d the Nash-Equilibrium for the following payoff

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Assignments
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Date: 21st April, 2015
Course “Computational Intelligence in Games”
Prof. Dr. Sanaz Mostaghim
Intelligent Systems Group
layer%game.%Both%of%the%players%have%mixed%strategies:%
Faculty of Computer Science
1 1
x player 1 = ( , )
Assignment 1: 2 2
Consider a 2-Player
1 1game. Both of the players have mixed strategies:
xplayer
2
xplayer
1
=( , )
12 12
1 1
= ( , ) xplayer 2 = ( , )
2 2
2 2
heir%expected%payoff%values,%when%cooperaLng%or%
using%the%following%payoff%matrix:%%
a) Compute their expected payoff values, when cooperating or defecting using the following
payoff matrix:
 C2%%%%%%%%%D2%
C1%%%0, 0 1, 1
A = D1%
0, 0 1, 1
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ment%2%
b) What kind of a strategy can the player 1 play to increase his payoff?
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c) What kind of a strategy can the player 2 play to increase his payoff?
Assignment 2:
Consider a 2-Player prisoner’s Dilemma game with the following payoff matrix:
e%prisoner’s%dilemma%payoff%matrix:%
C2%%%%%%%%%D2%
3, 3 0, 5
A = C1%%%
D1% 5, 0 1, 1
nment%3%
What strategy profile is a Nash-Equilibrium?
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egy%profile%is%the%Nash@Equilibrium?%
Assignment 3:
e%Nash@Equilibrium%for%the%following%payoff@matrices:%%
Find the Nash-Equilibrium for the following payoff matrices:
C2%%%%%%%%%D2%

est%Response%
1, 1
A = C1%%%
esponse%to%s-i%if%and%only%if%
D1%
⇡si ,s
i
⇡s0 ,s
1, 0
2, 1 0, 4
8s0 2 S
i
Sanaz%Mostaghim%
 %%%%%%C2%%%%%%%%%%%%%%%%%D2%
2, 1
1, 1
C = C1%%%
D1% 1, 1
1, 2
©%Sanaz%Mostaghim%
C2%%%%%%%%%D2%

Nash%Equilibrium%%
1, 1 1, 0
profile%S = (s1B
,...,=sn)C1%%%
%is%a%Nash%Equilibrium%if%and%only%if%for%%%%
D1% 1, 1 0, 0
i,%his%strategy%si%is%a%best%response%to%s-i%
Assignment 4:
a) Write a payoff matrix for scissors-rock-paper game.
b) Write a payoff matrix for rock-paper-scissors-lizard-spock game with the following rules:
• Scissors cuts Paper
• Paper covers Rock
• Rock crushes Lizard
• Lizard poisons Spock
• Spock smashes Scissors
• Scissors decapitates Lizard
• Lizard eats Paper
• Paper disproves Spock
• Spock vaporizes Rock
• (and as it always has) Rock crushes scissors
Assignment 5:
Which of the following payoff matrices has the “C” as an ESS? Why?
C  C D  C D 5, 5 2, 5
B = D 5, 2 1, 1
C Biomass
6, 6 0, 5
A 30
= D 5, 0 1, 1
25
 C D  C D C C 1,
1
2,
1
1, 1 0, 1
C 15
= D = 1,Evolutionary
15
324 10 D 1, 2
ecology D 1, 0
1, 1
0
10
20
Density
Resources
3
Nutrient
Sunlight
1000
2
Assignment
6:
1
500
Analyze
the following replicator equations:
0
0
0.8
15
0
0
5
100
200
Normal
Cancer
10
300
400
500
15
600
700
800
Resources
Strategy
0.7
2400
x˙ 2200
0.6
0.5
2000
0.4
1800
10
15
Time
0
100
200
300
400
500
600
700
800
2Figure 10.2 The solution obtained in the previous example is found to be convergent stable.
Normal
Strategy
1600
x˙
0
1.5
0
1
−0.1
0.5
0
−0.2
5
Cancer
100
200
300
400
Time
500
600
700
800
G(v,u,x* )
Figure
10.8 After evolutionary constraints have been removed, cancer develops
−0.3
rapidly in the first year.
−0.4
−0.5
10.3 Plant ecology
−0.6
10.3.1 Flowering time for annual plants
Cohen (1971, p. 10) developed a model for modeling the flowering time for a
−0.7
single annual flowering plant. During the growing season of length T , the plant