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All the examples in the current Nash equilibrium article seem to be "one-shot" games (is there a better term?) -- as opposed to repeated games.
Strategies such as Tit for Tat don't work for "one-shot" games.
The Robert Aumann article mentions
Does that mean that
? If that's true, the article should mention it.
-- DavidCary 13:40, 11 October 2005 (UTC)
On this page, the phrase
resulting in strategy profile
indicates that a strategy profile is simply a vector of strategies, one for each player. I agree. However, if one follows the link, a strategy profile is defined as something that "identifices, describes, and lastly examines a player's chosen strategy". It is conceivable that someone somewhere defined strategy profile in this way, but the "vector"-meaning of the term is much more common. So, at least the link, if not the page linked to is misleading.
Also there is something slightly wrong in the notation concerning strategy profiles on this page itself. You say that
S is the set of strategy profiles.
Okay, but then each element of S is a vector of strategies. Still, you write
where is a single strategy. I think you want to write
is the set of strategy profiles
and
.
Bromille 11:50, 31 May 2006 (UTC)
There are some issues in the "Occurrence" section.
First, whatever the stated conditions are, they can only be sufficient, not necessary for equilibrium play. To see this, observe that nothing prevents a bunch of completely non-rational agents from playing the Nash equilibrium, not because it is a rational thing to do, but because non-rational agents can do anything they want.
Second, conditions 1. and 5. together say that agents are rational and they all believe that the other agents are also rational. But these are not sufficent conditions on rationality for equilibrium play. A classical counterexample is Rosenthal's centipede which has a unique Nash equilibrium as assumed. In this game, if you and I play the game and I believe you are rational and you believe I am rational, we still might not play the equilibrium, if, for instance, I do not believe that you believe that I am rational. This could be the case, even if I do believe that you are rational. What is needed to ensure that the equilibrium is played in the case of Rosenthal's centipede is common knowledge of rationality (CKR) which means that
A: I am rational, B: You are rational, C: I believe B, D: You believe A, E: I believe D, F: You believe C, G: I believe F, H: You believe E, etc.. etc..
The difference may seem like splitting hairs but is actually the key to understanding the discrepancy between how Rosenthal's centipede is actually played by rational-seeming people and the equilibrium play.
Reference: Robert Aumann, Backward induction and common knowledge of rationality, Games and Economic Behavior 8 (1995), p. 6--19.
Third, for general games, even if CKR is assumed, this usually only ensures that a correlated equilibrium is played, not a Nash equilibrium. The fact that a unique Nash equilibrium is assumed may make this ok - I'm not sure about this.
Bromille 13:54, 1 June 2006 (UTC)
Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:49, 26 September 2006 (UTC)
Ed: This is an incomplete example that does not illustrate. If I choose 9 and you choose 0, I'm out two bucks. If I choose 9 and you choose 7, you make $9 and I still make $5 - why is this not superior?
The system of strategies 9-0 is not a Nash equilibrium, because the first player can improve the outcome by choosing 0 instead of 9. The system of strategies 9-7 is not a Nash equilibium, because the first player can improve the outcome by choosing 9 instead of 7. The only Nash equilibrium of this game is 0-0, as stated. AxelBoldt 15:04 Aug 27, 2002 (PDT)
The second player can chose 9 and he can get 11 while the other one will get only 8. That is why 10-10 is not at Nash equilibrium -- AM
I think the thing to keep in mind here is that it is a competition game. Both players are trying to beat the other's "score". This example would work better, I think, using points instead of dollars. Yes, there is a benefit to both players if they both win some money, but no benefit if they both win some points and they are each trying to beat the other's score. -truthandcoffee /ed: Does anyone else confirm/deny this? I would like to see this example changed from dollars to points, as I think it is much clearer and makes much more sense this way. I'd like some feedback before I go ahead and make the change. -- Truthandcoffee 04:27, 13 November 2006 (UTC)
If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 11 Nash equilibria.
Does the Nash equilibrium not suppose that everybody is selfish and rational? So, if you suppose that the other player is selfish, you can savely assume that he/she chooses 10. IMHO that's the only equilibrium. Or does the Nash equilibrium not presuppose selfishness? Andy 12:53, 26 September 2006 (UTC)
I am curious about the relationship between NE and government regulation. Consider the case of airbags in cars. Without regulation requiring airbags, normal supply and demand will dictate the number of airbags produced. If air bags are mandated by the government, the unit cost will be much lower, and more consumers will choose to purchase cars with airbags and receive the consumer surplus from the purchase. The consumer was able to move to a better outcome through the intervention of a third party. In this case it is hard to say how the auto manufacturer is impacted. It is possible that they benefited because they can now offer new cars with more value at a lower cost than without regulation. More people buy a new car vs. a used car to benefit from the air bag. Measuring if there was a benefit to the auto company is impossible, but let's assume there is. Did the regulation help both parties move out of a NE to a better outcome? 65.198.133.254 20:36, 30 November 2006 (UTC)David Wilson
Under the heading specified above, it claims that "...an NxN matrix may have 0 or N Nash Equilibriums." Shouldn't it be 1 or N NE, since at the beginning of the article it says that Nash proved the existence of equilibria for any finite game with any number of players?
This section provides a rule for finding NEs. It fails to discuss either the rule's history - who discovered it and when - or why (the reader should believe) it works. While there's a place for pragmatic rules in an encyclopedia, the article should at least indicate where its theoretical underpinnings may be found. yoyo 23:09, 24 December 2006 (UTC)
I have seen in some papers the notation used to describe a set of strategies for all players other than . This cleans up notation greatly; for example,
becomes
This seems more compact and conveys the point more clearly. SanitySolipsism 22:27, 6 January 2007 (UTC)
Does anyone know anything about this? How about what journal the article is in, or anything? I couldn't find anything, the ip of the anon who added the information is at U of Toronto, and another, erilly similar ip address claims at User talk:128.100.53.169 to be familiar with Goldstein, although not think that it belongs in the lead. As unverifiable, I'm still for removing the mention, but loath to do it without mentioning it here, per WP:1RR. Smmurphy( Talk) 17:55, 14 March 2007 (UTC)
OK, what to do? I put this on my watchlist now. ~ trialsanderrors 17:20, 6 March 2007 (UTC)
Hello all - The article currently uses two inconsistent citation style (APA and wiki-ref style). We need to be consistent. I personally prefer APA-inline citations, but I know wikipedia generally seems to be leaning toward the footnote style. Anybody have any druthers? --best, kevin kzollman][ talk 23:19, 29 April 2007 (UTC)
Governing Dynamics, i.e. the way an object or phenomenon will occur or behave under a given set of circumstances is not Nash's idea, so why when one types the phrase into the search box, does one get diverted to his page? It's ludicrous.
Governing dynamics is a centuries old theory which simply refers to the way an e.g. object or phenomenon will behave given a set of prevailing circumstances. If anything it is a philosophical rationale more than a scientific theory. By understanding the laws which governs said object in said circumstances, one is essentially able to understand the true nature of all matter and occurrence and abstracts are erased. The principle of governing dynamics serves to found and object or occurrence in its pure and absolute state because it is essentially a malleable law, under which nothing is fixed nor finite, that is to say X will behave as Y under Z, but that does not imply X will also behave as Y, because X is not fixed and will not always be under condition Z. Under condition V, X will behave as A and not Z, because it will reflect the new conditions and parameters it finds itself operating in. Thus by understanding the laws, the governing dynamics of phenomenon, one is ultimately able to understand the true and original nature of the object or phenomenon under inspection, because one essentially has a series or set of laws which encompass all possible existence and not simply a single law under which an object or phenomenon is expected to eternally function.
I therfore suggest that for accuracy these two pages should be split and a seperate page developed which specefially deals with the Law of Governing Dynamics in its historical and philosphical contexts.
Andrew Woollock 14:08, 13 July 2007 (UTC)
This is a notification of an intention to delist this article as a good article. In February this year at Wikipedia_talk:WikiProject_Game_theory#CotM, the consensus was that this article is B-Class (at best) from the point of view of WikiProject Game theory. I agree with this assessment, and last month I also rated it as B-Class from the point of view of WikiProject Mathematics. The article has not improved since these assessments, despite being Game theory collaboration of the month for two months.
The lead is inadequate as a summary of the article. The article is missing a history section, which is surely vital in this case (and is partly covered in the lead). The accessibility of the article could be much improved, the references are poor, and citation is both inadequate and inconsistent. I will attempt to improve the article myself over the next day or two, but I don't think I will be able to raise the article to GA standard myself. I hope others will contribute to ensuring that this article meets with criteria, otherwise I will have to delist it. An alternative would be to take the article to Good article review and any editor is welcome to do that. Geometry guy 21:28, 12 July 2007 (UTC)
I think there is consensus that this is not yet a good article. The lead is weak, the prose poor. Citation is extremely weak. Improve and renominate: good luck!
Geometry guy
20:35, 14 July 2007 (UTC)
I hope the specialists among us don't mind my attempt at defining Nash equilibrium in a lay person's language. If you are tempted to quibble with my definition, just remember how awfully vague the (non)explanation of Nash equilibrium was in that bar scene in the film of A Beautiful Mind. Such a fundamental and elegant idea deserves to be explained in words everybody can understand. -- Rinconsoleao 19:46, 16 July 2007 (UTC)
I have added a comma after the <forall> i, 194.223.231.3 15:26, 17 July 2007 (UTC)
I think it is weird that the proportion between the players' payoffs are not taken into account. Surely I would, in a two-player game, prefer weakening both myself and my opponent if I weakened him more. Consequently, I would, from the situation A,B (25,40) being player one, change to C (15,5) to force the opponent into also doing C (10,10). That is a truly stable position, in which no player can earn anything by changing strategy. Isn't that actually the only of the three "stable" combinations that satisfies both stability criteria? —Preceding unsigned comment added by 77.40.128.194 ( talk) 21:25, 15 January 2008 (UTC)
I've noticed that although the article makes use of the terms strong & weak Nash equilibria, they are not defined anywhere. -- ricmitch 08:17, 21 April 2008 (UTC)
History of Nash equilibrium. I just changed this section. I can develop it a bit more and add some discussion. The concept of a Nash equilibrium in pure strategies (to use the current name) was clearly in use in oligopoly theory in the 19th Century and was well known: the writings of Edgeworth, Hotelling, Stackelburg to name a few of the better known names. I do not know for sure, but it quite possible that Oscar Morgenstern did not know much about this, since it is left out of the Theory of games and economic behavior (as far as I remember it). Von Neuman was primarily interested in card games and developed the idea of a mixed strategy (bluffing) in this context. In the Theory of Games, he showed that all zero-sum games posses a mixed strategy equilibrium. The modern form of the Nash equilibrium (where the strategies are in generla mixed) was the contribution of Nash: he extended Von Neuman's concept to all games with a finite number of actions. There have of course been various extensions since then: in particular Dasgupta and Maskin provided an existence Theorem for the non-finite case (when the strategic variables such as price are continuous) in their 1987 Review of Ecoonomic Studies articles and so on, but much of this is secondary. Zosimos101 ( talk) 18:04, 25 April 2008 (UTC)
Oh well, the bank run might be as good as well... feel free to revert me if you think so... but i think a cartel is something more people are prone to be familiar with than a bank run?... and in the cartel example all players get a boost in profits wheras in a bank-run only some get more than theit share of the banks assets value, Gillis ( talk) 21:38, 31 May 2008 (UTC)
Gentlemen, this article is going to get an ugly citation template if you continue without proper citations. This is a serious topic, not pop culture fan-cruft, so we expect better from you. It's been a year since it was shown that this was a problem, yet there are still only two in-line citations. Please try to rectify this as soon as possible! Note, the citations need not be Internet accessible - good, old fashion book citations are more than sufficient. -- Dragon695 ( talk) 03:01, 8 July 2008 (UTC)
Article states, in the "Stability" section: "Note that stability of the equilibrium is related to, but distinct from, stability of a strategy."
I thought it should be relatively easy to track down the meaning of "stability of a strategy". Unfortunately, a visit to the Stability disambiguation page and the linked Stability (probability) page have left me none the wiser.
So, unless the reader can find some meaning in this phrase, I suggest that leaving the quoted sentence in the article is both unhelpful and potentially discouraging. Should we remove it? yoyo 23:02, 24 December 2006 (UTC)
I think that the current section on stability is rather weak. In general, we know that in a mixed strategy NE, all actions played with a strictly positive probability have the same expected payoff (i.e. if they were played with probability 1 they would yield the same payoff as the equilibrium mixed strategy. Hence if more than one strategy is played in equilibrium it must be unstable in sense 2. Only pure strategy NE can be strict equilibria. Zosimos101 ( talk) 17:41, 25 April 2008 (UTC)
I am a layman who is merely interested in mathematics, and whose formal education stopped at more or less high-school level. From reading this article, I do not understand why or how the Nash equilibrium is important, although I am persuaded (by other sources) that it is so. To give you an idea of how mathematically (il)literate I am, I have what I imagine is a reasonably accurate (albeit non-technical) understanding of the significance of Gödel's first incompleteness theorem, although I didn't get it from Wikipedia but from reading various books on the subject. I can only assume that this article faithfully represents Nash's result, but all I can tell you is that it explains the result in terms that mostly mathematicians will understand. As an encyclopedia article, designed to explain the equilibrium and its importance in layman's terms, it is a failure. The lead paragraph helps me to understand the nature of the result, and because I have read a few introductory books about game theory I think I get how it builds upon earlier work, but it doesn't tell me why it matters. Since Nash won the Nobel for this work, I assume the result must be very significant, but all this article seems to do is give a precis of the result that would help a math undergraduate help with coursework. It does not explain to non-mathematicians why Nash deserved a Nobel for this work. Sorry to be a pain, but I am not mathematically literate enough to improve the article, since I still don't understand the significance of the result. Lexo ( talk) 23:39, 10 July 2008 (UTC)
This article needs a good "Applications in economics" section, although I'm not really qualified to do it. The real-world application here is rather light. II | ( t - c) 00:34, 12 July 2008 (UTC)
Section states:
A strategy profile is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable, that is
but I think that it would be more accurately to say:
A strategy profile is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is
since some other player could certainly increase his payoff if player (foolishly) changes his strategy. Am I right? -- Obradović Goran (talk 01:17, 25 July 2008 (UTC)
I've argued with the editor who added this section. I can definitely understand the impulse, but I think the addition as it is doesn't improve the article. It tries to provide some more context for application of NE. On reviewing the article, I agree it needs that early on, perhaps in the lead, perhaps in an early section. It also attempts to flesh out further the prisoner's dilemma/tragedy-of-the-commons type game application, which could possibly use a little extra development in the prisoner's dilemma section.
On the downside, the addition puts too much emphasis on this specific kinda of dilemma game, while NE is much more broadly applicable (and NE isn't needed to solve such games). It's also a bit essayish with the risk of including POV and inaccuracies (about whether people follow their own best interest in fact, and about what NE assumes about it).
I'm going to pull it out (again), and this it should remain out until the important ideas can be better integrated, possibly via discussion here. CRETOG8( t/ c) 16:33, 7 January 2009 (UTC)
ROUGH DRAFT of Applications section:
COMMENT: I have now added the Applications section. Maybe I should point out that I wrote it with nonspecialists in mind. Economists and game theorists and mathematicians already know "why it's useful", so the people that need to see an Applications section are others. But therefore I have NOT tried hard to distinguish between applications of Nash equilibrium per se and applications of game theory in general. And I have not taken care, in my list of examples of specific applications, to distinguish between examples using Nash equilibrium per se and examples that use refinements or generalizations of Nash equilibrium. -- Rinconsoleao ( talk) 15:44, 25 February 2009 (UTC)
There is an incosistent use of the acronym "NE" throughout the article. I encourage us not to use the acronym at all. It's just a two-word long name after all.-- Forich ( talk) 04:51, 16 March 2009 (UTC)
The infobox states that RPS is an example of a game with a NE. Can someone enlighten me as to what that strategy is supposed to be? Neither this article nor the RPS article mentions it, and all the strategies I can think of offhand will cause an opponent to change strategy... so not sure what is meant here. Gijs Kruitbosch ( talk) 23:31, 22 March 2009 (UTC)
Could the authors of the Nash Equilibrium article say something about WHY a NE is not always Pareto Optimal? Is it because the conditions for a NE game are no cooperation between the players and so they can get stuck, not having control over the whole strategy vector but only their own part of it? Clejan ( talk) 23:51, 1 June 2009 (UTC)
Nash equilibrium may not be a specific goal of the major entities in the economic environment at all times, but the usual state of the economy is much closer to a Nash equilibrium than not.
It will generally not be possible for the actions of an individual to affect the market's equilibrium. After he is in a stable situation, he won't be able to improve his situation by his own actions alone. He must form alliances or take other actions in cooperation with others, in order to improve his situation.
Usually he will form an alliance with one or more of the entities, called employers, that are part of the economy. After making the best decisions he can with the income he has, he will be able to improve his situation further only if he acts as a cohort with other agents. Even as an employee, he must consult with either his present employer, other employers, or employment counselors in order to improve his situation.
This whole area of economics seems to be woefully unknown. Yet it should be known thoroughly, for millions of new agents join the largely stable economic system each year. Their living standards, even their very lives, depend on a win-win economic relationship with others. How does a new graduate successfully join a stable economy? How does he maintain a sense that he can act effectively in charge of his own fate? Why is knowledge of successful strategies coveted and privileged information, not widely known, as if the market economic system did not actually plan for the number of people entering it each year? Or are there only a few possible ways to use successful entry strategies? —Preceding unsigned comment added by SyntheticET ( talk • contribs) 02:12, 1 October 2009 (UTC)
It seems that this page needs to mention the case when strategies are intransitive. In the number game for example, a comparison of 0 vs. 4 would give an additional equilibrium of (4,4), but choosing 3>4, 3>2, 2>1, and 1>0. This situation seems to apply to several cases such as the Traveler's Dilemma and Centipede game where the Nash equilibrium is not the result of play and not the best strategy. I don't know of any references and am not educated in game theory, but explicitly addressing this could make these cases where the Nash equilibrium is not chosen much less confusing. —Preceding unsigned comment added by Sapiens scriptor ( talk • contribs) 03:22, 14 December 2009 (UTC)
Am I the only one who finds the bomb example really confusing? Why would one person kill the other if they have money? (It doesn't even mention at the beginning that the people have money.) Unless I'm missing something totally obvious, I think the example should be modified to be more clear. -- Lasindi ( talk) 11:15, 5 July 2008 (UTC)
In voting theory, Nash equilibria are much too common, while trembling-hand equilibria are often incalculable. For instance, in the simple election between Gandhi and Hitler, it is a Nash equilibrium for everyone to vote for the universally-despised Hitler, because any one person changing their strategy is not enough to improve the result. Peter de Blanc (not me) has proposed a more restrictive equilibrium, the Cabal equilibrium, in a blog post. Basically, it's an equilibrium where no set of players can strictly improve the result for all of them by simultaneously changing strategies. I believe this is a very useful concept, and wonder if it has been proposed before, or if anyone can find (or create :) citations to establish its WP:Notability. Homunq ( talk) 17:24, 28 May 2010 (UTC)
In the Proof of Existence there seems to be a bit of loose ends in terms of notation. For instance in the first half proof, the σ-i is introduced with a mention that somewhere earlier this is defined. I was not able to locate this definition and so even if its somewhere in the article, it is not easy to find. It doesn't hurt to define all the symbols within the scope of the section.
Also in the next proof I find the same issues with the σ notation and also that the u in the G=(N,A,u) is also undefined. I find this particularly annoying as I would like to learn the proofs myself. If knows what to do about this, I feel this is an easy fix. -Xian —Preceding unsigned comment added by Dragonflare82 ( talk • contribs) 21:14, 20 May 2011 (UTC)
Comments should go on the talk page, rather than in the article itself. The current numbers for the network traffic/Braess's Paradox section are correct.
If this still isn't clear, see if the main article on Braess's Paradox is clearer to you. The bit on this page is meant to be short. If the main article is still confusing, then maybe it will need tweaking for clarity. CRETOG8( t/ c) 23:03, 18 August 2009 (UTC)
Thanks, will need to brush up! —Preceding unsigned comment added by 92.9.84.41 ( talk) 16:33, 21 August 2009 (UTC)
The article states that "Every driver now has a total travel time of 3.25." As you noted above, in the article's example every driver has a total travel time of 3.75, and removing the B->C route would reduce the total travel time to 3.50. So the numbers aren't right, in the sense that there's a typo that incorrectly says the total travel time in the ABCD model is 3.25. 216.175.89.152 ( talk) 18:32, 26 May 2012 (UTC)MMHerbst
A few things...
1) The section on Prisoner's dilemma says: each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. Assuming strategy 1 is cooperate and strategy 2 is defect (and it makes even less sense the other way around) this is not true. The players only improve their situation (originally both cooperating and both get 3) if only one of them makes the switch (then 5 for the improved defector, 0 for the poor cooperator), if both of them make the switch they worsen both their positions (both get only 1). Besides that, the use of 'strategy 1' and 'strategy 2' is confusing, just say 'Cooperate' and 'Defect'.
2) The notation 'C > A > D > B' seems very odd and I still haven't figured out what that is supposed to mean. We had strategy A and B in the coordination game, which this section refers to C (cooperate) and D (defect), but I don't see how they are related. Rewrite for clarity...
3) The formatting for the title is a bit messed up (both IE8 and Firefox 5). The section title should follow the figure for the driving game and be aligned on the left. — Preceding unsigned comment added by 128.244.9.9 ( talk) 16:55, 1 July 2011 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
All the examples in the current Nash equilibrium article seem to be "one-shot" games (is there a better term?) -- as opposed to repeated games.
Strategies such as Tit for Tat don't work for "one-shot" games.
The Robert Aumann article mentions
Does that mean that
? If that's true, the article should mention it.
-- DavidCary 13:40, 11 October 2005 (UTC)
On this page, the phrase
resulting in strategy profile
indicates that a strategy profile is simply a vector of strategies, one for each player. I agree. However, if one follows the link, a strategy profile is defined as something that "identifices, describes, and lastly examines a player's chosen strategy". It is conceivable that someone somewhere defined strategy profile in this way, but the "vector"-meaning of the term is much more common. So, at least the link, if not the page linked to is misleading.
Also there is something slightly wrong in the notation concerning strategy profiles on this page itself. You say that
S is the set of strategy profiles.
Okay, but then each element of S is a vector of strategies. Still, you write
where is a single strategy. I think you want to write
is the set of strategy profiles
and
.
Bromille 11:50, 31 May 2006 (UTC)
There are some issues in the "Occurrence" section.
First, whatever the stated conditions are, they can only be sufficient, not necessary for equilibrium play. To see this, observe that nothing prevents a bunch of completely non-rational agents from playing the Nash equilibrium, not because it is a rational thing to do, but because non-rational agents can do anything they want.
Second, conditions 1. and 5. together say that agents are rational and they all believe that the other agents are also rational. But these are not sufficent conditions on rationality for equilibrium play. A classical counterexample is Rosenthal's centipede which has a unique Nash equilibrium as assumed. In this game, if you and I play the game and I believe you are rational and you believe I am rational, we still might not play the equilibrium, if, for instance, I do not believe that you believe that I am rational. This could be the case, even if I do believe that you are rational. What is needed to ensure that the equilibrium is played in the case of Rosenthal's centipede is common knowledge of rationality (CKR) which means that
A: I am rational, B: You are rational, C: I believe B, D: You believe A, E: I believe D, F: You believe C, G: I believe F, H: You believe E, etc.. etc..
The difference may seem like splitting hairs but is actually the key to understanding the discrepancy between how Rosenthal's centipede is actually played by rational-seeming people and the equilibrium play.
Reference: Robert Aumann, Backward induction and common knowledge of rationality, Games and Economic Behavior 8 (1995), p. 6--19.
Third, for general games, even if CKR is assumed, this usually only ensures that a correlated equilibrium is played, not a Nash equilibrium. The fact that a unique Nash equilibrium is assumed may make this ok - I'm not sure about this.
Bromille 13:54, 1 June 2006 (UTC)
Members of the Wikipedia:WikiProject Good articles are in the process of doing a re-review of current Good Article listings to ensure compliance with the standards of the Good Article Criteria. (Discussion of the changes and re-review can be found here). A significant change to the GA criteria is the mandatory use of some sort of in-line citation (In accordance to WP:CITE) to be used in order for an article to pass the verification and reference criteria. Currently this article does not include in-line citations. It is recommended that the article's editors take a look at the inclusion of in-line citations as well as how the article stacks up against the rest of the Good Article criteria. GA reviewers will give you at least a week's time from the date of this notice to work on the in-line citations before doing a full re-review and deciding if the article still merits being considered a Good Article or would need to be de-listed. If you have any questions, please don't hesitate to contact us on the Good Article project talk page or you may contact me personally. On behalf of the Good Articles Project, I want to thank you for all the time and effort that you have put into working on this article and improving the overall quality of the Wikipedia project. Agne 05:49, 26 September 2006 (UTC)
Ed: This is an incomplete example that does not illustrate. If I choose 9 and you choose 0, I'm out two bucks. If I choose 9 and you choose 7, you make $9 and I still make $5 - why is this not superior?
The system of strategies 9-0 is not a Nash equilibrium, because the first player can improve the outcome by choosing 0 instead of 9. The system of strategies 9-7 is not a Nash equilibium, because the first player can improve the outcome by choosing 9 instead of 7. The only Nash equilibrium of this game is 0-0, as stated. AxelBoldt 15:04 Aug 27, 2002 (PDT)
The second player can chose 9 and he can get 11 while the other one will get only 8. That is why 10-10 is not at Nash equilibrium -- AM
I think the thing to keep in mind here is that it is a competition game. Both players are trying to beat the other's "score". This example would work better, I think, using points instead of dollars. Yes, there is a benefit to both players if they both win some money, but no benefit if they both win some points and they are each trying to beat the other's score. -truthandcoffee /ed: Does anyone else confirm/deny this? I would like to see this example changed from dollars to points, as I think it is much clearer and makes much more sense this way. I'd like some feedback before I go ahead and make the change. -- Truthandcoffee 04:27, 13 November 2006 (UTC)
If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 11 Nash equilibria.
Does the Nash equilibrium not suppose that everybody is selfish and rational? So, if you suppose that the other player is selfish, you can savely assume that he/she chooses 10. IMHO that's the only equilibrium. Or does the Nash equilibrium not presuppose selfishness? Andy 12:53, 26 September 2006 (UTC)
I am curious about the relationship between NE and government regulation. Consider the case of airbags in cars. Without regulation requiring airbags, normal supply and demand will dictate the number of airbags produced. If air bags are mandated by the government, the unit cost will be much lower, and more consumers will choose to purchase cars with airbags and receive the consumer surplus from the purchase. The consumer was able to move to a better outcome through the intervention of a third party. In this case it is hard to say how the auto manufacturer is impacted. It is possible that they benefited because they can now offer new cars with more value at a lower cost than without regulation. More people buy a new car vs. a used car to benefit from the air bag. Measuring if there was a benefit to the auto company is impossible, but let's assume there is. Did the regulation help both parties move out of a NE to a better outcome? 65.198.133.254 20:36, 30 November 2006 (UTC)David Wilson
Under the heading specified above, it claims that "...an NxN matrix may have 0 or N Nash Equilibriums." Shouldn't it be 1 or N NE, since at the beginning of the article it says that Nash proved the existence of equilibria for any finite game with any number of players?
This section provides a rule for finding NEs. It fails to discuss either the rule's history - who discovered it and when - or why (the reader should believe) it works. While there's a place for pragmatic rules in an encyclopedia, the article should at least indicate where its theoretical underpinnings may be found. yoyo 23:09, 24 December 2006 (UTC)
I have seen in some papers the notation used to describe a set of strategies for all players other than . This cleans up notation greatly; for example,
becomes
This seems more compact and conveys the point more clearly. SanitySolipsism 22:27, 6 January 2007 (UTC)
Does anyone know anything about this? How about what journal the article is in, or anything? I couldn't find anything, the ip of the anon who added the information is at U of Toronto, and another, erilly similar ip address claims at User talk:128.100.53.169 to be familiar with Goldstein, although not think that it belongs in the lead. As unverifiable, I'm still for removing the mention, but loath to do it without mentioning it here, per WP:1RR. Smmurphy( Talk) 17:55, 14 March 2007 (UTC)
OK, what to do? I put this on my watchlist now. ~ trialsanderrors 17:20, 6 March 2007 (UTC)
Hello all - The article currently uses two inconsistent citation style (APA and wiki-ref style). We need to be consistent. I personally prefer APA-inline citations, but I know wikipedia generally seems to be leaning toward the footnote style. Anybody have any druthers? --best, kevin kzollman][ talk 23:19, 29 April 2007 (UTC)
Governing Dynamics, i.e. the way an object or phenomenon will occur or behave under a given set of circumstances is not Nash's idea, so why when one types the phrase into the search box, does one get diverted to his page? It's ludicrous.
Governing dynamics is a centuries old theory which simply refers to the way an e.g. object or phenomenon will behave given a set of prevailing circumstances. If anything it is a philosophical rationale more than a scientific theory. By understanding the laws which governs said object in said circumstances, one is essentially able to understand the true nature of all matter and occurrence and abstracts are erased. The principle of governing dynamics serves to found and object or occurrence in its pure and absolute state because it is essentially a malleable law, under which nothing is fixed nor finite, that is to say X will behave as Y under Z, but that does not imply X will also behave as Y, because X is not fixed and will not always be under condition Z. Under condition V, X will behave as A and not Z, because it will reflect the new conditions and parameters it finds itself operating in. Thus by understanding the laws, the governing dynamics of phenomenon, one is ultimately able to understand the true and original nature of the object or phenomenon under inspection, because one essentially has a series or set of laws which encompass all possible existence and not simply a single law under which an object or phenomenon is expected to eternally function.
I therfore suggest that for accuracy these two pages should be split and a seperate page developed which specefially deals with the Law of Governing Dynamics in its historical and philosphical contexts.
Andrew Woollock 14:08, 13 July 2007 (UTC)
This is a notification of an intention to delist this article as a good article. In February this year at Wikipedia_talk:WikiProject_Game_theory#CotM, the consensus was that this article is B-Class (at best) from the point of view of WikiProject Game theory. I agree with this assessment, and last month I also rated it as B-Class from the point of view of WikiProject Mathematics. The article has not improved since these assessments, despite being Game theory collaboration of the month for two months.
The lead is inadequate as a summary of the article. The article is missing a history section, which is surely vital in this case (and is partly covered in the lead). The accessibility of the article could be much improved, the references are poor, and citation is both inadequate and inconsistent. I will attempt to improve the article myself over the next day or two, but I don't think I will be able to raise the article to GA standard myself. I hope others will contribute to ensuring that this article meets with criteria, otherwise I will have to delist it. An alternative would be to take the article to Good article review and any editor is welcome to do that. Geometry guy 21:28, 12 July 2007 (UTC)
I think there is consensus that this is not yet a good article. The lead is weak, the prose poor. Citation is extremely weak. Improve and renominate: good luck!
Geometry guy
20:35, 14 July 2007 (UTC)
I hope the specialists among us don't mind my attempt at defining Nash equilibrium in a lay person's language. If you are tempted to quibble with my definition, just remember how awfully vague the (non)explanation of Nash equilibrium was in that bar scene in the film of A Beautiful Mind. Such a fundamental and elegant idea deserves to be explained in words everybody can understand. -- Rinconsoleao 19:46, 16 July 2007 (UTC)
I have added a comma after the <forall> i, 194.223.231.3 15:26, 17 July 2007 (UTC)
I think it is weird that the proportion between the players' payoffs are not taken into account. Surely I would, in a two-player game, prefer weakening both myself and my opponent if I weakened him more. Consequently, I would, from the situation A,B (25,40) being player one, change to C (15,5) to force the opponent into also doing C (10,10). That is a truly stable position, in which no player can earn anything by changing strategy. Isn't that actually the only of the three "stable" combinations that satisfies both stability criteria? —Preceding unsigned comment added by 77.40.128.194 ( talk) 21:25, 15 January 2008 (UTC)
I've noticed that although the article makes use of the terms strong & weak Nash equilibria, they are not defined anywhere. -- ricmitch 08:17, 21 April 2008 (UTC)
History of Nash equilibrium. I just changed this section. I can develop it a bit more and add some discussion. The concept of a Nash equilibrium in pure strategies (to use the current name) was clearly in use in oligopoly theory in the 19th Century and was well known: the writings of Edgeworth, Hotelling, Stackelburg to name a few of the better known names. I do not know for sure, but it quite possible that Oscar Morgenstern did not know much about this, since it is left out of the Theory of games and economic behavior (as far as I remember it). Von Neuman was primarily interested in card games and developed the idea of a mixed strategy (bluffing) in this context. In the Theory of Games, he showed that all zero-sum games posses a mixed strategy equilibrium. The modern form of the Nash equilibrium (where the strategies are in generla mixed) was the contribution of Nash: he extended Von Neuman's concept to all games with a finite number of actions. There have of course been various extensions since then: in particular Dasgupta and Maskin provided an existence Theorem for the non-finite case (when the strategic variables such as price are continuous) in their 1987 Review of Ecoonomic Studies articles and so on, but much of this is secondary. Zosimos101 ( talk) 18:04, 25 April 2008 (UTC)
Oh well, the bank run might be as good as well... feel free to revert me if you think so... but i think a cartel is something more people are prone to be familiar with than a bank run?... and in the cartel example all players get a boost in profits wheras in a bank-run only some get more than theit share of the banks assets value, Gillis ( talk) 21:38, 31 May 2008 (UTC)
Gentlemen, this article is going to get an ugly citation template if you continue without proper citations. This is a serious topic, not pop culture fan-cruft, so we expect better from you. It's been a year since it was shown that this was a problem, yet there are still only two in-line citations. Please try to rectify this as soon as possible! Note, the citations need not be Internet accessible - good, old fashion book citations are more than sufficient. -- Dragon695 ( talk) 03:01, 8 July 2008 (UTC)
Article states, in the "Stability" section: "Note that stability of the equilibrium is related to, but distinct from, stability of a strategy."
I thought it should be relatively easy to track down the meaning of "stability of a strategy". Unfortunately, a visit to the Stability disambiguation page and the linked Stability (probability) page have left me none the wiser.
So, unless the reader can find some meaning in this phrase, I suggest that leaving the quoted sentence in the article is both unhelpful and potentially discouraging. Should we remove it? yoyo 23:02, 24 December 2006 (UTC)
I think that the current section on stability is rather weak. In general, we know that in a mixed strategy NE, all actions played with a strictly positive probability have the same expected payoff (i.e. if they were played with probability 1 they would yield the same payoff as the equilibrium mixed strategy. Hence if more than one strategy is played in equilibrium it must be unstable in sense 2. Only pure strategy NE can be strict equilibria. Zosimos101 ( talk) 17:41, 25 April 2008 (UTC)
I am a layman who is merely interested in mathematics, and whose formal education stopped at more or less high-school level. From reading this article, I do not understand why or how the Nash equilibrium is important, although I am persuaded (by other sources) that it is so. To give you an idea of how mathematically (il)literate I am, I have what I imagine is a reasonably accurate (albeit non-technical) understanding of the significance of Gödel's first incompleteness theorem, although I didn't get it from Wikipedia but from reading various books on the subject. I can only assume that this article faithfully represents Nash's result, but all I can tell you is that it explains the result in terms that mostly mathematicians will understand. As an encyclopedia article, designed to explain the equilibrium and its importance in layman's terms, it is a failure. The lead paragraph helps me to understand the nature of the result, and because I have read a few introductory books about game theory I think I get how it builds upon earlier work, but it doesn't tell me why it matters. Since Nash won the Nobel for this work, I assume the result must be very significant, but all this article seems to do is give a precis of the result that would help a math undergraduate help with coursework. It does not explain to non-mathematicians why Nash deserved a Nobel for this work. Sorry to be a pain, but I am not mathematically literate enough to improve the article, since I still don't understand the significance of the result. Lexo ( talk) 23:39, 10 July 2008 (UTC)
This article needs a good "Applications in economics" section, although I'm not really qualified to do it. The real-world application here is rather light. II | ( t - c) 00:34, 12 July 2008 (UTC)
Section states:
A strategy profile is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable, that is
but I think that it would be more accurately to say:
A strategy profile is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is
since some other player could certainly increase his payoff if player (foolishly) changes his strategy. Am I right? -- Obradović Goran (talk 01:17, 25 July 2008 (UTC)
I've argued with the editor who added this section. I can definitely understand the impulse, but I think the addition as it is doesn't improve the article. It tries to provide some more context for application of NE. On reviewing the article, I agree it needs that early on, perhaps in the lead, perhaps in an early section. It also attempts to flesh out further the prisoner's dilemma/tragedy-of-the-commons type game application, which could possibly use a little extra development in the prisoner's dilemma section.
On the downside, the addition puts too much emphasis on this specific kinda of dilemma game, while NE is much more broadly applicable (and NE isn't needed to solve such games). It's also a bit essayish with the risk of including POV and inaccuracies (about whether people follow their own best interest in fact, and about what NE assumes about it).
I'm going to pull it out (again), and this it should remain out until the important ideas can be better integrated, possibly via discussion here. CRETOG8( t/ c) 16:33, 7 January 2009 (UTC)
ROUGH DRAFT of Applications section:
COMMENT: I have now added the Applications section. Maybe I should point out that I wrote it with nonspecialists in mind. Economists and game theorists and mathematicians already know "why it's useful", so the people that need to see an Applications section are others. But therefore I have NOT tried hard to distinguish between applications of Nash equilibrium per se and applications of game theory in general. And I have not taken care, in my list of examples of specific applications, to distinguish between examples using Nash equilibrium per se and examples that use refinements or generalizations of Nash equilibrium. -- Rinconsoleao ( talk) 15:44, 25 February 2009 (UTC)
There is an incosistent use of the acronym "NE" throughout the article. I encourage us not to use the acronym at all. It's just a two-word long name after all.-- Forich ( talk) 04:51, 16 March 2009 (UTC)
The infobox states that RPS is an example of a game with a NE. Can someone enlighten me as to what that strategy is supposed to be? Neither this article nor the RPS article mentions it, and all the strategies I can think of offhand will cause an opponent to change strategy... so not sure what is meant here. Gijs Kruitbosch ( talk) 23:31, 22 March 2009 (UTC)
Could the authors of the Nash Equilibrium article say something about WHY a NE is not always Pareto Optimal? Is it because the conditions for a NE game are no cooperation between the players and so they can get stuck, not having control over the whole strategy vector but only their own part of it? Clejan ( talk) 23:51, 1 June 2009 (UTC)
Nash equilibrium may not be a specific goal of the major entities in the economic environment at all times, but the usual state of the economy is much closer to a Nash equilibrium than not.
It will generally not be possible for the actions of an individual to affect the market's equilibrium. After he is in a stable situation, he won't be able to improve his situation by his own actions alone. He must form alliances or take other actions in cooperation with others, in order to improve his situation.
Usually he will form an alliance with one or more of the entities, called employers, that are part of the economy. After making the best decisions he can with the income he has, he will be able to improve his situation further only if he acts as a cohort with other agents. Even as an employee, he must consult with either his present employer, other employers, or employment counselors in order to improve his situation.
This whole area of economics seems to be woefully unknown. Yet it should be known thoroughly, for millions of new agents join the largely stable economic system each year. Their living standards, even their very lives, depend on a win-win economic relationship with others. How does a new graduate successfully join a stable economy? How does he maintain a sense that he can act effectively in charge of his own fate? Why is knowledge of successful strategies coveted and privileged information, not widely known, as if the market economic system did not actually plan for the number of people entering it each year? Or are there only a few possible ways to use successful entry strategies? —Preceding unsigned comment added by SyntheticET ( talk • contribs) 02:12, 1 October 2009 (UTC)
It seems that this page needs to mention the case when strategies are intransitive. In the number game for example, a comparison of 0 vs. 4 would give an additional equilibrium of (4,4), but choosing 3>4, 3>2, 2>1, and 1>0. This situation seems to apply to several cases such as the Traveler's Dilemma and Centipede game where the Nash equilibrium is not the result of play and not the best strategy. I don't know of any references and am not educated in game theory, but explicitly addressing this could make these cases where the Nash equilibrium is not chosen much less confusing. —Preceding unsigned comment added by Sapiens scriptor ( talk • contribs) 03:22, 14 December 2009 (UTC)
Am I the only one who finds the bomb example really confusing? Why would one person kill the other if they have money? (It doesn't even mention at the beginning that the people have money.) Unless I'm missing something totally obvious, I think the example should be modified to be more clear. -- Lasindi ( talk) 11:15, 5 July 2008 (UTC)
In voting theory, Nash equilibria are much too common, while trembling-hand equilibria are often incalculable. For instance, in the simple election between Gandhi and Hitler, it is a Nash equilibrium for everyone to vote for the universally-despised Hitler, because any one person changing their strategy is not enough to improve the result. Peter de Blanc (not me) has proposed a more restrictive equilibrium, the Cabal equilibrium, in a blog post. Basically, it's an equilibrium where no set of players can strictly improve the result for all of them by simultaneously changing strategies. I believe this is a very useful concept, and wonder if it has been proposed before, or if anyone can find (or create :) citations to establish its WP:Notability. Homunq ( talk) 17:24, 28 May 2010 (UTC)
In the Proof of Existence there seems to be a bit of loose ends in terms of notation. For instance in the first half proof, the σ-i is introduced with a mention that somewhere earlier this is defined. I was not able to locate this definition and so even if its somewhere in the article, it is not easy to find. It doesn't hurt to define all the symbols within the scope of the section.
Also in the next proof I find the same issues with the σ notation and also that the u in the G=(N,A,u) is also undefined. I find this particularly annoying as I would like to learn the proofs myself. If knows what to do about this, I feel this is an easy fix. -Xian —Preceding unsigned comment added by Dragonflare82 ( talk • contribs) 21:14, 20 May 2011 (UTC)
Comments should go on the talk page, rather than in the article itself. The current numbers for the network traffic/Braess's Paradox section are correct.
If this still isn't clear, see if the main article on Braess's Paradox is clearer to you. The bit on this page is meant to be short. If the main article is still confusing, then maybe it will need tweaking for clarity. CRETOG8( t/ c) 23:03, 18 August 2009 (UTC)
Thanks, will need to brush up! —Preceding unsigned comment added by 92.9.84.41 ( talk) 16:33, 21 August 2009 (UTC)
The article states that "Every driver now has a total travel time of 3.25." As you noted above, in the article's example every driver has a total travel time of 3.75, and removing the B->C route would reduce the total travel time to 3.50. So the numbers aren't right, in the sense that there's a typo that incorrectly says the total travel time in the ABCD model is 3.25. 216.175.89.152 ( talk) 18:32, 26 May 2012 (UTC)MMHerbst
A few things...
1) The section on Prisoner's dilemma says: each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. Assuming strategy 1 is cooperate and strategy 2 is defect (and it makes even less sense the other way around) this is not true. The players only improve their situation (originally both cooperating and both get 3) if only one of them makes the switch (then 5 for the improved defector, 0 for the poor cooperator), if both of them make the switch they worsen both their positions (both get only 1). Besides that, the use of 'strategy 1' and 'strategy 2' is confusing, just say 'Cooperate' and 'Defect'.
2) The notation 'C > A > D > B' seems very odd and I still haven't figured out what that is supposed to mean. We had strategy A and B in the coordination game, which this section refers to C (cooperate) and D (defect), but I don't see how they are related. Rewrite for clarity...
3) The formatting for the title is a bit messed up (both IE8 and Firefox 5). The section title should follow the figure for the driving game and be aligned on the left. — Preceding unsigned comment added by 128.244.9.9 ( talk) 16:55, 1 July 2011 (UTC)