We will see that the long-term survival of the species requires diversity.

Here, the word “diversity” means the difference in personal preferences and, eventually, their divergence from rationality on the personal level. It might be related, or completely unrelated to the physical parameters, such as shape, size, social background, or skin/eye colour.

A logical, yet striking conclusion is that the species with 100% fully rational individuals (within the frame of their experience and current circumstances) are expected to die off. An alternative approach that we tend to follow naturally – it ensures our survival as the species – is the probability matching. A simple example of the probability matching is as follows: suppose there are 2 possible outcomes, one that occurs with the probability of 95% and the other one that occurs with the probability of 5%; then we optimize for the first outcome with the frequency of 95% and for the second outcome with the frequency of 5%.

The ideas below come from Professor Andrew Lo‘s book Adaptive Markets, which I warmly recommend. Even though its subject is a financial markets theory, it discusses a wide range of subjects. One of them is a simple mathematical model that shows that long-term survival of a species requires diversity (see pages 190-205).

The Model: The Valley vs. Mountains

Only valley tribbles will survive

In the model presented in the book, a tribble is a simple creature that lives only for one season, produces 3 offsprings, then dies of old age, unless it dies of a natural disaster earlier.

Any tribble can choose where to live: in the valley or in the mountains. Once their choice is made, it cannot be changed.

Each season, a natural disaster strikes. If the weather is sunny, which is more likely, the draught in the mountains will kill all tribbles that live there. Those who are in the valley will survive.

If it is not sunny, it will rain, and the valley will be flooded. All tribbles who have chosen to nest in the valley will drown. Those who are up in the mountains will survive.

Individual Survival vs. Species Survival: Quick Conclusions

Only mountain tribbles will survive

Obviously, on the individual level, it makes sense to live in the valley, because the sunny weather is more likely, and thus the chances of survival of those who live in the valley are greater.

However, if all tribbles were making this rational (on the individual level) choice, the whole population would be wiped out during the first rainy season.

Therefore, individual irrationality (the “unhealthy obsession” with the mountains?) of a part of the population seems to be a prerequisite of species survival. In other words, the eccentricity of mountain tribbles seems to be an adaptation feature on the level of the population as a whole.

The Optimal Number of Irrational Tribbles

Naive Approach: Maximizing The Expected Number

We need to define what “optimal” means; this is not trivial.

If we were wishing to maximize the expected number of tribbles, then it is easy to show that there should be 0 irrational tribbles. However, with a substantial probability, the population of tribbles will be wiped out. Thus, this is a very risky strategy.

Let’s put some numbers on this claim. Let

s be the probability that it is sunny; this implies that the probability of rain is 1-s.
v be the proportion of tribbles who choose the valley; it means that 1-v out of all tribbles will choose the mountains.
k be the number of children (k for kids) each tribble has.
G be the number of generations that we set as the horizon for our study.
N be the number of tribbles at the beginning of our study.

So, if there are 0 irrational tribbles (v=1), the population will grow by the factor $k^G \text{ with probability } s^G$ and will be $0 \text{ with probability } 1-s^G$
If each tribble has 3 children, the chances of rain are 5% and we have the horizon of 100 generations,

with the probability of 99.4%, the tribbles will become extinct;
with the probability of 0.6%, there will be approximately 515377520732011000000000000000000000000000000000 times more tribbles that there were initially.

So, even though this strategy maximizes the expected number of tribbles, it is too risky to be accepted because of the high probability of tribbles’ extinction.

Thus, we need to find a balance between the expected size of the population and the risk. The “optimal strategy” is, in fact, a subjective concept, its choice depends on the risk appetite of the decision maker(s).

Approach From The Book: Maximizing The Expected Growth Rate

Now let us consider optimizing the expected logarithmical growth rate of population growth, as suggested in the book. If it is sunny, the number of tribbles in the next generation will grow by the factor of $k v$

and if it rains, the number of tribbles in the next generation will grow by the factor of $k (1-v)$

Therefore, the expected logarithmical rate of population growth per generation will be $log(k) + s \ log(v) + (1 - s) log(1-v)$

Solving its first derivative w.r.t. v yields that the maximal rate of growth is reached when $v = s$

That is, if the chances of sunny weather are 95% (s = 95%), then 95% of tribbles should nest in the valley (v = 95%) in order to maximise the expected growth rate of the population. As mentioned in the book, this strategy is called probability matching.

Reasons For Maximizing The Expected Growth Rate

There is no immediate reason to optimize the expected population growth rate: unlike the expected population size, it has no “physical” meaning. However, we would expect the risk to be much lower since the strategy is much less extreme than the “always choose the best” strategy we have seen earlier. Also, it is very convenient from numerical/computational point of view.

As one could have expected, a similar problem arises in finance, in money management studies. Optimization of the growth rate in the context of money management is called the Kelly criterion. In the original paper, published by John Larry Kelly Jr. in 1956, the author shows that choosing to optimize the growth rate helps to avoid ruin, but he does not explore the alternatives. As per this discussion, it is claimed that there is no reason why optimizing for the growth rate is à priori optimal. In fact, this is just one of many criterions that (implicitly in this case) include the penalty on risk.

To decide on the optimal strategy, we need measures that factor in both expected growth and the risk of drastic reduction/extinction of the population. The science of finance, that deals with similar problems (maximize the profit and minimize the risk of default) offers us several measures that we could study.

Penalty On Risk

There are several possible explicit measures of risk that can be used for defining the penalty:

Volatility: roughly speaking, this is the average deviation from the mean;
Expected shortfall: this is the average over α% worst outcomes, α being a user-defined threshold, typically 1% or 0.1%;
Maximum drawdown: it is the maximum loss from a peak to a trough before a new peak is attained;
Lower partial moments, a.k.a. downside deviation: roughly speaking, this is the average return over all returns that are below a certain user-defined threshold.

The first two measures only take the final size of the population as the input, whereas the other two require the information about the evolution of the population size. Since we are only interested in the final number of tribbles, the first two measures are more relevant for our study.

The Formulae For The Number of Tribbles

One can show that the number of tribbles can be defined using the binomial distribution (over a number of sunny days). We can use it to derive several closed-form formulae.

We have a slight technical difficulty because of the discretization. Indeed, suppose there are 50 tribbles, and 5% are expected to go to the mountains. 5% of 50 is 2.5 tribbles, and we can only expect an integer number of tribbles to go to the mountains! So we will do the roundings randomly, with the probability of rounding up or down such that the expectations are matched!

With this assumption, the probability of having exactly g rainy reasons is
$\frac{G!}{g! (G-g)!} (1-s)^g s^{G-g}$
The number of tribbles after g rainy reasons (and (G-g) sunny seasons) is
$N k^G (1-v)^g v^{G-g}$
The expected number of tribbles is
$N (k (s v + (1 - s) (1-v)))^G$
The variance of the number of tribbles is
$N^2 k^{2G} ((s v^2 + (1 - s) (1-v)^2)^G - (s v + (1 - s) (1-v))^{2G})$
The log growth of the number of tribbles is
$G (\log k + (s \log v + (1 - s) \log (1-v)))$

We can compute the expected shortfall as a sum of a series. The probability that there will be R or more rainy days is
$\sum_{g=R}^G \frac{G!}{g! (G-g)!} (1-s)^g s^{G-g}$
The expected number of tribbles is there were R or more rainy days is
$N k^G \sum_{g=R}^G \frac{G!}{g! (G-g)!} ((1-s)(1-v))^g (vs)^{G-g}$
To compute the shortfall, we will first need to find R such that the resulting probability matches the expected shortfall we had defined, then use this R to compute expected number of tribbles is there were R or more rainy days – this will be our expected shortfall.

The Optimal Number of Irrational Tribbles: Numerical Model

My simulation results agree with Professor Lo’s statement about the optimality of probability matching for logarithmic growth maximization.

I have studied the dynamics of tribbles population after 100 seasons, for a range of rain probabilities and probabilities to choose the mountains. I have dropped the number of tribble’s children because this parameter only affects the results as a static multiplier, same for all rain and mountain probabilities.

To start, I have ploted the distribution of the number of sunny days, and the number of tribbles given the number of sunny seasons. Next, I have plotted initial measures of strategy success: expected number of tribbles, and expected number of tribbles over standard deviation of the number of tribbles (this measure was inspired by Sharpe ratio, the bigger the better). The conclusions we can draw from these two measures are contradictory: the former would suggest to choose the valley all the time, whereas the other suggest that splitting the choice 50%-50% is the best!

This contradiction is due to a fact that, when we compute the expected number of tribbles, the biggest contribution comes from the valley-almost-always strategies; these are very low probability strategies (if the probability of rain is not so low) with extremely high values.

If we compute the expected number of tribbles excluding the extremely favourable scenatios (this measure is insprired by expected shortfall risk measure. We can observe that, while the probability matching is fully optimal in all settings, it appears to be a “good enough” choice, thus making it a great rule of the thumb for the situations when a more sophisticated analysis is not available.

Finally, I have plotted the expected growth rate, which indeed peaks when the probabilities of rain and choosing mountains are equal.

The Python code I have used to get the numbers, as well as the results, can be found here.

Additional Information

“It is not the strongest of the species that survives, nor the most intelligent that survives. It is the one that is most adaptable to change.” Supposedly from On the Origin of Species, this quote is all over the web. However, Charles Darwin never said that. This quote comes from a management studies text.
As per Adaptive Markets, people find the probability matching strategy seems intuitive. Indeed, we can even use it wrongly when a simpler, “always choose the best”, strategy is optimal, thus creating a cognitive bias.