Talk:Game theory/Archive 2

External link removed
I just removed a link to a "concise" intro to GT. Unfortunately, in the very first page there are already two non-negligeable mistakes: "... Nash equilibrium condition and 'subgame perfection'. Before we can define either of these criteria, we need to define the concept of ‘dominant strategy’" and "If there is only one rationalisable strategy, it is the dominant strategy". --Fioravante Patrone en (talk) 05:54, 20 May 2008 (UTC)


 * For the benefit of those less familiar with game theory, perhaps you could kindly explain what you consider to be the mistakes in the two statements you cite. Thank you.Ranger2006 (talk) 17:24, 28 May 2008 (UTC)


 * Frankly speaking, having seen your contributions, I feel that you are the author of those notes, or at least that the author belongs to the group for which you have been advertising so much on wiki. Moreover, sorry: you added that link. So, if you think that the link is relevant, please make the effort to convince readers here that it is free of relevant mistakes. I have done already my job. --Fioravante Patrone en (talk) 06:41, 29 May 2008 (UTC)

Laffer curve (Trialsanderrors)
The Laffer curve has no citations, but can someone (Trials?) verify that this really is a game theoretic analysis? It seems to me to be merely a weighted average driven effect from the new paragraph. 14:49, 17 April 2007 (UTC)


 * I agree. The Laffer curve idea is not really based on game theory as it lacks a strategic component. It's just plain ol' individual-level incentives (to the extent it holds). The sentence should be removed.radek (talk) 06:21, 19 December 2007 (UTC)


 * Did this go away and then come back again? Anyway, I'm pulling it out again, it's really far afield. Cretog8 (talk) 04:48, 30 July 2008 (UTC)

Aumann 1987 Quote
I think this is a horrible quote to include on this page. It abuses the term 'unified field' theory and adds extremely little to the article. I propose eliminating the quote entirely from the introductory section.

Bkessler (talk) 08:12, 15 August 2008 (UTC)


 * (moved to end page to respect the usual chronologiacl order) --Fioravante Patrone en (talk) 13:43, 15 August 2008 (UTC)


 * I like the quote. The 'unified field' bit might be a bit silly, but it's in quotes and it might help some people in understanding. In any case, it is helpful in identifying the importance of game theory, and it comes from a good source. I've heard/read similar sentiment elsewhere, so maybe there's a better quote which captures the same idea, if you can fish it up. Cretog8 (talk) 15:36, 15 August 2008 (UTC)

I agree...please don't include this quote. Vextron (talk) 19:15, 25 September 2008 (UTC)

New subsection on use of game theory in radio networks ???
I propose to add a subsection on use of game theory in radio networks research, under the section "Applications and Challenges".

My proposal would be as follows:

Game theory has recently become a useful tool for modeling and studying interactions between cognitive radios envisioned to operate in future communications systems. Such terminals will have the capability to adapt to the context they operate in, through possibly power and rate control as well as channel selection. Software agents embedded in these terminals will potentially be selfish, meaning they will only try to maximize the throughput/connectivity of the terminal they function for, as opposed to maximizing the welfare (total capacity) of the system they operate in. Thus, the potential interactions among them can be modeled through non-cooperative games. The researchers in this field often strive to determine the stable operating points of systems composed of such selfish terminals, and try to come up with a minimum set of rules (etiquette) so as to make sure that the optimality loss compared to a cooperative - centrally controlled setting- is kept at a minimum. Omer182 (talk) 15:39, 2 October 2008 (UTC)
 * The reference does not appear to be published. Can you fix this? Also, it's reasonable to expect that topics included in our articles are widely regarded as important. Unless someone other than the author has chosen to comment on the use of game theory in radio networks, it may not be appropriate for this article (though it does sound interesting). EdJohnston (talk) 20:41, 9 October 2008 (UTC)


 * I'm sorry I didn't reply to your suggestion earlier, Omer182. To me it looks like it's a bit too narrow a focus for a general article. That's reinforced by the impression that the actual application is still speculative. If there's more references, maybe this would fit in an article on the more specific tools they use? C RETOG 8(t/c) 20:45, 9 October 2008 (UTC)


 * Hi, actually the reference is published. I will fix it in case you agree that the section should stay. There are also tons of different publications on the subject. The more important question, you (both) have raised though, is whether this application is a good fit to put in this article. Frankly I was encouraged by the presence of computer science applications section. The correct observation that you have made is that the use of game theory in communications is at an "infant" stage, meaning people are using it as a "tool" to understand how systems would behave when we have selfish, smart (cognitive) radios around. So I know it is pretty important for the research community, however I am not aware of any systems out there made of selfish radios yet (except for "cheater" users in WiFi which try to have more access to channel through adjusting their transmission parameters). Those said, I have no problems if you remove the section. In case you want it to stay, I can provide more references. Omer182 (talk) 21:41, 12 October 2008 (UTC)
 * I would say that such applications are getting more and more important, and also more widespread. I consider too much to have a chapter on cognitive radios here, but I would like to see a special page devoted to game theory and telecommunication, of which the cognitive radios could be a chapter. --Fioravante Patrone en (talk) 04:58, 13 October 2008 (UTC)
 * You might find this link interesting, which is a call for papers from a well known journal in communications research, for a special issue on game theory in communications. In any case, I will now remove the section I had added here previously, as it doesn't seem to have too much support. I will, hopefully soon, start working on a special page as the one mentioned by Fioravante. Omer182 (talk) 13:14, 13 October 2008 (UTC)

Swedish National Bank Prize?
A recent edit renamed 'Nobel Prize' to 'Swedish National Bank Prize' throughout this article, even coining 'Swedish National Bank Prize Laureat' for a holder. My concerns are:

1) This differs from the standard wikification of the prize as the 'Nobel' (common usage though slightly misleading).

2) It doesn't appear as any of the names given on the "Nobel Prize in Economic Sciences'.

3) It appears to be a neoligism, violating multiple AVOID guidelines (for instance pushing POV with loaded atypical language).

4) The edit message doesn't acknowledge multiple viewpoints, calling this simply a "correction". It may not have been fully considered.

Wragge (talk) 13:32, 17 October 2008 (UTC)
 * inclined to agree w Wragge. I suggest discussion over at an appropriate WPPROJ econ page.  Let consensus within that group decide what to do here. Pete.Hurd (talk) 14:02, 17 October 2008 (UTC)

Unified field theory
Um hey I don't know anything about this but in the intro, the "'unified field' theory" comment should be deleted. As a physics person I can testify that a field theory is a concept describing force lines and how they can be seen as a playing field upon which forces move and act. A unified field theory is a term refering to a field theory on which all four forces(gravity, electromagnetic, strong, and weak) may 'play'.75.65.50.89 (talk) 04:47, 19 February 2009 (UTC) Thankyou Blaque
 * I strongly disagree. First, it is not a "registered tademark" of physicists. Second, my guess is that Aumann was purposedly pointing at an analogy with physics. --Fioravante Patrone en (talk) 13:17, 19 February 2009 (UTC)
 * I was about to agree with Fioravante Patrone en, but took a closer look, and find I agree with both of you. I agree that it's OK to use "unified field theory" there but only because/if that's the expression Aumann used--his authority lends notability to a metaphor which is strictly inaccurate. I think (don't have Palgrave available) that he did use that term, and so it can be used but should be in a larger quote by Aumann. As it is written now, it reads too much as if it's just an assertion of the article. C RETOG 8(t/c) 17:45, 19 February 2009 (UTC)
 * It looks like the issue was just a typo in this article. The test is a direct quote from Aumann (can see here). Somehow the opening quotation mark was missing, so I put it back. Given that way, as a quote, I think it's useful and the use of "unified field theory"--while an inaccurate metaphor--is useful in describing how many game theory folks feel about it. C RETOG 8(t/c) 21:24, 19 February 2009 (UTC)
 * I'm inclined to side with the Anon IP here, we don't *have* to quote Aumann. We could paraphrase an equivalent sentence from any of many undergraduate textbooks, and not have to shift uncomfortably in our seats at the slight abuse of terminology in the WP article. Pete.Hurd (talk) 22:09, 19 February 2009 (UTC)
 * of course we do not need to quote Aumann. But I think that the phrase, and the intriguing reference to "unified field theory", gives the feeling of what game theory is/aims to be for the social sciences. A unifying viewpoint, a kind of "universal tool". I would keep it somewhere in the page, if not in the prologue. --Fioravante Patrone en (talk) 09:16, 22 February 2009 (UTC)
 * I like the quote (though I disagree with it), and also lean towards thinking it should remain on the page somewhere. I also think the *sentiment* belongs in the intro. So, using that quote for the sentiment seems efficient, but I'd be OK moving the quote and re-phrasing the sentiment. C RETOG 8(t/c) 16:48, 22 February 2009 (UTC)


 * I deleted it and replaced it with something that is less a kind of advocacy of an enthusiast. It is rhetorically cute but unprofessional. Bracton (talk) 07:42, 21 March 2009 (UTC)

solution concepts
This edit rephrases the lead. The rephrasing of what an equilibrium is seems reasonable to me. There's another bit which I assume was meant to just rephrase but actually introduces inaccuracies: "How an equilibrium is reached depends on the rules of each different game and the motivations of the participants, although equilibrium concepts from different games often overlap or coincide. Attempts to generalise equilibrium concepts from one game to another are subject to criticism...".

This new text conflates equilibrium concepts with games. Equilibrium concepts can overlap or coincide (A subgame perfect Nash equilibrium is a Nash equilibrium, but that's (at least in the technical sense intended) independent of the game they're applied to)--I'm not sure what the clarification was that the editor had in mind, but we should figure it out here. C RETOG 8(t/c) 18:04, 7 January 2009 (UTC)

The sentence "How an equilibrium is reached..." implies dynamics to me, which Nash equilibrium are blind to... I suggest reverting the article and editing the paragraph here on the talk page. Pete.Hurd (talk) 22:22, 7 January 2009 (UTC)

"Traditional applications of game theory attempt to find equilibria in these games&mdash;in an equilibrium all of the players of the game have adopted a strategy in their behaviour which they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. How an equilibrium is reached depends on the rules of each different game and the motivations of the participants, although equilibrium concepts from different games often overlap or coincide. Attempts to generalise equilibrium concepts from one game to another are subject to criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally."

I've tweaked the opening of the paragraph, and reverted the rest for the moment, so here's what's there currently:

"Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea.  These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide.  This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally."

I don't know yet how to address the concerns which led to the edit, because I'm not sure what it was trying to clarify, so I'm leaving it be pending discussion. C RETOG 8(t/c) 02:04, 8 January 2009 (UTC)

I am also having difficulty understanding the clarification introduced by the edit. However there is a point worth making in the text which is not captured by the current version: The studies of the equilibria and the algorithms that are used to reach these equilibria are different and they are independent. For instance you might have an equilibria that the system might not actually converge through distributed algorithms (So the system would stay stable if somehow it was taken to the equilibrium state by an invisible hand but would not converge there if the players went through best response dynamics given an initial state). I believe this is an important point to make as NE is most often confused to be confined to be a stable state that could be achieved through best response dynamics or some other distributed algorithm. Omer182 (talk) 15:27, 9 February 2009 (UTC)

"Equilibrium" does not imply "dynamics". That is using it as though it was only about dynamic disequilibrium phenomena, but for mathematicians it is about the geometry of the utility function. Bracton (talk) 07:47, 21 March 2009 (UTC)

Continuous Games should be Uncountably infinite games
The use of the terminology continuous games is incorrect. A game with a finite or countable number of options can still be continuous, but that does not communicate the point. a function f:X->Y is continuous iff the inverse image of any open set in Y is open in X under f inverse. check out the Discrete Topology for an example. I think the intention here was to say games with uncountably infinite number of options for strategies.

RyanHLewis (talk) 18:29, 11 December 2008 (UTC)


 * To me the phrase uncountably infinite game makes it sound as though it lasts for an uncountable number of moves. I don't know that such games have been studied much.  A game that lasts past the first infinite ordinal, &omega;, is called a long game, and their relationship to large cardinals and inner model theory has been studied a bit by Neeman, but I don't know that he's found any use for ones that go to the first uncountable ordinal &omega;1 or beyond. --Trovatore (talk) 19:05, 11 December 2008 (UTC)


 * I don't understand. The article says, "Continuous games allow players to choose a strategy from a continuous strategy set." That pretty much provides the definition of a continuous game. But then, I also don't understand how a finite number of strategies (>1) can have a continuous strategy set. C RETOG 8(t/c) 20:34, 11 December 2008 (UTC)


 * The term "continuous" game is used to refer to games in which "moves" occur on a real axis, such as time, and "positions" also map into real axes, such as the movement of missiles and targets in a 3-dimensional battlefield. Within such games there may be a finite or infinite number, countable or uncountable, of moves, strategies, or players. Bracton (talk) 07:56, 21 March 2009 (UTC)

discrete / continuous / differential
I trimmed the discrete vs continuous games section a good bit. There were a few concerns: It would be great if someone can flesh out the differential games article. Cretog8 (talk) 00:06, 27 June 2008 (UTC)
 * I think differential games are just one class of continuous games.
 * I don't think it's reasonable to make the generalization that most continuous games are concerned with time (it may be for differential games).
 * I doubt that a lot of continuous games are big in engineering and physics--I'm eager to be proven wrong with references.

I did that by citing to Rufus Isaacs book, which has been reprinted by Dover. I read it back in college.

There is plenty of research in the field, but most of it doesn't use the phrase "game theory". Look for things like "optimal control", "servomechanism", "process control", or any number of other terms. The underlying math is essentially the same but the research flies diverse flags. Bracton (talk) 08:02, 21 March 2009 (UTC)

Revert #1
Reverted recent edits 1) "field of play" is not typical to the game theory literature, "A field of play and set of rules for how moves may be made on that field." is highly idiosyncratic. 2) "A utility function that evaluates the payoffs of the outcomes of play" is also odd, the payoffs are typically taken as is, not interpreted through a utility function, 3) "The problem of a game is to find the optimal strategy for each player, which is a set of rules that..." confuses "rules" and "strategy", 4) "There are also mixed cooperative-competitive games like the Prisoner's Dilemma" the PD is a non-cooperative (not sure what is meant here by "competitive" but these terms cooperative and noncoperative refer to fundamental assumptions about the world the game is played in, not whether the players are helping each other out or not... Pete.Hurd (talk) 23:40, 20 March 2009 (UTC)

Not if you are really familiar with the literature. You seem to be fixated on finite game theory at an early stage in the history of game theory. The subject is vastly larger than you seem to be aware of, nor are my edits "idiosyncratic". Just do web searches on the phrases and "game theory". Let us examine each point using your numbering:

1. "Field of play". Here are some examples:
 * Craig A. Lindley, The Semiotics of Time Structure in Ludic Space As a Foundation for Analysis and Design
 * Frank Lantz & Eric Zimmerman, Rules, Play and Culture: Towards an Aesthetic of Games
 * Bonds, The Mission Statement
 * Lynn M. LoPucki and Walter O. Weyrauch, A Theory of Legal Strategy, 49 Duke L. J. 1405 (2000)

2. "Utility function". Some examples:
 * Entry in the Stanford Encyclopedia of Philosophy
 * Another entry in the Stanford Encyclopedia of Philosophy
 * David K. Levine, What is Game Theory
 * Wiktionary definiton of Utility Function
 * Utility function

3. The key point is that a strategy is a function on all possible states of the game which defines (if only a random) move at that point. A function can be described as also a kind of rule, mapping game states to moves, without confusing it with the rules of the game.

4. Problem of finding the best strategy.
 * Game Theory Based Risk and Impact Analysis
 * Robert Matthews, Don't get even, get mad
 * Sequential Games
 * Game Theory; – Finding the best strategy
 * A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006)

5. "competitive" is synonymous with "non-cooperative", and is the common term. To be more precise, it is strategies that are competitive or cooperative, or a mix of the two, but it is common parlance to characterize a game as competitive or cooperative if the optimal strategy for all players is. If it is competition for some and cooperation for others, perhaps depending on the moves made by other players, then it is mixed.
 * Entry in the Stanford Encyclopedia of Philosophy
 * Game Theory

Moreover, the previous version confuses the matter by treating the strategy as part of the representation of a game. It is not. a game is specified in the way I outlined.

I am going to revert to my version. If you have specific objections to specific edits then please make them here in the Talk section. Bracton (talk) 05:48, 21 March 2009 (UTC)


 * Here's what I think ought to be changed back
 * The part where you say that a game is composed of four elements: one of which is "* A field of play and set of rules for how moves may be made on that field." and another is "* A set of moves available to those players.". You have "moves" specified twice here which is confusing, do you have a reference for this four element definition of a game?
 * another of the elements you list is "A utility function that evaluates the payoffs of the outcomes of play." you've produced a number of links to utility functions above, but none of which address my concern, which is that payoffs *are* utility.  There is no additional utility function imposed onto the payffs to evaluate them.
 * If you don't have an academic reference defining a game according to these four elements, then I think you should change it back to the way it was.
 * Where you say "Strategies may be competitive (non-cooperative), cooperative, or mixed. A game for which all the strategies for all the players is competitive is conveniently characterized as a competitive game. Similarly for cooperative or mixed games." this totally confabulates the topics of Cooperative game and Non-cooperative game with characterizations of strategies within games like the PD. Where you write "An example of a mixed cooperative-competitive game is the Prisoner's Dilemma, in which the optimal strategy for each player may differ from what is the optimal strategy for the players as a group." you confuse the distinction between cooperative games, non-cooperative games and Pareto optima. I really think this ought to be changed back
 * Where you write "Games may also have, or lack, complete or perfect information. In other words, the players may not have complete or accurate knowledge of the state of the game, the number or moves of the other players, the intermediate outcomes of play, or may be deceived or bluffed by other players." I think this confuses the distinct issues of Complete information and Perfect information and should be reverted.
 * I think "The problem of a game is to find the optimal strategy for each player, which is a function on all possible states of play that defines the move to be made for that state and moves of other players." is just not really well written, the latter half of the sentence really isn't going to inform a lay reader.
 * and others, I the article was more correct, and clearer, before. Pete.Hurd (talk) 08:19, 21 March 2009 (UTC)


 * I agree with most of Pete.Hurd's points above, so I've mostly re-reverted. I think the changes you propose need more discussion here before they go into the page--both for deciding if they're correct and making them clear.
 * I partially put back in your addition about one-player & many-player games. I've never heard of zero-player games, and the idea seems counter to the construction of game theory. I'm not sure if the section is helpful, and it needs some cleanup, but it's not evidently wrong.
 * Some of your references above are to "ludic" game studies--which has a big overlap with, but is something other than game theory. The "field of play" may be a standard notion in ludic studies, and will sometimes come up in specific game theory applications, but isn't a major--and certainly not a necessary or definitional--part of game theory. C RETOG 8(t/c) 09:09, 21 March 2009 (UTC)


 * The "infinitely long games" section deserves a comment. These changes weren't simply clarifications, but changed the meaning. I'm not a mathematician, so I could be off. However, as odd as it may be most infinitely-long games that I've seen are still effectively determined by the "final" payoff, even though the game doesn't end--just typical infinity weirdness. The second paragraph went from being specific (talking about the axiom of choice and importance to descriptive set theory) to very vague. What benefit is there to that vagueness? C RETOG 8(t/c) 09:47, 21 March 2009 (UTC)


 * I am a mathematician and computer scientist who has worked with game theory professionally, which is why I find the changes I made to be accurate and important. I also object to reverting everything whe n you are only objecting to a few things. I have heard no objection to the paragraph about James Madison using game-theoretic concepts in 1787, nor to the additional cites and internal links. I'm putting those back.


 * You also don't seem to be reading or understanding the several documents to which I've provided links, or to do a web search on "game theory" plus " " which confirm my positions. You have not backed your positions with such cites. Until you do mine should stand. Let's examine your points further:


 * Specifying moves twice. There is a broader set of rules that may be made during the course of a game, and the subset of them available to particular players at particular points. For example, in a game of chess the initial move of a pawn may be one or two spaces, but once a pawn has been moved one space it is no longer an available move to move it two, and if a player tries to move a pawn past another the other may take it en passant, a move that he will only have after a certain kind of opposing move, and only immediately after. These are two different sets, but they are both needed to define the game.


 * Payoffs and utility. Utility is defined by a function. Its domain is the game state. Its values are the payoffs, which may be defined at each point in play (for nonterminating games) as well as at the end. That is clearly brought out in the documentation I have provided, and you can easily confirm further with web searches. Yes, in very simple, finite games sometimes the payoffs are just shown without mentioning the utility function, but from a deeper knowledge of game theory you will find there is always a utility function. I made it clear by using the term of art utility function, not just utility, to cover games of all kinds.


 * Competitive and cooperative. You are missing the distinction between the general strategy for the game as a whole and the strategy at each point in the game.


 * Incompleteness and imperfection of information are often combined. For example, one player can strategically reveal accurate information in a way that is deceptive, so the situation is both imcomplete and imperfect (in being misleading). But those subjects need further development themselves.


 * You are welcome to proposed a clearer definition of strategy, but the reversion away from my version is not such an improvement.


 * Ludism is just a name some people use for applying game-theoretic ideas to other fields such as philosophy. It is not a kind of game.


 * You will find that every game definition presumes, if it does not make explicit, a field of play. That is just a convenient way to refer to, for example, the "board" in board games, or physical space in evasion games.

I am going to revert, again, the parts to which there have been no objections. As for the rest, I challenge you to learn more about the subject by actually reading standard texts in the field and key research papers, both those cited and those that can easily be found with web searches. Bracton (talk) 15:03, 21 March 2009 (UTC)

Revert #2

 * I'm really uncomfortable with the "One, two, and many-player games" section. I'm glad to see that zero player games are no longer in there.  Where I come from, one player games are simply called "optimizations" rather than games, and decision theory is not a term that gets used at all.  I'm uncomfortable with the use of a webpage on Combinatorial game theory to buttress a claim on this page.  There a a wide gulf between game theory and combinatorial game theory, they are totally different topics, and webpages really aren't satisfying references.  I'm also not totally sure what is meant by "playing against the "board"", it suggests that a "scramble competition" (game against the field) is involved, and I don't think that is what is meant here.  The second reference in this paragraph is another weblink to the table of contents for a conference proceedings volume, containing the paper "Cooperative games with infinite number of players, Projective systems, and Cores".  Note that the paper is on cooperative games.  Cooperative games treat the number of players very differently, and abstractly as coalitions, from non-cooperative games, in which each player is an independent agent.  Like I say, where I come from, a player competing against an infinite number of opponents, would be called a scramble competition.  It's not at all clear to me that Bracton understands these distinctions or not, but his edits blur many important distinctions that really ought not to be blured. Pete.Hurd (talk) 16:45, 21 March 2009 (UTC)


 * As for "You will find that every game definition presumes, if it does not make explicit, a field of play." I think it is OR to insert this explicitly into the formal components of a game. The original version, stating that a game was composed of a set of players, a set of strategies or moves, and a set of payoffs, is how this is typically stated in text books (far more complicated formal definitions certainly do exist, but those three are typical).  I don't know who wrote the original version (my guess would be User:Kzollman or User:trialsanderrors, both academics specializing in game theory).   Bracton states above that he is "mathematician and computer scientist who has worked with game theory professionally," and I'm not suggesting that he's making this up, or that he's asking that we take his statements on faith, but the version he is proposing strikes me to be ideosyncratic, unlike the typical specification presented in standard texts.  If he can provid an explicit reference for this explicit formulation, that we could evaluate, then I'd feel a lot better about it being OR. Pete.Hurd (talk) 16:45, 21 March 2009 (UTC)


 * This is going to take more time than I can devote now, unfortunately. So I won't be able to help as I'd like, or be as circumspect as usual. Bracton, I'm not sure how much of our troubles are in your prose (which is frankly hard to follow) versus in your ideas (which do appear nonstandard). That's why I suggest figuring things out here on the talk page before integrating them. Unfortunately, I'll have to bow out on taking part in that. Various comments:
 * Sorry about the large reversion, but it's par for the course. When a lot of edits get made at once, and it seems like many are problematic, sometimes good ones will get reverted along with bad ones. I tried to fix that with the one/many players bit, but missed the Madison bit. I won't object to your putting that back, at least for now, but would like it to include a reference not to Madison himself, but to someone's pointing out Madison was using game theory.
 * Pete.Hurd, the decision theory/game theory distinction & overlap is pretty standard in economics. Some people talk about decision theory as dealing with "degenerate games" (i.e. one-player games), for instance. I'm not sure the best way to make that clear in the article, but I think it does belong.
 * Combinatorial game theory is technically a part of game theory broadly, but is different enough in focus (and in the people who concentrate on it), that it's very distinct in practice. That's probably also something which should be made clear in the article. I don't think it's appropriate for this article to include many notions from combinatorial game theory, other than to mention it exists, and refer people over there.
 * Bracton, Ludism/Ludology/Ludic studies can mean various things, and much of it isn't game theory.
 * Talking about deceit in games is *at best* a very deep topic, and the way you included it is wrong. Keep in mind that in standard analysis with Nash equilibria, deceit is impossible.
 * I skimmed a couple of your links, and in doing so didn't find anything which satisfied me. If the ideas you're proposing are standard, then you should be able to point to authoritative sources which are quite explicit about them. For instance, I just checked the indices of 7 texts on game theory which I have on my desk at the moment, and none of them includes "field of play".
 * Good luck, guys. C RETOG 8(t/c) 18:26, 21 March 2009 (UTC)

It was a central theme in The Trap, a documentary by famous film maker Adam Curtis
I don't see a section where it could be added though as information. --AaThinker (talk) 18:16, 22 March 2009 (UTC)


 * Probably best not to, per WP:TRIVIA. —David Eppstein (talk) 22:09, 22 March 2009 (UTC)
 * 'In popular media' is common. --AaThinker (talk) 22:19, 22 March 2009 (UTC)
 * Yes. A common way to add listcruft to articles, that generally accumulates large amounts of unsourced original research (aka trivia) until someone rips it out and the cycle can begin anew. —David Eppstein (talk) 22:41, 22 March 2009 (UTC)

Revert #3
This section has gotten too long so I broke it out.

This discussion is made difficult by the apparent lack for both Pete.Hurd and C RETOG 8 of extensive domain knowledge of this topic. The latter reports having some books on the subject on his desk but does not say which ones they are so we can evaluate how well they represent the field. The former shows this with his statement that there is a "wide gulf between game theory and combinatorial game theory", which indicates to me that he doesn't really grasp what game theory, or any mathematical field, is.

A mathematical field like game theory is a body of formal methods of analysis and problem-solving. It falls within the "analysis" side of the "analysis-algebra" divide. Its domain is anything to which those methods may be usefully applied, whether under the same name or not. That is how we get the inclusion of zero-, one- or infinite-player games as branches of the field, or decision theory as a branch that mainly focuses on one-player games. It is not limited to what some author might have focused on in a paper or treatise he wrote. Pete.Hurd would seem to be treating the same methods as different theories if different authors applied them using different terms (even though some other authors did use the same terms). It is not OR to recognize that the method of fluxions applied by Isaac Newton to the motion of celestial bodies involves methods mathematically equivalent to what Gottfried Leibniz called infinitesimals and applied to algebraic functions derived from other phenomena, especially since numerous mathematicians have shown they are equivalent. "Addition" and "sums" are the same method.

The expression "playing against the board" is mainly a figure of speech used in one-player games in which "nature" (the board) may seem like an opposing player. But a lot depends on how the game is defined. Some will prefer to define the rules so that they include the "field of play". Thus, they might define the rules of chess to include a rule that the board is an 8x8 array of alternating colored squares, and another rule that pieces are placed in the middle of the squares. Others may prefer to separate such rules from others that provide that bishops may move any number of squares along a diagonal provided they do not exceed the boundaries of the board, or go beyond a piece of their own color, or do other than take an opposing piece on the diagonal by occupying that square. Either approach can define the game. In a general discussion of games it is better to allow for both.

It is, however, erroneous to say "rules or strategies" in a statement about defining a game. A strategy is not part of the definition. It is something for the players to use, and perhaps to find, but the game is defined by the field, the rules, the players, and the utility functions of each.

Competitive games do not necessarily treat the number of players differently from cooperative games. A game has the number of players it has. Depending on the utility functions of each, they may play competitively or cooperatively or both. There are even utility functions that make the distinction meaningless, with each of several players maximizing his utility in ways that have little effect on the utilities of the others, other perhaps than to require them to use different strategies. Thus, on a roadway driving on one side or the other is the way to get to the end. Choose to drive on the right and opposing traffic will be advised to drive on their right, but it costs the opposition nothing to choose differently (and a lot if they make the wrong choice). Played right it is a positive-sum game, and a cooperative one, but erect a barrier so that neither player can cross over to the other side and it ceases to be cooperative. Then it becomes a game of "playing against the road" to avoid colliding with a barrier or going off the side.

Most equilibria, such as Nash equilibria, are about games with complete and perfect information. That section was about games without one or the other or both. Granted it could be expanded. But what do you think happens in the game of Poker?

Jumping to another section, it is not really meaningful to say the result was derived from the axiom of choice. That axiom is very basic, and can usually be found buried in the proofs of most propositions in set theory, but to single it out as the source of the proposition is mathematically and didactically incompetent. That is like saying the proposition is derived from the definition of a set as a subclass of the Universal Class. Well, yes, trivially, but there is a lot more in between to get to that proposition that is more important.

I will have to break it off for now. Have dinner plans with friends. In the meantime I suggest you really explore the field much more thoroughly. Bracton (talk) 22:48, 21 March 2009 (UTC)


 * I'm sorry that you feel Cretog8 & I are too unschooled and ignorant for you to deal with productively. You claim professional expertise in game theory, but seem unable to provide reliable sources to back up what Cretog8 & I find problematic.  Cretog8 is doing a pHD in this topic, and I know the academic background of the two major writers of the article prior to your edits (User:Kzollman and User:trialsanderrors).  It may be coincidence that I know their credentials, where they did their PhDs & post-docs etc, and I also find all of their edits square neatly with what I know about game theory.  I don't really know your credentials, but that is not important.  What is important is that you don't seem to be able to provide reliable sources to back up the statements that don't square with the article as it was before, or what I know.  I won't put words into Cretog8's mouth, scrolling upwards lets him have his say.


 * I'll admit that "Game theory" does not appear on my business card (I just checked), and I admit to knowing diddly squat about Combinatorial Game Theory (which is outside my research area), but it is my memory that the topic does not appear in any of the game theory texts in my library (unlike Cretog8, I don't have access to my books, they are in my office, and I off on sabbatical). One of the webpages you cite to support your edits here, has under the references section: "David Eppstein  has an excellent page on combinatorial game theory", (I didn't realize David Eppstein did CGT) so I asked User:David Eppstein if he had time to look over this discussion.


 * Now, I think you are totally wrong about many of the things you said above. You havn't done anything to convince me that any of my previous comments was wrong.  I will rebut just one new point here, where you say "Most equilibria, such as Nash equilibria, are about games with complete and perfect information." this just totally wrong.  For some examples of games without perfect information that have Nash Equilibria see: Hurd, 1995, Hurd 1997, Hurd & Enquist, 1998, Enquist, Ghirlanda & Hurd, 1998, Hurd 2006.
 * Pete.Hurd (talk) 06:33, 22 March 2009 (UTC)


 * I believe that classical game theory (linear programming, mixed strategies, Nash equilibria, etc; simultaneity of player moves as in rock-scissors-paper; prisoner's dilemma issues and payoff matrices) is almost completely disjoint from combinatorial game theory (players move one at a time like chess or go, everything is knowable in principle ahead of time so there is no point in mixed strategies, nim-sums, temperature theory, octal games, misere play, etc). They're both motivated by the fact that humans play games, and I've seen papers and grant proposals that confuse the two because they have similar names, but beyond that superficial similarity they go in completely different directions. But I don't know what this distinction adds to this discussion, how it relates to the context of the discussion or to Bracton's tl;dr comment which seems to be entirely about classical game theory as near as I can tell. —David Eppstein (talk) 07:48, 22 March 2009 (UTC)
 * Yeah, a couple of things:
 * This is the most important one: It's true that it's not original research to talk about the underlying commonalities between Newton's and Leibniz's approach. But that's because it's been done before. In fact it's been a commonplace for well over two centuries, meaning (IMO) that it doesn't even need a citation.  Finding commonalities that are not commonplaces, if not cited, absolutely is is is original research and is rigorously banned' from WP.  In my opinion this is one of the most pernicious sorts of OR and one that we have to be extremely vigilant about excluding.
 * Actually point 1 is the main one. I have more to say but I don't want to dilute the most important point. --Trovatore (talk) 10:19, 22 March 2009 (UTC)

I just want to note my support for Pete Hurd's and CRETOG8's version here. In the most charitable interpretation Bracton is introducing some highly peculiar terminology to replace more commonly used terms. For example, at least in economics (and in mathematics too - from what I've read) "field of play" is never used, but rather "strategy space". Likewise there is no "board" (which has confusing connotations in any case as it makes people think of board games) but rather a player called "nature".radek (talk) 06:41, 23 March 2009 (UTC)

Books 0pen for inspection

 * Roger B. Myerson, Game Theory: Analysis of Conflict, Harvard University Press (September 15, 1997). ISBN 0674341163
 * Eric Rasmusen, Games and Information: An Introduction to Game Theory, Wiley-Blackwell; 4 edition (December 5, 2006). ISBN 1405136669
 * Harold William Kuhn, ed., Classics in Game Theory, Princeton University Press (January 17, 1997). ISBN 0691011923
 * James N. Webb, Game Theory: Decisions, Interaction and Evolution, Springer; 1 edition (October 31, 2006). ISBN 1846284236
 * Fernando Vega-Redondo, Economics and the Theory of Games, Cambridge University Press (July 28, 2003). ISBN 0521775906
 * Ein-Ya Gura and Michael Maschler, Insights into Game Theory: An Alternative Mathematical Experience, Cambridge University Press; 1 edition (December 15, 2008). ISBN 052187422X
 * Rufus Isaacs, Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Dover Publications (January 20, 1999). ISBN 0486406822
 * Steve Alpern and Shmuel Gal, The Theory of Search Games and Rendezvous, Springer; 1 edition (November 2002). ISBN 0792374681
 * Robert Axelrod, The Evolution of Cooperation, Basic Books; Revised edition (December 4, 2006). ISBN 0465005640
 * Josef Hofbauer and Karl Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press (June 13, 1998). ISBN 052162570X
 * Jozsef Beck, Combinatorial Games: Tic-Tac-Toe Theory, Cambridge University Press (April 21, 2008). ISBN 0521461006
 * The Beck book is neither about classical game theory nor combinatorial game theory, although it has a connection to the actual games of tic-tac-toe and Set. It is, rather, pure combinatorics. —David Eppstein (talk) 06:54, 23 March 2009 (UTC)

Revert #4
Just a couple of points for now. I am adding a section of books on the subject with links to amazon.com that allow one to open and read at least the table of contents, usually the first few pages, where definitions are often found, and a terminological basis is often laid. The list is not complete, and I request patience as more are added.

One of the things one can notice is that different authors with different focus and backgrounds will often use quite different terms. For example, some will use the term "strategy space" to cover everything that others break out as "field of play", "game rules", and "moves" or "actions". Some speak in terms of utility theory, utility functions or payoff functions, and others just in terms of payoffs. Some use the term strategy to include the definitions of all the moves, and define the only choice as a choice of a strategy after which all the moves are both defined and made as the game unfolds. That can work as a formalism but is not entirely useful for a full range of games in which the strategy is something to be guessed at from one move to the next.

I can recognize Pete.Hurd as a biologist from the way he approaches the subject. (Incidentally, it does not really work to cite to abstracts of papers one cannot read without paying a fee.) My approach is that of a mathematician and computer scientist who has actually written programs involving games of all kinds, so I see a body of analytic methods that translates into a lot of reusable code that can be applied as well to classical games as to differential, combinatoric, or whatever.

As a biologist I'm sure you can appreciate the problem of using a textbook with the title of a broad subject to define the coundaries of that field when the author, instead of providing a comprehensive treatment, provides an introductory treatment which focuses on some branch of it. It would be a similar problem if one described the boundaries of "algebra" with only reference to a high-school textbook, then wondered what to do with a text by the title "Algebra" written by MacLane and Birkhoff, about category theory.

As for Nash equilibria, note my use of the term "most". Of course it is possible to speak abstractly of such equilibria for games of incomplete or imnperfect information, but seldom usefully. Such games usually require the use of statistical methods and the appropriate extension of a concept like the Nash equilibrium to something like a Bayes-Nash equilibrium. Bracton (talk) 06:26, 23 March 2009 (UTC)


 * I see again the careless use of the word "combinatoric". I repeat: Combinatorial Game Theory is not the same as the subject of this article and should not be mixed into it. Both subjects are motivated by human game playing, but then, so was a lot of early probability theory, and yet we succeed in keeping game theory and probability theory as distinct subjects. And computer game programming (which you seem to be largely focused on) is yet again a different subject: it is motivated by the minimax theory of zero-sum games but the inability to explore the complete game tree of most serious games, the need to make up "payoffs" out of whole cloth at non-terminal positions, and the nonexistence of mixed strategies gives it quite a different flavor. This is not an article about game programming and should not be reworked into one. By the way, please do not link to Amazon for the books: just supply the ISBN, and the wiki software will do the rest. (And that goes double if you hope to link to Amazon using associate links; you are likely to get banned as a spammer for doing that.) If we're showing off our expertise, by the way: I have written and taught computer game programming, written and published a very small amount of combinatorial game theory research, maintain web page concerning both combinatorial game theory and complexity-theoretic aspects of human games, and participated in the peer-review of but not myself produced research in the complexity-theoretic aspects of classical game theory (e.g. how hard is it to find a Nash equilibrium, given as input a payoff matrix?). —David Eppstein (talk) 06:51, 23 March 2009 (UTC)
 * Going off on a slight tangent here: You've done research in "combinatorial game theory", so hopefully you have a definition for the term.  My understanding of it is that it's the theory that derives from the project started by On Numbers and Games, by Conway, therefore including the surreal numbers, but that it has essentially nothing to do with the theory of winning real-world games.  Does that jibe with your understanding? --Trovatore (talk) 07:50, 23 March 2009 (UTC)


 * "Of course it is possible to speak abstractly of such equilibria for games of incomplete or imnperfect information, but seldom usefully. Such games usually require the use of statistical methods and" your comments seem more and more ill-informed with each successive missive. Aside from the fact that you continue to glibly lump together the totally distinct topics of imperfect information and complete information, the idea that one needs statistical methods to deal with games of imperfect information is ludicrous...  I really think this is a waste of my time.  You are consistent wrong on every substantive point you make, have yet to produce a single reliable source to back up your wild statements, and rely on your vague claims of professional expertise and these lengthy confabulations. I, for one, am done feeding. Pete.Hurd (talk) 07:56, 23 March 2009 (UTC)

What we have here is disagreement on the proper boundaries of this article. What many of you seem to be trying to do is to confine this article, entitled Game theory to classical game theory, for which there is not presently an article, and to assert that combinatorial game theory is an entirely different field with no commonalities other than the name. I suppose we could rename the article to Classical game theory and make this article, Game theory, a disambiguation or umbrella page. However, for many researchers "game theory" is a body of analytic methods that are useful for many kinds of things that we can characterize usefully as "games". Having done work on most of the branches of the field and found essentially equivalent methods usable in all of them, I don't accept that treatment of the subject.

Consider, for example, this from Wolfram Mathworld;


 * Game theory has two distinct branches: combinatorial game theory and classical game theory.

David K. Levine, Department of Economics, UCLA, says in What is Game Theory?:


 * In addition to game theory, economic theory has three other main branches: decision theory, general equilibrium theory and mechanism design theory. All are closely connected to game theory.


 * Decision theory can be viewed as a theory of one person games, or a game of a single player against nature. ...


 * General equilibrium theory can be viewed as a specialized branch of game theory that deals with trade and production, and typically with a relatively large number of individual consumers and producers. ...


 * Mechanism design theory differs from game theory in that game theory takes the rules of the game as given, while mechanism design theory asks about the consequences of different types of rules. Naturally this relies heavily on game theory. ...

Mark Voorneveld in The possibility of impossible stairways and greener grass:


 * In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function.

From the abstract of Game theory by James Webb:


 * Covering the basic ideas of decision theory, classical game theory, and evolutionary game theory, this book provides an unified account of three types of decision problem.

Clearly, many writers are using the term "game theory" more broadly than some editors seem to want to make this article cover.

We also have such branches as evolutionary game theory, Quantum game theory. Bracton (talk) 05:42, 24 March 2009 (UTC)


 * This article uses 'game theory' in the most common, most usual sense. I suggest creating categories for more esoteric usages and placing these in the respective articles.radek (talk) 07:37, 24 March 2009 (UTC)


 * First of all, while MathWorld is occasionally a useful resource, its treatment of terminology and nomenclature is — to put this extremely politely — "idiosyncratic". It is not a reliable source for this sort of thing.
 * In the standard usage of experts in the field, what is typically called "game theory" does not include combinatorial game theory. Conway, the founder of combinatorial game theory, to my knowledge never claimed to be making a contribution to game theory.  And the concerns of combinatorial game theory are quite distinct from those of game theory; as far as I can tell they have almost no overlap.
 * If it ever becomes standard to treat game theory and combinatorial game theory as branches of the same subject, then the article can be amended to reflect that. Right now it is not.  For good reason, in my opinion, though my opinion on that, and your opinion on it, are both irrelevant.  --Trovatore (talk) 07:38, 24 March 2009 (UTC)


 * Who is deciding here what is "standard" and what is idiosyncratic". and using what method? Can you cite to leading experts who say what is "standard"? Or are you merely conveying your impression from reading a few sources? I submit that some writers are referring to classical game theory without using the qualifier "classical", and others are not. As editors we need to recognize such abbreviated usage. Bracton (talk) 16:02, 24 March 2009 (UTC)

That the article was intended by many its editors to be about the "umbrella" term and not just about classical game theory is made clear in the opening paragraph and by the inclusion in the article of many branches and subtopics, many of which are not "classical". The problem seems to come in the "Representation" section, which presents only one version of the classical branch, as though that were the only version and as though only the classical branch counts. I suggest that the section be clarified by adding the word "classical" before "game theory" wherever it appears. Bracton (talk) 16:02, 24 March 2009 (UTC)
 * The following standard game theory text books have no section on combinatorial game theory, and contain no indenx entry for the term combinatorial game theory:
 * Fudenberg & Tirole: Game theory
 * Osborne & Rubenstein: A course in game theory
 * Gibbons: A primer in game theory
 * Kreps: Game theory and economic modelling
 * Gintis: Game theory evolving
 * Also, the following books (that I wouldn't consider to be textbook, but may be considered influential texts in the history of game theory) also contain no section on combinatorial games, or entry in the index for the term
 * Dresher: The mathematics of games of strategy
 * Luce & Raiffa: Games & decisions
 * Williams: The compleat strategyst
 * Rapoport: Two-person game theory the essential ideas
 * In summary, in the real world, "game theory" is taken to mean game theory and "combinatorial game theory" is taken to mean combinatorial game theory. Textbooks on game theory are called textbooks on "game theory" not textbooks on "classical game theory".  Wikipedia ought to reflect how people use the terms in the real world, and not invent neologisms to conform with the misconceptions of the confused and uninformed. Pete.Hurd (talk) 16:47, 24 March 2009 (UTC)

Boundaries
From the Stanford Encyclopedia of Philosophy we get:


 * Game theory is the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences (or utilities) of those players...

From the Britannica Online Encyclopedia we get a similarly broad definition:


 * Branch of applied mathematics devised to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests.

From Lycos retriever we get:


 * Game Theory is the study of situations where multiple decision-makers influence one another.

From a course taught at North Carolina State University we get:


 * Game theory is a branch of logic which deals with cooperation and conflict in the context of negotiations and payoffs. The theory of games can elucidate the incentive conditions required for cooperation, can aid understanding of strategic decisions of nations or actors in conflict, and can help in the development of models of bargaining and deterrence.

From WordIQ we get:


 * Game theory is a branch of mathematics that uses models to study interactions with formalised incentive structures ("games").

From MoneyTerms we get:


 * Game theory is a branch of mathematics that provides a framework for analysing what choices rational individuals will make, when the outcome ("payoff") depends on both their choice and the choices of other "players".

From Science Daily we get:


 * Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns.

From ThinkQuest we get:


 * Game theory is the mathematical analysis of a conflict of interest to find optimal choices that will lead to a desired outcome under given conditions.

From StratGaming we get:


 * ...game theory is the science of strategic thinking. A branch of applied mathematics and economics, game theory is used to analyze interactive situations with two or more “players” whose choices are interdependent. What one does affects what another will want to do, and vice versa, and the combination of their choices determines their respective “payoffs.”

From GameTheory.net we get:


 * Game theory is a branch of applied mathematics that studies "rational behavior in interactive or interdependent situations."

From Theodore L. Turocy & Bernhard von Stengel (London School of Economics) Game Theory:


 * The object of study in game theory is the game, which is a formal model of an interactive situation. It typically involves several players; a game with only one player is usually called a decision problem. The formal deﬁnition lays out the players, their preferences, their information, the strategic actions available to them, and how these inﬂuence the outcome.

This last one would seem to indicate the most comprehensive definition for our use in the article.

But the Principia Cybernetica only considers non-cooperative situations:


 * Game theory is a branch of mathematical analysis developed to study decision making in conflict situations.

Likewise Wolfram MathWorld ignores cooperative games:


 * Game theory is a branch of mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests).

Branches
David K Levine divides game theory into two branches What is Game Theory?:


 * There are two main branches of game theory: cooperative and noncooperative game theory.

From StratGaming we get:


 * Leonid Hurwicz, Eric Maskin and Roger Myerson won the 2007 Nobel prize for their work in mechanism design theory, a branch of game theory that extends the application of game theory to how different types of rules, or institutions, align individual incentives with overall social goals.

From Game Studies we get:


 * ... game theory [is] the systematic study of the relationship between rules, choice and outcome in competitive situations. Two main branches exist. Analytical game theory is the analysis of games played by non-empirical players; ... On the other hand, behavioural game theory is the study of actual human players as they are confronted with precisely defined games.

So we have many credible sources that define the boundaries broadly and the branches inclusively. You can list books selected for the focus they take, but others can find other writings with a different focus. We as editors should report on this diversity and not try to impose our own views on how the boundaries or branches should be restricted. Bracton (talk) 02:33, 25 March 2009 (UTC)

James Madison
The source says "His answers to this rhetorical question took the form of what we now recognize as game theory, a mode of analysis that did not formally exist at the time." I don't see how this amounts to a contribution to game theory (per Bracton's edit summary), especially if 2007 is the first time anyone characterized his reasoning as being game theoretical (I can't find anyone making that point other than the History Now article Bracton cites). Pete.Hurd (talk) 23:56, 26 March 2009 (UTC)


 * Jack Rakove is a leading constitutional historian and is regarded as a leading expert on James Madison and his writings. Madison is not less worthy of regard than the other writers who provided contributions to the concept antecedent to John von Neumann and Oskar Morgenstern. Bracton (talk) 03:24, 27 March 2009 (UTC)
 * Except that von Neumann & Morgenstern originated the theory of games, and James Madison sorta thought something through in a clever way, rather like Jean-Jacques Rousseau did with the Stag hunt. Can you find anyone (other than you) that has suggested that Madison's thought had *any influence on* the development of game theory, preferably in a reliable source?  Jack Rakove doesn't claim that he influenced game theory. Pete.Hurd (talk) 03:33, 27 March 2009 (UTC)
 * That is not essential to establishing historical antecedents. We have no evidence before us that any of the others listed in this section before von Neumann influenced the development, except perhaps in some general cultural way. They appear to be largely isolated developments. In particular, I doubt any of them influenced von Neumann. What we have is historians discovering antecedents long after the fact. Such developments need not form a chain of influence to be historic. Bracton (talk) 04:12, 27 March 2009 (UTC)


 * Cournot's analysis is still used today, and frames a lot of similar game theory with continuous strategies. Waldegrave may or may not have influenced later developments, but is now standardly recognized in histories of game theory. One f'rinstance.
 * As an aside, what would be useful to add to the history is (at least) Bernoulli, Bertrand, and (more) Borel.
 * But I agree Madison doesn't belong. C RETOG 8(t/c) 07:48, 27 March 2009 (UTC)


 * Also discussed in Iain McLean, "Before and after Publius: the sources and influence of Madison’ political thought", Paper for conference on James Madison’ 250th Birthday, Department of Political Science, University of California, San Diego, March 2001. Link

Bracton (talk) 04:05, 27 March 2009 (UTC)