1. GAME THEORY AND DYNAMICAL SYSTEMS
LECTURER: DR. MALGOSIA GUZOWSKA
ATALAY KOÇAK

2. Games in Extensive Form
The extensive form of a game is a complete description of:
A game tree
A list of players
Who moves when and what their choices are
A set of allowable actions at each node
The players’ payoffs as a function of the choices that are made.

4. In the first game tree we can see how player 1 is the first to decide, while player 2 will make a decisions after observing what player 1 has decided. The payoffs represented at the end of each brand represent all possible outcomes. For instance, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs will be p1A,p2B.

5. Player 2 Player 1 Player 1 Up Down Player 2 Player 2 - - - - - Left
Right
Left
Right
3,1
0,0
2,2
2,2
Player 2
Left
Right Up
3,1
0,0
Player 1
Down
2,2
2,2

6. Games in Extensive Form
(3,-1)
(2,2)
(3,4)
(1,3)
(0,-1)
(-2,0)
(5,-2)
(3,8)
(4,2)
(1,2)
(0,4)
(1,-2) II D S
(0,1)
(5,5)
(2,-8)
(7,-3)

7. DRAUGHTS / CHECKERS Rules of Game:
This game is a two player board game.
Each players must choice one type of stone (white or black stone) to play
Both players must move on only dark squares on the board.

8. Players take turns to move a piece of their own colour.
Stones must move only diagonally into an unoccupied square.
If the players’ stone is next to the other players’ stone and the black square behind players’ stone is open, players must jump over it.
After jump the other players’ stone remove from the board and the players captured the other players’ stone.

9. Winning the game :
A player wins the game when they capture all of the other players’ stones or when the player block other player completely and the other player cannot move.
Players may agree to a drow if nether player can win the game.

11. 6 5 8 4 2 1 7 3 …… Winner: Winner: 3 ->4 3 -> 5 6 ->5 4->6
1 ->3
2 ->3
8->6
7->6
Winner:
Winner:

12. 6 5 8 4 2 1 7 3 …… …… …… Winner: 3 ->4 3 -> 5 6 ->5 4->6
1 ->3
2 ->3 …… ……
5->1
Winner:

13. 3 4
3 5
6 5
6 4
5 6
1 3
2 3
1 3
4 6
2 3
7 5
5 1
8 6
winner
8 4
8 6
76
4 2
76
winner
winner
winner
1 3
2 3
5 7
4 8
winner
winner
5 1
4 2
winner
winner
winner

14. The Definition of Centipede Game
The Centipede Game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion.  

15. Centipede Game Either stop or pass to other player
If pass to other player, the total size of the pie(sum of payoffs) doubles
If stop, get 4/5 of pay and the other player gets 1/5

16. Perfect Information
All players know the game structure.
Each player, when making any decision, is perfectly informed of all the events that have previously occurred.

17. Imperfect Information
All players know the game structure.
Each player, when making any decision, may not be perfectly informed about some (or all) of the events that have already occurred.

18. Perfect and imperfect information
Game 2
Imperfect Information
Game 1
Perfect Information I II
4, 6
1, 0
3, 2
Information Set I II
3, 2
4, 6
1, 0 A B A B A B A B A B A B
3, 2
4, 6
1, 0 II I

19. Perfect and Imperfect InformationII
2, 1
5, 2
4, 3
6, 1
Information Set T B L R L R T
4, 3
6, 1 B
2, 1
5, 2 II I

20. I S D II II S D S D I I I I S D S D S D S D
5, 2
6, 1
3, 2
5, 4
-1, 6
0, 0
-3, 4
5, -2 S D
S S
5, 2
3, 2
S D
D S
D D II I
6,1
5,4
-1,6
-3,4
0,0
5,-2

21. Perfect Memory
A game is of perfect memory if each player remembers his previous actions and informations
Imperfect Memory
Each player may not be perfectly remembers about some(or all) events

22. Perfect and imperfect memoryB II II
3, 2 I
3, 1
0,4 C D E F I
Information Set
3, 2 G H G H
3, 2
5,1

23. Complete information
The complete information in games knowledge about other market participants or players is available to all participants.
Every player knows the payoffs and strategies available to other players.

24. Incomplete information
If some players don’t know the rules of the game, e.g.,
players’ preferences
available actions
identity or number of players
ordering of decisions

25. Complete and incomplete information
1/3
2/3 I I S D S D II II II II S D S D S D S D
a, b
c, d
e, f
g, h
i, l
m, n
o, p
q, r S D II I
1/3 a + 2/3 i, 1/3 b + 2/3 l
1/3 c + 2/3 m, 1/3 d + 2/3 n
1/3 e + 2/3 o, 1/3 f + 2/3 p
1/3 g + 2/3 q, 1/3 h + 2/3 r

26. Complete and incomplete information
1/5
4/5 I I S D S D II II II II S D S D S D S D
2, 5
7, -2
0, 3
-1, 4
5, 5
3, 2
-3, 1
0, -2 S D II I
1/52 + 4/55, 1/55 + 4/55
1/57 + 4/53, 1/5(-2)+ 4/52
1/50 + 4/5(-3), 1/53 + 4/51
1/5 (-1) + 4/50, 1/54 + 4/5(-2)

27. Resources:
Game Theory Course
(Jackson,Leyton-Brown & Shohom
Game Theory Online
Game Theory.net
Mastergames.com
Introduction to Game Theory
(Steve Schecter)