## Tuesday, May 20, 2008

### Natural Selection and the Gambler's Ruin

I mentioned in my comments with Mighty Pile the Gambler’s Ruin. The GR occurs when a gambler runs completely out of money. There are two aspects of the GR that impact our understanding of Natural Selection. First is the fact that if you are at a numerical disadvantage, then even if you have a statistical advantage in gambling you will often hit GR first simply because the other person can take “more damage” before he reaches it. Thus, just because one individual gains a favorable mutation does not mean that that mutation will be automatically chosen for due to the sheer number of competitors that the individual would have to compete with.

But more importantly is the fact that Natural Selection, in order to work at all, is an All-Or-Nothing proposition. That is, favorable traits must be selected for while unfavorable traits must die out. In one of his comments, Mighty Pile said:

Some traits DO confer an advantage to a particular organism and its progeny. While fit individuals certainly do die sometimes and unfit individuals certainly do live sometimes, the fit organisms would outcompete the unfit ones in large numbers. One antelope's chance vs another antelope's chance may be a 49%-51% split. But in a whole herd, the one that gets eaten will almost always be of the slower variety, or of the sick or injured variety. This is, of course, supposition based on logic. I don't know how I'd prove it right now; it seems obvious in any context I can come up with for it. The difference between fit and unfit would probably be very small most times, setting up a sort of tipping point situation. I don't have to outrun the bear, I only have to outrun you, right?
In response, I pointed out that one would be foolish to wager everything he owned for a chance to win a billion dollars if he only had a 51% chance of winning it and a 49% chance of losing it. This did get me to thinking a bit further, however, and I developed the following.

Suppose you start with 100 individuals. Each begins with \$100. Each wagers \$100 in order to gain \$100. The odds are 51% win and 49% lose each bet, but with the following stipulation: as soon as you hit \$0, you’re out of the game. You cannot continue. This is important to mimic Natural Selection, because as soon as you die you can no longer reproduce. It’s over. So you need a final set point.

How long will it take for a person to reach \$1,000 given this structure? And how many people will hit GR before that occurs?

I made up an Excel spreadsheet to show this to me (click here for graphic). It assumes a literal 51% - 49% split for each round (in other words, I don’t randomize the data; this is the “ideal”). The vertical axis is how much money people have; the horizontal axis is the number of rounds. The number plotted in each cell is how many people remain for each row (i.e., how many people have whatever money is in that row). The bottom line beneath the graph simply sums how many people remain (i.e., those who did not hit GR). The cell at the far right of the 0 line is the grand total of those who hit GR.

Thus, we begin with 100 people holding \$100. After the first round, 49 people are bankrupt and 51 people have \$200. In the second round, 51% of those 51 people (26.01) at \$200 will gain another \$100 for a total of \$300, while 49% (24.99) lose \$100 to go back to \$100 total. For the third round, we again calculate each group: 51% of the 26.01 (13.27) at \$300 will go to \$400; simultaneously, 49% of the 26.01 (12.74) drop back to \$200. In the meantime, 51% of the 24.99 (12.74) who dropped to \$100 will gain \$100 and make it back to \$200. They combine with the 12.74 who lost \$100 to drop down to \$200 to make 25.49. Finally, 49% of the 24.99 (12.25) who were at \$100 will go bust.

Note that unlike in real life (where a whole number of people either win or lose), these calculations are made with the decimal points from the previous numbers still intact. In fact, I used dependent formulas for each cell. If we were rounding before we did the math, the answers would vary slightly.

And the results: It takes 11 rounds for the first person to hit \$1,000 (and that’s only if you round 0.58 up to 1; the line does reach in round 9, but the value would round down to 0). In the meantime, 75.93 people have gone bust. That means that in order to get one person from \$100 to \$1,000, 76 people have to go bankrupt. And that’s starting with 100 people. A 1% advantage does not provide much of an advantage at all under these circumstances.

Natural Selection falls to the same principal. Just because a favorable mutation may confer a 1% advantage onto an antelope does not mean that the antelope really has that much more of an advantage than other antelope. And I should point out that living systems are actually far more complex than even this illustrates.

The key to why this works this way is because the chart is capped at 0. Once you hit 0, it’s over. That provides a literal line in the sand that has a huge impact. Because in Natural Selection death is such a line in the sand, this demonstrates that even a 1% advantage holds no real benefit to the furtherance of a trait in the species.

In reality, survival rarely comes down to a single trait though. Chance encounters are almost always going to outweigh any mutational advantage of a single trait. Consider all the following that mitigate against the classical view of Natural Selection:

* An antelope is born with 1% more speed than any other antelopes who have been born. However, when the antelope is a newborn, he is not as fast as the adults. As a result, despite being 1% faster than all other newborns, he is still slower than the slowest adult; therefore, he remains a preferential target for predators. If he is near adults at the edge of the herd when lions attack, they will go after him rather than the adults. This brings to mind the second point:

* As Mighty Pile pointed out, there is an oft repeated joke that one need only be faster than the slowest prey when a predator attacks. This, however, ignores the fact that if you are faster than me, but you are five feet away from a hungry bear while I am a quarter mile away from the hungry bear, the bear will catch you before you can run far enough to surpass me and make me a target.

* Sometimes pure dumb luck happens. A ram may be the fittest ram ever, but if he slips and breaks his leg, he’ll be eaten. And accidents happen quite often in nature. And even aside from nature. A highly specialized and advanced snake in Baghdad might happen to get hit by a mortar round fired from an insurgent that was not intended to strike the snake, but did. Or a random lightning strike could kill an elk in the forest who was “superior” to the other elk. When it comes to random events, traits have no bearing on survivability. There is no survivability trait for bad luck.

* For that matter, the strongest bull may be cut down by a viral infection that attacks only strong animals, leaving the weak bulls alive. The weak bulls are “more fit” (by definition, since they survived) but once the infection runs its course the herd would have been better off with the stronger bulls.

* A mutation for greater intelligence might occur in a sheep that’s also the least hearty sheep in the herd. Despite the fact that this intelligence trait would benefit the herd as a whole, the sheep dies of an illness before reproducing.

So survival rarely is about any one trait anyway. Instead, to have the best chance at surviving, organisms need to have a wide range of traits, any one of which may or may not be relevant at any particular time. But some traits are mutually exclusive. Because evolution must be blind (in a materialistic world) it cannot predict which trait will be needed in the future. And because it cannot predict what is needed (after all, it is non-teleological; and furthermore, even intelligent agents like weathermen cannot predict what will happen in the environment tomorrow), the random forces of nature will far outweigh any slight statistical advantage that individuals in a herd have.

So the only way to have beneficial mutations that avoid the GR problem is if they grant a far greater than 1% chance upon the individuals (after all, think of mutations, which convey far more than a 1% disadvantage to the individuals and therefore are seen!), or if they occur more often than random mutations would enable them to occur so that more individuals get the trait (remember, we started the above graph with 100 individuals already having \$100, and 76 of them went bankrupt before a single person reached \$1,000; if you had 1,000 people to begin with, 760 would go bankrupt…but you’d have 10 make it to the \$1,000 mark, so clearly having more individuals get the same mutation would help), or the mutation would have to occur in an individual that is already “more fit” due to other traits to begin with (and that brings up the converse: a detrimental mutation can occur in those who are “more fit” due to other traits and therefore be “selected for” simply because it’s riding along with the system; whereas a “less fit” organism might evolve a wonderful trait that cannot overcome the aspects that make it “less fit” and therefore that trait is not “selected for”).

That’s a lot of front-loading you need before you can get the system going. Living systems are far too complex to be affected greatly by any slight advantage in a single trait.