The Effect of Payoffs on Evolutionary Dynamics
in Games with Indirect Invasions
Julian Sie
15 July 2017
Student Number: 10373381 Supervisor: Matthijs van Veelen
MSc Economics: Behavioral Economics and Game Theory Faculty of Economics and Business, University of Amsterdam
Abstract
This paper studies stochastic evolutionary dynamics for games that have a neutrally stable strategy that can be indirectly invaded. Such games generally involve matrices that have two or more neutral mutants and one mutant that has a selective advantage against one of these neutral mutants. Fixation probabilities and expected conditional fixation times are calculated for different payoffs of the invading mutant. This paper finds that a change in this payoff has negligible effects on fixation times, but important effects on fixation probabilities. In models with very low mutation rates, the time it takes to reach the ESS is completely dependent on fixation probabilities. Moreover the number of neutral mutants in the invasion chain almost completely determines this time.
Statement of Originality
This document is written by Julian Sie who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
1
Introduction
Fixation probabilities and fixation times are often quantities of importance in evolutionary game theory. Antal and Scheuring (2006) investigated these quan-tities for general cases of 2 by 2 matrices, such as bi-stable or coexistent strate-gies, as well as matrices with one dominant strategy. Other studies focused on these stochastic dynamics for a variety of games, including, for instance, the Prisoner’s Dilemma and Rock-Paper-Scissors. (Traulsen and Hauert, 2009).
This paper will focus on games which have a series of neutral mutants that can be indirectly invaded by a mutant that has a selective advantage. More specifically, a strategy that can successfully be invaded indirectly in this manner can be classified as not meeting the criteria to be called Robust Against Indirect Invasions (RAII) (van Veelen, 2012). This research intends to explore and quantify the fixation probabilities and times of matrices of this type of game. To what extent the payoff of the invading mutant affects these quantities of interest is the main question.
This paper therefore aims to answer the following research question:
What is the effect of a change in payoff of an indirectly invading mutant on evolutionary game dynamics?
This paper looks at the total expected time to reach the ESS in a matrix with indirect invasion. In order to do this, conditional fixation times of a game with two neutral mutants are compared to a game where an ’invading’ mutant has a selective advantage. I find that the payoff of the invading mutant affects its conditional fixation time, but that this effect is negligible when compared to the expected conditional fixation time of the two neutral mutants.
However, in a model that includes the time waiting for mutants to appear, all fixation times become irrelevant and the only quantity of importance is the fixation probability. This paper finds a relationship between the payoff of the invading mutant and the fixation probability. For large population sizes however, the fixation probability of a neutral mutant is significantly lower than
the fixation probability of the invading mutant, implying that a payoff change of the invading mutant still does not significantly affect the total time to reach the ESS in these kinds of games.
The next section contains the literature review, where the field of evo-lutionary game theory is discussed in more detail. This sections also gives a complete overview of the evolutionary processes used in this paper. Section 3 contains the Methodology. Section 4 will show the main findings of this research and in section 5 these results will be summarized.
2
Literature Review
2.1
Evolutionary Game Theory
The concept of evolutionary game theory was first proposed by John Maynard Smith and George Price in the 1970s. In contrast to traditional game theory, evolutionary game theory does not look at players’ utility functions but in-stead considers individuals’ reproductive fitness as derived from payoffs in games (Traulsen and Hauert, 2009). It describes populations of individuals instead of just two players. Individuals using successful strategies have a higher chance of reproduction and thus, spread in the population. This leads to evolutionary dynamics of games which can be studied.
Evolutionary game theory mostly has applications in the field of bi-ology, where it can be used to describe the evolution of bacteria and viruses, as well as cancer. It is also used often to explain strategies in animal conflicts or the behavior of prolonged human interactions (humans are also animals, of course) (Dawkins, 1976). Another interesting application is language evolution (Nowak, 2006).
Perhaps the most important concept of evolutionary game theory is the evolutionary stable strategy (ESS), as first defined by Maynard Smith (1982). It is a powerful concept that has become the standard method of describing evolutionary dynamics of games with individuals that play different types of
strategies (Traulsen et al, 2007). In general, populations converge to the ESS, because it can be considered a stable equilibrium. The exact mathematical dy-namics of this convergence has been described by both deterministic differential equations for infinite populations (Weibull, 1995), as well as stochastic formu-lations for finite popuformu-lations (Nowak, 2006). Fixation probabilities and times are insightful quantities in games with finite populations. Therefore, this paper focuses on finite populations and uses the Birth-Death process as described by Nowak. Before explaining this process, I will give a brief outline of the important concepts of evolutionary game theory used in this paper.
2.2
Evolutionary Stability of Strategies
We can make a distinction between different strategies with respect to their evolutionary stability. The most stable kind of strategy is an ESS. Once a population has reached an ESS, it has reached a stable fixed point. In an evolutionary setting, this does not necessarily imply that all individuals are of the same type. There are definitely games in which the ESS is a mixture of types of individuals (Smith, 1988), but the games in this paper exclusively contain ESS’es where all individuals are of the same type.
1 1 1 2
The game above is meant to illustrate the concept of an ESS. Here e2, which means that all individuals in the population are of type 2 and thus always play strategy 2, is an ESS. Deviating from this strategy with e1 gives you a lower payoff (1), therefore individuals of type 1 have a lower reproductive fitness and will eventually seize to exist in an evolutionary setting. It is expected that the population will consist fully of type-2 individuals given enough time.
A neutrally stable strategy (NSS) can be considered less stable than an ESS but is still an important concept in evolutionary game theory because such a strategy can still have a basin of attraction, which means that a population can
converge to this strategy. If a strategy is NSS, it means that no other strategy can invade and obtain a higher payoff than the NSS, but there is another strategy that earns an equally high payoff. The following game serves to exemplify such a situation. 1 1 0 1 1 0 0 0 1
Here, e1 is NSS. There is no other strategy that can invade a population of
type-1 individuals and subsequently earn a higher payoff. An individual of 2 however can thrive in a population of 1 individuals because a type-2 individual will always do equally well (earn a payoff of 1). Therefore e2is also NSS, as well as any mix of individuals that are either of type-1 or type-2.
Note that a single 3 individual cannot invade a population of type-1 players, or type-2 players or any mix thereof, because the type-3 individual will always earn a payoff of 0 because it is unable to meet other type-3 individuals. Yet e3is an ESS. This shows that it is possible for an ESS to exist alongside an
NSS in the same game. Whether the population settles at the NSS or the ESS depends solely on the initial distribution of individuals. In fact, in this game, if more than half of the initial population consists of type-3 individuals, dynamics are expected to lead the population to the ESS. If more than half of the initial population consists of either type-1 or type-2 individuals, then dynamics are expected to lead the population to the NSS.
By slightly adapting the previous game, I will explain the concept of Robustness Against Indirect Invasions (RAII) (van Veelen, 2012). Note that in the previous game the NSS was relatively stable. If the population settles in the NSS, it is not expected to move from there because deviations (e3) from the
NSS would lead to a lower reproductive fitness. RAII is a concept that allows us to make a distinction between different NSS’es on the basis of their evolutionary stability. The NSS in the previous game could be considered RAII, but the NSS
in the following game cannot. 1 1 0 1 1 0 0 2 1
In this game, 3 individuals still obtain a payoff of 0 if they meet a type-1 individual, but now they have a selective advantage against type-2 players. Type-3 players will have a high reproductive fitness and thrive in the presence of type-2 players.
In the previous game we have seen that any mix between type-1 and type-2 individuals can be considered NSS. The same does not apply here. If the share of type-2 individuals in the population is large enough, it metaphorically opens the door for type-3 individuals to invade. There is a critical point where the share of type-2 individuals is just large enough to still be considered NSS. This critical point is dependent on the payoffs. With a type-3 individual earning a payoff of 2 against a type-2 individual, we find that this critical point is exactly halfway. In other words, if half of the population consists of type-1 individuals and the other half consists of type-2 individuals, a type-3 individual can invade. All mixtures of strategy 1 and 2 up until but not including a 50/50 mix between the two strategies can be considered NSS. Formally, this looks like this: α 1 - α 0 With 0.5 < α ≤ 1
In a stochastic process such as the Birth-Death Process, dynamics can make the population drift between any mix of type-1 of type-2 individuals, given that the population started at the NSS. Once the population has fixated with type-2 individuals, the appearance of a type-3 mutant could lead the population towards the ESS.
2.3
The Birth-Death Process
The Birth-Death process describes evolutionary dynamics for a population size that is fixed but finite. The population size is denoted by the constant N. The Birth-Death Process is an example of a Moran process. The simplest possible Moran process does not depend on reproductive fitness. It is useful to explain this, since it also describes the case of neutral drift, where neither individual has a selective advantage. Such is the case for instance when two types of individuals always gain the same payoff in a matrix, like the NSS in the game shown in the previous section. Neutral drift is an important part of games that involve a strategy that is not RAII.
The Moran process assumes that every single individual in the popu-lation plays the same game. According to the matrix of the game, individuals receive payoffs depending on what type of player they meet. They are equally likely to meet any of the other individuals in the population. In this paper it is restricted to only two of the same types of individuals at the same time. The Moran or Birth-Death process consists of two steps: First, one individual is chosen for reproduction. Then, any individual, except for the new offspring, is chosen for death. Thus the same individual can be chosen for reproduction and death. Both of these events depend only on frequency in the case of neutral drift. Therefore, if we have individuals of either type 1 or type 2, and the amount of individuals of type 2 is described by the variable i, then the probability of a type-2 player being chosen for reproduction is i/N , where N denotes the popu-lation size. The probability of a type-1 player being chosen for reproduction is then (N − i)/N .
In a general Birth-Death process, the fitness of individuals plays a role as well. This paper assumes a strong intensity of selection, which implies that fitness depends solely on the relative payoffs of the two types of individuals (Nowak, 2006). It works as follows: Individuals that obtain a payoff of 0 will never be chosen for reproduction. An individual earning a positive payoff will have a certain probability to be chosen for reproduction dependent on his payoff
relative to the total payoff of all individuals combined in the population. The offspring is always of the same type as its parent who was chosen for reproduc-tion. Every individual is still equally likely to be chosen for death.
The Birth-Death process can be seen as a game between two types of individuals. Let us consider a situation with two arbitrary types, for instance type-1 and type-2. We could define a state as any moment in the Birth-Death process where there are a certain amount of type-2 individuals. In the games discussed in this paper there are then two absorbing states; all other states are transient. These two absorbing states are in this case: all individuals are of type-1, and all individuals are of type-2. Once any of these two states is reached, the system will no longer move to any other state. We can say that the either type-1 or type-2 individuals have fixated. In any of the transient states in between, the population has not reached a fixated state yet because there are still individuals of both types in the population.
A population that has reached a fixated state consists only of one type of individuals. As mentioned, the Birth-Death process can no longer lead the population into any other state. A mutation event could however accomplish this. Mutations occur at reproduction; one of the individuals in the fixated population is chosen for reproduction, but instead of creating an offspring who is of the same type as the parent, a mutation occurs, such that the new offspring is of a different type. We can call the new offspring a mutant and the initial population residents. Note that this mutation event has led the population to a different state: the population now contains one mutant and (N-1) residents. 2.3.1 Fixation Probability
We are interested in the probability of a single mutant taking over the popula-tion. The variable describing the number of mutants currently in the population is i. The number of residents in the population is therefore N − i. Mutations always create just one mutant, thus we always start a Birth-Death process with i = 1. There are two absorbing states in this process, one where i = 0 and one
where i = N. When i = N, the whole population consists of individuals that are playing the mutant strategy. We can say in this case that the mutant has fixated. If i = 0, we can say that the mutant has failed to reach fixation, such that the population only consists of residents once more.
The fixation probability is defined as the probability that a single mu-tant will reach fixation and can be calculated as follows (Nowak, 2006):
φ1= 1 1 +PN −1 j=1 Qj k=1 Ti− Ti+ (1)
Where φ1 is the fixation probability of a single mutant.
Ti− is the probability of going from the state with an i number of mutants to the state with an (i - 1) number of mutants.
Ti+ is the probability of going from the state with an i number of mutants to the state with an (i + 1) number of mutants.
Ti−
Ti+ is the Up/Down-Ratio. If this ratio is relatively small for a certain state,
it implies that a mutant has a better chance of reaching the next state (i + 1). This ratio is dependent only on the relative fitness of mutants and residents in the current state. A mutant with a selective advantage against residents will have a higher relative fitness, which subsequently increases this mutant’s fixation probability.
2.3.2 Expected Conditional Fixation Time
Another variable of interest is the fixation time. Given that a mutant reaches fixation, how long does it take, on average, for this mutant to reach fixation? In this context, time could be seen as the number of steps in the Birth-Death process. The expected conditional fixation time can be calculated as follows (Traulsen & Hauert, 2009):
t1= N −1 X k=1 k X i=1 φi Ti+ k Y m=i+1 Tm− Tm+ (2)
2.4
Birth-Death Process and the Indirect Invasion
Now let us consider how a Birth-Death process works in a game involving an indirect invasion. An example of such a game is shown below:
1 1 0 1 1 0 0 2 1
As mentioned before, the Birth-Death process is a process between two types of individuals in the population. The game above however is a 3 by 3 matrix; there can be three types of individuals in the game at any point in time. A model that allows for interactions between all three types of individuals is very complicated and outside the scope of this paper, but it is also not needed, given a few assumptions.
If we assume that the population has already managed to fixate at the NSS in the game above, such that every individual is of type-1, this allows to then analyze the path to the ESS using the Birth-Death process without including interactions between all three types of individuals. Another necessary assumption is, that mutations, at a reproduction event, always create just one mutant, never two. This is important because a single mutant of type-3 has no chance of invading a population of type-1 residents, but two of them could hypothetically do so. Finally, whilst the Birth-Death process is still ongoing (neither the mutants nor residents have fixated yet), the possibility of another mutation happening during any of the reproduction events is excluded. This last assumption is necessary because otherwise we could again arrive in a situation with three different types of individuals (if the second mutant is of a different type than the first mutant). The exact consequences of this assumption are discussed in more details in the Methodology.
A population where all individuals are currently of type-1 could, with mutations and given enough time, eventually arrive in a state where all individu-als are of type-3. This is because type-3 individuindividu-als can indirectly invade type-1
individuals because they have a selective advantage against type-2 individuals and type-2 individuals are neutral mutants of type-1 individuals. The path of this indirect invasion is as follows: First, a type-2 mutant appears through a mutation at a reproduction event. This type-2 mutant has a certain fixation probability and given that the mutant fixates it takes a certain amount of time for this mutant to do so. Dependent on the fixation probability, some or many mutants do not fixate but given enoguh time one will eventually make it such that we arrive in a situation where all individuals are of type-2. Then we wait for a mutation creating a mutant of type-3. Once a type-3 mutants fixates, we end up in the ESS. The population structure will then no longer change, because a mutant of type-1 or type-2 has no chance of invading a population of type-3 residents, because they will always receive a payoff of 0 in this game.
3
Methodology
This section is split in two parts. First, the specific questions that this pa-per aims to answer will be discussed. Afterwards, we will go into more detail regarding mutation and its effects on models of total fixation time.
3.1
Analyses
In this paper we will calculate fixation probabilities and expected conditional fixation times for a set of different games and compare the results between these games. Specific comparisons are done in order to answer a few questions of interest, which will be discussed in more detail in this section.
Since stochastic processes such as the Birth-Death process become quite complicated for most games, such calculations cannot be done by hand. An exception is the neutral game. The derivations for this game are found in the Appendix. For other games, calculations were done by programming the formulas for fixation probability and expected conditional fixation time in Mi-crosoft Excel using Visual Basic. In order to adequately capture population size
effects, these quantities were calculated for all population sizes until N = 1000. 3.1.1 Effect of a Payoff Change of the Mutant against the Resident The first analysis is a comparison between different payoffs of the indirectly invading mutant. For this analysis I will use the matrix shown earlier:
1 1 0 1 1 0 0 2 1 (Matrix 1)
The path to the ESS can be broken down in separate steps, which can be il-lustrated by 2 by 2 sub-matrices. The first sub-matrix below shows the step where a population of type-1 residents is invaded by a type-2 mutant. In this sub-game, no type of individual has a selective advantage, therefore it is iden-tical to the case of neutral drift. It has also been called the Neutral Game (Antal & Scheuring, 2006). The sub-matrix on the right shows the step where a population of type-2 residents is invaded by a type-3 mutant.
1 1 1 1 (Sub-Matrix N) 1 0 2 1 (Sub-Matrix 1.2)
The invading mutant obtains a payoff of 2 here. We are interested in finding the effect of a change in this payoff on the fixation probability and expected fixation time. Therefore we will compare Sub-Matrix 1.2 with the game below, where the invading mutant has payoff 3. We will also try to find a general relationship between this payoff and the fixation probability and fixation time. For this purpose, we will look at the matrix with payoff a below.
1 0 3 1 (Sub-Matrix 1.3) 1 0 a 1 (Sub-Matrix 1.a)
relative fitness. This has two expected effects: the mutant with higher fitness will have a relatively larger fixation probability and its conditional fixation time will be relatively smaller.
3.1.2 Effect of a Payoff Change of the Mutant Against Itself
The games described in the previous paragraph are just a few examples of possible games involving indirect invasions. All of these matrices involved a type-3 mutant with a selective advantage against type-2 individuals, but a payoff of 1 against itself. One could wonder what effect this payoff of 1 has on the fixation probability as well as the conditional fixation time. To answer this question, we will investigate a game of the following type:
1 1 0 1 1 0 0 2 2 (Matrix 2)
The main difference here is the bottom-right entry of the matrix. A type-3 mutant now receives a payoff 2 when facing another type-type-3 mutant. In this game, an indirect invasion is still possible. e1is still NSS but not RAII, e3is still
ESS and type-3 individuals can still eventually indirectly invade a population of type-1 individuals through mutational events.
The 3 by 3 game can again be broken up into smaller sub-games. Note that the game between type-1 and type-2 individuals is the same as before and can be described by the Neutral Game (Sub-Matrix N). The game between type-2 and type-3 individuals is shown below on the left. Finally, since we are interested in finding the general effects of a change in this payoff, we will also examine a game where both of the bottom row payoffs are replaced with a.
1 0 2 2 (Sub-Matrix 2.2) 1 0 a a (Sub-Matrix 2.a)
receive a high payoff against type-2 individuals, they also receive a higher payoff against themselves now. We therefore expect the fixation probability to be relatively higher and the expected fixation time to be relatively shorter as a result of the higher fitness.
3.1.3 Effect of a Longer Chain of Neutral Mutants
A strategy cannot be considered RAII if there is at least one neutral mutant that can be invaded by a different strategy that has a selective advantage, even if this neutral mutant is part of a longer chain of neutral mutants. An example of such a situation is depicted below (taken from van Veelen, 2012).
1 1 1 1 1 1 1 1 0 1 1 1 0 0 2 2 (Matrix 3)
Here, e1 is NSS but not RAII for the following reason: e2 is a neutral mutant
of e1, and e3 is a neutral mutant of e2. But e3 can be directly invaded by e4,
because e4has a selective advantage against e3. We therefore expect that, given
enough time, the population will settle at the ESS (e4).
We are interested in the effects of a longer chain of neutral mutants. Of course, the total expected time to reach the ESS is likely to be longer as a result of a larger game; After all, we need to cycle through more strategies before the population settles in e4. The argument that we are trying to make however is the following: A game between two neutral mutants is expected to take many times longer than a game where one mutant has a selective advantage. This implies that in the game above, we will spend most of the time somewhere in between e1, e2 and e3. The final step from e3 to e4 is expected to take only a
3.2
Mutation
3.2.1 Mutation Rate
The Mutation Rate should be sufficiently low, because the appearance of a second mutant while the process has not fixated yet would greatly distort the Birth-Death process itself. Another mutant of the same type appearing while the population is still fixating would imply ’skipping’ a step; we would go from state i to i + 1 not as a general result of the Birth-Death process, but instead because of a mutation. An even larger issue would occur if the new mutant was neither the same type as the previous mutant, nor the same type as the resident. This would lead to a very complicated situation with not two, but three types of players interacting. The analysis, based on the Birth-Death process, will then no longer be valid. An important assumption of the model is therefore that whilst the process is still fixating another mutant cannot arise.
The mutation rate is defined as the chance per step of a mutant arising. This rate should be sufficiently low for the previous assumption to make sense. The rate is set at such a level that the probability of a mutant arising in a given fixation process is below 1%. However, the expected fixation times vary for the different transitions. We therefore base the mutation rate on the longest expected transition, which is the transition between two neutral mutants. Since expected fixation times are mostly based on population size, the mutation rate is made flexible and a function of population size. The exact derivation of the mutation rate can be found in the Appendix.
4
Results
First some general notes on the quantities of interest with respect to population size: All expected conditional fixation times are increasing in N, whereas all fixation probabilities are decreasing in N. After all, a larger population size implies that mutants have to reproduce more often in order to reach fixation. The process will therefore take longer if the population size is larger. A similar
argument applies for the fixation probability: if the population size is larger, it is more difficult for a single mutant to fixate regardless of payoff, therefore the fixation probability is decreasing in population size.
4.1
Mutation Rate and its Implications
A sufficiently low mutation rate turns out to have a large impact on the analysis. When the population has fixated, we must wait for a new mutant to arise. Because of the low mutation rate, this waiting time is actually is very large, many orders larger than the expected conditional fixation times of any of the sub-games discussed before. This has important implications for our analyses.
Since expected fixation times of single mutants are relatively insignif-icant when compared to the time waiting for a new mutant to appear, we can ignore these times and focus only on the waiting time. The only determining factor of the total time it takes to reach an ESS in a certain game is then the fixation probabilities of its different sub-games. This is because the fixation probability tells us the expected amount of mutants it takes to reach fixation.
Even though expected fixation times are insignificant when compared to mutant waiting times, we would still like to compare the fixation times of different sub-games because it could lead to interesting conclusions.
4.2
Effect of a Payoff Change of Selective Advantage
To examine these effects, a comparison was first done between the following two sub-games: 1 0 2 1 (Sub-Matrix 1.2) 1 0 3 1 (Sub-Matrix 1.3)
A mutant in Sub-Matrix 1.2 is expected to take longer to fixate than a mutant in Sub-Matrix 1.3. After all, the individual obtains a slightly higher payoff in the latter game, such that the probability of going one state up (a mutant be-ing chosen for reproduction and a resident bebe-ing chosen for death) is slightly
higher in nearly every individual state. Because the fixation time exponentially increases as the population size becomes larger, the difference in expected con-ditional fixation times of a the two mutants becomes larger as N increases. This can be seen in more detail in Figure 1 in the Appendix section. For general payoffs a, the expected fixation time decreases as a becomes larger, regardless of population size. 1 0 a 1 (Sub-Matrix 1.a)
For small population sizes, the fixation probability is large and close to 1. As the population sizes become larger however, fixation probabilities tend to converge to a certain value and a correlation can be found. It turns out that, for large N, the fixation probability of a single mutant depends on the payoff a, such that:
φ1=
a − 1
a (3)
This is valid for all a larger than 1. For values of a between 0 and 1 the fixation probability converges to 0. The case a = 0 can be excluded because a mutant would not be able to invade in this case.
Now that we have found a general relationship between a and the fixation probability, we can analyze the effect of a payoff difference on the total expected time to reach an ESS in the larger Matrix 1 with payoff a.
1 1 0 1 1 0 0 a 1
(Matrix 1 with payoff a) 1 1 1 1 (Sub-Matrix N)
The first step towards the ESS is the game between the two neutral mutants of type-1 and type-2. It is depicted by Sub-Matrix N above. For the Neutral Game, the fixation probability of a single mutant is:
φ1=
1
N (4)
And the expected conditional fixation time of a single neutral mutant is:
t1= N (N − 1) (5)
The exact derivations can be found in the Appendix section. Important things to note are that as the population size N becomes very large, the fixation probabil-ity becomes very small. As N goes to infinprobabil-ity, the fixation probabilprobabil-ity converges to 0. Another point to mention is the fact that the fixation time for the Neutral Game is significantly larger than the fixation time for Sub-Matrix 1.a, as long as a is larger than 1. What this means is that mutants with a selective advantage (a larger than 1) are expected to fixate much faster than mutants without an advantage (neutral mutants). A comparison between the fixation time of the Neutral Game and those of sub-matrices 1.2 and 1.3 is shown in Figure 2 in the Appendix.
If we include time waiting for mutants however, the total time it takes to reach the ESS, starting from e1 and ending in e3, is only dependent on the
fixation probabilities. For large N, we can then make the argument that total time to reach the ESS is irrelevant of the payoff a, as long as a > 1. This is because the fixation probability of a neutral mutant (φ1 = N1) is significantly
smaller than the fixation probability of a mutant with a selective advantage such as in Sub-Matrix 1.a (φ1 = a−1a ) for large N. We can conclude that, for large
population sizes, the payoff of a mutant indirectly invading an NSS (as long as it has a selective advantage) does not significantly impact the total time it takes to reach the ESS in games with indirect invasions.
4.3
Effect of a Payoff Change of Mutant Against Itself
In order to answer this question, a comparison was done between the following two sub-games: 1 0 2 1 (Sub-Matrix 1.2) 1 0 2 2 (Sub-Matrix 2.2)
We find that, in line with expectations, an invading mutant in Sub-Matrix 2.2 takes on average less time to reach fixation than an invading mutant in Sub-Matrix 1.2 (see figure 3 in the Appendix). The fixation probability for a mutant in Sub-Matrix 2.2 is slightly higher for small population sizes, but as N becomes larger, this probability converges to 0.5, which is identical to a mutant in Sub-Matrix 1.2 (see figure 4 in the Appendix). Thus for large N the fixation probability of both matrices are identical. In fact, for a payoff of a, we find the same relationship as before between this payoff and the fixation probability of a single mutant.
φ1=
a − 1
a (6)
Since the fixation probability is the driving factor for the total time it takes to reach the ESS in these games, we can say that these total times are identical in Matrix 2 and Matrix 1. This result led to a suspicion which was subsequently tested in this research. Perhaps the payoff that the mutant receives against itself does not affect the fixation probability of said mutant. To test this, we looked at the following sub-matrix:
1 0 a b (Sub-Matrix ab)
The value of b does impact the fixation time, but it does not affect the fixation probability for large N (for all a larger than 1 and b larger than 0). This result should be refined a bit more. The payoff b should be larger than 0, because
if not the bottom row strategy would no longer be a dominant strategy and e3 would seize to be an ESS in this game. Thus, for games with a dominant strategy (b > 0), the payoff of a is the only deciding factor for the fixation probability of the dominant mutant in large population sizes.
4.4
Effect of a Longer Chain of Neutral Mutants
1 1 1 1 1 1 1 1 0 1 1 1 0 0 2 2 (Matrix 3)
We found that neutral mutants have much higher expected fixation times, as well as much smaller fixation probabilities if compared to mutants with a selective advantage. We could say, in terms of total time to reach the ESS, the sub-game between the neutral mutants is the bottleneck. Thus we can argue that a longer chain of neutral mutants will significantly increase the time it takes to reach the ESS.
5
Conclusion
This paper finds that when mutations are rare, conditional fixation times of individual mutants are insignificant. Instead, total fixation time depends only on the expected number of mutants needed to reach fixation. The fixation probability is therefore the only determining factor of total fixation time.
Because the fixation probability is significantly lower for a neutral mu-tant than a mumu-tant with a selective advantage, the bottleneck in terms of total fixation time in an indirect invasion is the series of neutral mutants. The pay-off of the invading mutant, as long as it maintains its selective advantage, is therefore relatively unimportant for total fixation time.
neces-sarily rare. Such a model is expected to be significantly more complicated, but could definitely lead to some interesting conclusions.
6
Appendix
6.1
Neutral Game: Derivations
1 1 1 1
In the neutral game as shown in the matrix above the Up/Down-Ratio always equals 1. This makes it possible to derive fixation probabilities and expected conditional fixation times for single mutants by hand. I use the formula for the fixation probability of a single mutant provided by Nowak (2006):
φ1= 1 1 +PN −1 j=1 Qj i=1 Ti− Ti+ (7)
Where φ1is the fixation probability of a single mutant. Since we know that the
Up/Down-Ratio for this game equals 1 in any state i, this becomes: φ1= 1 1 +PN −1 j=1 Qj i=11 (8) φ1= 1 1 + (N − 1) (9) φ1= 1 N (10)
For the expected conditional fixation time of a single mutant I use the formula as defined by Traulsen and Hauert (2009), albeit with some slight notation differences: t1= N −1 X k=1 k X i=1 φi Ti+ k Y m=i+1 Tm− Tm+ (11) Where t1 is the expected conditional fixation time of a single mutant and φi
the fixation probability of mutants in a given state i (i signifies the number of mutants).
The term φi Ti+ can be simplified. φi=Ni and Ti+ = i(N −i) N2 . Therefore we get: φi Ti+ = N N − i (12)
Taking into account that the Up/Down-Ratio still equals 1, the formula can be written as: t1= N −1 X k=1 k X i=1 N N − i k Y m=i+1 (1) (13)
The product of 1 can be ignored: t1= N −1 X k=1 k X i=1 N N − i (14)
The combined sums written out looks as follows: t1= N N − 1+ ( N N − 1+ N N − 2) + ... + ( N N − 1+ N N − 2 + · · · + N 2 + N 1) (15) Note that the first term occurs in every single term after that as well, such that it occurs (N-1) times in total. A similar argument applies for the second time, which occurs (N-2) times. Using this fact gives us:
t1= (N − 1) ∗ N N − 1+ (N − 2) ∗ N N − 2+ ... + 2 ∗ N 2 + N 1 (16) Every single term cancels out, leaving only N every time.
t1= N (N − 1) (17)
6.2
Mutation Rates
As mentioned before, the mutation rate in this paper’s model is defined as a function of population size. It is derived by looking at the ’longest’ expected transition, that of the invasion of a neutral mutant. The probability of another mutant arising whilst the process is still fixating should be below 1%. This leads to the following equation:
µ = 1 − 0.991/(N (N −1)) (18) Where µ is the mutation rate per step. N(N-1) is the expected conditional fixation time of a neutral mutant.
6.3
Figures and Tables
Figure 2: Fixation Times for Sub-Matrix N, Sub-Matrix 1.2 and Sub-Matrix 1.3
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