Game Theory P . v. Mouche Wageningen University 2020, Period 4 - - PowerPoint PPT Presentation

Organisation Motivation Games in strategic form Games in extensive form

Game Theory

P . v. Mouche

Wageningen University

2020, Period 4

Organisation Motivation Games in strategic form Games in extensive form

Outline

1

Organisation

2

Motivation

3

Games in strategic form

4

Games in extensive form

Organisation Motivation Games in strategic form Games in extensive form

Organisation

Lecturers: P . v. Mouche (room 0108) and H.-P . Weikard. Web page: Brightspace or http://pvmouche.deds.nl/gametheory.html. Content: a short introduction to game theory. Non-cooperative game theory (week 1,2) and cooperative game theory (week 3). Course is designed for all students interested in decision-making in strategic situations. Starting level: ’no’ required knowledge. We focus on concepts and shall try to avoid (too difficult) mathematics.

Organisation Motivation Games in strategic form Games in extensive form

Organisation (ctd.)

Meetings 14:00–15:30, Monday in C317 and Friday in C211. Grading: short exam (± 105 minutes) and assignment; both count for 50%. In order to pass, the mark for the exam should be ≥ 5.5. The assignment, which may be conducted also with 2 people, concerns a short essay in which one describes a problem and set up and solve a model.

Organisation Motivation Games in strategic form Games in extensive form

Organisation (ctd.)

Meetings consist of a lecture and tutorial. Lecture: slides (will be updated). Tutorial: exercises. Please also read chapters 1 and 3 in the little book ‘Game Theory, a Very Short Introduction’ of K. Binmore.

Organisation Motivation Games in strategic form Games in extensive form

What is game theory?

Traditional game theory deals with mathematical mod- els 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 different concepts. However, the intelligence notion presupposes which type

• f rationality we are speaking about.

Aim of game theory is to understand how games are played.

Organisation Motivation Games in strategic form Games in extensive form

Nature of game theory

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

Organisation 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 about ‘winners’ and ‘losers’.

Organisation Motivation Games in strategic form Games in extensive form

Rationality

Because there is more than one player, rationality is a problematic and difficult notion. Here is a simple try: a rational player has well-defined preferences concerning the outcomes of the game. For example, what would You as player 1 play in the following bi-matrix-game: −1; −1 −3; 0 0; −3 −2; −2

• .

One player chooses a row, the other a column; first (second) number is payoff to row (column) player. This game is the classical prisoner’s dilemma game (of A. Tucker).

Organisation Motivation Games in strategic form Games in extensive form

Intelligence

Intelligence also is a not so easy notion. Intelligence depends on context: it refers to the (rational) goal of the player. Intelligence has to do with the way how the goal is approached.

Organisation Motivation Games in strategic form Games in extensive form

Tic-tac-toe

Notations: 1 2 3 4 5 6 7 8 9 Example of a play of this game:

Organisation 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.

Organisation Motivation Games in strategic form Games in extensive form

Hex

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

Organisation 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 later when we develop the theory.) perfect information – imperfect information complete information – incomplete information

Organisation Motivation Games in strategic form Games in extensive form

Perfect information

A game is with perfect information if each player knows at each moment when it is his turn to move how the game was played until that moment. A game is with imperfect information if it is not with 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).

Organisation Motivation Games in strategic form Games in extensive form

Complete information

A game is with complete information if heach player knows all payoff functions. A game is with incomplete information if it is not with 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, ...

Organisation 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 ... Common knowledge is a difficult notion; not so easy to formalise.

Organisation Motivation Games in strategic form Games in extensive form

Main formal types

Game in strategic form. Game in extensive form. Game in characteristic function form. (Third week.)

Organisation Motivation Games in strategic form Games in extensive form

Formalisation

We focus on concepts and shall try to avoid mathematics. However, there will be some abstractness in presentation.

Organisation 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. If xi ∈ Xi denotes a strategy for player i, then (x1, . . . , xn) is called a strategy profile . for each player i a payoff function fi(x1, . . . , xn). Interpretation: players choose simultaneously and independently a strategy. This leads to a strategy profile. Then the game can be played and payoffs can be calculated.

Organisation Motivation Games in strategic form Games in extensive form

A game in strategic form is called finite if each strategy set Xi is finite. In the case of two players a finite game in strategic form can be represented as a bi-matrix game. Many parlor games are zero-sum games , i.e. the total payoff is zero.

Organisation Motivation Games in strategic form Games in extensive form

Some concrete games (ctd).

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

Organisation Motivation Games in strategic form Games in extensive form

Some concrete games (ctd).

Cournot-duopoly : n = 2, Xi = [0, mi] or Xi = R+ fi(x1, x2) = p(x1 + x2)xi − ci(xi). p: price function, ci: cost function. Transboundary pollution game : n arbitrary, Xi = [0, mi] fi(x1, . . . , xn) = Pi(xi) − Di(Ti1x1 + · · · + Tinxn). P: production function, Di damage cost function, Tij: transport matrix coefficients

Organisation Motivation Games in strategic form Games in extensive form

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. 1 2 3 4 5 · · · n Two players simultaneously and independently choose a vertex. If player 1 (2) chooses vertex x1 ( x2), then the payoff fi(x1, x2)

• f player i is the number of 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.

Organisation Motivation Games in strategic form Games in extensive form

Some concrete games (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

Organisation Motivation Games in strategic form Games in extensive form

Some concrete games (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

Organisation Motivation Games in strategic form Games in extensive form

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 by taking as strategy set for a player its set of completely elaborated plans of play. For example, chess and tic-tac-toe: fi(x1, x2) ∈ {−1, 0, 1}.

Organisation Motivation Games in strategic form Games in extensive form

Fundamental notions

Conditional payoff function f (z)

i

• f player i: fi as a function
• f xi for fixed strategy profile z of the opponents.

Best reply correspondence Ri of player i: assigns to each strategy profile z of the opponents of player i the set of maximisers Ri(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.

Organisation Motivation Games in strategic form Games in extensive form

Solution concepts

Value : we define later. Strictly dominant equilibrium : strategy profile where each player has a strictly dominant strategy. Nash equilibrium : strategy profile such that no player wants to change his strategy in that profile.

Organisation Motivation Games in strategic form Games in extensive form

Example

Example Determine (if any), the strictly dominant equilibrium and the nash equilibria (if any) of the game     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. Nash equilibria: (1, 1), (1, 2), (2, 2) and (2, 3) (i.e. row 2 and column 3).

Organisation Motivation Games in strategic form Games in extensive form

Example

Example Question: determine (if any), the strictly dominant equilibrium and (if any) the nash equilibria of the game   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 game has one nash equilibrium: (3, 3).

Organisation 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). Often, in the case of continuous strategy sets, can be determined by the first-order conditions ∂fi ∂xi = 0 (i = 1, . . . , n)

Organisation Motivation Games in strategic form Games in extensive form

Example

Question: consider the Hotelling bi-matrix game in the case n = 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:   3/2; 3/2 1; 2 3/2; 3/2 2/1; 1 3/2; 3/2 2; 1 3/2; 3/2 1; 2 3/2; 3/2   . Nash equilibria: (0, 1), (1, 0), (1, 1), (1, 2), (2, 1).

Organisation 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 his strategy set Xi. 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. We shall not further deal with mixed strategies. (May be see Chapter 2 in the text book.)

Organisation Motivation Games in strategic form Games in extensive form

Other fundamental notions

If x = (x1, . . . , xn) and y = (y1, . . . , yn) are strategy profiles, then one says: y is a unanimous pareto-improvement of x if fi(y) > fi(x) for each player i. A strategy profile x is weakly pareto-efficient if there does not exist an unanimous pareto-improvement of x. A strategy profile x is weakly pareto-inefficient if there exists an unanimous pareto-improvement of x. Remark: there is another notion of pareto-efficiency, also called strong pareto efficiency . This notion is the usual efficiency notion in economics.

Organisation Motivation Games in strategic form Games in extensive form

Example Given the following bi-matrix-game:     3; 8 4; 8 2; 3 1; 7 2; 6 8; 1 3; 4 4; 4 2; 2 1; 1 3; 7 1; −1     . Determine the weakly pareto-efficient strategy profiles. Answer: Weakly: (1,1), (1,2) (2,3), (3,2).

Organisation Motivation Games in strategic form Games in extensive form

Other fundamental notions (cont.)

Fully 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. Each Fully cooperative strategy profile is weakly pareto efficient.

Organisation Motivation Games in strategic form Games in extensive form

Existence of nash equilibria

There exist various results that guarantee equilibrium existence for a game in strategic form. Just to mention one: the Nikaido-Isoda theorem. We shall not elaborate further on this as such results require some not elementary mathematics like the Brouwer’s fixed point theorem.

Organisation Motivation Games in strategic form Games in extensive form

Antagonistic game

Consider an antagonistic game : two players and f1 + f2 = 0. 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).

Organisation Motivation Games in strategic form Games in extensive form

Little test

Are the following statements about games in strategic form true

• r false?
• a. If each player has a strictly dominant strategy, then there

exists a unique Nash equilibrium.

• b. A player has at most one strictly dominant strategy.
• c. The 2 × 2-bi-matrix-game:

4; 0 2; −2 0; 1 1; 0

• has a strictly dominant equilibrium.

Organisation Motivation Games in strategic form Games in extensive form

Little test (ctd.)

• d. If each strategy profile is a Nash equilibrium, then each

payoff function is constant.

• e. Each fully cooperative strategy profile is pareto efficient.
• f. In a zero-sum game each strategy profile is pareto efficient.

Organisation 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.

Organisation 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 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.

Organisation Motivation Games in strategic form Games in extensive form

Perfect information (ctd.)

Solution concept: Nash equilibrium. Note: Games in strategic form are games with imperfect information.

Organisation 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.

Organisation 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 21 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 21/7 has remainder 0, 21 is a loosing position and player 2 has a winning strategy.

Organisation 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 Chapter 3. But: a nash equilibrium is not necessarily a backward induction strategy profile. The issue here relates to subgame perfection (See for more on this Chapter 3 in the text book.)

Organisation 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 .