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 Motivation 2 Games in strategic form 3 Games in extensive form 4
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 optimisation theory. ‘Traditional’ because of rationality assumption. ‘Rationality’ and ’intelligence’ are different concepts. However, the intelligence notion presupposes which type of 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 1 2 3 Notations: 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 X X O O X X O X X O X X O O O X 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, oligopoly 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) X i . If x i ∈ X i denotes a strategy for player i , then ( x 1 , . . . , x n ) is called a strategy profile . for each player i a payoff function f i ( x 1 , . . . , x n ) . 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 X i 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 , 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 ) . p : price function, c i : cost function. Transboundary pollution game : n arbitrary, X i = [ 0 , m i ] f i ( x 1 , . . . , x n ) = P i ( x i ) − D i ( T i 1 x 1 + · · · + T in x n ) . P : production function, D i damage cost function, T ij : 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 . n 0 1 2 3 4 5 · · · Two players simultaneously and independently choose a vertex. If player 1 (2) chooses vertex x 1 ( x 2 ), then 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.
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
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