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A synthesis of game theory and simulation techniques gives a better understanding of evolutionary processes: Two models are be
Page 1
Enhancing game theory with coevolutionary simulation models of honest signalling
Dave Harris & Seth Bullock
Informatics Research Institute, University of Leeds, Leeds LS2 9JT, UK.
Tel: +44 (0)113 233 5322, Fax +44 (0) 113 2335468, Email: dharris@comp.leeds.ac.uk, seth@comp.leeds.ac.uk
Abstract - Game-theoretic models provide a rigorous
mathematical modelling framework, but tractability
considerations keep them simple. In contrast, Evolutionary
Simulation Models (ESMs) may be complex, but can lack
rigour. We demonstrate that careful synthesis of the two
techniques provides improved insights into the processes
underlying the evolution of cooperative signalling systems.
I. INTRODUCTION
Evolutionary systems are of interest to both biologists and
computer scientists. For the former, they represent
challenging objects of study. For computer scientists, they are
both a source of inspiration and increasingly a domain within
which computational techniques are being profitably applied.
Traditionally, biologists have tended to use mathematical
methods to model biological systems. In the field of
behavioural ecology, for example, game theory has been the
preferred tool since it was first championed by Maynard
Smith and Price in the early 1970s [1].
Computer scientists have used simulation models to
explore similar systems with some success [2]. However,
their uptake within the biological community has been
limited by the perceived lack of rigour associated with
simulation in comparison to equational modelling techniques
[3]. Formal models are expected to explicitly present all of
their assumptions and to provide an intelligible account of the
moves leading to the model's conclusions. Simulations, on
the other hand, are often
opaque,
even to their designers [4].
Unlike the interpretation of mathematical models,
understanding why a particular simulation model produces
the results that it does is often a significant undertaking. As a
result, even when the actual details of the implementation are
made clear (which they are often not) by providing the source
code, for example, simulation models are difficult to assess
and appreciate. This sometimes leads to artefactual claims
[5]-[6].
Despite the success of game theory modelling, it suffers
from the same tractability limitations as all mathematical
modelling techniques. While simple games can be analysed
fairly easily, including more detail often renders models
effectively insoluble. This has not been a severe problem in
behavioural ecology partly because theoreticians have been
interested in quite simple games with few equilibria. These
simple models have thrown light on many important
questions, but further exploration often requires addressing
more complicated scenarios.
We believe that this is best accomplished by using game-
theoretical models as the basis for evolutionary computer
simulation modelling. Wedding the two approaches may
serve to overcome the limitations of each-the perceived lack
of rigour in the simulation design, and the tractability
constraints on mathematical modelling.
One might consider a game-theoretic model to represent
the
pure selective force
driving an evolutionary system. This
is because game theory assumes evolution takes place in a
world where even the smallest of selective forces will
eventually overcome any limitations [7]-[8]. The only
constraints on evolutionary change imposed by a game-
theoretic model are that strategies must be drawn from the
pre-defined strategy set. There are no genetic constraints, no
developmental limitations, no noise, no fitness landscapes to
traverse, etc., there are only evolutionarily stable strategies
(An ESS is defined as a strategy, or mixtures of strategies,
that, when prevalent, cannot be invaded by any others).
In contrast, an evolutionary simulation model immediately
introduces evolutionary constraints, many of which might be
described as "logistic" factors. These factors would include
stochastic effects such as noisy fitness functions or sampling
errors, and genetic constraints, which govern how far one
strategy is from another across a fitness landscape. In
addition, population structure, direct vs. indirect costs, life
history strategies and other influences are also often brought
to bear upon the evolutionary process being modelled. It is,
of course, possible to include these kinds of constraints in a
game theory model, but only at the cost of greatly
complicating the mathematics required to obtain a solution.
Importantly, a simulation modelling approach also enables
a researcher to study any
non-ESS
, transitory phenomena.
This is particularly important with systems that have non-
stable or multiple equilibria, as well as potentially revealing
system behaviours that might give an indication of an
evolutionary trajectory toward an ESS. Just because a
behaviour is not an ESS does not necessarily mean that it will
rarely be observed. Commonly occurring transitory
phenomena also require explanation.
In the remainder of the paper we will apply both game
theory and evolutionary simulation modelling techniques to
an important problem in biology. Our primary aim is to
demonstrate that the synthesis of both techniques provides a
better understanding of the problem than game theory alone,
but we also aim to contribute to the biological literature on
the evolution of honest signalling systems.
II. THE EXAMPLE PROBLEM
One problem that has been studied extensively through
both game-theoretic [9]-[10] and evolutionary simulation
modelling [11] approaches is the discrete action-response
game, which models communication between individuals
who may or may not have a conflict of interest. In the
majority of cases, only one modelling technique was used.
Page 2
We will take Hurd's game theoretic model of the evolution
of honest handicap signalling and by implementing it as an
evolutionary simulation, demonstrate that the game theory
does not tell the whole story of the system's behaviour. To
give an idea of the basis of the model, we present the
important equations of the game theory below.
The popularity of the problem is due to the difficulty that
modellers have had with the concept of the
handicap
principle
, as proposed by Zahavi [12]. Despite widespread
interest in Zahavi's idea, it took around 15 years from its first
publication for a successful model to prove its validity.
Zahavi suggested that for signallers to be honest where
conflicts of interest exist, signals must be costly. Grafen's
game-theoretical proof demonstrated this and also required
the condition that signals must be more costly for low-quality
individuals to make, where quality is of interest to the signal
receiver [13]-[14]. For example, this could be the genetic
quality of a displaying male for a female choosing a mate.
A competing view of signal system evolution, which has
fallen out of favour since the rise of the handicap principle,
proposes that signallers and receivers may settle upon cheap
conventions with which to convey information to their
mutual advantage, e.g., red insect colouration might indicate
unpalatability to birds [15]. There is nothing to prevent these
conventions from being abused by free-loaders, mimics and
cheats, e.g., red insects which nevertheless are tasty food for
birds. As the prevalence of these abusers grows, the value of
the signal deteriorates until the signalling system collapses
under the weight of its own infidelity. Partly as a result of the
inherent instability featured in this account, it has been
neglected in favour of the handicap principle's stable
equilibria.
A signaller,
S
, is given an internal state,
z
, which only they
know but which is of interest to a receiver,
R
. This state will
either be
H
, representing high quality, or
L
, representing low
quality. On the basis of this state, the signaller makes a
signal,
s
, to which the observer makes a response,
r
. The
signal will either be East (
E
) or West (
W
) and the response
will be either Up (
U
) or Down (
D
).
An example of this system might be male blackbirds
signalling to females in the hope of eliciting a mating
opportunity. Imagine a female blackbird flying about and
suddenly observing a male blackbird sitting in a tree. He also
observes her and can do one of two things. He could remain
inactive (East) and hope she flies over or he can start singing
in the hope that she will be attracted by it (West). The female
having observed the male's action could either ignore him
(Down) or fly over and mate with him (Up).
Note that there is no reason why, for example, the cost of
signalling East should not be higher than signalling West or
vice versa, or that High state males should find signalling
East cheaper than Low state males or
vice versa
.
The game can be played in many ways, but Hurd
concentrated upon the version in which the signallers always
want to get an Up response from the observer, but the
observer only wishes to give that response to a signaller with
a High internal state. This generates a conflict of interest that
makes the solution non-trivial.
The participants in the game are given a fitness score
determined by how successful they have been. In game
theoretical terms these would be:
)
,
(
)
(
s
z
c
r
v
w
S
-
=
(1)
)
,
(
r
z
v
w
R
=
(2)
In equation 1
w
S
is the fitness score of the signaller,
v
is the
value of the receiver's response, and
c
is the cost of the
signaller's action given his internal state. In equation 2
w
R
is
the fitness score of the receiver, and
v
is the value of the
receiver's response given the signaller's internal state.
The relative cost of signalling can be defined as:
)
,
(
)
,
(
E
H
c
W
H
c
C
H
-
=
(3)
)
,
(
)
,
(
E
L
c
W
L
c
C
L
-
=
(4)
This defines
C
H
as the difference in cost for a High quality
signaller between signalling West and East.
C
L
is the
equivalent for the Low quality signaller. Honest signalling is
defined arbitrarily as High state signallers making the West
signal while Low state signallers make the East signal. Hurd
went on to show that honest signalling is only an ESS when:
H
L
C
V
C
>
>
(5)
Where
V
is the value of the receiver's Up response to the
signaller. Other models have explored cases in which the
value of the Up response is not the same for High and Low
state signallers, but in our model
V
is insensitive to signaller
state. This is represented graphically in Figure 1.
The importance of equation 5 is that it contradicts Zahavi's
claim that signals must be costly in order to remain honest.
According to (5), signalling could remain stable despite
C
H
being negative. This indicates, that as long as Low state
signallers cannot afford to make an advertisement, it can
remain honest, even if High state signallers pay nothing to
make it, or are even better off making the advertisement than
not.
III. OUR MODEL
In order to explore more fully the evolutionary dynamics of
the action-response game, we constructed an evolutionary
simulation that faithfully captures the structure of Hurd's
model. Due to the exclusivity of the signaller and receiver
behaviours, a two population model is appropriate. A
population of 100 signallers was coevolved against a
population of 100 receivers over 1000 generations. At birth,
each signaller was either determined to be High or Low
(p=0.5). Each signaller's strategy was defined by a two-bit
genotype, where the first bit represented which action to
make when Low, and the second which action to make when
Page 3
Fig 1. Graph showing where honest signalling is an ESS in Hurd's model
High. The receiver's two-bit genotype represented which
response to give when observing East and which to give
when observing West. At the outset of coevolution, all genes
were initially set randomly to either 1 or 0.
The available signaller and receiver strategies are named in
relation to Hurd's arbitrary honest signalling scenario in
which signallers signal West when High and East when Low,
while receivers respond Up to West and Down to East. A
Cynic never signals West, while a Bluffer always does.
Honest signallers signal East when in the Low state and West
in the High state, while dishonest signallers signal West in
the Low state and East in the High state. A Generous receiver
always responds Up, while a Mean receiver never does. A
Believer responds Up to West and Down to East, while a
Non-Believer behaves in the opposite manner.
Each generation, every signaller was paired at random with
a receiver. After playing the action-response game once, they
were awarded scores according to the fitness functions
defined above. Reproduction was determined by tournament
selection with tournament size 2. During reproduction, each
gene had a 3% chance of mutation.
For each coevolutionary run, several parameters defined
the cost and value of signalling and responding (see Figure
1). Receivers were given a score of 100 if they gave an Up
response to a High state signaller or Down to a Low state
signaller, otherwise they received a zero score. The value of
an Up response for a signaller was set to 100, while a Down
response had zero value. The cost of both East and West
behaviours for both High and Low state signallers was varied
across runs from ­100 to +100 in steps of 25. Each of the
6561 combinations of cost parameters was replicated 20
times. We have described the decisions involved in choosing
these experimental parameters and the consequent effect on
results in another paper [16].
For each run, the number of signallers in the final
generation utilising each of the four possible strategies was
counted. If the signaller population was behaving randomly,
we would expect them to be distributed equally across the
potential strategies with a binomial distribution, p=0.25,
n=100. This gives a 95% confidence interval of 10-37. If the
number of signallers using a particular strategy lies outside of
this interval, we can deduce that the distribution is non-
random. For instance, if we observe that more than 37 Cynics
in the final generation then we can be confident that there
was some selection for this particular strategy. Conversely, if
we observe less than 10 Cynics, we can infer that there has
been selection
against
this course of action. For each of the
20 runs carried out for each combination of cost parameters,
the signaller and receiver populations were scored for each
possible strategy (positive selection, negative selection, and
no selection).
It is common in signalling theory papers to concentrate on
the signaller and to pay little or no attention to receiver
strategy. This may be due to the perception that it is the
signaller's behaviour that is of interest; it is the phenomenon
of honest
signalling
that is under investigation. However, it
should be remembered that this is a coevolutionary problem
and the receiver's strategy is of equal importance in
determining the behaviour of the signalling system. In
recognition of this, the receiver strategy will be discussed
before the signallers.
IV. RESULTS
A. Receiver
strategy
We found that the mean and generous strategies are only
ever used at a low level and there is no region of the
parameter space within which they approach an ESS. This is
a little surprising given the presence of areas where signallers
are predominantly using the Cynic or Bluffer strategy. One
might imagine that the Mean or Generous receiver strategies
might do as well as any other in these areas.
The believer and non-believer distributions are shown in
Figure 2. These distributions are rotationally symmetrical.
One item of note is that the area of the parameter space that
results in significant numbers of Believer strategists not only
coincides with the area that game theory predicts should
favour Honest signallers, as one would expect, but is actually
more widespread. Likewise, Disbelief is not limited to the
area of the parameter space predicted by game theory. Why
are receiver strategies sometimes favoured in the absence of
the signalling strategy that justifies their existence?
Confronted with, for example, a uniform population of
cynics, receivers cannot glean any information concerning
signaller state from signaller behaviour. Regardless of
whether signallers are High or Low, they always move East.
In such a situation each of the four receiver strategies are, on
average, equally successful, scoring 100 roughly half the time
and zero for the remainder. Given such a scenario, in the
perfect world of game theory (where idealizations such as
infinite population size and fitness assessment in the limit are
assumed to hold) each receiver strategy should be represented
equally in the population.
However, in a stochastic simulation with finite population
size in which individuals are assessed over a small number of
0
V
C
H
V
C
L
= C
H
C
L
Area in which honest
signalling can occur
Page 4
Figure 2.Graphs showing the distribution of receiver strategies with respect
to C
H
(X-axis) and C
L
(Y-axis). Increasingly heavy shading represents an
increasing number of runs that display positive selection pressure for a
strategy. Left to right: Believer, Non-believer
trials, the likelihood that the behaviour of a population of
signallers will convey no information about internal state is
very low, since in each generation, mutation will generate a
small number of non-ESS signallers. Given that Believers
and Non-believers are the only strategists capable of
exploiting any correlation between signaller internal state and
behaviour, it is perhaps not surprising to see them outperform
Mean and Generous strategists.
But why should believers predominate above the line
CH=CL with Non-believers below? It must be the case that
the cost parameters above the line CH=CL are, to some
perhaps quite small extent, biased in favour of Honesty (even
outside the area of the parameter space admitting of an
Honest signalling ESS). Likewise, below the line CH=CL,
Dishonesty is favoured to some extent. Even a small number
of either signaller strategy would be enough to tip the
receiver population in favour of either Belief or Non-Belief.
B. Signaller strategy
Figure 3 clearly shows that honesty is observed under the
ESS conditions predicted by the game-theoretic model. We
can also appreciate which conditions favour the other three
strategies. Note that considering the four graphs together,
these results are also rotationally symmetrical, with honesty
and dishonesty being ESS's on opposite sides of the chart.
This demonstrates that given appropriate signalling costs,
East and West are effectively interchangeable as one would
expect. Near the centre of the parameter space, there exist
regions of that are not dominated by any particular strategy.
The game-theoretic model suggests that absolute signalling
costs are irrelevant to the stability of honest signalling.
Rather it is the difference between the cost of East and West
that is critical to stability. Each point on the graphs in Figure
3 pools several runs with different cost parameters. Are there
significant differences
within
these pooled results? Figure 4
shows that there are not. Varying the cost of signalling East
when in the Low state has no effect on the number of
signallers using the honest strategy. As predicted, it is the
relative cost, (
C
L
) which determines this. Absolute costs are
unimportant. This also holds for variation in the other
absolute cost parameters. In combination with the rotational
symmetry displayed by the graphs in Figure 3, this result
allows us to greatly reduce the number of different cost
values that we need to explore.
Figure 3. Graphs showing the distribution of signaller strategies with respect
to relative signalling costs C
H
(X-axis) and C
L
(Y-axis). Increasingly heavy
shading represents an increasing number of runs that display positive
selection pressure for a signalling strategy. Clockwise from top-left: Cynic,
Honest, Bluffer, Dishonest.
0
2
4
6
8
10
12
14
16
18
2 0
- 2 0 0 - 17 5
- 15 0
- 12 5
- 10 0
- 7 5
- 5 0
- 2 5
0
2 5
5 0
7 5
10 0
12 5
15 0
17 5
2 0 0
C L
- 10 0
- 7 5
- 5 0
- 2 5
0
2 5
5 0
7 5
10 0
Figure 4. A graph showing the effect of
C
L
on strategy frequency whilst
varying the cost of signalling East.
C
H
was held constant at 0 for all runs.
Figure 4 also highlights an area of particular interest to us.
The game theory suggests that we should see a sharp "phase
transition" separating honest signalling from no honest
signalling. Instead we see a gradual increase in the number of
populations significantly using the honest strategy as we
cross the predicted boundary in the parameter space. What is
responsible for the low level of Honesty?
The distribution of strategies over the graphs is
summarised in Figure 5, with regions lacking a dominant
strategy labelled NS. A closer examination of these areas
suggests that honest signalling occurs under these conditions,
but is not stable over time due to the presence of Bluffers.
In order to gain a better idea of what was happening in the
areas designated NS, we re-ran the experiment with a finer
resolution. For this higher resolution experiment, the cost of
signalling East was set to zero and the cost of signalling West
varied between 0 and +100 in steps of 5 for each state. Figure
6 shows the higher resolution graphs, which confirm the
Page 5
Figure 5. Graph showing the distribution of signaller strategies with respect
to C
L
and C
H
.
Figure 6. Graphs showing a higher resolution of the distribution of signaller
strategies with respect to C
H
(X-axis) and C
L
(Y-axis). Clockwise from top
left: Cynic, Honest, Bluffer. The last graph shows the corresponding area on
the original plots
0
10
20
30
40
50
60
70
80
90
100
0
200
400
600
800
1000
Generations
Honest
Bluffer
Believer
L
CI
Figure 7. Graph showing the number of individuals using the Honest, Bluffer
and Believer strategies during one simulation run in the NS region.
presence of a non-ESS signalling strategy. This appears to
take the form of a mixture of honest and bluffing strategies.
This is of particular interest as it has often been theorised that
such mixed strategies occur. Despite the fact that they are not
ESS strategies, these transitory phenomena still occur
frequently in this model.
Figure 7 shows the number of individuals utilising a
particular strategy over the course of a simulation run with
the signal cost parameters
C
L
=75,
C
H
=25, which corresponds
to the middle of the area marked NS on Figure 3. One can see
that the numbers of Bluffers, Believers and Honest strategists
are correlated. The initial increase in the frequency of the
Believer strategy is probably due to stochastic effects, since
against the background of predominant Cynicism the
Believer strategy has the same fitness as the other three
response strategies. However, once the frequency of Belief
rises slightly above chance, this favours both Honest and
Bluffer strategists, since they get an Up response from
Believers. The increase in frequency of Honest strategists
gives Believers a slight edge over their competitors, boosting
the numbers of both. However, Bluffers are able to piggy-
back on this trend. Eventually, the increasing numbers of
Bluffers will devalue the Honest signal, leading to a crash.
Receivers abandon the believer strategy, causing a
corresponding reduction in the number of signallers using
both the Honest and Bluffing strategies. The system returns
once more to the non-signalling equilibrium at which
signallers are Cynics and receivers are equally well off
whatever strategy they pursue.
V. DISCUSSION
The results clearly show that an area of the parameter space
outside of that predicted by game theory contains some
honest signalling (Figure 8). The fact that it appears to be
evolutionarily unstable explains why. However, it should be
pointed out that although it is unstable it does appear in
between 50% and 75% of the populations within the area. If
one were to rely purely on the game-theoretic results, then it
would be predicted that no honest signalling could occur in
that area of the parameter space-signals are too cheap for
the Low quality signallers, they will invade and destabilise
any nascent signalling system. We show that significant
periods of honesty will occur in these scenarios, but that it
will be mixed with signallers using the Bluffer strategy, and
that each evolved signalling system will be relatively short-
lived. It is important to realise that this is not just an arbitrary
anomaly. As discussed earlier, within behavioural ecology,
short-lived signalling systems of this type had been theorised
to exist, but the notion of this type of cheap, but fragile
signalling system has been displaced by theories of more
stable handicap signalling systems.
The extended areas of the Believer and Non-Believer
strategies also require some explanation. The most likely
reason is that as long as a number of signallers are Honest or
Dishonest, signaller behaviour will provide
some
information
about signaller state. If a source of this information exists,
V
V
Honest
Cynic
Bluffer
NS
NS
-V
-V
0
C
L
C
H
C
L
= C
H
Dishonest
Page 6
Figure 8. Graph showing where honest signalling occurs in our model,
including an area not predicted by game theory.
despite its unreliability, it will allow receivers to make a
better than random choice. The question of how reliable such
a signal must be remains open.
VI. CONCLUSION
We have shown the utility of evolutionary simulation
models in combination with game theory in relation to
discovering potentially important results. The existence of
ephemeral honest signalling will be of particular interest to
biologists. Current dogma states that signals cannot be
considered reliable unless they are costly. Our model shows
that not only can relatively cheap signals allow some level of
honest signalling, but that receivers are willing to tolerate
some level of "cheating" by non-honest signallers.
Biologists have become accustomed to only considering
behaviours that have been shown to be an ESS. This is
largely due to the success of game theory. However, now that
the limits of the ability game theory to shed new light on the
mechanisms involved in behavioural ecology may be in sight,
it is time to consider additional theoretical methods.
One can consider the discoveries due to the use of game
theory to be a framework, we now need to begin filling in the
gaps between the supports. Merely knowing the eventual
evolutionary outcome of selective forces provides an
unbalanced picture, which may be misleading when applied
to the real world. By using simulation techniques we can
discover both the evolutionary trajectories that may be
followed and also the nature of ephemeral phenomena. Such
equilibria may be more common than currently assumed.
How many seemingly stable signalling systems in the natural
world are actually transitory in nature? The use of simulation
modelling may help empiricists to find out by both opening
their eyes to the possibility and giving them some idea of
which conditions favour these phenomena.
Whilst we have concentrated upon the enhancement of
game theory using simulation, there is no reason why this
process should not work in reverse. This would entail a
simulation being reverse engineered to produce a game-
theoretic model, which would show which parts of the
simulation result are due to the selective forces acting on the
phenomenon under investigation and which are due to the
manner in which the simulation has been implemented.
VII. Further Work
Having demonstrated that logistics are important in signal
evolution, there is ample scope to expand upon this work.
Potential areas include exploring multiple signalling bouts,
different receiver pay-off matrices, the effect of having the
value of receiver response vary with signaller state, allowing
the signaller's internal state to vary over a lifetime and an
examination of a continuous version of the model.
VIII. Acknowledgements
The authors would like to thank Jason Noble, John
Cartlidge and Tom Carden for their helpful comments during
the writing of this paper.
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