A Review of Tournament Selection in Genetic Programming

Abstract

This paper provides a detailed review of tournament selection in genetic programming. It starts from introducing tournament
selection and genetic programming, followed by a brief explanation of the popularity of the tournament selection in genetic
programming. It then reviews issues and drawbacks in tournament selection, followed by analysis of and solutions to these
issues and drawbacks. It finally points out some interesting directions for future work.

Full text

A Review of Tournament Selection
in Genetic Programming
Yongshen g Fa n g 1and Jun Li2
1Department of Finance, Anhui Polytechnic University,
Wuhu City, Anhui, P.R. China
2Department of Information and Computing Science, Anhui Polytechnic University,
Wuhu City, Anhui, P.R. China
fbmy@ahpu.edu.cn
Abstract. This paper provides a detailed review of tournament selec-
tion in genetic programming. It starts from introducing tournament se-
lection and genetic programming, followed by a brief explanation of the
popularity of the tournament selection in genetic programming. It then
reviews issues and drawbacks in tournament selection, followed by anal-
ysis of and solutions to these issues and drawbacks. It finally points out
some interesting directions for future work.
1 Motivation
Accurately and objectively identifying the economic situation of students is the
most important step of funding poor students in tertiary educational organiza-
tions. For effectively implementing the funding policies, it requires a workable,
reasonable and scientifically sound assessment approach. Campus OneCard, the
platform of Digital Campus, accumulates large volume of complicated consump-
tion data from students. How to mine these consumption data in order to help ac-
curately and objectively identifying the economic situation of students becomes a
hot research topic. Genetic programming (GP) [1], one of the metaheuristic search
methods in Evolutionary Algorithms (EAs) [2], is based on the Darwinian natu-
ral selection theory. Its special characters make it a very attractive algorithm for
many real world problems, including data mining, finical prediction, image recog-
nition, and symbolic regression. Therefore, we intend to use GP in the student
consumption data mining project. To be able to apply GP in mining the student
consumption data successfully, it is necessary to understand the status of the cur-
rent research results of the most important operator — selection — in GP.
Selection is a key factor of affecting the performance of EAs. Commonly used
parent selection schemes in EAs include fitness proportionate selection [3], rank-
ing selection [4], and tournament selection [5]. The most popular parent selection
method in GP is tournament selection. Therefore, this paper focus on tournament
selection and gives a detailed review of tournament selection in GP in order to pro-
vide a thorough understanding of selecting behaviour, features and drawbacks in
tournament selection, as well as possible directions for our research work.
Z. Cai et al. (Eds.): ISICA 2010, LNCS 6382, pp. 181–192, 2010.
c
Springer-Verlag Berlin Heidelberg 2010

182 Y. Fang and J. Li
This paper is organised as follows: Section 2 briefly introduces GP; Section 3
introduces standard tournament selection and explains why it is so population
in GP, as well as formal modellings which describe its sampling and selection
behaviour; Section 4 presents issues related to the sampling strategy in standard
tournament selection, together with corresponding analyses and clarifications;
Section 5 discusses drawbacks related to selection pressure control in standard
tournament selection, followed by corresponding solutions; Section 6 concludes
the paper and shows possible directions for future work.
2 Genetic Programming
GP is a technique of enabling a Genetic Algorithm (GA) [3] to search a poten-
tially infinite space of computer programs, rather than a space of fixed-length
solutions to a combinatorial optimisation problem. These programs often take
the form of Lisp symbolic expressions, called S-expressions. The idea of applying
GAs to S-expressions rather than combinatorial structures is due originally to
Fujiki and Dickinson [6, 7], and was brought to prominence through the work of
Koza [1]. The S-expressions in GP correspond to programs which a user seeks to
adapt to perform some pre-specified tasks. The fitness of an S-expression may
therefore be evaluated in terms of how effectively it performs this task. GP with
individuals in S-expressions is referred to as tree-based GP.
GP also has other categorises based on the representations of individual, for
instance, linear structure GP [8–11] and graph-based GP [12–15]. Linear struc-
ture GP is based on the principle of register machines thus programs can be
linear sequences of instructions. Graph-based GP is suitable for the evolution of
highly parallel programs which effectively reuse partial results [16].
Briefly, to fulfill a certain task, GP starts with a randomly initialised pop-
ulation of programs. It evaluates each program’s performance using a fitness
function, which generally compares the program’s outputs with the target out-
puts on a set of training data (“fitness cases”). It assigns each program a fitness
value, which in general represents the program’s degree of success in achieving
the given task. Based on the fitness values, it then chooses some of the programs
using a stochastic selection mechanism, which consists of a selection scheme and
a selection pressure control strategy. After that, it produces a new population
of programs for the next generation from these chosen programs using crossover
(sexual recombination), mutation (asexual), and reproduction (copy) operators.
The search algorithm repeats until it finds an optimal or acceptable solution,
or runs out of resources. A much more comprehensive field guide to GP can be
found in [16].
3 Tournament Selection
The standard tournament selection randomly samples kindividuals with re-
placement from the current population of size Ninto a tournament of size k
and selects the one with the best fitness from the tournament. Therefore, the

A Review of Tournament Selection in Genetic Programming 183
selection process of the tournament selection consists of two steps: sample and
then select. Commonly used tournament sizes are 2, 4 and 7. In general, since
the standard breeding process in GP produces one offspring by applying mu-
tation to one parent and produces two offspring by applying crossover to two
parents, the total number of tournaments needed is Nat the end of generating
all individuals in the next generation.
Tournament selection has the following features compared with other selection
schemes:
–Its selection pressure can be adjusted easily.
–It is simple to code, efficient for both non-parallel and parallel architecture
[17].
–It does not require sorting the whole population first. It has the time com-
plexity O(N)[18].
The last two features make tournament selection very population in GP: 1) GP
is very computationally intensive, requiring a parallel architecture to improve its
efficiency; 2) It is common to have millions of individuals in a population when
solving complex problems [19], thus sorting a whole population is really time
consuming. The linear time complexity in tournament selection is therefore very
attractive for GP.
As the selection process of standard tournament selection consists of sampling
and selecting, there are a large number of research focusing on different sampling
and selecting strategies [20–24]. In addition to these practical studies, there are
many theoretical studies that model and compare the selection behaviour of a
variety of selection schemes [18, 25–29], as well as many dedicated theoretical
studies on standard tournament selection [17, 30, 31].
Based on the concept of takeover time [18], B¨ack [25] compared several selec-
tion schemes, including tournament selection. He presented the selection prob-
ability of an individual of rank jin one tournament for a minimisation task1,
with an implicit assumption that the population is wholly diverse (i.e., every
individual has distinct fitness value), as:
N−k((N−j+1)
k−(N−j)k)(1)
In order to model the expected fitness distribution after performing tournament
selection in a population with a more general form, Blickle and Thiele extended
the selection probability model in [25] to describe the selection probability of
individuals with the same fitness. The model is quite abstract although it is quite
elegant. They defined the worst individual to be ranked 1st and introduced the
cumulative fitness distribution,S(fj), which denotes the number of individuals
with fitness value fjor worse. They then calculated the selection probability of
individuals with rank jas:
S(fj)
Nk
−S(fj−1)
Nk
(2)
1Therefore the best individual is ranked 1st.

184 Y. Fang and J. Li
In order to demonstrate the computational savings in backward-chaining evolu-
tionary algorithms, Poli and Langdon [31] calculated the probability that one
individual is not sampled in one tournament as 1 −1
N, then consequently the
expected number of individuals not sampled in any tournament as:
NN
N−1−ky
(3)
where yis the total number of tournaments required to form an entire new
generation.
In order to illustrate that selection pressure in standard tournament selection
is insensitive to population size in general for populations with a more general
situation (i.e., some programs have the same fitness value and therefore have
thesamerank),Xieet al. [32] presented a sampling probability model that any
program pis sampled at least once in y∈{1, ..., N }tournaments as:
1−N−1
NNy
Nk
(4)
and a selection probability model that a program pof rank jis selected at least
once in y∈{1, ..., N }tournaments as:
1−⎛
⎜
⎝1−j
i=1 |Si|
Nk
−j−1
i=1 |Si|
Nk
|Sj|⎞
⎟
⎠
y
(5)
where |Sj|is the number of programs of the same rank j.
4 Issues Related to Sampling Strategy
There are two commonly recognised issues in standard tournament selection.
These two issues are closely related to each other because they are both caused
by the sampling with replacement scheme in standard tournament selection.
4.1 Multi-Sampled Issue
One issue is that because individuals are sampled with replacement, it is possible
to have the same individual sampled multiple times in a tournament, which is
referred as multi-sampled issue [33].
To address this issue, Xie et al. [33] analysed the other form of tournament
selection described in [5] and termed it as no-replacement tournament selection.
The no-replacement tournament selection samples individuals into a tournament
without replacement, that is, it will not return a sampled individual back to the
population immediately thus no individual can be sampled multiple times into

A Review of Tournament Selection in Genetic Programming 185
the same tournament. After the winner is determined, it then returns all individ-
uals of the tournament to the population. Therefore, no-replacement tournament
selection is a solution to the multi-sampled issue.
Xie et al. [33] gave mathematical models describing the sampling and selection
behaviour in no-replacement tournament selection as follows: if Dis the event
that an arbitrary program is drawn or sampled in a tournament of size k,the
probability of Dis:
P(D)= k
N(6)
For a particular program p∈Sj,ifEj,y is the event that pis selected at least
once in y∈{1, ..., N }tournaments, the probability of Ej,y is:
P(Ej,y )=1−⎛
⎜
⎜
⎜
⎝
1−1
|Sj|⎛
⎜
⎜
⎜
⎝
j
i=1 |Si|
k
N
k−j−1
i=1 |Si|
k
N
k⎞
⎟
⎟
⎟
⎠
⎞
⎟
⎟
⎟
⎠
y
(7)
They then used three selection pressure measures, namely loss of program di-
versity [29], selection frequency [27], and their own measure selection probability
distribution, to compare no-replacement tournament selection with standard one
based on simulations of four populations with different fitness rank distributions.
Their simulation results showed that there exist few difference between the two
tournament selection schemes. To further investigate underlying reasons, they
presented a model to describe the relationship between population size, tourna-
ment size, and confidence level in order to answer the following question:
“ for a given population of size N, if we keep sampling in-
dividuals with replacement, then after how many sampling
events, will we have a certain level of confidence that there
will be duplicates amongst the sampled individuals?”
They finally showed that for common tournament sizes 4 or less, it is not
expected to see any duplicates in anything except very small populations. Even
for tournament size 7, it is not expected to see duplicates for populations less
than 200. For most common and reasonable settings of tournament sizes and
population sizes, the multi-sampled issue seldom occurs in standard tourna-
ment selection. Further, since duplicated individuals do not necessarily influence
the result of a tournament when the duplicates have worse fitness values than
other sampled individuals, the probability of significant difference between stan-
dard tournament selection and no-replacement tournament selection will be even
smaller. Therefore eliminating the multi-sampled issue in standard tournament
selection is very unlikely to significantly change the selection performance. They
concluded that the multi-sampled issue generally is not crucial to the selection
behaviour in standard tournament selection.

186 Y. Fang and J. Li
4.2 Not-Sampled Issue
The other issue is that because individuals are sampled with replacement in
standard tournament selection, it is also possible to have some individuals not
sampled at all when using small tournament sizes, which is referred as not-
sampled issue [34].
This issue was illustrated initially through an experimental work by Gather-
cole [35]. He showed the selection frequency of each individual and the likelihoods
of not-selected and not-sampled individuals in tournament selection of different
tournament sizes through 1000 simulations on a sample population of size 50.
In his simulation, only one child is produced by crossover or mutation, thus the
total number of tournaments required to generate the next entire population
is a function of the crossover rate, the mutation rate and the population size,
instead of being just the same as the population size. His experimental results
are interesting and useful, however it is not clear whether the sample population
was fully diverse or not. This issue was then theoretically described by [31].
Two earliest known attempts to address the not-sampled issue are unbiased
tournament selection by Sokolov and Whitley [24] and fully covered tournament
selection by Xie [36]. A further detailed analysis of the issue was given recently
by Xie et al. [34].
In [24], Sokolov and Whitley believed that the potential of better individuals
not getting chosen for recombination due to the random sampling is the bias
presented in standard tournament selection. Therefore, they developed their un-
biased tournament selection that “lines up two different permutations of the
population and performs a pairwise comparison” with a constraint, which forces
compared individuals to be distinct. Consequently, their method can ensure that
every individual is sampled at least once. Tournament size 2 was used to test the
unbiased tournament selection on three problems: one with permutation-based
solution representation and two under bit encodings. Although the advantage of
a generational GA using the unbiased tournament selection varied for different
population sizes on the three problems, they concluded that the impact of the
bias is significant, and the unbiased tournament selection provides better perfor-
mance than other selection methods, including standard tournament selection,
a rank based selection and fitness proportionate selection.
In [36], Xie also mentioned that since the sampling behaviour in standard tour-
nament selection was random, the individual with bad fitness could be selected
multiple times and the individual with good fitness could never be selected. He
presented fully covered tournament selection which excludes the individuals that
have been selected for next selection to ensure every individual have an equal
chance to participate tournaments. He tested the fully covered tournament se-
lection on two symbolic regression problems and concluded that the method is
effective, implying that not-sample issue is worth of addressing.
Later, Xie et al. provided a detailed algorithm termed round-replacement tour-
nament selection which seems based on the fully covered tournament selection
and extended the analysis of the not-sampled issue. The algorithm is as follows:

A Review of Tournament Selection in Genetic Programming 187
1: Initialise an empty population T
2: while need to generate more offspring do
3: if population size <k then
4: Refill: move all individuals from the temporary population Tto the
population S
5: end if
6: Sampling kindividuals without replacement from the population S
7: Select the winner from the tournament
8: Move the ksampled individuals into the temporary population T
9: return the winner
10: end while
They described that in the round-replacement tournament selection any program
will be sampled exactly ktimes during the selection phase thus there is no need
to model the sampling probability. They gave a model to describe the selection
probability as follows: for a particular program p∈Sj,ifWjis the event that p
wins or is selected in a tournament of size k, the probability of Wjis:
P(Wj)= k
n=1 1
n|Sj|−1
n−1j−1
i=1 |Si|
k−n
N
k(8)
They used two measures, namely loss of program diversity and selection prob-
ability distribution, to compare round-replacement tournament selection with
standard one based on simulations of three populations with different fitness
rank distributions. Their simulation results showed that there exist some differ-
ence between the two tournament selection schemes. They then tested the ef-
fectiveness of the round-replacement tournament selection on the even-6-parity,
a symbolic regression, and a binary classification problems using three differ-
ent tournament sizes, namely 2, 4 and 7. However, the experimental results
showed that the improvement of the round-replacement tournament selection
is statistically significant only when the tournament size is 2 for the symbolic
regression and the binary classification problems but practically the differences
are small. They finally concluded that although there are some different selection
behaviour in the round-replacement tournament selection comparing with the
standard tournament selection, the different selection behaviour leads to better
GP search results only when tournament size 2 is used for some problems; overall
solving the not-sampled issue does not appear to significantly improve a GP sys-
tem for the given tasks; and the not-sampled issue in the standard tournament
selection is not critical.
The clarifications from the literature are very useful. The results show us that
in order to improve tournament selection, simply tackling different sampling
with or without replacement strategies is not sufficient and should not be our
research focus.

188 Y. Fang and J. Li
5 Drawbacks Related to Selection Pressure Control
5.1 Finer Level Selection Pressure Control
Controlling selection pressure in tournament selection could be done by chang-
ing tournament size. However, tournament size can only be an integer number.
This drawback limits the ability of tournament selection to influence selection
pressure at a coarse level. In order to make tournament selection be able to tune
selection pressure at a fine level, Goldberg and Deb [18] presented an alterna-
tive tournament selection, which uses an extra probability p. When conducting
a tournament between two individuals, the individual with higher fitness value
can be selected as a parent with the probability p, while the other is with the
probability 1 −p. By setting pbetween 0.5 and 1, it is possible to tune the se-
lection pressure continuously between the random selection and the tournament
selection with tournament size two. Recently, Hingee and Hutter [37] showed
that every probabilistic tournament is equivalent to a unique polynomial rank-
ing selection scheme.
Another attempt to address this drawback is a fine grained tournament selec-
tion by Filipovi´cet al. [21] in the context of GAs for a plant location problem.
They argued that standard tournament selection does not allow precise setting
of the balance between exploration and exploitation [30]. In their fine grained
tournament selection method, the tournament size is not fixed but close to a
pre-set value. They claimed that the fine grained tournament selection makes
the ratio between exploration and exploitation be able to be set very precise,
and that the method solves the plant location problem successfully.
5.2 Automatic and Dynamic Selection Pressure Control
In general, the larger the tournament size, the higher the selection pressure;
by using different tournament sizes, the selection pressure can be changed to
influence the convergence of the genetic search process. However, it will not work
as we expected due to two existing drawbacks during population convergence.
One drawback is when groups of programs having the same or similar fitness
values, the selection pressure between groups increases regardless of the given
tournament size configuration, resulting in “better” groups dominating the next
population and possibly causing premature convergence [38]. The other draw-
back is when most of programs in population have the same fitness value, the
selection behaviour effectively becomes random [39]. Therefore, tournament size
itself alone is not adequate for controlling selection pressure.
In fact, according to [38], these drawbacks are part of a more general issue: the
evolutionary learning process itself is very dynamic. At some stages, it requires
a fast convergence rate (i.e., high parent selection pressure) to find a solution
quickly; at other stages, it requires a slow convergence rate (i.e., low parent
selection pressure) to avoid being confined to a local maximum. These require-
ments could be achieved by changing tournament size dynamically in standard
tournament selection. However, standard tournament selection is not aware of

A Review of Tournament Selection in Genetic Programming 189
the dynamic requests. In order to pick correct tournament size, it should col-
laborate with an extra component that can reveal the underlying dynamics and
determine the requests.
The known attempts to implicitly and partially address the drawbacks in-
clude bucket tournament selection by Luke and Panait [22] and clustering tour-
nament selection by Xie et al. [40]. They are implicit and partial solutions to the
drawbacks because the original goals of the alternatives were not to solve the
drawbacks. Instead, the buckets tournament selection is used to apply lexico-
graphic parsimony pressure on parent selection for problem domains where few
individuals have the same fitness, and the clustering tournament selection is a
consequence of a fitness evaluation saving algorithm which clusters a population
using a heuristic called fitness-case-equivalence.
Fortunately, later on, Xie et al. re-analysed and re-test their clustering tour-
nament selection explicitly for the drawbacks and adopted two additional pop-
ulation clustering methods, including phenotype-based2and genotype-based3
[38]. They concluded that different population clustering methods have different
impact on different problems but the clustering tournament selection can auto-
matically and dynamically adjust selection pressure along evolution as long as
the population is clustered properly. They further presented mathematical mod-
els and simulation results to explain why the clustering tournament selection is
effective [32].
From these research results, we think that clustering tournament selection
is a promising direction to improve tournament selection as it is an automati-
cally biased parent selection scheme that is needed by the dynamic evolutionary
process: when most of the population are of worse fitness ranks and evolution
encounters a danger of missing good individuals, it tends to increase selection
bias to better individuals, hoping to drive the population to promising regions
quickly; when the population tends to converge to local optima and evolution
encounters a danger of losing genetic material, it tends to decrease selection bias
to better ones, hoping to keep the population diverse. We also think it would be
challenging to choose or develop an appropriate population clustering method
for a given problem but it is certainly an interesting research topic.
6 Conclusions
This paper reviewed the current status of research of tournament selection in GP.
It clearly showed that different sampling with or without replacement strategies
have limited impact on the selection behaviour in standard tournament selection.
It also pointed out a promising research direction that is about population cluster-
ing and clustering tournament selection. As the student consumption data is very
complex, we should follow these research results to implement and/or to improve
the tournament selection operator in order to obtain good mining results.
2A population is clustered based on fitness value.
3A population is clustered based on exact program structure and content.

190 Y. Fang and J. Li
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