USING A GAME THEORETICAL APPROACH FOR
EXPERIMENTAL SIMULATION OF BROOD REDUCTION
Conflict and co-operation, effect on brood size with limited resources
Fredrik Åhman, Lars Hillström
Department of mathematics natural and computer sciences, University of Gävle, 801 76 Gävle, Sweden
Keywords: ESS, agent, brood reduction, fitness, intra-familiar conflicts
Abstract: A number of hypothesis have been presented to explain the complex interactions occurring during brood
reduction, but few simulation models successfully combines hypothesis together necessary to describe
evolutionary stable strategies. In our solution we present a simple experimental simulation for brood
reduction for which each sibling act as an autonomous agent that has the ability to initiate actions for co-
operation and competition against others chicks within the same brood. Agents have a limited set of actions
that can be activated during the onset of some environmental condition. Parameters for food distribution are
determined on a basis of a former known theory for maximizing inclusive fitness. During the experimental
simulations we have studied size and fitness measures with varying degree of asynchrony and prey density
for siblings within the artificial brood. Results from the experimental simulation shows interesting
similarities with brood reduction in a real world setting. Agents within the artificial brood respond with
competitiveness whenever resources are limited. Simulated later hatching also showed a lower rate of
survival because of natural size hierarchy to co-siblings within the simulated brood.
1 INTRODUCTION
A number of hypothesis have been presented to
explain causes of brood reduction but few simulation
models successfully combines important parameters
from each hypothesis together necessary to describe
evolutionary stable strategies. In intra-familiar
conflicts there is always a trade off between degree
of selfishness and altruistic behaviours serving the
benefits to the next sibling in the brood (Krebs,
Davies, 1984). This trade-off is also well known
from the Hamilton equation (1) (Bergstrom ref to
Hamilton, 2000) where relatedness ( r ) and benefits
(b) is compared to cost for co-operation (c).
0>− crb
(1)
Mock et al, 1998 describes an interesting
optimization problem for fitness in which the degree
of selfishness is balanced against the interest of
maximizing inclusive fitness for the whole brood.
Mocks theory describes a simplistic relationship
between total parental investment (M) and portions
of (M) for siblings (m
A
), (m
B
). If portions of prey
could do twice as much use for one of the co-
siblings within the same brood its in the dominant
chicks best interest to pass that portion on to the next
sib. Dependent on relatedness between chicks the
portion of the meal passed on can vary in size. Mock
et al, 1998 shows optimisation for full siblings
which means that relatedness is always 0.5.
In a brood with two siblings portions of parental
investment (PI) can be split into p*M for the
dominant chick A and (1-p)*M for the subordinate
chick B. Mock tries to find a value (p) that maximize
fitness for the most dominant chick. The fitness
curve f(m) (2) describes an exponential relationship
between parental investment (m) and fitness f(m).
Increase is very in the beginning of the curve but
will slow down as the amount of investment
approaches maximum amount (M).
min))(1exp(1)( mmkmf −
−
−
=
(2)
Mock believe that there is a value of portion (p)
in which the remains (1-p) comes in better use for
the co-sibling B. The equation describes a
relationship between the first partial order derivate
where increase of fitness for the dominant A-chick
should be equivalent to the double increase of fitness
for the B-chick (3).
220
Åhman F. and Hillström L. (2005).
USING A GAME THEORETICAL APPROACH FOR EXPERIMENTAL SIMULATION OF BROOD REDUCTION - Conflict and co-operation, effect on
brood size with limited resources.
In Proceedings of the Seventh International Conference on Enterprise Information Systems, pages 220-225
DOI: 10.5220/0002539502200225
Copyright
c
 SciTePress

)('2/1)('
BA
mfmf =
(3)
1.1 Conflicting versus co-operative
behaviours
Combining conflicting behaviours with co-operating
behaviours such as portioning foods to co-siblings
seems to be an appealing game theoretical approach.
In order to maximize fitness individual interests are
likely to be balanced to the benefit of the whole
brood. Environmental changes such as variation in
prey density and climate variation demands good
strategies for survival (Temme, Charnov, 1987).
When food abundance is low resources must be
carefully invested to maximize reproductive output
(Ploger, Mock, 1986).
Controlled brood reduction through hatching
asynchrony is often initiated by parents when
environmental conditions are less beneficial
(Vinuela, 2000; Lamey, Mock, 1991). Hatching
asynchrony plays a major role in sibling size
hierarchy where the first hatched chick often has
total control of food resources due to its large size.
Sibling competition will be less frequent because of
the natural dominance of the elder chick (Lamey,
Mock, 1991).
1.2 Theoretical framework for the
experimental simulation
Our aim with this work is to build an experimental
simulation model for brood reduction. The
underlying theory for the game theoretical approach
is based on Mock & Parkers earlier work on intra-
familiar parent-offspring conflict (Mock et al, 1998).
Our experimental model is based on a number of
assumptions that is originally inspired by the food
and egg viability hypothesis that effects hatching
asynchrony (Vinuela, 2000).
We view each sibling is as an agent able to make
its own decisions for competition and food
ingestion. The agent behaviours is controlled by its
characteristics such as aggressiveness and strategy
selection for competitive games.
2 METHODS
We have chosen to evaluate the agents response to
three different variables namely access to prey,
hatching asynchrony and rankings within the sibling
hierarchy. Each agent is activated by starting a new
thread for the corresponding chick. In our game
simulation sibling hierarchy is dependent on the
number of winnings in conflicts. Chicks winning
many conflicts will conquer favourable positions
during feeding. The dominant chick is always
controlling the amount of food distributed similar to
(Mock et al, 1998).
After each feeding data is stored about the chicks
size, fitness, minimum mass for starvation and
ranking. Feeding is repeated iteratively until chick is
ready for fledging or may have deceased due to
starvation. The simulation procedure is repeated ten
times for each parameter setting.
2.1 The simulation platform
We have chosen to implement the brood simulation
platform in Java. The Java language is object
oriented and supports threading which is needed if
we want agents within our simulation environment
to act independently. A maximum number of three
siblings act simultaneously in our nestling
environment. Each sibling can initiate two basic
behaviours namely consume() and compete().
Winning a competition means higher ranking.
Higher ranking means higher probability for food.
Figure 1: Agent interaction with two basic behaviours
{consume, compete}
From a game theoretical perspective we can view
the consumption of food as a simple
consumer/producer problem where we have a
number of consumers that are dependent on a
limited resource. In our simulated environment food
items is delivered regularly. Upon delivering food
the most dominant contestant will take its share first
and then left the remains its co-siblings.
2.2 Game theory and strategy
selection
For each game contestants can choose to alter
between different strategies for survival. Common
strategies are transgress or retreat. Each chosen
strategy has its obvious payoffs and costs dependent
on the outcome of the game.
USING A GAME THEORETICAL APPROACH FOR EXPERIMENTAL SIMULATION OF BROOD REDUCTION -
Conflict and co-operation, effect on brood size with limited resources
221

0
)](
2
1
)(
2
1
)([
)(
2
1
)(
2
1
)(),,(
=
∂
++∂
++=
p
mfmfmf
mfmfmfmmmf
CBA
CBACBA
In a game where each opponent can choose
between two strategies four combination of payoffs
are possible as seen in (Krebs, Davies 1984). If one
player choose to retreat ( R ) the cost for that player
will be zero or minimum even if he looses the game.
The same goes for the opponent who is winning the
game. However if both players choose to transgress
the cost for each player will affect their immediate
fitness with cost –c1, -c2. Gains during winning are
described as g1, g2.
A winning in our simple game means higher
ranking and therefore better positioning within the
nest whenever prey is delivered. A dominant chick
in our game determines how food portions to pass on
to other subordinate chicks in order to maximize its
own inclusive fitness. This works exactly the same
as the method used for portioning prey as seen in
(Mock et al 1998).
Figure 2: Payoffs in two-player games with fixed
strategies
The following equations is taken from (Mock et al
1998).
1. For two sibling games :
A:s portion of prey is (p)
B:s portion of prey is (1-p)
(4)
(5)
2. For three sibling games :
A:s portion of prey: m
A
=Mq
B:s portion of prey : m
B
=Mp(1-q)
C:s portion of prey : m
´C
=M(1-p)(1-q)
(6)
In extension to the two above equations a logistic
sigmoid function was added to calculate mass m(w)
(7) with respect to intake of energy (w) which is the
transfer function for most birds (Ricklefs 1969).
Both (L) and k are constants representing bias and
increase in growth.
(7)
wk
L
e
wm
*
1
1
)(
−
+
=
3 THE PROBABILISTIC MODEL
As our model is based on former known theories
about sibling rivalry and intra-familiar conflicts we
need to state predictions about how the agent will
respond to certain conditions within the
environment. Such events could be lower resource
such as lower prey intensity, lower size advantage
etc.
3.1 Sibling size hierarchy and
likelihood to win a conflict
Siblings that have a size advantage is more likely to
win a conflict because of less resistance from the
minor chick (Smith, Graeme, Crosswell 2001)..
Sibling size hierarchy is also of major importance in
hatching asynchrony where younger chicks are
sometimes doomed to starvation by selfish older and
much bigger chicks as seen in (Vinuela 2000). In
our simulation we put this as a proportional measure
(8) between the sum of chicks total weight
m(i)+m(j) in relation to the contestants weight m(i)
)(
2
1
)(),(
BABA
mfmfmmf +=
)()(
)(
),(
jmim
im
jipw
+
=
(8)
0
)](
2
1
)([
=
∂
+∂
p
mfmf
BA
Larger chicks will have a natural size advantage
to minors in proportion pw(i,j) (8). Using the
probability measure as input to our stochastic
process we rank each chick accordingly to winnings
or loss for each game. Rankings are later used to
determine most dominant chick upon the event of
feeding.
Probability for ranking as most dominant (9) is
determined by the dominant chicks ranking in
relation to the remaining chicks within the same
brood. Rankings are always set at a bias level in the
ICEIS 2005 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
222

beginning of the simulation. Bias is needed to insure
non-negative ranking values due to increased loss.
Growth for chick A,B,C with prey density 0.25 of
maximum (M)
0
200
400
600
800
1000
1
11 21 31 41 51 61 71 81 91
101 111 121
Days
Mass (g)
Serie1
Serie2
Serie3
Expected fitness for chick A,B,C with prey
density 0.25 of maximum (M)
0
0,5
1
1,5
1
14 27 40 53 66 79 92
105 118
Da ys
Fitness
Ser ie1
Ser ie2
Ser ie3
)()(
)(
),(
jrir
ir
jipdom
+
=
(9)
If stochastic process for determining domination
gives advantage to chick (i) in both cases {chick A-
Chick B} and {Chick A-Chick C} then chick (i) will
win domination game and get the best position for
next feeding round. If no chick is winning more than
one domination game the chick with the highest
ranking will be ranked as dominant.
4 RESULTS
4.1 Evaluation of Parkers
optimisation algorithm for
inclusive fitness
The purpose of the first test aims to show how food
is distributed by the dominant chick holding the
criteria for maximization of inclusive fitness. When
the first dominating chick is leaving for fledging the
simulation shifts from three sib co-operation to two
sib co-operation.
Treshold for fledging is set to 800 grams. Initial
weight for each chick is 50 grams. The time scale
represents the number of days within the nest until
fledging. Domination stays the same throughout the
whole test. Chick A is dominating food distribution
from start. Simulation were performed with altered
prey density. Graph 4.1.1 shows growth and fitness
rate with prey density 0.25. Graph 4.1.2 shows
growth and fitness rate with prey density 0.5.
4.1.1 Observations during the test
In the first test with prey density 0.25 Chick C is
likely to suffer from starvation considering the slow
growth rate. As expected the mass curve for chick B
is much closer to A than C. However mass and
fitness curves for prey intensity 0.5 shows higher
increase of mass for chick C. As food abundance
increase portions will be shared more equally
between chicks.
Graph 4.1.1: Growth and fitness for Chick A, B, C with
low prey density k= 0.25
.
Growth for chick A,B,C with prey density 0.5
0
200
400
600
800
1000
1 7
13 19 25 31 37 43 49 55 61
Da ys
Mass (g)
Serie1
Serie2
Serie3
Expected fitness for chick A,B,C with prey densit
y
0.5 of maximum (M)
0
0,2
0,4
0,6
0,8
1
1,2
1 6
11 16 21 26 31 36 41 46 51 56 61 66
Serie1
Serie2
Serie3
Graph 4.1.2: Growth and fitness for Chick A,B, C with
medium prey density k=0.5
4.2 Simulating synchronized hatching
with medium prey density (k=0.5)
In this test algorithms for conflicting behaviours
were activated which means that rankings were
altered dependent on winnings through out the
simulation. Shifted rankings also means that
domination were altered amongst sibs during
feeding. Low food abundance resulted in starvation
shifting from 10-30% amongst siblings within the
brood. Graph 4.2.1-Graph 4.2.2 shows mass, fitness
and rankings for all three siblings. Note that
deviation listed in the table represents a mean value
for all deviations acquired during all ten simulations.
Table I: Synchronized hatching with medium prey density
k=0.5. Running 10 simulations
Chick Fledging
(days)
Fledged Starved Ranking
(mean)
Ranking
(dev)
A 45.75 8 2 20.5 1.97
B 48.67 7 3 19.49 1.243
C 48.56 9 1 20.48 2.06
Growth for synchronized birth with
medium prey density (K = 0.5)
0
500
1000
1 5 9
13 17 21 25 29 33 37 41
Days
Mass (g)
Serie1
Serie2
Serie3
Fitness for chick A,B,C synchronized birth with
medium prey density (K = 0.5)
0
0,5
1
1,5
1 6
11 16 21 26 31 36 41
Days
Fitness
Ser ie 1
Ser ie 2
Ser ie 3
Graph 4.2.1: Growth and fitness for synchronized hatching
with medium prey density k= 0.5
USING A GAME THEORETICAL APPROACH FOR EXPERIMENTAL SIMULATION OF BROOD REDUCTION -
Conflict and co-operation, effect on brood size with limited resources
223

Graph 4.2.2: Rankings for synchronized hatching with
prey density k=0.5
4.3 Simulating hatching asynchrony
with medium prey density (k=0.5)
In this test each agent was activated asynchronous
with a time delay of 3 tics in between each chick. In
practical terms this means starting threads in Java
asynchronous. Each tic corresponds to one day in a
reality setting. The results shows a definite
starvation for the latest hatched chick C which also
has the smallest size advantage in proportion to A, B
upon hatching. As expected chick A gets the highest
ranking followed by B and C. Growth, fitness and
rankings are shown in graph 4.3.1-4.3.2.
Table II: Hatching asynchrony with medium prey density
k=0.5. Running 10 simulations
Chick Fledging
(days)
Fledged Starved Rank
(mean)
Ranking
(deviation)
A 46,9 10 0 21,28 2,1183
B 49,9 10 0 19,74 1,8052
C (-) 0 10 19,50 0,786
Growth Chick A,B,C - Asynchronized
hatching, medium prey density (K = 0.5)
0
500
1000
1 8
15 22 29 36 43 50
Days
Mass (g)
Serie1
Serie2
Serie3
Fitness Chick A,B, C - Asynchronized
hatching, medium prey density (K =
0.5)
0
0,5
1
1,5
1 7
13 19 25 31 37 43 49
Days
Fitness
Serie1
Serie2
Serie3
Graph 4.3.1: Growth and fitness for hatching
asynchrony with medium prey density k=0.5.
Rankings Chick A,B,C - Asynchronized
hatching, medium prey
density (K = 0.5)
0
20
40
1 7
13 19 25 31 37 43 49
Days
Ranking
Serie1
Serie2
Serie3
Graph 4.3.2: Rankings for hatching asynchrony with
medium prey density k=0.5
4.4 Simulating hatching asynchrony
with high prey density (k= 0.7)
In the fourth test hatching asynchrony was
performed for higher prey density 0.7. With higher
food abundance all chicks reached to the state of
fledging. The growth rate was also higher for all
three siblings. Fledging for chick A only took 30,2
days, chick B 34,2 days and chick C 39,6 days on
average. This is of course just a theoretical measure.
Growth, fitness and rankings are shown in graph
4.4.1-4.4.2.
Rankings for Chick A, B, C with synchronized birth
and prey intensity 0.5
0
10
20
30
1 5 9
13 17 21 25 29 33 37 41
Days
Ranking
Serie1
Serie2
Serie3
Table III: Hatching asynchrony with high prey density
k=0.7. Running 10 simulations
Chick Fledging
(days)
Fledged Starved Ranking
(mean)
Ranking
(deviation)
A 30.2 10 0 21.66 1.514
B 34.2 10 0 20.31 1.461
C 39.6 10 0 17.34 2.197
Growth Chick A, B, C, Asynchronized hatching
high prey density (K = 0.7)
0
200
400
600
800
1000
1 4 7
10 13 16 19 22 25 28 31 34 37
Days
Mass (g)
Serie1
Serie2
Serie3
Fitness Chick A,B,C Asynchronized hatching,
high prey density (K = 0.7)
0
0,2
0,4
0,6
0,8
1
1,2
1 4 7
10 13 16 19 22 25 28 31 34 37
Days
Fitness
Serie1
Serie2
Serie3
Graph 4.4.1: Growth and fitness for hatching asynchrony
with high prey density k=0.7
Rankings for chick A,B,C, Asynchronized hatching,
high prey density (K = 0.7)
0
5
10
15
20
25
30
1 4 7
10 13 16 19 22 25 28 31 34 37
Days
Ranking
Serie1
Serie2
Serie3
Graph 4.4.2: Rankings for hatching asynchrony with high
prey density k=0.7
5 DISCUSSION
Mocks theory relies on the assumption that the most
dominant chick controls food distribution amongst
co-siblings. The theory also assumes that dominance
hierarchy is fixed through out the simulation. In our
simulation we have chosen to allow alterations of
dominance between siblings as a result of winnings
and loss. It is consistent with the results that there is
a correlation between rankings and the amount of
parental investment that the chick receives. In cases
of simulated hatching asynchrony the latest hatched
chick is likely to get a lower ranking because of the
natural size hierarchy and lower chances of winning
a food/dominance contest.
In a realistic setting it is likely that parents will
have a greater influence in portioning food partly
affected by begging patterns produced by each
offspring (Kölliker 2001). Begging models needs to
be considered in order to get a more accurate results
ICEIS 2005 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS
224

for actual distribution. The influence of begging
models is also a stressed in (Mock, Parker 1998) .
6 CONCLUSIONS
From the first simulations with three sib and two sib
co-operation it follows that Mocks principles of
inclusive fitness behaves as expected. When food
abundance is low the chick with the lowest rank is
likely to suffer from starvation. When agents were
started in synchrony performed simulations shows
that any chick may suffer from starvation if food
resources are poor. However, when agents were
started in asynchrony it is also consistent with the
results that the latest hatch sibling dies due to
starvation in most cases.
Our first experimental model for brood reduction
has shown some interesting results that could be
useful for simulation of incubation behaviours such
as hatching asynchrony but of course the model
needs further refinement in describing state and
policy variables for each agent. Further research is
needed to describe dominance/ranking and learning
processes for each agent. In our simulation we have
not yet considered important component of learning
strategies for competition. Conflicting behaviours
have a degree of learning and adaptation. Adaptation
to conflicting behaviours needs to be considered if
the experimental model for brood reduction should
be representative for any real world situation.
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Conflict and co-operation, effect on brood size with limited resources
225