Game Theoretical Models in
Biology 2nd Edition
ST
SOLUTIONS
UV
IA
MANUAL
_A
PP
Mark Broom
RO
Jan Rychtář
────────────────────────────────────────────────────
Comprehensive Solutions Manual for
VE
Instructors and Students
D?
© Mark Broom & Jan Rychtář. All rights reserved. Reproduction or distribution without
permission is prohibited.
?
© DreamsHub
, Solutions Manual for Game-Theoretical Models in Biology (2nd Edition)
Mark Broom & Jan Rychtář
ISBN: 9781003024682
ST
UNIT 1: FOUNDATIONS OF GAME THEORY IN BIOLOGY
1. Introduction
2. What Is a Game
3. Two Approaches to Game Analysis
UV
UNIT 2: CLASSICAL AND MATHEMATICAL GAME STRUCTURES
4. Some Classical Games
5. The Underlying Biology
6. Matrix Games
IA
7. Nonlinear Games
UNIT 3: COMPLEX AND MULTI-AGENT GAME FRAMEWORKS
_A
8. Asymmetric Games
9. Multi-player Games
10. Extensive Form Games and Other Concepts in Game Theory
UNIT 4: EVOLUTIONARY DYNAMICS AND POPULATION
PP
STRUCTURE
11. State-Based Games
12. Games in Finite Populations and on Graphs
13. Evolution in Structured Populations
RO
14. Adaptive Dynamics
UNIT 5: COOPERATION, SOCIALITY, AND MATING STRATEGIES
15. The Evolution of Cooperation
16. Group Living
VE
17. Mating Games
18. Signalling Games
UNIT 6: ECOLOGICAL, EPIDEMIOLOGICAL, AND MEDICAL
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APPLICATIONS
19. Food Competition
20. Predator–Prey and Host–Parasite Interactions
21. Epidemic Models
22. Evolutionary Cancer Modelling
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UNIT 7: SYNTHESIS AND FUTURE DIRECTIONS
23. Conclusions
© DreamsHub
, Solutions Manual for Game-Theoretical Models in
Biology, 2e by Mark Broom, Jan Rychtář (All
Chapters)
2
Chapter 2
ST
2.1. We can define a pure strategy as a choice to play Rock, Scissor or Paper
at any position of the game. If the position of the game is determined by the
current score (number of rounds won, irrespective of the order of results), then
a pure strategy is a six-tuple (xp1 ,p2 ) where p1 ∈ {0, 1, 2} is the score of Player
1 and p2 ∈ {0, . . . , 2 − p1 } is a score of Player 2. If a position of the game
UV
is given by the current score and the round number, then a pure strategy is
a countable collection (xp1 ,p2 ,n ) p1 ∈ {0, 1, 2, 2} is the score of Player 1 and
p2 ∈ {0, . . . , 2 − p1 } is a score of Player 2 and n = p1 + p2 , p1 + p2 + 1, . . . is
the round number.
2.2. Let CP be the cost of colicin production, CI be the cost of being immune
IA
ad H > CP + CI be the harm. We then have
Producing Immune Neither
Producing −(CP + CI ) −(CP + CI ) H − (CP + CI )
Immune −CI −CI −CI .
_A
Neither −H 0 0
Adding CP + CI to the first collum and CI to the second column yields
Producing Immune Neither
Producing 0 −CP H − (CP + CI )
PP
Immune CP 0 −CI .
Neither CP + CI − H CI 0
2.3. Let pXY be payoff to Paul if he went to bar
X while John
went to bar
pAA pAB
RO
Y . The payoff matrix to Paul is then given by where, pAA >
pBA pBB
pBB > pAB ≈ pBA . Similarly for John.
2.4. Let E be the option for Paul to go somewhere else. Let jXY be the payoff
to John
if he went to bar X while Paul went to bar Y . The payoff for John is
jAA jAB jAE
then where jAA > jBB > jAB ≈ jBA ≈ jAE ≈ jBE . The
VE
jBA jBB jBE
pAA pAB
payoff to Paul is given by pBA pBB where pBA = pAB > pEA = pEB >
pEA pEB
pAA = pBB .
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, 3
(0, 0, 1)
(0, 1/3, 2/3)
(0, 1/2, 1/2)
(1/3, 1/3, 1/3)
(1/10, 3/5, 3/10)
(2/5, 2/5, 1/5)
ST
2.5. (1, 0, 0) (0, 1, 0)
P
2.6. If a σ-strategist
P plays in the population αj δpj where pj is a mixed
strategy
P p j = p S
i j,i i , then the σ-strategist plays against Si with probability
α p
j j j,i . It is thus the same as playing in the population δp̄ .
UV
2.7. E[p, q] = pAqT = (−p2 + p3 )q1 + (p1 − p3 )q2 + (−p1 + p2 )q3 = p1 (q2 −
q3 ) + p2 (q3 − q1 ) + p3 (q1 − q2 ). Let m = max{q2 − q3 , q3 − q1 , q1 − q2 }. The
best reply to q is (1, 0, 0) if q2 − q3 = m; it is (0, 1, 0) if q3 − q1 = m; and it is
(0, 0, 1) if q1 − q2 = m.
IA
2.8. (i) generic, (ii) technically non-generic but does not affect the analysis,
(iii) non-generic, (iv) generic, (v) technically non-generic but does not affect
the analysis, (vi) technically non-generic but does not affect the analysis, (vii)
non-generic, (viii) non-generic.
_A
2.9. By Exercise 2.6, the payoffs are the same as if the game is played against
an individual playing p = 0.4(0.5, 0.5) + 0.3(1, 0) + 0.3(0.2, 0.8) = (0.56, 0.44).
We thus get E[(0.5, 0.5); Π] = (0.5, 0.5)A(0.56, 0.44)T = 0.4 and similarly for
other strategies, giving payoffs 0.56 and 0.464.
PP
2.10. For the matrix games, the mean strategy is as in Exercise 2.9. p =
0.4(0.5, 0.5) + 0.3(1, 0) + 0.3(0.2, 0.8) = (0.56, 0.44). If the opponent is se-
lected based on its probability of playing S1 the mean strategy is pe =
0.5 1 0.2
0.5+1+0.2 (0.5, 0.5) + 0.5+1+0.2 (1, 0) + 0.5+1+0.2 (0.2, 0.8).
RO
Chapter 3
β=0 β=1 β=10 β=100
0.5 0.5 0.5 0.5
VE
0.3
p
0.3
p
0.3
p
0.3
p
0.1 0.1 0.1 0.1
0 1 2 3 0 5 10 0 50 100 0 500 1000
3.1. Generation Generation Generation Generation
D?
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Biology 2nd Edition
ST
SOLUTIONS
UV
IA
MANUAL
_A
PP
Mark Broom
RO
Jan Rychtář
────────────────────────────────────────────────────
Comprehensive Solutions Manual for
VE
Instructors and Students
D?
© Mark Broom & Jan Rychtář. All rights reserved. Reproduction or distribution without
permission is prohibited.
?
© DreamsHub
, Solutions Manual for Game-Theoretical Models in Biology (2nd Edition)
Mark Broom & Jan Rychtář
ISBN: 9781003024682
ST
UNIT 1: FOUNDATIONS OF GAME THEORY IN BIOLOGY
1. Introduction
2. What Is a Game
3. Two Approaches to Game Analysis
UV
UNIT 2: CLASSICAL AND MATHEMATICAL GAME STRUCTURES
4. Some Classical Games
5. The Underlying Biology
6. Matrix Games
IA
7. Nonlinear Games
UNIT 3: COMPLEX AND MULTI-AGENT GAME FRAMEWORKS
_A
8. Asymmetric Games
9. Multi-player Games
10. Extensive Form Games and Other Concepts in Game Theory
UNIT 4: EVOLUTIONARY DYNAMICS AND POPULATION
PP
STRUCTURE
11. State-Based Games
12. Games in Finite Populations and on Graphs
13. Evolution in Structured Populations
RO
14. Adaptive Dynamics
UNIT 5: COOPERATION, SOCIALITY, AND MATING STRATEGIES
15. The Evolution of Cooperation
16. Group Living
VE
17. Mating Games
18. Signalling Games
UNIT 6: ECOLOGICAL, EPIDEMIOLOGICAL, AND MEDICAL
D?
APPLICATIONS
19. Food Competition
20. Predator–Prey and Host–Parasite Interactions
21. Epidemic Models
22. Evolutionary Cancer Modelling
?
UNIT 7: SYNTHESIS AND FUTURE DIRECTIONS
23. Conclusions
© DreamsHub
, Solutions Manual for Game-Theoretical Models in
Biology, 2e by Mark Broom, Jan Rychtář (All
Chapters)
2
Chapter 2
ST
2.1. We can define a pure strategy as a choice to play Rock, Scissor or Paper
at any position of the game. If the position of the game is determined by the
current score (number of rounds won, irrespective of the order of results), then
a pure strategy is a six-tuple (xp1 ,p2 ) where p1 ∈ {0, 1, 2} is the score of Player
1 and p2 ∈ {0, . . . , 2 − p1 } is a score of Player 2. If a position of the game
UV
is given by the current score and the round number, then a pure strategy is
a countable collection (xp1 ,p2 ,n ) p1 ∈ {0, 1, 2, 2} is the score of Player 1 and
p2 ∈ {0, . . . , 2 − p1 } is a score of Player 2 and n = p1 + p2 , p1 + p2 + 1, . . . is
the round number.
2.2. Let CP be the cost of colicin production, CI be the cost of being immune
IA
ad H > CP + CI be the harm. We then have
Producing Immune Neither
Producing −(CP + CI ) −(CP + CI ) H − (CP + CI )
Immune −CI −CI −CI .
_A
Neither −H 0 0
Adding CP + CI to the first collum and CI to the second column yields
Producing Immune Neither
Producing 0 −CP H − (CP + CI )
PP
Immune CP 0 −CI .
Neither CP + CI − H CI 0
2.3. Let pXY be payoff to Paul if he went to bar
X while John
went to bar
pAA pAB
RO
Y . The payoff matrix to Paul is then given by where, pAA >
pBA pBB
pBB > pAB ≈ pBA . Similarly for John.
2.4. Let E be the option for Paul to go somewhere else. Let jXY be the payoff
to John
if he went to bar X while Paul went to bar Y . The payoff for John is
jAA jAB jAE
then where jAA > jBB > jAB ≈ jBA ≈ jAE ≈ jBE . The
VE
jBA jBB jBE
pAA pAB
payoff to Paul is given by pBA pBB where pBA = pAB > pEA = pEB >
pEA pEB
pAA = pBB .
D?
?
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, 3
(0, 0, 1)
(0, 1/3, 2/3)
(0, 1/2, 1/2)
(1/3, 1/3, 1/3)
(1/10, 3/5, 3/10)
(2/5, 2/5, 1/5)
ST
2.5. (1, 0, 0) (0, 1, 0)
P
2.6. If a σ-strategist
P plays in the population αj δpj where pj is a mixed
strategy
P p j = p S
i j,i i , then the σ-strategist plays against Si with probability
α p
j j j,i . It is thus the same as playing in the population δp̄ .
UV
2.7. E[p, q] = pAqT = (−p2 + p3 )q1 + (p1 − p3 )q2 + (−p1 + p2 )q3 = p1 (q2 −
q3 ) + p2 (q3 − q1 ) + p3 (q1 − q2 ). Let m = max{q2 − q3 , q3 − q1 , q1 − q2 }. The
best reply to q is (1, 0, 0) if q2 − q3 = m; it is (0, 1, 0) if q3 − q1 = m; and it is
(0, 0, 1) if q1 − q2 = m.
IA
2.8. (i) generic, (ii) technically non-generic but does not affect the analysis,
(iii) non-generic, (iv) generic, (v) technically non-generic but does not affect
the analysis, (vi) technically non-generic but does not affect the analysis, (vii)
non-generic, (viii) non-generic.
_A
2.9. By Exercise 2.6, the payoffs are the same as if the game is played against
an individual playing p = 0.4(0.5, 0.5) + 0.3(1, 0) + 0.3(0.2, 0.8) = (0.56, 0.44).
We thus get E[(0.5, 0.5); Π] = (0.5, 0.5)A(0.56, 0.44)T = 0.4 and similarly for
other strategies, giving payoffs 0.56 and 0.464.
PP
2.10. For the matrix games, the mean strategy is as in Exercise 2.9. p =
0.4(0.5, 0.5) + 0.3(1, 0) + 0.3(0.2, 0.8) = (0.56, 0.44). If the opponent is se-
lected based on its probability of playing S1 the mean strategy is pe =
0.5 1 0.2
0.5+1+0.2 (0.5, 0.5) + 0.5+1+0.2 (1, 0) + 0.5+1+0.2 (0.2, 0.8).
RO
Chapter 3
β=0 β=1 β=10 β=100
0.5 0.5 0.5 0.5
VE
0.3
p
0.3
p
0.3
p
0.3
p
0.1 0.1 0.1 0.1
0 1 2 3 0 5 10 0 50 100 0 500 1000
3.1. Generation Generation Generation Generation
D?
?
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