The Aldas and the Lecters

Those of you who aren't bored by now may have noticed that Sean and I have been carrying on a very interesting and enjoyable comment thread about evolution, the genetic origins of cooperative behavior, and fun stuff like that. Quoth he:

Without the ability to know the modeling, I don't think I can take this any further...

So, just to be a pill, here is a simple Excel model to think about. REMINDER: I am not in any way an expert in this subject, nor even qualified, so don't cite me in your homework.

Description

For this model, we have a tribe of cave persons. These cave persons come in two genetic stripes: Aldas and Lecters.

• Aldas are kind, cooperative, brave, loyal, trustworthy, altruistic, and generally the kind of people you'd like to have as neighbors. They help out other tribe members, even when it costs them. Aldas are good hunters and good providers, because they work well together and share their takings equitably.
• Lecters are intelligent sociopathic libertarians. The Lecter creed is "every cave-person for himself". A Lecter will only help out another tribe member if he/she thinks there's something of equal or greater value for him/her in the deal. Lecters are less effective providers. However, they're clever and selfish and they know how to take advantage of the Aldas (and each other). Thus, they have a slight reproductive advantage. 

Simply put, being an Alda is much better for the tribe, but being a Lecter is a little bit better for you, personally.

Terms

The spreadsheet uses the following terms and equations.

• Replacement rate is the number of children each tribesperson generates who (a) survive, and (b) share his genetic signature. A replacement rate of 1 indicates perfect population stability. By assumption, Lecters have a slightly higher replacement rate than Aldas, because they don't risk their genes or waste their resources on helping others. 
• Carrying capacity is the number of tribespersons each adult can support by hunting, gathering, macrame, performance art, etc. A carrying capacity of 1 indicates that each individual can support one person. By assumption, Aldas have a much higher carrying capacity, because they look out for each other and cooperate effectively.
• Each generation is a single row in the spreadsheet. For simplicity, I use a total-replacement model: at each generation, all the adults die off and are replaced by their surviving children.
• The number of Aldas born in each generation is the Alda replacement rate × the number of Alda parents, and similarly for the Lecters.
• Not all of these children may survive (sorry). The number of survivors is the sum of # of Alda adults × Alda carrying capacity + # of Lecter adults × Lecter carrying capacity.
• Die-off of children is distributed evenly between Aldas and Lecters. (Aldas may cooperate better to care for their children, but Lecters are heartlessly efficient in protecting their individual genetic progeny.) That is, if 70 out of 100 children are Aldas, and 30 children die because of insufficient carrying capacity, 21 of them (70%) are Aldas and 9 (30%) are Lecters.

Results

I started with these values:

	Alda	Lecter
Replacement rate	1.5	1.6
Carrying capacity	2	0.9

Note that the Lecter carrying capacity is actually less than 1. That is, Lecters are so quarrelsome and inefficient that they can't even support their own numbers. Thus, Lecters must have Aldas around them to survive. The Lecter reproductive advantage is comparatively small.

With an inital population of 50 Aldas and 50 Lecters, the first few generations look like this:

Generation	Aldas	Lecters	Tribe Size	% Lecter
1	50	50	100	50%
2	70	75	145	52%
3	97	111	208	53%
4	133	161	294	55%
5	179	232	410	56%
6	238	328	566	58%
7	312	459	771	60%

(Calculation columns omitted for clarity.)

After only 7 generations, the tribe is much larger, but it's 60% Lecter. Looking at further generations, the Lecter advantage only grows:

18

2524

7561

10086

75%

23

3946

16324

20271

81%

37

3750

38284

42033

91%

Generation 37 is the high point of the tribe's success, in terms of sheer numbers. After this, the number of conniving, squabbling, backstabbing, selfish Lecters is so large that the tribe can't sustain itself. But the fraction of Lecters keeps growing, even as the tribe shrinks ...

45

1980

33883

35864

94%

55

617

20128

20745

97%

65

151

9367

9517

98%

The last of the Aldas dies off at generation 101. The tribe goes extinct at generation 162.

Comments

This is obviously a very naive and hyper-simplified model, with many gross assumptions. Perhaps the most glaring one is that infant mortality is distributed randomly between Alda and Lecter children. 

Nonetheless, it does illustrate--no more than that--the paradox that population geneticists grapple with. 

• One the one hand, if a given trait is even slightly advantageous within a group, then it can be expected to come to dominate that group even if it makes the group as a whole weaker. 
• On the other hand, real-world hunter-gatherers are much more like the Aldas. Even among "civilized" folk, we don't see sociopaths dominating society anywhere in the world (except in YouTube comments).

Playing around with the parameters changes the shapes of the curves but doesn't change the basic outcome: as long as they have a higher replacement rate, Lecters dominate. If the Lecter carrying capacity is set to 1, the population eventually stabilizes (at 100% Lecter) and neither grows nor shrinks. If the Lecter carrying capacity is > 1, the population grows without limit, although the growth rate flattens out once the Aldas are gone.

Also, the Alda/Lecter imagery is irrelevant to the basic math, so don't get hung up on this particular concept. It doesn't matter if the trait distinguishes between cooperative and selfish individuals, or if it's something else--height, or extra fingers, or trombone-playing ability. (For example: in this tribe the trombone is considered super duper sexy, and therefore the trombonists get a lot of sweet lovin'; but all the noise attracts saber-toothed carnivorous music critics, so some people get randomly eaten.) Any time you get a trait that's good for its owner, but bad for its group, this model can be used.

Update, 6/15: I was curious to see what happened if we change the assumption that deaths are equally likely among the Alda and Lecter children. I introduced a setting for the presumed Alda advantage. An Alda advantage of 0.8, for example, indicates that only 80% as many Alda children die, with the balance being Lecters.

Somewhat to my surprise, this does change the shape of the curve.

• With an Alda advantage of 0.8, the fraction of Lecters in the population stabilizes at about 65%, around generation 67+, and the tribe doesn't die out. 
• With an Alda advantage of 0.9, the tribe does begin dying out, and the Lecter % keeps going up, but it's much slower than in the base case. 
• With an Alda advantage of 0.2, the initial 50/50 balance is unchanged. At this level, 89% of all child deaths are Lecters. So an Alda advantage of 0.2 is necessary to offset the effect of the Lecters' higher replacement rate of 1.6.