CSC5350 Game Theory in Computer Science
Tutorial 2
CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905
(Exercise 18.2)
Formulate a first price auction as a strategic game and analyze its Nash equilibria. In particular, show that in all equilibria player 1 obtains
CSC5350 Game Theory in Computer Science
Tutorial 8
CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905
Outline
Please come and take your assignment in my office SHB 905. ffi SHB 905 Extensive game with imperfect information ga Strategies
Pure Mixed Behavioral
Nash
CSC5350 Game Theory in Computer Science
Tutorial 6
CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905
Outline
Repeated Games Iterated Prisoners Dilemma
Repeated Games
Let G be a strategic game. An infinitely repeated game game of G is an extensive game with perfe
CSC5350 Game Theory in Computer Science
Tutorial 4
CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905
Outline
Extensive game with perfect information Examples
D
H 0, 1 (1-c), (1-c)
T3 - Q.1(a)
D H
, 1, 0
Let (b*, b*) be the mixed strategy Nash equilibrium
u1(e(D)
CSC5350 Game Theory in Computer Science
Tutorial 1
CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905
Outline
Strategic game A real life example Nash equilibrium Examples
Strategic game
Consists of
A set of players N A set of actions Ai for each player i N For ea
Bargaining Set
Recall the definition of bargaining sets.
I have an objection ( y , S ) against you to x ! But I have a counterobjection ( z , T ) to your objection ( y , S ) against me !
i
j
DEFINITION. The bargaining set of a coalitional game with transf
A Majority Game
There are three people. N cfw_1, 2, 3 . The worth of the teams is: v(cfw_1,2, 3) 1. v(cfw_1,2) 1 , v(cfw_2, 3) 1, v(cfw_1, 3) 1. v(cfw_1) 0 , v(cfw_2) 0 , v(cfw_3) 0 . Q: What should be the payoff profile?
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v(cfw_1,2, 3) 1. v(
A Majority Game
Three people form a coalition to get a treasure. A team of 3 people gets all (1). A team of any 2 people gets 3 . 5 A team of any 1 person gets none (0).
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Coalitional Games with Transferable Payoff
There are three players. N cf
Subgame Perfect Equilibrium
We recall the definition equilibrium for extensive information.
DEFINITION. extensive
N , H , P ,(
of subgame games with
perfect perfect
The subgame perfect equilibrium of an game with perfect information
i
) is the strategy p
Extensive Games with Imperfect Information
Extensive games with imperfect information are extensive games in which the players are imperfectly informed about some or all of the choice that have already been made.
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EXAMPLE. 1 B l 1,2 r 0,0
L A
Infinitely Repeated Games
We recall that an infinitely repeated game of G N , ( Ai ),( i ) is an extensive game with perfect information and simultaneous moves N , H , P ,( in which H cfw_ (t 1 At ) A P( h) N for each nonterminal h H
* i
* i
)
is a prefe
Repeated Games
Consider the game of Iterated Prisoners Dilemma, in which players repeatedly engage in the Prisoners Dilemma (the constituent game).
Suppose the prisoners encounter for T round. What are Nash equilibria? Subgame perfect equilibria?
Page 1 o
Extensive Games
Some times games are played by n players, in the following manner: The game starts. Player i1 chooses and takes an action. Player i2 chooses and takes an action. Player i3 chooses and takes an action. The game continues until there is an o
Battle of the Sexes Revisited
Wife Boxing Husband Boxing Opera 2, 1 0, 0 Opera 0, 0 1, 2
There are two Nash equilibria: (B, B) and (O, O).
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Battle of the Sexes Revisited
Wife Boxing Husband Boxing Opera 2, 1 0, 0 Opera 0, 0 1, 2
What if the w
The Prisoners Dilemma
Player 2 Cooperate Defect Cooperate Player 1 Defect freedom & award freedom imprisonment imprisonment remission freedom & award remission freedom
If you are one of them, what will YOU do?
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Strategic Games
Prisoners Dilemm
P age 1 of 4
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CSC 5350 Assignment 3 1. (a) This game can be modeled as an extensive game with incomplete information as follows:
A
Y
R
ch pB B R R Y pC C Y R Y B pB
ch pC C R Y
(0,0,1)
(1,1,0)
(0,1,0)
(1,0,1)
(1,1,0)
(0,0,1)
(1,0,1)
(0,1,0)
i. N = cfw_A, B, C ii. H =
CSC 5350 Assignment 2 1.
2.
3. (a) N= cfw_1, 2 A1 = [1/2, 1] (The action means that player 1 cuts the cake into 2 pieces, and is the proportion of the larger piece. Suppose the cake is 1 unit.) A2 = cfw_L, S (L: player 2 chooses the larger piece. S: playe
CSC5350Assignment1
1. (a).
(b). (a1,a2,a3) (c) i.(a1,a2,a3) ii. Assume that an outcome which maximizes the total utilities of all players is not Pareto optimal. Then, there must be another outcome , such that at least one player i gets a higher payoff in
CSC 5350 Assignment 3
Due date: 7 December 2009 1. Amy, Betty and Cindy play the following game using one red box and one yellow box on the table. First, Amy puts a coin in either the red box or the yellow box. Betty and Cindy are not allowed to see which
CSC 5350 Assignment 2
Due date: 9 November 2008
1. Subgame perfect equilibrium can be defined using the one deviation property if the game has finite horizon. Show that this property does not hold for infinite horizon games. 2. Say that a finite extensive
CSC 5350 Assignment 1
Due date: 12 October 2009 1. Consider a 3-player strategic game G N ,( Ai ),(ui ) . Each player has three available actions: a1 , a2 and a3 . That is, A1 A2 A3 cfw_ a1 , a2 , a3 . Suppose in G , player 1 always receives a payoff of
CSC5350 Game Theory in Computer Science
Tutorial 9
Chen Wenhao whchen@cse.cuhk.edu.hk SHB 905
Outline
Beliefs Assessment Sequential equilibrium Coalitional Games Feasible payoff and Core
Beliefs
At an information set that contains more than one history
pl