Representation of Games 14.12 Game Theory Muhamet Yildiz 1 Game: - PDF document

Lecture 3 Representation of Games 14.12 Game Theory Muhamet Yildiz 1

Game: Ingredients • Who are the players (decision makers)? • What moves are available to each player and when? • What does each player know at the time of each of his decisions? • What are the outcomes and payoffs at the end? 2

Road Map 1. Extensive form representation 2. Strategy 3. Normal form representation 4. Mixed strategy 3

Extensive- form representation Definition: A tree is a set of nodes connected with directed arcs such that 1. There is an initial node; 2. For each other node, there is one incoming arc; 3. each node can be reached through a unique path. 4

A tree / ... -- ... - , r I / ll.. lIe I \ Non-terminal rm ___ ina _ I_ N _ o _ d_ es .... no d es I \ , , \ I I I , I , -, 5

Extensive form - definition Definition: A game consists of - a set of players - a tree - an allocation of each non-terminal node to a player - an informational partition (to be made precis e) - a payoff for each player at each terminal node. 6

Information set An information set is a collection of nodes such that 1. The same player is to move at each of these nodes; 2. The same moves are available at each of these nodes. An informational partition is an allocation of each non-terminal node of the tree to an information set. 7

A game 1 L R 2 (2,2) r I u 1 (0,0) 1 A p p (1,3) (3, 1) (3,3) (1, I) 8

Another game 1 x T B 2 L R R L 9

The Same Game x 1 B T L R 10

What is wrong? 1 x T B Up L R R L 11

What is wrong? 1 x B T 3 2 L R R L 12

What is wrong? 3 A B 13

Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). 14

Matching pennies with perfect information 2's Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if I plays Head, Head Tail Tail if 1 plays Tail; TH = Tail if 1 plays Head, 2 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1 , 1) (1,-1) (1,-1) (-1,1) 15

Matching pennies with perfect information 2 1 HH HT TH TT Q Q Q Q Head (-1,1) -1,1) (1,-1) (1,-1) Q Q Q Q (1,-1) -1,1) (1,-1) (-1,1) Tail Head Tai 2 2 head head tail (-1 , 1) (1,-1) (1,-1) (-1,1) 16

N ormal- form representation Definition (Normal form): A game is any list , , uJ G = ( Sp ... ' S n; u p " where, for each i E N = {1,2 , . .. , n} , • S; is the set of all strategies available to i, : SI x·· · X Sn ---t 9t is the VNM utility function of • u i player i. Assumption: G is "common knowledge". Definition: A player i is rational iff he tries to maximize the expected value of U ; given his beliefs. 17

.~ ~. , ~- ~, ~ Chicken Chicken :;:: ::..... -: (-1,-1) (1 ,0) (-1,-1) (1,0) (0,1) (0,1) (1/2,1/2 ) Image by MIT OpenCourseWare. 18

Matching pennies Tail Head (-1,1) (1,-1) Head Tail (1,-1) (-1,1) 19

Extensive v. Normal Forms • Extensive to Normal: - Find the set of strategies for each player - Every strategy profile s leads to an outcome z( s), a terminal history - Utility from s is u(z(s)) • Normal to Extensive: many possibilities 20

Matching pennies with perfect information 2 1 HH HT TH TT Q Q Q Q Head (-1,1) -1,1) (1,-1) (1,-1) Q Q Q Q (1,-1) -1,1) (1,-1) (-1,1) Tail Head Tai 2 2 head head tail (-1 , 1) (1,-1) (1,-1) (-1,1) 21

Matching pennies with imperfect information 1 2 I Head Tail Head Tail Head (-1,1) (1,-1) head tail Tail (1,-1) (-1,1) (1,-1) (1,-1) (-1,1) (-1,1) 22

~ ,-,-,- A game 1 A 2 a 1 a (1,-5) d D (4,4) (5 ,2) (3 ,3) 23

A game with nature (5,0) Left 1 Head 112 Right (2,2) Nature 0 (3,3) 112 Left 2 Tail Right (0 , -5) 24

Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {Sil ,Si2" .. ,Sik} A mixed strategy: cr i: Si --* [0,1] S.t. cri(Sij) + cri(Si2) + ... + crlsik) = 1. If the other players play S_i =(Sj, ... , Si_j,si +j"",sn), then the expected utility of playing cri is crlSij)Ui(Sij,SJ + crlsi2) UlSi2,SJ + ... + cri(Sik) UlSik,SJ. 25

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