Talk:Evolutionarily stable strategy

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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)

This is still a major problem with the article. An example in the opening paragraph would be ideal. TitaniumDreads (talk) 19:03, 5 August 2013 (UTC)

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)


It seems someone ( 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)

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 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)

ESS & Nash equil[edit]

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)

Merge Bishop-Cannings theorem?[edit]

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)

Good article nomination[edit]

  • 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)

GA review[edit]

This review seems somewhat superficial to me: there are deeper issues with this article than the number of occurences of "strategy" words (and "References" do not have to be div small). The lead is weak as a stand-alone summary of the article, and the article is much less accessible than it could be. It seems to me that there is some historical content here that deserves a separate section in the article, and there is missing data in some of the references. It may be possible to fix these issues easily, and I will attempt to do some of this myself over the next couple of days. If we can't fix these problems, I will delist the article so that it can be renominated. An alternative is to take the article to Good article review, and any editor is welcome to do that. Geometry guy 21:54, 12 July 2007 (UTC)

GA review (see here for criteria)
  1. It is reasonably well written.
    a (prose): b (MoS):
  2. It is factually accurate and verifiable.
    a (references): b (citations to reliable sources): c (OR):
  3. It is broad in its coverage.
    a (major aspects): b (focused):
  4. It follows the neutral point of view policy.
    a (fair representation): b (all significant views):
  5. It is stable.
  6. It contains images, where possible, to illustrate the topic.
    a (tagged and captioned): b (lack of images does not in itself exclude GA): c (non-free images have fair use rationales):
  7. Overall:
    a Pass/Fail:

I wasn't able to fix the evident problems with this article. The lead is still an inadequate as a summary of the article, and the prose is poor and inaccessible. The article does not seem to cover the topic adequately, or explain it to a sufficiently wide audience. Geometry guy 19:37, 14 July 2007 (UTC)

I have a few questions about the checklist. What are the manual of style violations that garnered the red minus? And why is the article not broad in its scope? --best, kevin [kzollman][talk] 18:28, 16 July 2007 (UTC)
Sure, and sorry for the slow response. Not all MoS issues are considered too carefully at GA level, but the lead and jargon are. The lead is supposed to be a stand-alone summary of the article: any important information in the article should be mentioned in the lead, and any ideas in the lead should be discussed in the article. Tim has already suggested that there is missing historical material in the article. Other examples: the lead mentions the evolution of altruism, whereas the body instead mentions sociopathy; the lead talks about rational foresight, but the body does not; the lead does not seem to refer to the section on "Evolutionary stable states". I'm also not sure about the Bishop-Cannings theorem - how important is it?
As for jargon, it is fine to use technical terms, as long as they are explained or there is a good wikilink for them (which preferably gives the reader the gist of the idea in the first sentence). In my view, there are problems here with marginally technical terms such as "invaded", "functionally equivalent" (I'm not sure what that is trying to say, or I would have fixed it in my recent edit), "mutant strategy", and "disturbance". Clarifications, explanations, definitions on the fly,... would all help: sometimes the addition of "in this sense", or "such a" could be enough to reassure the reader.
Finally coverage: the criteria ask that a good article "addresses the major aspects of the topic"; it is not required to be comprehensive. Now, I am not a domain expert: I cannot be sure whether the article covers the major aspects, so let me assume that it does. However, in my view, there are several topics that it mentions (e.g. in the lead) but which are not adequately addressed. One example is the role of "rational forsight" which can distinguish ESSes from Nash equilibria. There are also not enough explanations. I think more words are needed. I have tried to show this in the "Nash equilibria and ESS" section with a few additions. I don't know if my edits are an improvement overall, but hope that they indicate some of the issues I am getting at.
I have the impression that a big difficulty in this field is that although the math is fairly straightforward, conveying the motivation, meaning, and intuition is quite hard work. I think this is partly because of the variety of applications (economics, biology etc.) of the same basic ideas, and the corresponding variety of terminology. Good luck with getting this back up to GA - I hope these comments help! Geometry guy 18:21, 17 July 2007 (UTC)


  • The lead devotes a paragraph to the history of this subject, but this is not mentioned further in the article. A "History" section would be a good addition, didn't somebody get a Nobel prize for some of this work?
  • "It also allows for a natural definition of other concepts like a weak ESS or an evolutionarily stable set" Unfortunately if there are no articles on these terms, they will have to be defined in the text.
  • The example could be improved by connecting it to a real-life situation.
  • I've tried to rewrite the section on "ESS vs. Evolutionarily Stable State" but am still not really sure what this section intends to do. Is it just to define two commonly-confused terms? It isn't very clear and I may not have improved it much by tinkering! :)
Hope this helps, Tim Vickers 16:04, 15 July 2007 (UTC)

Thanks Tim. I'll take a look at your edits on the "ESS vs. Evolutionarily Stable State" section. No Nobel prize, but JMS's Crafoord press blurb is certainly very close. Pete.Hurd 19:33, 16 July 2007 (UTC)


I like the idea of a easy to access example in the intro, but I'm worried that sexual selection/handicapping is probably not it (I'm not sure how many people will be familiar with handicapping, but not ESS especially in the economics g.t. world). I must say I'm at a bit of a loss for an informal example. Perhaps the biologists have something handy? --best, kevin [kzollman][talk] 18:25, 16 July 2007 (UTC)

Best exampes from biology would be either sex ratio or hawk-dove game: The potential problems with these two examples is that:
  1. sex ratio is very tightly linked to a lot of (very interesting) empirical data. My experience with using real world examples to explain theoretical points on wikipedia is that the article will quickly sprout a whole bunch of if-ands-and-buts dealing with the intricacies of Trivers-Willard hypothesis Bateman's principle and Sex allocation etc. The information in the article will become more "correct" as editors tweak on more specific counter-examples until the value of the example is buried under a huge section of empirical noise.
  2. The hawk-dove game is the most common example used in undergrad biology, perhaps because it's simple enough, yet totally abstract; so that facts don't get in the way. If the major problem with the game theory GA articles is that they're not comprehensible to high-school level readers, then I don't think that hawk-dove is a good candidate.
These are pretty minor problems though, I think the biggest problem with using a specific example in the intro is having one that exemplifies an ESS (rather than exemplifying a mere Nash equilibrium), you know what I mean?. Pete.Hurd 19:19, 16 July 2007 (UTC)
I agree that the handicapping example is not ideal. I could just about guess what it was about, but my guess wasn't confirmed until I got to the sentence "Receivers know that the signal indicates quality because inferior quality signallers cannot afford to produce such wastefully extravagant signals." at the end of the long first paragraph of handicap principle. It is also not clearly explained why this is an example of an ESS: for instance, why can't the invaders obtain an advantage by not performing the wastefully extravagant display? Why are the invaders weaker? Why not perform a different display? And so on...
I also agree that it would be nice to have an example illustrating the difference between Nash and ESS. I think you could take an example from the real world, as long as you adjoin a phrase like "in a simplfied model of sex ratios" (or whatever) to try to prevent the invasion of this strategy by the "ifs and buts". Geometry guy 17:36, 17 July 2007 (UTC)
Yeah, I'm also inclined towards sex ratio plus reality firewall, but the ESS vs Nash angle won't happen there. I'm drawing a blank on concrete examples for that (sleep deprivation, or maybe simply because the rationale is traditionally presented as a need to avoid rational foresight in biological models, rather than the shortcomings of a concrete example of a Nash that would not otherwise be stable in biology), Kevin? Pete.Hurd 18:54, 17 July 2007 (UTC)
Maybe something in the evolution of altruism might work here. Perhaps the green-beard effect or the invasion of tit-for-tat in a population of all defect in a repeated Prisoner's dilemma (basically the same idea). In these cases defection is not an ESS, but it is a NE (invaders do equally well against the population, but better against themselves). Our article on the GBE says that the Red imported fire ant exhibits a green beard effect, so perhaps that would be a good example. What do you think? --best, kevin [kzollman][talk] 23:02, 17 July 2007 (UTC)
Upon reflection, I think sex ratio is the way to go, it's a one shot game (unlike the IPD) and it's a whole lot simpler than the Green Beard, and it can be used to highlight the ESS vs. Nash distinction. I'll put it on my to do list... Pete.Hurd 21:49, 18 July 2007 (UTC)
Cool, I'll let you handle that. I only know the super toy examples and I would probably screw something up. --best, kevin [kzollman][talk] 02:52, 19 July 2007 (UTC)

Adoption versus evolution[edit]

The lead is rather problematic. I always assumed that the evolutionarily referred to the fact that the strategy has evolved rather than adopted as is mentioned. Shyamal 09:22, 17 July 2007 (UTC)

I'll check, but I think "adopted" is the standard language used by JMS (I'll ponder a better alternative/wording tweak). I think it's worth amplifying in the article that an ESS is a stable (evolutionarily, meaning JMS' second condition) Nash, but that it is not necessarily an attracting state, one that is evolvable in the EGT dynamics sense. The standard biological interpretation is that it has evolved, but there's nothing in the game theoryness that requires it to be a strategy that is evolvable to, if you see what I mean. Work on this point:
  • Nowak, M. 1990. An evolutionarily stable strategy may be inaccessible. Journal of Theoretical Biology, 142, 237–241.
  • Hamblin SR & Hurd PL. in press. Genetic algorithms and non-ESS solutions to game theory models. to appear in Animal Behaviour.
Pete.Hurd 14:08, 17 July 2007 (UTC)
Yeah, "adopted" is the word used in the one sentence definition of ESS on p14 of Evolution and the Theory of Games, it's also the word used in the definition in what I would think of as the current standard textbook (p7 of Chapter 1 "The evolution of behavioural ecology" 4th edition of Krebs & Davies' Behavioural ecology: an evolutionary approach). "Adopted" is also used in the first sentence of the definition of ESS in The dictionary of ethology and animal learning (MIT press, 1986), and is used in the one-sentence definition in other well-informed works such as Houston & McNamara's (1999) Models of adaptive behaviour: an approach based on state (p159), and p74 of Dawkins' The Selfish Gene. It's not quite universal however, e.g. "A strategy is an ... ESS if, when fixed in a population, there is no alternative strategy that can invade as a rare mutant" (p140 Milinsky & Parker "Competition for resources" in 3rd ed Behavioural Ecology: an evolutionary approach) has a bit more of a population genetics feel, but says the same thing (ie, makes no claim about it being an attracting state, or require anything at all about how it got to be pervasive in the population). Pete.Hurd 22:11, 17 July 2007 (UTC)
The Milinsky & Parker definition definitely sounds better. Adoption suggests volition, and this may be applicable to plastic behavioural traits, but it appears odd when ESS is applied to life-history strategies (where that kind of plasticity is absent Example: A parasitoid adopts a strategy of parasitizing a certain percentage of the host population). Perhaps the definitions that use "adoption" are meant for ethology. Shyamal 01:42, 18 July 2007 (UTC)
Yeah, but Milinski and Parker are definately unusual. If virtually all reliable sources, including the original definer of the term, use the word "adopt" then I'm disinclined to say wikipedia should not use it because it's somehow incorrect. What I think ought to be done is to give some explanation of why it's correct. The reason it's correct is that an ESS is a definition (in the realm of game theory, not biology) which defines a locally stable version (a "refinement") of a Nash equilibrium. The biology baggage of "fixation" language, and stronger (incorrect) interpretations of being an attracting state that is evolved to, are in the realm of biology (and require picking the concept up from the shoebox of theoretical model landia, and putting it down in the shoebox of population genetics, a biological application land where it does not apply without a bunch of shims and wrappers. I hope that made sense, I'm a bit low brain sugar at the moment. Cheers, Pete.Hurd 04:22, 18 July 2007 (UTC)
Ok, maybe the verbatim definition should be quoted to make it clear that this is not editorial interpretation. Shyamal 04:49, 18 July 2007 (UTC)
Works for me I suppose, but I havn't seen any other wikipedia entries beginning with a verbatim quote for a definition sentence. Pete.Hurd 05:44, 18 July 2007 (UTC)
Does not have to start with it, but can be suitably located. Shyamal 06:00, 18 July 2007 (UTC)

Sentence for the lead[edit]

We should add a sentence to the lead about the history. I wrote this, but realized I don't really know what the proper names for all of the biology fields are. Can you fix it up, Pete?

First developed in 1973, the ESS has come to be widely used in behavioural ecology and economics, and has been used in evolutionary psychology, philosophy, and political science.

Thanks --best, kevin [kzollman][talk] 03:38, 27 August 2007 (UTC)

Looks fine to me, perhaps I'd have left EP out, considered it a subset of Behav. Ecol., but I see no need for tweaks. Pete.Hurd 18:08, 27 August 2007 (UTC)

Differences with NE[edit]

I don't quite follow the discussion in the lead about the differences between ESS and NE. Pete can you say more about what you have in mind? --best, kevin [kzollman][talk] 03:52, 27 August 2007 (UTC)

Sure, since an ESS is a tweaked Nash I thought the rationale for the tweaking needed to be explained. A can be a Nash if B scores as well against A as A does against itself. If B is a reasonably good strategy against itself but otherwise a lousy strategy (say it gets whallopped by C) then this is no problem for a Nash, (players stick to A knowing not to drift into B because they'll get creamed by C) but for an explanation of evolved behaviour it's not acceptable. Evolution is a trial and error process, it only knows not to play B after being sent packing by C, by which point A is demonstrably unstable. Maynard Smith's point in making the refinement was to exclude these cases, and only accept strong Nash (condition 1), or cases in which B evolves back to A before any question of C exploitation arises (condition 2). Now, how to say that in the lead? (or put that in somewhere later? I think it ought to go in the lead, IMHO). Pete.Hurd 18:23, 27 August 2007 (UTC)
Okay, I think I've got it. You have a case like this in mind:
A 1, 1 0, 1 0, 0
B 1, 0 2, 2 0, 5
C 0, 0 5, 0 3, 3
Here (A, A) and (C, C) are Nash, but only (C, C) is an ESS. The underlying motivation is that while (A, A) is Nash, it can be invaded by B, which can in turn be invaded by C. So, I take it your saying that (A, A) is a Nash because people have the rational foresight not to switch to B (since C does better against B). I hadn't thought of it that way, but it has a nice ring to it. I agree wholeheartedly that something about the difference between ESS and NE should go in the lead, and this looks like a good candidate. I do think we need to find a better way to say it, but I don't have an alternative on hand. Let me give it a think. --best, kevin [kzollman][talk] 20:49, 27 August 2007 (UTC)
Ummm (thinking through headache here) yes, I think you're right that A & C are Nash, but only C is an ESS (but it's ESS because it's a strong Nash). I'll take a stab at tweaking this example so C is an ESS by JMS' second condition (once the acetaminophen has kicked in). Cheers, Pete.Hurd 21:17, 27 August 2007 (UTC)
I removed the sentence describing this example from the lead. I think we should still discuss this example, but every attempt I made involved many sentences (too long for the lead). Once we have a good example hammered out, I think we should add a section about ESS/NE and rational foresight. Sound good? --best, kevin [kzollman][talk] 03:56, 31 August 2007 (UTC)
Sounds great. Pete.Hurd 19:22, 31 August 2007 (UTC)

How about, two matrices: in top A is NE but not ESS, in bottom A is ESS by 2nd condition.

A 2, 2 0, 2
B 2, 0 4, 4
A 2, 2 0, 2
B 2, 0 1, 1

Pete.Hurd 22:05, 31 August 2007 (UTC)

I there must be a typo on the second one. A is not a ESS, right? E(AA) = E(BA) = 2 (condition #1 is violated) but E(AB) = 0 < E(BB) = 1 (violating condition #2). Maybe something like this:
A 2, 2 3, 2
B 2, 3 1, 1
This example also occurred to me:
A 2, 2 0, 2
B 2, 0 2, 2
While (A,A) and (B,B) are Nash only (B,B) is ESS (again an example of condition #2).
I also likes your example with three strategies. I thought it did a nice job of elucidating your point about the difference between rational foresight underwriting an equilibrium and natural processes underwriting them. --best, kevin [kzollman][talk] 06:32, 1 September 2007 (UTC)
*sigh* yes I muffed it up...
A 2, 2 1, 2
B 2, 1 0, 0
for the second table, so then
A 2, 2 1, 2
B 2, 1 2, 2
for the first... Pete.Hurd 21:13, 3 September 2007 (UTC)
Okay, I took a whack at it. I'm not particularly happy with it, so I would be much obliged if others would hack it to death ;) --best, kevin [kzollman][talk] 04:05, 6 September 2007 (UTC)
"hack to death" I can probably manage, "improve" I'm not so sure ;) Pete.Hurd 04:09, 6 September 2007 (UTC)

interpreted as an equilibrium refinement[edit]

Hey Kevin, I don't quite get this sentence: "While technically a constraint on strategies, the ESS concept is often interpreted as an equilibrium refinement of the Nash equilibrium" Pete.Hurd 01:02, 7 September 2007 (UTC)

I've been toying with how to express this idea. The ESS is a property of individual strategies, that is, a strategy satisfies MS's definition or not. A Nash equilibrium is a property of strategy sets (specifications of strategies for each player). It is natural to extend the concept of ESS to be about strategy sets (i.e. a strategy set is an strategy-set-ess if and only if every strategy in that set is an ESS) formally the ESS is not about strategy sets. Since ESS is about strategies and NE about strategy sets, ESS (as defined by MS) doesn't refine the NE, only the extension does. Maybe this is a stupid technical point, that isn't worth mentioning... --best, kevin [kzollman][talk] 19:50, 7 September 2007 (UTC)
Hmmm, right, now I know exactly what you mean. Nash equilibria are traditionally thought of as strategy profiles (with the implication that these consist of two different roles), whereas ESSes are traditionally thought of as single strategies since biological problems have a strong tradition of using symmetric games(* some exceptions apply, see below). ESSes get applied to asymmetric games all the time, sometimes by Seltenizing the game (and making strategies specify which move to make when playing either strategy, and forcing the m*n game to become (m×n)*(m×n) symmetric), or simply explicitly applying the ESS conditions to an asymmetric strategy profile solution (I'm pretty sure I used such a definition in one of my papersnope, not in the published version, at least). It used to bother me when I heard people parroting the definition of an ESS as "a strategy which is the best when played against itself" because such a definition obviously can't be applied to signalling games (where ESSes get spoken of a lot). I see what you mean, and I can almost certainly dig up published exceptions, but either way I don't think it's going to pay it's rent living in the lead section. Pete.Hurd 20:37, 7 September 2007 (UTC)
Yeah, I don't know why I put it there. I must have been feeling overly pedantic that day... --best, kevin [kzollman][talk] 22:04, 7 September 2007 (UTC)
Well, if there's an appropriate place to go when feeling overly pedantic, it's just gotta be wikipedia! Pete.Hurd 03:04, 8 September 2007 (UTC)

"Prisoner's dilemma and ESS"[edit]

I get the feeling that this section is using the PD to illustrate a larger point, that the topic at hand isn't necessarily the PD. I'm inclined to chop it out... Thoughts? Pete.Hurd 22:46, 13 September 2007 (UTC)

tit for tat is no ESS[edit]

Kenneth G. Binmore and Larry Samuelson. "Evolutionary Stability in Repeated Games Played by Finite Automata." Journal of Economic Theory, 1992, pp. 278-305.

argue, that tit for tat is no ESS, because if the entire population is t.f.t. than invaders that use "always cooperate" are superior, because this strategy yields the same result as t.f.t. but is more simple. t.f.t. has a strategy branch that is never used in equilibrium (defect). "alway cooperate" doesn't. If "beeing unneccessarily complex" is an evolutionaly disadvantage, then t.f.t. will be wiped out by "always cooperate". And of course "always cooperate" will be wiped out by almost any other strategy being the next invading strategy.

If one follows this line of argument, the next question arising is, if the idea of ESS makes sense at all.

Winkelhake Oct. 13

—Preceding unsigned comment added by (talk) 09:02, 13 October 2007 (UTC)

Yeah, in an IPD tft may be a Nash equilibrium, but it's not an ESS. That's because allC scores equally well and therefore can invade neutrally. The two strategies score equally well against each other, so Maynard Smith's second condition doesn't keep allC out. AllC is not stable, other strategies can exploit it, so tft isn't a good candidate for an evolutionarily stable outcome, given formal techniques. All of that comes without factoring in a penalty for "unnecessary complexity" of a strategy. I think the problem with the ESS definition here is that the strategy space for the IPD is just too big to be managable. An ESS is a very powerful concept when the number of possible strategies is small, but when it's infinite... That's my fallible 2c worth. Pete.Hurd 19:22, 15 October 2007 (UTC)

Pete, thanks for the reply. As I'm only dabbling in game theory, this might be perfunctory, but I couldn't think of a repetitive game with a significantly smaller strategy space than IPD. IPD has only two choices, which is the minimum for any decision. If none of the choices is instantly lethal to the player, every strategy space in a repetitive game must be more or less infinite - musn't it? With two choices (A and B) in any game, a possible strategy could be: "choose A for n times and then switch to B for m times".

Maybe also perfunctory: How can a game that has a Nash Equilibrium can have an ESS? If Nash equilibrium means, that nobody changes his/her behaviour although s/he would be able to, then there must always be a more simple strategy to this : the strategy with the same behaviour but with the missing ability to change the behaviour. This strategy could at least successfully invade the population and even crowd out the original strategy if being complex is a disadvantage c.p. It's not very likely that this invading strategy would be stable, because if it always behaves the same way and survives against other invadors, this would mean, that there is no difference in the payoff of the choices.

This would mean that any game with a Nash equilibrium cannot have an ESS. This sounds very strange. Where is my error in reasoning?

--Olaf Winkelhake —Preceding comment was added at 20:38, 19 October 2007 (UTC)

Yes, while the unit game for the IPD has only two strategies, cooperate and defect, when repeated the number of possible (formal) strategies increases geometrically, we might as well think of them as infinite and the IPD as infinitely repeated. I think the definitions of Nash equilibria and ESSs are of heuristic, rather than formal, value when talking about repeated games.
I'm having a bit more difficulty understanding your second question. I think the "ability to change behaviour" has two meanings in your question. The first is in the definition of the equilibrium, which we can put as "if the population is playing a strategy, and you are about to play that same strategy also, would you like to change your mind at the last minute?" If the answer is "No" then it's an equilibrium (ignoring temporarily the difference between Nash & ESS). That's the first sense of "nobody changes his/her behaviour although s/he would be able to". I think the second, where you say "there must always be a more simple strategy to this : the strategy with the same behaviour but with the missing ability to change the behaviour" I think you mean that the strategy is a conditional strategy, "if my opponent does X then I will do Y, but if they do W then I will to Z". I can't see any other sense of removing the player's ability to change their behaviour that doesn't take away the essence of game theory, that individuals choose a strategy in a manner that is sensitive to what other individuals are doing, or likely to do. The first sense of "choose behaviour" is a fundamental assumption of game theory, the second is the assumption that a single strategy may contain more complex "if then" contingencies.
"any game with a Nash equilibrium cannot have an ESS" all ESSs are Nash equilibria, by definition, but not all Nash equilibria are ESSs, because some Nash equilibria have equal scoring alternative best responses which make them unstable if rational foresight is not assumed.
I'm happy to try again if I've misunderstood your question. Best regards, Pete.Hurd 21:30, 19 October 2007 (UTC)

Thanks for your patience!

Maybe I can describe the problem better, if I refer to the IPD setting. I was talking about strategies like "always cooperate". These "strategies" _cannot_ change their behaviour although there are alternatives. They simply cannot use them by genetic design. So strictly speaking, for these strategies there is no decision. You are right here. But they might very well be part of a game theory problem. These strategies are "idiots", that will not live for long, but dangerous idiots to the more complex strategies in a settled equilibrium in a repeated game.

If an ESS is a Nash Equilibrium, equililbrium means, that every player behaves alike. In a repetitve game again and again.. In IPD they either all defect or all cooperate.Right? Over and over again. Right?

Now if all players behave alike over and over again in a Nash equilibrium (whatever strategy yields this result) there _must_ be an "idiot strategy" that behaves like the strategy in the Nash Equilibrium but is less complex and thus invades the population.

If there is a penalty on being complex, all branches of a strategy must be in use in an ESS. Right? For instance if the players use tft and always defect, the "cooperate branch" is never used and thus superflous. Vice versa if all playser "always cooperate" they never use the "defect branch". In both cases, an idiot strategy can invade.

BUT: if a strategy that is an ESS needs to use all branches every now and then, these branches MUST result in different behaviour.

In "IPD-language": An ESS needs to "defect" AND to "cooperate". Not just in theory (like tit for tat), but also in practice. If only one branch is used, the strategy is invaded by one of the two idiot stragegies.

I could imagine, that the crucial point is my assumption is, that "all players behave alike over and over again in a Nash equilibrium". Am I talking about a strange subclass of Nash Equilibria here? (the prisoner dilemma being part of this subclass) If so, it seems to me, that in this subclass of Nash equilibria there _cannot_ exist an ESS because (I repeat myself, sorry...) any conceivable strategy would successfully be kicked out by an invading idiot "strategy" once the equilibrium is settled. - Which - I totally agree with you, sounds bizarre.

A neccessary condition for an ESS in IPD would be that it _must_ "defect" _and_ "cooperate". If it never defects or never cooperates, it cannot keep the "idiot strategies" away. Thus "equilibrium" could not mean "all players behave alike over and over again". My knowledge is too limited if this could still be called "Nash Equilibrium".

I hope I've made my point a bit clearer. I shouldn't have used the German translation of the game theory textbooks way back then ;) I'm sorry for repeating myself over and over again, but actually this is a bit beyond the border of my communication skills in Game-Theory-English  ;)

--Olaf Winkelhake —Preceding comment was added at 09:57, 20 October 2007 (UTC)

Hi Olaf,
"equililbrium means, that every player behaves alike. In a repetitve game again and again.. In IPD they either all defect or all cooperate.Right? Over and over again. Right?" No, the equilibrium strategy maybe something complicated and contingent like:

on the first move, play C with 85% probability, then play C with 16% probability if the opponent played D last move, and with 87% probability if they played C last move. But if the other player did C for the last 3 moves in a row then play C with probability 1, but if they played C four times in a row, then play D with proability 1. If it's a Tuesday and the weather is rainy, then play ... etc etc etc.

Any strategy that can be specified has the potential to be an equilibrium strategy in the IPD. There is no formal requirement that the moves all be the same under any condition, or that the strategy be easily deducible from the player's behaviour.
"If there is a penalty on being complex, all branches of a strategy must be in use in an ESS. Right?" Ohhh! a most excellent question! (Note: the property of all branches of a strategy being used when the strategy plays against itself is called "pervasiveness"). What you write is true, regardless of the assumption about a penalty on strategy complexity: a strategy cannot be an ESS unless it is pervasive. The reason for this is precisely per your logic here when you say "if the players use tft and always defect, the "cooperate branch" is never used and thus superflous. Vice versa if all playser "always cooperate" they never use the "defect branch". In both cases, an idiot strategy can invade.". If branches of a strategy are not used at the equilibrium, then it cannot be an ESS because it can be invaded by neutrally scoring alternatives that differ in those unused (aka "unsupported") branches (but it can be a member of an Evolutionarily Stable Set -aka ES Set- providing an example of the ESS concept being of intuitive, but not formal use. (see Hamblin & Hurd 2007 Genetic algorithms and non-ESS solutions to game theory models. Animal Behaviour 74: 1005-1018, for more on this topic than I can type right now).
"BUT: if a strategy that is an ESS needs to use all branches every now and then, these branches MUST result in different behaviour." Hmmmm, I'm not convinced that this is true. I will have to think about this one a bit (not much sleep last night, not feeling super-smart).
"the crucial point is my assumption is, that "all players behave alike over and over again in a Nash equilibrium"", and that's not true... but it doesn't make the previous point any less interesting.
"I hope I've made my point a bit clearer." yes, you definately have. I hope my answer is convincing, in short, it is very unlikely that an ESS exists for a big complicated game (of which iterated games are but one example), but other forms of equilibrium strategies (such as ES Sets) may exist.
Best regards Pete.Hurd 17:53, 20 October 2007 (UTC)
Sorry to chime in so late, but I think it might be possible to prove that there is no ESS in an indefinitely iterated PD (following Pete's suggestion). Since unconditionally playing C or D is not an ESS, any possible ESS strategy will have a counterpart which is behavioristicly identical in equilibrium. (We might have to restrict ourselves to deterministic strategies.) All-C is in a population of TfT is an example of this phenomenon, but it is very general. The general problem occurs because one can always invent a strategy that behaves identically in equilibrium but responds differently to contingencies that don't occur (since the iPD has an infinite game tree, there must be branches which are unexplored in equilibrium, assuming deterministic strategies). There is a neat example from Eric Maskin, which addresses some questions above. Consider the strategy Alt which starts with C and then plays alternating D and C on the next rounds. If the counterpart plays the same way, it continues to alternate, otherwise it plays D forever. Alt (like TfT) invades a population of All-D. But it a much lower payoff than TfT (although TfT cannot invade). One can construct even worse alts. Suppose Alt-2 that plays CDDCDDC. This invades All-D, but gets an even worse payoff. This can go on (although I think there is a point at which TfT might invade). Anyway, this shows that the phenomenon of large games can be very complex, and ESS are (a) hard to find and (b) perhaps not all that interesting. Dynamic stability concepts might be more appropriate here. But the large strategy space provides a problem there too. --best, kevin [kzollman][talk] 18:59, 20 October 2007 (UTC)
"Dynamic stability concepts might be more appropriate" but "the large strategy space provides a problem" is an understatement, EGT (and the same goes for Adaptive Dynamics) require a dimension of space for each possible strategy. Solving the dynamics of an hugely (maybe infinite) dimensional space is just not on. Pete.Hurd 19:27, 20 October 2007 (UTC)
Yeah, yeah, yeah. Well at least it's not this big :) Seriously, what I had in mind was things like neighborhood stability and neutral stability. Not actually solving the dynamical systems. --best, kevin [kzollman][talk] 21:29, 24 October 2007 (UTC)
Okay, I have tried to make this section more clear and demonstrate the concerns discussed here. There is, of course, a lot to say. Hack away! :) --best, kevin [kzollman][talk] 23:46, 27 October 2007 (UTC)

sex ratio redux[edit]

Is there somewhere on wikipedia where we talk about Hamilton's sex ratio model? I noticed someone asked about it recently at WP:RD/S, but every article they were referred to failed to mention the underlying theory. We used to talk about it here, but it's been removed. There really isn't much mention at sex ratio. Is it somewhere I'm missing? Perhaps we should put it back here or at sex ratio. --best, kevin [kzollman][talk] 21:14, 28 October 2007 (UTC)

No, not that I can find right now. The closest article to do that seems to be Bateman's principle, which I keep meaning to fix, but the original Bateman article isn't on-line, and I'm lazy... good intentions, bad habits... Ummm, Parental investment and Sex allocation need work too. An article on Sex ratio evolution linking all of these together with Trivers-Willard hypothesis etc., would be a Very Good Thing. Pete.Hurd 23:01, 28 October 2007 (UTC)

"Evolutionary stable strategy" vs. "Evolutionarily stable strategy"[edit]

A search through the ISI database shows many uses of the phrase "Evolutionary stable strategy" in the abstracts of peer reviewed papers. I can't see how we can call this "incorrect", we're just documenting the term as it is actually used by researchers in their work. Pete.Hurd 15:33, 7 November 2007 (UTC)

It should be obvious from the grammar of the term itself that it's a mistake. The "strategy" is "evolutionarily stable". It is not a "stable strategy" of the "evolutionary" variety, whatever that would mean. People make the mistake because they are used to hearing about "evolutionary stability". But when you change "stability" to an adjective, you have to change "evolutionary" to an adverb. This is simple English grammar. In addition, Maynard Smith and Dawkins have both said that it's wrong. For example, quoting Dawkins from "The Extended Phenotype", the glossary, pg. 293:

evolutionarily stable strategy (ESS) [Note: 'evolutionarily' not 'evolutionary'. The latter is a common grammatical error in this context.] A strategy that does well in a population dominated by the same strategy. This definition captures the intuitive essence of the idea (see Chapter 7), but is somewhat imprecise; for a mathematical definition, see Maynard Smith, 1974.

Bueller 007 14:24, 13 November 2007 (UTC)
I agree with you that it *ought* to be "Evolutionarily stable strategy", but the demonstrable fact is that many, fairly authoritative, people do use "Evolutionary stable strategy" in papers published in peer reviewed journals. Since scientists do often called it an "Evolutionary stable strategy", we ought to have that term here when people come to look it up and see what it means. It's Wikipedia's job, as a tertiary source, to report upon how the word is used in the real world and not to enforce propper grammar (as much as I would like to ban "Drive Thru", "Late Nite", "Lite Beer" and the egregious crime against English that is the inappropriate use of "utilize" in place of "use" from the face of the Earth, as encyclopedists, it' is our duty to document all of these). We could comment upon existing debate over the appropriateness of each form, but when you say "In addition, Maynard Smith and Dawkins have both said that it's wrong." do you actually mean that they explicitly say "Evolutionary stable strategy is wrong" or do they merely use "Evolutionarily stable strategy" exclusively? If they do the former, then dig up the references for where they said that, and we can add it to the article. But my suspicion is that they merely used the grammatically correct form without commenting on the other. Best regards, Pete.Hurd 14:54, 13 November 2007 (UTC)
Well, the Dawkins reference is right above. He quite clearly states that it's wrong. But personally, I don't consider that "authoritative", since he was not the coiner of the term. I don't have the Maynard Smith reference handy, but I'm sure I've seen him say it in print or in person. (And I can almost guarantee that he used "evolutionarily" exclusively.) Nevertheless, I don't see a problem in marking it "incorrect", or at the very best "unorthodox". It may be our duty to document something, but that doesn't mean that we can't point out obvious grammatical errors where they exist. Bueller 007 15:58, 13 November 2007 (UTC)
Yes, Dawkins does say that (pg 286 in my edition), and he does indeed point out that it's a common misspelling. I'll add that it's a "typo" so common that it's used in the titles, abstracts and authors keywords of peer reviewed publications by such luminaries as Alan Grafen (e.g. Grafen, A. 2007. JOURNAL OF EVOLUTIONARY BIOLOGY 20: 1243-1254) and Peter Hammerstein (e.g. Laubichler MD, Hagen EH & Hammerstein P, 2005. BIOLOGY & PHILOSOPHY 20: 1041-1050) indicating that it's a acceptable term of art. Just look at the pages of hits ISI returns for that spelling. Pete.Hurd 17:41, 13 November 2007 (UTC)

Revisiting the grammar question[edit]

I think Dawkins was probably trying to be provocative when he said that "evolutionary" was a "common grammatical error." It is only a grammatical mistake if you hyphenate the two modifiers into a compound modifier (i.e., it is a mistake to use "evolutionary-stable strategy"). As is, "evolutionary stable strategy" means an "evolutionary strategy" (which defines the kind of game being played) and a "stable strategy" (which defines the equilibrium). The space between the two terms just indicates that the order of the adjectives cannot be reversed. Arguably, the order can be reversed, and the term could be accepted as "evolutionary, stable strategy", but then that allows for "stable, evolutionary strategy" which is inconsistent with the acronym ESS. Regardless, the two modifiers are both adjectives that modify "strategy", and that is perfectly OK. Consult Wikipedia's own WP:MOS for more information about modifiers and hyphenation. Furthermore, "evolutionarily stable strategy" is arguably the bigger grammatical mistake because "evolutionarily" isn't a word in many dictionaries. Consequently, I'm removing comments about grammar from main document. —TedPavlic (talk/contrib/@) 19:03, 1 July 2011 (UTC)

Polymorphic equilibrium[edit]

I'm failing to find a description of "polymorphic equilibrium". I strongly suspect such a description belongs here?Cretog8 (talk) 04:49, 30 April 2008 (UTC)

Prisoner's Dilemma and ESS[edit]

...even though a population of Always Cooperate and Tit-for-Tat can coexist, in case an attack on the population by a group of Always Defect, the selective pressure is against Always Cooperate, and in favour of Tit-for-Tat. This is due to the lower payoffs of cooperating than those of defecting in case the opponent defects.

I have not cited a reference for this statement, and in the strictest sense it may be called original research. But it appears to me as a self-evident mathematical fact. Please provide comments.

Geeteshgadkari (talk) 06:16, 19 June 2009 (UTC)

Dr. Schlag's comment on this article[edit]

Dr. Schlag has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

The definition of ESS due to Thomas is missing the condition that T has to be in a neighborhood of S. Note that it is then equivalent to the definition of Maynard Smith and Price, at least in the classic setting with linear payoffs.

I would mention evolutionarily stable sets and possibly other generalizations of ESS. I would also mention finite population ESS.

Definitely missing is a reference to the replicator dynamics. In fact, many have abandoned the concept of ESS in favor of looking at explicit dynamics. Evolution is not shaped by formal concepts but by evolutionary pressures on fitness.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Schlag has published scholarly research which seems to be relevant to this Wikipedia article:

  • Reference : Sjaak Hurkens & Karl H. Schlag, 1999. "Communication, coordination and efficiency in evolutionary one-population models," Economics Working Papers 387, Department of Economics and Business, Universitat Pompeu Fabra.

ExpertIdeasBot (talk) 18:58, 27 June 2016 (UTC)

Dr. Heller's comment on this article[edit]

Dr. Heller has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

1) History section should mention Nash's population interpretation to Nash equilibrium in his Dphil thesis from 1950 (didn't appear in the published papers), which, informally, captures many aspects of EE. See, e.g., Section 1.2 in "Nash equilibrium and evolution by imitation" with J. Björnerstedt, in K. Arrow et al. (eds.), The Rational Foundations of Economic Behaviour, London: Macmillan, 155-171, 1996.

2) The part in which the definitions of ESS is given (currently called "Nash equilibria and ESS"), should open with an explanation that ESS is usually applied to symmetric (two-player) games, and it is a refinement of a symmetric Nash equilibrium. Some later Section at the end may discuss applications of ESS to asymmetric games, and to symmetric games in which a playe rmay condition his behaivor on his role in the game (see, e.g., Sections 2.7 in the textbook "Evolutionary Game theory, Jorgen Weibull, MIT Press, 1997).

3) The following cited paragraphs from ("Nash equilibria and ESS"), cited from the Wikipedia article, are inaccurate and misleading:

"There is also an alternative, stronger definition of ESS, due to Thomas.[10] This places a different emphasis on the role of the Nash equilibrium concept in the ESS concept. Following the terminology given in the first definition above, this definition requires that for all T≠S

E(S,S) ≥ E(T,S), and E(S,T) > E(T,T) In this formulation, the first condition specifies that the strategy is a Nash equilibrium, and the second specifies that Maynard Smith's second condition is met. Note that the two definitions are not precisely equivalent: for example, each pure strategy in the coordination game below is an ESS by the first definition but not the second.

In words, this definition looks like this: The payoff of the first player when both players play strategy S is higher than (or equal to) the payoff of the first player when he changes to another strategy T and the second players keeps his strategy S. *AND* The payoff of the first player when only his opponent changes his strategy to T is higher than his payoff in case that both of players change their strategies to T.

This formulation more clearly highlights the role of the Nash equilibrium condition in the ESS. It also allows for a natural definition of related concepts such as a weak ESS or an evolutionarily stable set.[10]" " (end of quote from wikipedia's article)

Thomas (1985, On Evolutioanry Stable sets, J. Math Biology, Page 107), the second requirement in Thomas's definition refers only to strategies T in a small punctured environment around S, while the Wikipedia article states it as if referring to any strategy T. The original Thomas's definition is equivalent to the Manyard-Smith and Price definition (at-least in the standard setup in which the sets of pure strategies is finite). The way that the Wikipedia's article states and interprets Thomas' definition is wrong and misleading.

I think that the Sections 2.1-2.2 in the textbook "Evolutionary Game theory, Jorgen Weibull, MIT Press, 1997) has excellent explanation for the two equivalent definitions for ESS (pages 36-37), and the third definition of uniform invasion barrier (page 42-43). I suggest that the editors of the article should completely revise the section with the definitions of ESS ("Nash equilibria and ESS"), and may find it helpful to base the new part on Sections 2.1-2.2 in Weibull's textbook.

4) "In most simple games, the ESSes and Nash equilibria coincide perfectly." I don't think that "most" is sensible. Perhaps change to "some"/many?

5) "However, only B is an ESS (and a strong Nash)" - the correct notion for "strong Nash" is "strict Nash" (strong Nash refer to robustness against a joint deviation of a coalition of players).

6) It is important to add to the article an example a simple game without an ESS, such as, Rock-paper-scissors game in which a tie gives more than the average of victory and loss).

Best wishes, Yuval Heller.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Heller has expertise on the topic of this article, since he has published relevant scholarly research:

  • Reference : Heller, Yuval, 2013. "Language, Meaning, and Games: Comment," MPRA Paper 49375, University Library of Munich, Germany.

ExpertIdeasBot (talk) 19:52, 1 July 2016 (UTC)

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