Hi everyone!
A while ago over at AtBC I had the notion of using dice to try and show how fitness and luck combine during the selection process. I (with the Help of Wes Elsberry) thought that using dice might be a good way to explore this and other concepts such as mutation, sexual selection etc.
So.. here we go.
Imagine a die. It has 6 sides. Let’s call these [A,B,C,D,E,F]. In the dice we all know and love each side has a number associated with it, let’s write that as [1,2,3,4,5,6]. So far so good? – so the “B” face is 2. The “E” face is 5.
Now let’s imagine the life of dice is gladiatorial and to “win” a die must roll higher than another competing die on a single trial of one on one dice combat. If he (I think of them like little people, I can’t write “it” any more) rolls a high number, his odds are better, so fitness clearly scales with numbers. So for our normal die, We’ll call him Norm, his fitness (total) is 1+2+3+4+5+6=21 and his average fitness is 21/6 sides = 3.5. Let’s imagine the Lord of the Dice is a nerdy dungeonmaster called KeithS. He hasn’t decided what happens if two dice roll “The dreaded tie” yet. Maybe dice purgatory? I just really wanted KeithS in this.
Now there’s no reason why a dice has to be [1,2,3,4,5,6]. Norm is, but let’s compare him to the other challengers:
Norm [1,2,3,4,5,6]
Uptight BiPolar [1,1,1,6,6,6]
Low SD [3,3,3,4,4,4]
IDist [1,1,4,5,5,5]
They all have the same average fitness (3.5) as I described above. How do you think they fair in head to head match ups? I’ll reveal the answers via the power of EXHAUSTIVE ENUMERATION in a while. If no one cares then You’ll never know and KeithS wins. No-one wants that!
Let’s Roll!
Rich,
Is that as powerful as cumulative selection?
keiths,
It always gets the best result… although for large search spaces you may need a couple of (universal) lifetimes..
I think Uptight Bipolar. High deviation from the mean gets selected for. Just my educated gambler instinct.
IDist kicks ass
Oooh-errr, nevermind.
That’s very cool!
I should add, If anyone wants to design a new combatant, we can add them.
Dice rules:
Integers only
minimum value of 1
Average fitness of all sides for this competition of 3.5
What do you think would happen if we allowed 7s and 8s or should we max at 6?
Norm [1,2,3,4,5,6] vs UpBiPolar [1,1,1,6,6,6]
= in terms of Norm 0 +1+1-1-1+0 = 0
Low SD [3,3,3,4,4,4] vs UpBiPolar [1,1,1,6,6,6] = int terms of Low SD
1 + 1 + 1 – 1 -1 -1 = 0
IDist [1,1,4,5,5,5] vs UpBiPolar [1,1,1,6,6,6] = in terms of IDist
0 + 0 + 1 – 1 – 1 – 1 = -2
Norm [1,2,3,4,5,6]vs Low SD [3,3,3,4,4,4] = in terms of Norm
-1 -1 +0+0+1+1 = 0
IDist [1,1,4,5,5,5] vs Norm [1,2,3,4,5,6] = in terms of IDist
0 – 1 + 1 + 1 + 0 – 1 = 0
Low SD [3,3,3,4,4,4] vs IDist [1,1,4,5,5,5] = in terms of Low SD
1 + 1 -1 -1 -1 -1 = -2
Net scores
Uptight = 0+0 + 2 = 2
Norm = 0 + 0 + 0 = 0
Low SD = 0 + 0 – 2 = -2
IDist = -2 + 0 + 2 = 0
Uptight Bipolar wins.
Did I implement the rules correctly?
This is somewhat related to the question of if Men and Women have the same mean intelligence, whether Men (on a level playing field) can field more scientists and CEOs. Answer, hypothetically “yes” if the Standard Deviation from the mean for men is more extreme (namely if they field larger numbers of imbeciles and geniuses).
Sal,
No, because you need to match each face of one die against each face of the other die.
Dungeon master is right 😉
I volunteer Matzke {1,1,2,5,6,6} as tribute…
ETA in place of Dawkins {1,1,3,4,6,6}
DNA_Jock,
Matzke has picked up an interesting quirk.. he can’t tie with Low SD.
Well, that would be the only thing he shares with Uptight Bipolar.
😮
Well, that and a few mammalian synapomorphies. 😉
How “win” happens not well explained, I had to guess what you meant by win.
Anyway, I tried.
Its a six by six matrix all possible outcomes, Win, Lose or tie. My explaining, as always, needs work.
Thanks Rich.
Side D,E,F appears for Upright Bipolar vs. Norm Dist.
I use +1 for win, -1 for loss, 0 for tie.
f(A) = f(6 vs 1) = 1
f(B) = f(6 vs 2) = 1
f(C) = f(6 vs 3) = 1
f(D) = f(6 vs 4) = 1
f(E) = f(6 vs 5) = 1
f(F) = f(6 vs 6) = 0
for Uptight Bipolar A,B,C
f(A) = f(1 vs 1) = 0
f(B) = f(1 vs 2) = -1
f(C) = f(1 vs 3) = -1
f(D) = f(1 vs 4) = -1
f(E) = f(1 vs 5) = -1
f(F) = f(1 vs 6) = -1
Composite score Norm Dist vs. Upright Bipolar = 0
I’m guessing Upright Bipolar vs. Low SD = 0
Upright Bipolar vs. IDist
f(A) = f(6 vs 1) = 1
f(B) = f(6 vs 1) = 1
f(C) = f(6 vs 4) = 1
f(D) = f(6 vs 5) = 1
f(E) = f(6 vs 5) = 1
f(F) = f(6 vs 5) = 1
for Uptight Bipolar A,B,C
f(A) = f(1 vs 1) = 0
f(B) = f(1 vs 1) = 0
f(C) = f(1 vs 4) = -1
f(D) = f(1 vs 5) = -1
f(E) = f(1 vs 5) = -1
f(F) = f(1 vs 5) = -1
composite score Uptight Bipolar = +2, Uptight wins on average vs. IDists.
Is that the right idea?
stcordova,
I was coming at it like this:
http://postimg.org/image/f0iss45pn/
Edit for link fix
Looking at those funky number-sets, my first thought was, “Non-transitive dice, huh?” But no, that’s not it. Only in one of the possible matchups does any dice have a probability of success greater than 50%—IDist has a 21/36 chance of defeating Low SD—and in a proper set of non-transitive dice, every die has at least one other die they’ve got a better-than-50% chance of defeating.
So.
It’s fairly trivial to set up a spreadsheet that covers all possible matchups, and i did that. The chance of Die X defeating Die Y is…
for Norm:
…against Norm, 15/36 chance of success
…against Uptight Bipolar, 15/36 chance of success
…against Low SD, 15/36 chance of success
…against IDist, 15/36 chance of success
for Uptight Bipolar:
…against Norm, 15/36 chance of success
…against Uptight Bipolar, 9/36 chance of success
…against Low SD, 18/36 chance of success
…against IDist, 18/36 chance of success
for Low SD:
…against Norm, 15/36 chance of success
…against Uptight Bipolar, 18/36 chance of success
…against Low SD, 9/36 chance of success
…against IDist, 12/36 chance of success
for IDist:
…against Norm, 15/36 chance of success
…against Uptight Bipolar, 12/36 chance of success
…against Low SD, 21/36 chance of success
…against IDist, 11/36 chance of success
Yes, I calculated the odds of success for a die battling its own type of die. Why wouldn’t I do that? Oh, and I didn’t bother with ties, but if you look at my results with a judicious eye, you should be able to determine the probabilities of ties for any possible matchup between those four dice.
cubist,
Awesome – I did the same here. I think averaged the win percentages against other dice:
I get this, but it needs a checking:
Win
Norm 42%
Uptight Bipolar 44%
Low SD 38%
Idist 38%
Matzke 43%
So I’m wondering:
/
Can you get a scissors | paper | stone type relationship?
Can we create an optimized die?
How good a measure of fitness *is* our fitness?
How much do competitors affect environmental fitness?
If we changed to best of three would it be different?
What about survive 5 trials to be successful? (which is maybe closer to some evolutionary niches)?
I feel like I’m almost sciencing.
Yes, you can—that’s what non-transitive dice are. Google for “non-transitive dice”, and be edified by the reading of the hits.
Sure, why not? Just decide what ‘target’ you’re optimizing for.
[shrug] Dunno. Good for what purpose? As a toy demonstration of evolutionary principles, I think did-DieX-roll-higher-than-DieY is a perfectly good measure of fitness. If you’re trying for a measure of ‘fitness’ which closely emulates all the funky weirdness we see in real biology, did-DieX-roll-higher-than-DieY probably isn’t what you want.
I don’t think there’s a one-size-fits-all answer to this question. It depends on the specifics of which competitors, and what sort of environment.
Yes. I think it would tend to exaggerate the probability of success. You may want to run a Monte Carlo-type simulation to see what happens.
“Survive 5 trials”? Hmm. If you mean win in five successive battles against the same die, that’s a recipe for extinction. The highest probability of success (IDist over Low SD) is 21/36; with that as the probability of winning any one bout, the chance of winning five bouts in a row is about 6.7%.
I think you should flesh out what you mean by “best of three” and “survive five trials”. Does only Complete, Total Success count? If less-than-Complete Success is a thing, what does it mean to say that one die had partial success over another?
cubist,
Thanks!
For all: https://en.wikipedia.org/wiki/Nontransitive_dice
‘Best of three’ would be win at least 2, ‘Survive 5’ would be win 5 (against random opponents)
Uptight Bipolar beats IDist, IDist beats Low SD.
All other pairings in the original set produce ties (including, critically, Uptight and Low SD).
However, Matzke (and to a lesser extent, Dawkins) beat IDist, while tieing everyone else…
Can I have a zero on my die?
No Junk DNA! 😉
Sure, why not? Nothing says a die must have positive integers on all faces.
😉
So… a leading light intelligent design mathematician can’t cope with high school mathematics.
Rich,
Thanks for the post.
As far as Uptight Bipolar, there has been an interesting question (which I don’t think is resolved) as to whether men are hypothetically under higher selection than women for intelligence because even assuming they have the same mean intelligence, they have a different Standard Deviation (SD) than women. Most presume the male population is like Uptight BiPolar, they have more imbeciles and geniuses than women, even though the mean intelligence is the same.
stcordova,
According to whose idea of intelligence are you basing this theory?
stcordova,
So now you believe in natural selection?
stcordova,
I saw this: http://www.iqcomparisonsite.com/sexdifferences.aspx
“The paper that supports the conventional wisdom is Jensen, A. R., & Reynolds, C. R. (1983). It finds that females have a 101.41 mean IQ with a 13.55 standard deviation versus males that have a 103.08 mean IQ with a 14.54 standard deviation.”
Although I believe IQ isn’t intelligence and learning opportunity, culture, environment etc. may be a factor.
If fitness is an after-the-fact assessment (which it is as fitness depends on reproductive success or lack thereof), then how can one combine it with luck during the “selection process” (whatever that is)?
We determine fitness after the process is finished. So what does this post have to do with biology?
Which it wasn’t. If you can read:
“fitness (total) is 1+2+3+4+5+6=21 and his average fitness is 21/6 sides = 3.5. ”
This is before any selection. Do you think it is positively or negatively correlated with survival?
Richardthughes,
I think it is you who can’t read (well).
What Frankie is saying is that in evolution, there is no such thing as fitness before the fact. So this has nothing whatsoever to do with evolution.
Luck IS fitness.
In biology fitness is an after-the-fact assessment for the reason provided. Ignoring that reason just exposes your desperation.
If that is true then my fitness measure should be orthogonal to survivability. Do you agree, Phoodoo?
Dice don’t need to survive
Frankie,
There are a few versions in biology:
https://en.wikipedia.org/wiki/Fitness_(biology)
Obviously this post is heading down this road:
https://en.wikipedia.org/wiki/Fitness_(biology)#Fitness_is_a_propensity
Richardthughes,
Not orthogonal! Did Patrick just teach you a new word?
That which survives is fit. Not that which is fit survives.
So that is your prediction? I already have the results.
Richardthughes,
I predict that you believe dice can turn into lobsters given a little more time.
Rich,
Hey thanks for the link. I was unware of those studies.
Not the most politically correct, but I thought it informative and compelling.
Sal
stcordova,
Do you believe in natural selection Sal? Be brave and answer the question.
It is still reproductive success, ie the propensity of certain genotypes and/ or phenotypes to leave more offspring- it is all after-the-fact AND changing. Also there is competition- is better eyesight fitter than better hearing? Is stronger better than faster? It is all contingent. That is why Dennett wrote that “there is no way to predict what will be selected for at any point in time”. Yet with your model we can predict as fitness was handed out from the start and is unchanging
Oh Phoodoo, are you afraid to make a prediction? That’s sad.
Frankie,
Yes we can predict. But not with certainty. A bit like life.
I believe it is possible. The issue of whether not the variation was accidental/ happenstance is still the big issue. Natural selection requires that the variation be due to accidental/ happenstances changes.
Your example has nothing to do with life.
Richardthughes,
You think dice reproduce. That’s why you think evolutionary algorithms on computers mean something.
You need help. Maybe you can try tweaking your nose?
Oh Phoodoo, are you afraid to make a prediction? That’s sad.
Richardthughes,
I predict your posts will continue to be even more mind numbingly boring than watching reruns of Mr. Bean.
Want to wager?