Motivation Games in strategic form Games in extensive form Advanced Microeconomics: Game Theory P . v. Mouche Wageningen University 2018
Motivation Games in strategic form Games in extensive form Outline Motivation 1 Games in strategic form 2 Games in extensive form 3
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 optimisation 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: 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.
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 1 2 3 Notations: 4 5 6 7 8 9 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:
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.
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, oligopoly 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) 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.
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
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 ) . 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 ) .
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: 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.
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
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: 2 + 1 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 (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 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 } .
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