上财高级微观范翠红课件七

History of game theory Definition of game theory Classification of games Description of games Objectives of game theory
Cuihong Fan
Introduction
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1. History
How to find a dominant strategy? What is an equilibrium in dominant strategies?
Cuihong Fan
Introduction
20/34
Strictly dominated strategies
In G = (I ; {Si , ui }n=1 ), the strategy si ∈ Si of player i is i strictly dominated, if there is another strategy si ∈ Si , such that for all s−i ∈ S−i , ui (si , s−i ) > ui (si , s−i ).
Cuihong Fan
Introduction
7/34
4. Description of games
Strategic (normal-form) games Definition (strategic form games) Players choose simultaneously their strategies; the combination of strategies chosen by all players determines a payoff for each player. Extensive form games Definition An extensive form game specifies the orders of play, what players can do at each of their opportunities to move, and what players know when they move.
1. Strategic form representation 2. Rule 3. Solution concepts:
3.1 Iterated elimination of dominated strategies (IEDS). 3.2 Nash Equilibrium (NE)
3.2.1 3.2.2 3.2.3 3.2.4 pure strategy NE mixed strategy NE Existence of NE Refinements of NE
Cuihong Fan
Introduction
21/34
Iterated Elimination of strictly dominated strategies
Assumption: This solution concept assumes that rationality is common knowledge Everybody is rational. Everybody knows that everybody is rational.
Cuihong Fan
Introduction
17/34
3.1.1. Iterated elimination of strictly dominated strategies (IESDS)
Def.: In G = (I ; {Si , ui }n=1 ), the strategy si ∈ Si is strictly i dominant, if for all strategies si ∈ Si , si = si , ui (si , s−i ) > ui (si , s−i ), for all s−i ∈ S−i .
Cuihong Fan
Introduction
10/34
1. Strategic form representation
A strategic form game consists of 3 components: (1). a finite set of players: I = {1, . . . , n } (2). a set of strategies (actions): S1 , . . . , Sn (3). payoff function (utility function): ui (s1 , . . . , sn ). G = (I ; {Si , ui }n=1 ) i
Cuihong Fan
Introduction
15/34
2. Rule
The players chooses their actions simultaneously. Player i chooses si , si ∈ Si . Each player obtains his utility, ui (s1 , . . . , sn ).
Cuihong Fan
Introduction
5/34
Basic definitions
Definition (complete information) Each player’s payoff function is common knowledge among all the players. Definition (common knowledge) A fact is common knowledge among players if each player knows the fact, and each player knows everyone else knows, and each knows everyone else knows everyone else knows, and so on.
Cuihong Fan
Introduction
12/34
Zero-sum game
Example: Matching Pennies Head Tail Head 1,-1 -1,1 Tail -1,1 1,-1
Strategic form representation: Properties of zero-sum game:
Cuihong Fan
Introduction
11/34
Notations
si : pure strategy of player i mi : mixed strategy of player i Si : the set of pure strategies of player i, si ∈ Si Mi : the set of mixed strategies of player i, mi ∈ Mi S = ×n=1 Si i s = ( s1 , . . . , sn ) ∈ S M = ×n=1 Mi i −i = j ∈ I \{i } ui : S → R:
Pioneers: Zermelo (1913) Von Neumann and Morgenstern (1944) John Nash (1950) Reinhard Selten (1965, 1975) John Harsanyi (1967-68)
Cuihong Fan
Introduction
Cuihong Fan
Introduction
8/34
5. Objectives of game theory
Formal description of games Solution concepts
Cuihong Fan
Introduction
9/34
I. Static games of complete information
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2. Definition of “game theory”
Definition (Game theory) Game theory is the systematic study of how rational agents behave in strategic situations. Rationality: Nonstrategic and strategic decisions:
Cuihong Fan
Introduction
6/34
Definition (Static games) All players choose simultaneously their actions (without knowing decisions of other players). Definition (dynamic games) Players make their decisions sequentially and more than once. Definition (incomplete information)
Cuihong Fan
Introduction
13/34
Non-zero sum game
Tobacco advisement: A A 27 ,27 NA 20,60 Strategic form representation: Properties of non-zero sum game: NA 60,20 50,50
Cuihong Fan
Introduction
4/34
3. Four groups of game theoretical models
(1) (2) (3) (4)
Static games of complete information Dynamic games of complete information Static games of incomplete information Dynamic games of incomplete information