Talk:Evolutionarily stable strategy

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The following users are actively contributing to this topic and may be able to help with questions about verification and sources: Kzollman (talk • contribs • email), Pete.Hurd (talk • contribs • email)

The concept was based on W.D. Hamilton's (1967) unbeatable strategy; the difference is that an unbeatable strategy is resistant to large migrations of different. -- different what? Martijn faassen 21:37, 9 May 2004 (UTC)

I guessed at 'strategies'. --Noisy 11:32, 26 Jun 2004 (UTC)

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If someone could add an example of an ESS to the page that would be great. -Owen Jones 11:05, Sep 24, 2004 (UTC)

Contents 1 Disputed 2 ESS & Nash equil 3 Merge Bishop-Cannings theorem? 4 Good article nomination

[edit] Disputed

It seems someone (65.217.214.3) essentially reverted my edit to the ESS and sex ratios section, calling it "incomprehensible drivel". As I'd described the original as "bogus" and "counterfactual", I suppose I have no reason to complain about the strong language. However, what we now have here is a genuine factual (mathematical, in fact) dispute.

Here's the (undisputed) lead-in to the section:

ESS is used a lot in evolutionary biology. For example, consider a herd of elephants. Male to female ratio in a herd is close to 50/50, even though only a small percentage of male elephants ever get to mate and have offspring; the rest of the males are not strong enough to acquire females of their own. Consequently, it would seem that giving birth to male elephants is "pointless".

Here's how I continued it:

However, those males that do manage to mate have proportionately more offspring. In fact, while one female will produce about the same number of offspring regardless of the sex ratio, the average number of offspring for each male is proportional to the ratio of females to males. At a 1:1 sex ratio the numbers are equal.

And here's the reverted version:

-- having a female gives one much better chances to pass on his genes.

So the core of the dispute is: If there is an equal number of males and females, but only a few of the males ever get to mate, do females have a better chance to pass on their genes than males?

My answer is: no. — Ilmari Karonen 18:15:05, 2005-08-23 (UTC)

How can you say no when EVERY female gets to mate but only few of the males do? It has nothing to do with the total number of offspring. It has to do with WHOSE genetic material is passed on, female's or male's. That's it. --Coontie 03:17, 26 August 2005 (UTC)

No, that's not it at all. The key variable is sex differences in the *variance* in reproductive success (the means will of course be equal - with the standard caveats about operational sex ratios). For true enlightenment go read some of the primary literature. I don't think we should be moving toward a correct treatment of this well understood topic by successive iterations of on-line approximation. Pete.Hurd 06:48, 26 August 2005 (UTC)

 : Here's the example, rephrased. A fact F, that in a given herd of elephants, almost all females mate while only about 10% of males do. Given F, what would you, as a mother elephant choose (assuming you CAN choose) to have as an offspring -- male or female? Your sole purpose in life is to pass your genes onto future generations. Are you more likely to do that by having a male offspring, who would need to be big and strong (gamble?) to sire children or would you rather just go with a daughter elephant and almost GUARANTEE her reproductive success? I hate to say this but OBVIOUSLY, you'd pick having a female offspring. Now, mother can't choose, obviously but assuming that a child's gender is genetically determined, after millions of years you'd expect to see a drift towards more females than males. Until there are so many females that they go childless. Swing the pendulum back. Keep swinging until the status quo -- voila: ESS. What is wrong with this line of reasoning??? Somebody called this naive. So? Perhaps. Is it wrong though? --Coontie 14:23, 26 August 2005 (UTC)

Yes, it's wrong. A male elephant might have only a 10% chance to pass on his genes. But if he's one of those lucky 10%, he'll sire ten times as many offspring as a female ever could.

Look at it this way: If you could choose, which one would you rather take: one dollar in cash, or a lottery ticket that has a 1% chance of winning $100? On the average, it makes no difference. — Ilmari Karonen 15:29:32, 2005-08-26 (UTC)

You say "it's wrong" - what is wrong? The line of reasoning? How? Sure $1=1% chance of winning $100. That's why female/male ratio is 50/50. The question is, does this example illustrate ESS? Yes, it does.

The example doesn't really illustrate an ESS, it just says it does. It states that a particular strategy is an ESS, but gives no rigorous explanation why. For that reason alone, it deserves to be removed. (I freely admit that my version wasn't much better in this regard.) And what little explanation it attemps, it gets wrong.

Before going into specifics, let me not that there are three different versions of this section out there: besides the one I wrote, there's both the current version and the original version. The current version is vague enough that it can be charitably interpreted to be almost right, although confusing and beside the point. The original, however, is longer, more detailed, and makes correspondingly more factual errors.

One flaw both versions share is that they describe dynamic stability against large perturbations, not the evolutionary stability described by Maynard Smith and Price. As Taylor & Jonker (J.Math.Biosc. 40, 1978) have shown, neither necessarily implies the other. Both versions also seem to imply an internal evolutionary pressure at the equilibrium state — the original even explictly claims that, in a 1:1 sex ratio population, "a mother is better off giving birth to a daughter" — whereas an ESS by definition must have none. The original version also seems to describe a completely unrealistic overshoot phenomenon. — Ilmari Karonen 19:34:30, 2005-08-26 (UTC)

Investing equally in each sex of offspring is an ESS. It meets the conditions specified by Maynard Smith for an ESS, all other investment ratios do not. The example does not explain this adequately (but then neither does the sex ratio page, which ought to). The sorts of efforts displayed here don't look like they will fix the sex ratio page. What is required is to explain why investing 50:50 in males & females will invade a strategy of investing 51:49, and 49:51. That argument will explain things in a sense which will involve replicator dynamics-like arguments, but the conditions of the ESS will demonstrably be met. Pete.Hurd 21:15, 26 August 2005 (UTC)

Yes, I've considered writing a proper treatment of this, but doing it well implies giving the whole ESS article a facelift. Currently it only properly describes linear games, and sex ratio is a nonlinear game. I'm thinking of doing this, and I have Taylor & Jonker's paper and some lecture notes here to base it on, but I really want dig up Maynard Smith's book before starting a full rewrite. — Ilmari Karonen 21:56:07, 2005-08-26 (UTC)

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Regardless of sex ratio, mating habits or anything, every offspring always has two parents: one female and one male. So the total number of offspring had by females (over a given period) is the same as the total number of offspring had by males (over the same period). If the sex ratio is 1:1, the average number of offspring per individual is clearly also equal for males and females. Thus, it does not matter if your offspring are male or female; in either case you'll have the same number of grandchildren (and great-grandchildren, etc.) on average (as long as the general population's sex ratio remain 1:1).

If you disagree with the reasoning above, please post your counterargument here. If no-one objects, I shall re-revert the disputed section in a couple of days. (I may try to make it less "incomprehensible", but will retain the core of my reasoning as outlined above.)

— Ilmari Karonen (formerly known as 85.76.76.123) 18:15:05, 2005-08-23 (UTC)

Actually, I see no reason to revert the section until some consensus is reached, as long as the {{dubious}} tag stays there.

— Ilmari Karonen 18:57:24, 2005-08-23 (UTC)

In the original example, males have the number of expected children as females. They are less likely to father children, but when they do, it's likely to be in quantity (ie, they "succeed"). So the two are balanced. Having said that, some genetic material pass from mother to daughter (mitochrondrial DNA) or from father to son exclusively (Y chromosome). Ascribing self-interest to these genes for rhetorical purposes, it would be in their interest to further animals of the corresponding sex.

Another issue is that certain lines might have a much greater chance of a successful male (eg, belonging to human nobility, for example). Then a strategy of supporting female children at the expense of male would be advantageous, if your males aren't likely to succeed, while the opposite would be true, if your males will succeed fairly easily. -- KarlHallowell 19:17, 23 August 2005 (UTC)

I think this section is really badly written, IMHO (as a bad writer myself). Sex ratio theory is worked out enough that there is no real excuse for this to remain 'disputed'. I think that what is needed is either or both of: 1) writing a sold description, well founded in sex ratio theory, 2) Finding a better example of an ESS, and leaving sex ratio to the sex ratio page. my 2c Pete.Hurd 22:20, 23 August 2005 (UTC)

I agree with Peter. I also think that the section on sex ratios should be removed as it is unclear, confusing (this should be obvious because of the dispute) and not a good example of the meaning of an ESS. But maybe Peter and I are a minority, so we could have a vote on it. --Anthony Liekens 01:39, 25 August 2005 (UTC)

Actually, I agree with the comments above. Sex ratio isn't a very good introductory example of an ESS, particularly since the definition of an ESS given in the article is based on replicator dynamics, which don't actually apply here. The fact that a naive application of the ESS conditions gives the correct equilibrium is something of a coincidence. So I vote we remove this section, at least for now. — Ilmari Karonen 05:54:07, 2005-08-25 (UTC)

Wait a sec here. This section should be removed, but I just looked through the requirements and we might have them all (the iffy one is inheritability of sex ratios). Maybe it's a coincidence and maybe it's not. I don't like the unsourced assumptions about inheritable sex ratios in elephants. It would be better to make this a more abstract model. OTOH, ESS doesn't explain deviations from the equilibrium. If one looks at human sex selection, you see that males have a slight edge (50.5% of naturally occuring births though sex might vary with the birth order) which shouldn't exist according to a naive application of this theory. -- KarlHallowell 21:52, 28 August 2005 (UTC)

There is no requirement that strategies be heritable in order for the definition of an ESS to be met. Heritability is a biological concept, and an ESS is a game theory concept, different areas. This discussion about the sex ratio example continues to confuse ESS with other issues. Sex ratio theory does not predict exact 50:50 sex ratios, it predicts 50:50 investment in each sex. If one sex is more expensive to raise to adulthood per individual then a 50:50 investment will not produce a 50:50 sex ratio. Again, this ought to be in sex ratio, along with some Trivers-Willard theory stuff. ESS, and game theory in general has nothing to say about behavior away from equilibria, that's just the way it goes... Pete.Hurd 01:31, 29 August 2005 (UTC)

[edit] ESS & Nash equil

Moved from main article

"As it turns out, every ESS corresponds to a pure strategy Nash equilibrium, but (as illustrated in the above examples) not every pure strategy Nash equilibrium corresponds to an ESS."

I think this isn't true (mixed ESSs are definitely ESSs, and not pure Nash). I'm wondering if there is a point here that I'm just not seeing. "All ESSs are Nash equilibria, but not necessarily vice versa" is true, is this the original point, or was it something more subtle? Pete.Hurd 20:45, 10 October 2005 (UTC)

[edit] Merge Bishop-Cannings theorem?

User:Trialsanderrors tagged Bishop-Cannings theorem for merging into Evolutionarily stable strategy on June 24. I seem to remember some discussion of the merits of this idea, but can't find it now. I think the best argument in favor of merging is that BCt is very stubby and stands poorly on its own. Arguments against are that, 1) if it were properly expanded (eg by incorporating the appropriate material from the external link on the BCt page) then it would e far longer than the ESS page. 2) The BCt applies equally to Nash equilibria as it does to ESS, and therefore putting it under ESS makes little sense (and neither does putting it under Nash equil, if only for historical reasons - it's always been discussed in terms of the ESS). Thoughts, comments? Pete.Hurd 17:46, 7 August 2006 (UTC)

It does look like there is a lot to say. I just skimmed the external link, so I don't know how much of that should be included in the article. But if most of it could be, I would vote for keeping it as its own article. In general, I don't mind keeping stubs around, so long as they can become longer. But I don't really feel strongly about the matter. --best, kevin [kzollman][talk] 18:07, 7 August 2006 (UTC)

I'll check see what User:Trialsanderrors thinks, maybe he's got some persuasive argument I ought to hear. Pete.Hurd 01:45, 8 August 2006 (UTC)

Thanks for the heads-up, Pete. I'm ok with keeping if someone is willing to properly expand it, but as a stub it has only very limited legitimacy as a stand-alone article. I think the proper protocol would be to put it in the ESS article temporarily and see if it gets expanded. As soon as it has enough detail in the ESS article it can easily be turned back into a stand-alone article. My issue is mostly that the link in the game theory topics box creates the false impression that we have anything substantial to say about it. ~ trialsanderrors 02:01, 8 August 2006 (UTC)

Makes sense to me. I wish I had time to expand the stub, but I don't, so I'll merge with the hope that it won't stay that way... Pete.Hurd 02:37, 8 August 2006 (UTC)

[edit] Good article nomination

• in section Bishop-Cannings theorem, the external link should be internal reference
• in section ESS vs. Evolutionarily Stable State, there are 9 "strategy" words, could you please somehow fix it?
• in that section, there are two Maynard Smith quotes without references
• in section Prisoner's dilemma and ESS + ESS and human behavior: there is an other external link which should be converted into internal reference
• References should have div small
• External links?

Anyway really great article. I read it with pleasure. NCurse _work 06:50, 10 September 2006 (UTC)

Thanks for your efforts Pete.Hurd, it's now a good article. Congratulations! :) NCurse _work 14:55, 12 September 2006 (UTC)