex, and otherwise it contributes wd/2.

Motivation Games in strategic form Games in extensive form

Some concrete games in strategic form (ctd.)

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

Motivation Games in strategic form Games in extensive form

Some concrete games in strategic form (ctd.)

Example n = 7 and w = 1. Action profile ( 2,6 ) : Payoffs: 1 + 1 + 1 + 1 + 1

2 = 4 1 2 1 2 + 1 + 1 + 1 = 3 1 2

Action profile ( 3,3 ) : Payoffs:

1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 = 4 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 = 4

Motivation Games in strategic form Games in extensive form

Some concrete games in strategic form (ctd.)

Example n = 5 and w = 1/4: Action profile ( 1,3 ) : Payoffs:

1 4 + 1 + 1 8 = 1 3 8 1 8 + 1 + 1 4 + 1 16 = 1 7 16

Action profile ( 1,4 ) : Payoffs:

1 4 + 1 + 1 4 = 1 1 2 1 4 + 1 + 1 4 = 1 1 2

Motivation Games in strategic form Games in extensive form

Normalisation

Many games can be represented in a natural way by normalisation as a game in strategic form. For example, chess and tic-tac-toe: n = 2, Xi is set of completely elaborated plans of playing of i, fi(x1, x2) ∈ {−1, 0, 1}.

Motivation Games in strategic form Games in extensive form

Some concrete games.

  0; 0 −1; 1 1; −1 1; −1 0; 0 −1; 1 −1; 1 1; −1 0; 0   Stone-paper-scissors

Motivation Games in strategic form Games in extensive form

Some concrete games (ctd).

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).

Motivation Games in strategic form Games in extensive form

Fundamental notions

Best reply correspondence Ri of player i: X1 × · · · × Xi−1 × Xi+1 × · · · × Xn ⊸ Xi. (Strictly) dominant strategy of a player i: (the) best strategy

• f 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.

Motivation Games in strategic form Games in extensive form

Fundamental notions (cont.)

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 deviate from it. Strictly dominant equilibrium: strategy profile where each player has a strictly dominant strategy.

Motivation Games in strategic form Games in extensive form

Fundamental notions (cont.)

If x and z are strategy profiles, then one says: z is a pareto-improvement of x if f(z) > f(x); z is an unanimous pareto-improvement of x if f(z) ≫ f(x). 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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

Nash equilibria

A strategy profile e = (e1, . . . , en) is a nash equilibrium if and

• nly if for each player i one has

ei ∈ Ri(e1, . . . , ei−1, ei+1, . . . , en). Sometimes (in economics even ’often’) can be determined by ∂fi ∂xi = 0 (i = 1, . . . , n)

Motivation Games in strategic form Games in extensive form

Example

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 1; 1 1; 2 5; 2

• . Thus the game does not have an

iteratively not strongly dominated equilibrium. Nash equilibria: (1, 1), (1, 2), (2, 2) and (2, 3) (i.e. row 2 and column 3).

Motivation Games in strategic form Games in extensive form

Example

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).

Motivation Games in strategic form Games in extensive form

Solution concepts

Theorem

• a. Each strictly dominant equilibrium is an iteratively not

strongly dominated equilibrium. And if the game is finite:

• b. Each nash equilibrium is an iteratively not strongly

dominated strategy profile. (So each nash equilibrium survives the procedure.)

• c. An iteratively not strongly dominated equilibrium is a

unique nash equilibrium. 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.)

Motivation Games in strategic form Games in extensive form

Mixed strategies

Some games do not have a nash equilibrium. Mixed strategy of player i: probability density over 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.)

Motivation Games in strategic form Games in extensive form

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 ∗

• q

1 − q

• ,

f 2(p, q) = (p, 1 − p) ∗ B ∗

• q

1 − q

• .

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

Example

Example Determine the nash equilibria in mixed strategies for −1; 1 1; −1 1; −1 −1; 1

• .

Answer: p = q = 1/2.

Motivation Games in strategic form Games in extensive form

Existence of nash equilibria

Conditional payoff function of player i: fi as a function of xi, 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 Rn,

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).

Motivation Games in strategic form Games in extensive form

Theorem of Nash

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

Motivation Games in strategic form Games in extensive form

Antagonistic game

Consider an antagonistic game: two players and f1 + f2 = 0. (Cfr. with Exercise 7.7 in the text book.) Theorem If (a1, a2) and (b1, b2) are nash equilibria, then f1(a1, a2) = f1(b1, b2) and f2(a1, a2) = f2(b1, b2). Proof. f1(a1, a2) ≥ f1(b1, a2) = −f2(b1, a2) ≥ −f2(b1, b2) = f1(b1, b2). In the same way f1(b1, b2) ≥ f1(a1, a2). Therefore f1(a1, a2) = f1(b1, b2) and thus f2(a1, a2) = f2(b1, b2).

Motivation Games in strategic form Games in extensive form

Little test

Are the following statements about games in strategic form true

• r false? Prove Your answer (may be by referring to a result on

the slides or in the book) or give a counter-example.

• 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

Motivation Games in strategic form Games in extensive form

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

Motivation Games in strategic form Games in extensive form

Appetizer

t-t-t chess 8 × 8 checkers hex value draw not known draw 1

• pt. 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.

Motivation Games in strategic form Games in extensive form

Hex

1

Invented independently by Piet Hein and John Nash.

2

http://www.lutanho.net/play/hex.html.

3

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’.

Motivation Games in strategic form Games in extensive form

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 endnodes. 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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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:

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.]

Motivation Games in strategic form Games in extensive form

Randomization

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

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

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.

Motivation Games in strategic form Games in extensive form

Nash

John Nash (1928 – 2015).

• Mathematician. (Economist ?)

Nobel price for economics in 1994, together with Harsanyi and Selten. Abel Price for mathematics in 2015. Just after having received it he was killed in a car crash. Got this price for his PhD dissertation (27 pages) in 1950. http://topdocumentaryfilms.com/a-brilliant-madne .