It’s still not ‘classical game theory’

Reflections, fixes, responses, valorization, suggestions

(~4500 words, reading time: 13-18 minutes)

Background

Before I posted ‘It’s not ‘classical game theory’’ I’d sent it out to two friends and colleagues to take a look at it. I don’t usually do this, in fact I don’t think I ever had a post “peer-reviewed” before by people other than Nóra. Here though, I’d wanted to get their opinions because (a) I consider both to be greater experts in evolutionary game theory than myself and thus, they might have had insights on the value of the term ‘classical’, and (b) the post touched on discussions we had before on using that very term.

Unfortunately, through no fault of theirs, the two reviewers, R1 and R2 did not really get a chance to improve the post. R1 was probably not super interested (for which I don’t blame them at all) but warned me jokingly that certain people might find it upsetting. R2 told me they’d get back to me later and at the time looked to me that they weren’t super interested either (again, I wouldn't have blamed them for this). Nóra, the hero that she is, did read it though. To my surprise, she liked it a lot, even though she doesn’t care about the argument whatsoever. So with that feedback, on March 12, roughly a week after I sent it out to my reviewers, I posted the thing.

R2’s super detailed feedback came in about an hour or so later. And it wasn't because they saw it posted; they probably had time for it over the weekend, I just wasn’t patient enough. I’d wanted to get it out of my system and online before yet another colleague from a different country visited me in Toulouse, because I thought I might talk about it with them too. The good news for me was that R2 had nothing against the main thesis and they didn’t offer any insights on why the term ’classical’ should be saved. The bad news was that, despite this, they really didn’t like the post. I’ll get more detailed alter but the short version is that they felt quite strongly that I was picking unnecessary fights and that evo game theory didn’t get depicted fairly.

I was (and I remain) super grateful for the feedback. Yet, I wasn’t at the time convinced, and was put in a mindset where I felt like I needed to defend the text as it was (since it went “out” already and almost everyone who would ever read it probably has), so we kind of agreed to disagree. Still, I was for sure hoping for a better reception from R2 and really regretting my impatience. I wouldn’t have sent out the post as it if I’d known they were going to dislike it as much as they did. So, I’m addressing their points now despite still not agreeing with them because (a) I think whatever audience is here should be know how a proper evo game theorist sees the post and (b) addressing them helps clarify a few points that were probably not expressed well in the original piece.

There are points, pertaining to my presentation of the history of evolutionary game theory that I won't address. My presentation is VERY subjective, was contested HOTLY, and I'm ready to defend why I wrote why I did (including the memes that are attempted jokes) but it's ultimately not super relevant for the main thesis. In any case, you should know that there was disagreement.

Meanwhile, on the public forums that I’d posted it (Zuckerbook and Musker), the reception was really positive! The comments and the small discussions that followed on these forums were warm and satisfying. The reactions really made me think that, even if the post was less than perfect, I had (1) captured a sentiment held by at least a measurable subset of my peers, and (2) was at least somewhat convincing to people who did not hold this sentiment up to that point. The community whose members gave the positive reactions was the exact same one that I secretly targeted, some of the foremost experts of both evo and non-coop game theory, the same community where I first met my current PI (though she herself did not comment, I was really contemplating to make her my R3 but then decided she’s probably too busy for such a thing).

In addition to this, there was some cool polite disagreement online as well. These claimed that I missed the mark in my definition of ‘classical game theory’. Then, when my visitor got here, they expressed an identical point. It’s a really good point and a big missing ingredient from the previous post, and I wanted to react to it.

So, these are the two objectives for today. I spice it up with a point on the value of this discussion and some ideas for transitioning away from the use of 'classical game theory'. On we go.

Response to R2

The main point of criticism made by R2 was that my representation of evo game theory is too simplistic. The only evo game theory model I show in any detail is a small (2-by-2), static game in normal form. What R2 I think felt important to point out in reaction to this, is that the basic model is not representative of what evo game theory is about, what the field is and should be known for, or what it is capable of doing.

And although I think these models already capture a LOT of what evo game theory is about (in terms of raw quantity of research output at least), and I think they’re extremely valuable, and indeed representative of the philosophy, the machinery, and the potential in applications, to the point that one can indeed judge a lot of the merit and limit of evo game theory just by looking at these models, I see completely why R2 felt they needed make this point. No self-respecting scientist or science enthusiast should read my post and consider themselves educated in terms of the current state of evo game theory based on it.

Whether or not small, static evo games are representative of evo game theory is not needed for my main thesis. I used them to showcase why evo game theory is indeed separate from (“mainstream”) non-cooperative game theory. Yet, there’s so much more that could have been said to highlight why it is distinct. My point in the original post should be taken as “even in the domain where non-cooperative and evo game theory are the most similar (=static games in normal form), the evo approach is distinct”.

The other point, which was more unfortunate, was me saying that “evo game theory is a special case of non-cooperative game theory, mathematically speaking”. There are many layers of this: The first is that in the post I don’t really attempt to show this, only for the featured, most simplistic class of evo games. The second is that saying “X is a special case of Y” is sometimes used as math speak for “it’s not new/interesting/valuable”, something I do not at all think is the case for evo game theory, and was hoping that the readers understand from context. In any case, I should have been more mindful. Mix in some disciplinary rivalries/prejudice and you have a conversation on your hands about whether or not economists know everything there is to be known about the world -- a bad way to spend one’s limited time on the Earth. The third is that even when I add the “mathematically speaking” to it, I may sound like I assign an identity to evo game theory that its practitioners don’t feel their own. Worse, it may make it seem like I want to colonize evo game theory for non-cooperative game theory or tell it to conform to something it may not feel like it wants to or needs to. For this, my apologies are hereby offered.

If I have a rebuttal on the first point, it is that the core of my claim is that there is an extremely close relatedness between non-cooperative game theory models and static evo games as put forth in evo game theory’s earliest iterations. In particular, the mathematical relatedness is much higher than the one between non-cooperative and cooperative. In my mind, a lot of evo game theory has its common root in static evo games in normal form which at least in part inherit this relatedness too. To make this point, I didn’t feel a need to give a holistic picture of modern evo game theory -- nor is this something I’m particularly well-suited to do.

While no one commented on this, this disclaimer also invites the parallel one: I didn’t give a holistic picture of modern non-cooperative or cooperative game theory either. In fact, if anything, I undersold those just as much as evo game theory, yet no one seemed to take issue with that (and my network neighborhood is populated with “non-coop people” and “coop people” more so than “evo people”). Frankly, I don’t see a holistic picture for either side to be necessary for my main thesis (=“don’t call it ‘classical’”), since (a) most people who insist on the word ‘classical’ aren’t doing it from a holistic point of view of cooperative and non-cooperative game theory, and (b) I don’t see how a more detailed look at the fields will make the term ‘classical game theory’ worthy to be used. If someone does, they’re welcome to show me how.

‘Classical’ meaning ‘coop’ and ‘non-coop’ together and why that’s wrong

The criticism that actually challenged the main thesis of my original post came from my visitor and a commenter online. They made me realize that ‘classical game theory’ can also be used differently than I had framed it in the first post. The next part of this post will be me reacting to this counterpoint because I think is worth addressing. So, if you think you’ve already had your fill in me ranting against the term ‘classical game theory’, this is the last call to get off.

The assertion is that ‘classical game theory’ means “non-evolutionary game theory”, including both cooperative and non-cooperative game theory. If true, this assertion completely destroys my first post with facts and logic: my call to use ‘non-cooperative’ instead of ‘classical’ can’t be valid if ‘cooperative’ is included in it.

The nature of this assertion, however, really touches on the crux of my problems with the term ‘classical’. While ‘non-cooperative’ and ‘cooperative’ game theory are very well defined and there is a pretty good consensus among fields’ practitioners on what they are, where they intersect, and where they end, this clearly can't hold for ‘classical game theory’, for it has no practitioners. It's not a recognized field by the people that some evo game theorists claim to be developing it.

In any case, several parallel definitions exist for ‘classical game theory’ that to me always felt informal and vague. Since it is invariably defined in relation to evolutionary game theory, in fact, its raison d’être is to highlight how evolutionary game theory is meant to be distinct from it, I show them in pairs. I think this is an exhaustive list but if you have a better one that is substantially different, don’t hesitate to let me know.

Classical game theory is… / whereas evolutionary game theory is…

a) where players are rational / where players are subject to evolution by natural selection,

b) where players’ strategies are chosen / where players’ strategies are inherited,

c) where we use concepts such as the Nash equilibrium / where we use concepts such as the ESS,

d) where we study economic interactions / where we study biological interactions,

e) everything in game theory that is not evolutionary game theory / everything in game theory that is not classical game theory.

Nothing seems to stop people from combining these definitions as many of them are either closely related or at least non-exclusive, but I’ll get to that later. In fact, in my first post, I sort of lumped (a), (b), and (c) together and wrote it with these definitions in mind. The main point was that these are attempting to describe non-cooperative game theory often without the user knowing it.

Definition (d), I didn’t touch on last time because I consider game theory to be first and foremost a mathematical field and the math doesn’t really care about the field of application (meaning that abstract mathematical objects, even when they prove useful, are just abstract mathematical objects, occupying a different realm of existence than the applications of our material world), but I’ll do so here in a bit.

Finally, (e) is the new one proposed by my visitor and a commenter online and the main one I’ll deconstruct here. But for reasons of completeness and pedagogy, I’d like to go through (a)-(e) and see which of cooperative and non-cooperative game theory they define.

The easiest to tackle is (b). Most cooperative games don’t have strategies and, particularly, the most studied cooperative framework, transferable utility (TU-)games, doesn’t have them. Not just as in “there is no mathematical object in the model called a ‘strategy’”, but also as in “most of the time, it doesn’t make much sense to talk about strategies at all”. The central issue in this framework isn’t “what the players do” but “what they are supposed to get”. So, if you use ‘classical game theory’ in the sense of (b), then I think your definition exactly defines non-cooperative game theory.

Next, (a), perhaps the most common one, the one by Maynard Smith. A LOT of cooperative game theory simply doesn’t care about the rationality of its players. With no strategies to pick from generally, players being rational would leave them very little to be rational about. The two most studied solution concepts in TU-games are the core (and variants) and the Shapley value (and variants). While the former may be interpretable as a set of outcomes of some payoff-maximization behavior, the latter doesn’t care about rational choice whatsoever. Even if we stick to the core, a lot of the time it equals the empty set. Then, when you want the game to be about rationality, you’re left with nothing. When, instead of individuals, a cooperative game’s players are network nodes, political parties, or features in an ML algorithm, rationality isn’t even in the vocabulary of the application. A lot of non-cooperative game theory cares about rationality, so definition (a) is also one that fits non-cooperative game theory (with the caveat that there are much better definitions than this) but doesn’t fit cooperative game theory.

I tackled definition (c) explicitly in the previous post. Here I only add that the Nash equilibrium isn’t defined for most cooperative games, so this also defines non-coop (again, not the best way to define it) but not coop.

Definition (d) is my least favorite. A lot of brilliant scientists have successfully shown that economic methods (feat. non-cooperative game theory) are useful in biology and vice versa (feat. evo game theory). They did so by taking risks and facing pushback from the more orthodox practitioners of their discipline. Interestingly, I haven’t seen any cooperative game theory in biology (yet!) so, in a twist, this definition actually comes closer to defining cooperative game theory than anything else.

So, finally take definition (e), where classical and evolutionary are only defined in relation to each other. To produce something like an absolute reference point, the next natural step is to define evo game theory as “games where strategies are inherited traits and the players are subject to evolution by natural selection” (notice the subtle incorporation of definition (b) and the ever-present lack of formalism). Then, ‘classical game theory’ can be defined as any field of game theory that doesn’t satisfy this, amounting to cooperative and (most of) non-cooperative game theory together.

I’ll be the first to admit that this definition of classical game theory is invulnerable to logic. But just because you can, doesn’t mean you should define classical game theory this way. The value and necessity of this definition is dubious at best, while it is fraught with problems. It’s akin to me saying that all European countries other than Hungary are now ‘Western Europe’. It’s not only pointless piece of nomenclature, it violates existing divisions of Europe, geographic logic, and bestows an identity to all non-Hungarian Eastern European countries that they probably don’t feel their own.

But even if people think it’s necessary, I question whether this is indeed what most users think of when they use the words ‘classical game theory’. In my experience, users of the term either don’t know or don’t care what subfields of game theory they’re grouping together, and in fact, most existing uses of the term can indeed be replaced by ‘non-cooperative game theory’ without any cost in precision to the intended point of the user.

When someone writes that, within the context of their model, “classical game theory would prescribe X”, or “classical game theory would imply X”, or “classical game theory would predict X” (cringe), they tend to mean that, “were this a non-cooperative game, X would be its Nash equilibrium”. When they introduce the novelty offered by the evolutionary approach in a paper introduction or a grant proposal and write “as opposed to classical game theory, in evolutionary games strategies are inherited and not chosen”, they may claim to be using it in the sense of definition (e) to hide behind its shield of logical invulnerability, but in reality, they’re using definition (b).

The only time that definition (e) is used consistently is when the user declares “I’m not interested in classical game theory”, which, given that the user uses an othering term that the field of their disinterest doesn’t recognize at all, is kind of a tautology. An ironic variant of this is when a user whips out the positive version “I’m interested in classical game theory”, which, instead of a tautology, becomes an oxymoron. Hopefully, that interest transforms into an interest in non-cooperative, or indeed, cooperative game theory, upon further exposure.

To illustrate how difficult it is to use (e) in practice, here’s probably by far the most honest and complete definition of “classical game theory” that I’ve seen in action, presented by my visitor on March 14 (I’ve deleted some accompanying figures to simplify the slide, but didn’t add anything). As the slide’s heading says, it was “Inspired by Péter Bayer”, a welcome and only slightly tongue-in-cheek reference to my post and the fact that we’ve disagreed on this before. I’ll treat this as an invitation to dissect it.

The slide’s definition is a combination of (a), (b), and (e), borrowing the concreteness of (a) and (b) and the logical invulnerability of (e). But the combination not only unites the strengths of the three definitions, it unites their weaknesses too. There isn’t much “rational decision making” or “choosing strategies” going on in most of cooperative game theory, so the “axiomatic approach” of Nash 1953 and the tons of literature it inspired just sort of hangs there, surprised to be invited to the party. On the two endpoints of the arrow sit two models that are very similar to each other (the static non-cooperative games of Nash 1951 and the static evolutionary games of Maynard Smith and Price), while on the left side of the arrow sit the earliest versions of non-cooperative and cooperative games which went in really different directions from each other (with poor “lapsed Nazi” Freiherr von Stackelberg probably being very surprised in the beyond to be called a game theorist though he probably has other concerns regarding his posthumous image).

To be clear, this isn’t a knock on my visitor, whom I love and respect dearly regardless of whether we’ll ever come to agree on this. I’ll probably never do anything in science that can hold a candle to what they have done already. But even great people can improve and I think not using ‘classical game theory’ would be an improvement in precision on this particular point. Whether that also leads to a payoff-improvement for them in the short run, I cannot judge.

Just semantics? On the value of this discussion

A recurring theme in the discussions that I got myself into, regardless of whether people agreed with me or not, was pointing out that this is “just a semantics question”. On which I fully agree. This is no more and no less than me wanting to discuss semantics. But of course, this point isn’t really trying to qualify the nature of the discussion, but rather, it is expressing judgement on its importance. Specifically, that it being a semantics discussion means the discussion is not important and I’m making a fuss about nothing.

On that I wildly disagree. But first, if this is just a semantics discussion and semantics questions aren’t important, then why do people engage with it at all, sometimes passionately so? The people to whom this was “just semantics” expressed pretty strong opinions at times. If it’s not important to you, why not just let me have this one?

To answer my own question, this is because of course semantics are important! Why people pretend they’re not is a puzzle to me, but somewhere deep down we all know they’re important. Some of the most important discussions we have are semantics discussions! What is marriage? What is race? What is gender? What is equality? What are rights? What actions constitute violations of rights? All semantics.

Whether it is ‘migrant’ or ‘refugee’ is a semantics debate. Whether it is ‘Covid-19’ or ‘Chinese Virus’ is a semantics debate. Whether it is ‘Russian invasion of Ukraine’ or ‘Special Military Operation’ is a semantics debate. Whether it is ‘telling people how it is’ or ‘being hurtful, inconsiderate, and stupid’ is a semantics debate. Mastering the meaning and perception of language whether it is with the intent to advance oneself or advance some larger subset of society, is literally an exercise in semantics. There is no aspect of social life where this is not incredibly useful. If language is used stupidly, it is inefficient. If it is used selfishly, it is dangerous. Kind of like science.

Now I’ll be the first to admit that ‘classic’ vs ‘non-cooperative’ is not (supposed to be) a dramatic discussion. Yet, the value of having this discussion is the same as the value of having semantics discussions in general. Even if we don’t agree at the end, the discussion makes us more aware and perhaps more careful users of language. Exempli gratia, whether or not your use of the term ‘classical game theory’ changes after reading my posts, you now know why some game theorists (perhaps not so few) won’t like it or will be puzzled by it.

Specifically, for the discussion on the topic at hand, ‘classical game theory’, this may be a niche topic for a niche audience, but that niche happens to be a scientific one and the precision of scientific language is crucial! That’s why we have science, its raison d’être! Whether we’re creating the optimal nomenclature that help our own fields grow, streamlining access between fields, or just trying to find a common language to educate or to inform policy, I think the term ‘classical game theory’ is suboptimal, if not downright harmful, and should be avoided.

Of course, resolving this alone won’t fix all the communication issues between non-cooperative and evo game theory, but you have to start somewhere.

How to make the change

Like any coordination problem, changing the word ‘classical’ to ‘non-cooperative’ comes with short-run costs to the field that uses it. If you make a pitch to evo game theorists, or just to people who know this language but have no idea what ‘non-cooperative’ means in this context, I can totally see why you’d prefer a familiar term over a precise one. While I’d argue that the short-run costs are far outweighed by the long-run benefits, we are only human and our time horizons are short. In academia, time horizons are perhaps even shorter. If there’s a position you want to get, a grant that funds you and your lab, a presentation that can get you noticed, you’re not going to take chances.

I have two solutions on how to minimize these short-run costs while making changes towards precise use of game theory terminology. The first I simply stole from R1: use ‘classical’ and ‘non-cooperative’ together. Their preferred form is ‘classical non-cooperative game theory’. Depending on your audience or your taste, you could even put one of those in brackets and/or quotation marks. My version would be ‘non-cooperative game theory (a.k.a., “classical game theory”)’. It’s a mouthful for sure, but just with a little effort you can make a small step towards a better world.

The second is the one I use in my talks. If I’m not afraid of backlash and want my audience to learn, I just say “non-cooperative game theory” and say to them that some of them may erroneously know it as “classical game theory”. If this is infeasible, I avoid the term ‘non-cooperative’ altogether because its literal meaning might trigger associations I don’t want to trigger in my audience (e.g., sentences like “But I want to study the evolution of cooperation so I can’t use non-cooperative game theory!”). Instead, when I introduce my game, I use the term “(normal-form) game” to describe it. “Normal-form” is a precise game theoretical term and neutral in regards of associations. Yet, anyone looking it up will immediately find themselves in “non-cooperative land”, exactly where I want them to find themselves in. Being an abstract mathematical object, a normal-form game by itself does not care about how players end up with their strategies and what the nature of the rewards is. Instead, they have a “classical” interpretation (paying my respects to Maynard Smith and the nomenclature he inspired) where strategies are chosen, and an “evolutionary” interpretation where strategies are inherited.

Let me thank all of my friends who have discussed with me on this: you’re not named in the post because I don’t want anyone else’s reputations be affected, but you know who you are and you (should) know that I value you greatly. As always, if you have a comment, correction, or question, or just want to further the debate, please feel free to reach out. I do enjoy discussing the topic in good spirits, independent of what side of the debate you are on, but I probably have said all I have to say on it, and more.

I’ll let Charles Dickens say my goodbyes.

“I see a beautiful [field of evo game theory] and a brilliant people rising from this abyss [of a fixation on classical game theory] and, in their struggles to be truly free, in their triumphs and defeats, through long years to come, I see the evil of this time and of the previous time of which this is the natural birth, gradually making expiation for itself and wearing out.

I see the lives for which I lay down my [mental health], peaceful, useful, prosperous, and happy in that [topic] which I shall see no more. I see that I hold a sanctuary in their hearts [though perhaps not always in a good way], and in the hearts of their descendants, generations hence. It is a far, far better thing that I do [on this blog], than I have ever done [the competition isn’t that strong]; it is a far, far better rest that I go to than I have ever known.”

April 29, 2023

Notes

• If one’s knowledge of cooperative game theory begins and ends with Nash’s 1953 paper ‘Two-person cooperative games’, they might disagree with the assertion that cooperative games don’t have strategies, as ‘strategies’ are the first of the primitives Nash introduces in that paper. Strategies, both implicitly and explicitly remain a part of the “bargaining” literature. Most of cooperative game theory, however, modeled itself after Nash (1953)’s “axiomatic approach”, itself modeled after von Neumann and Morgenstern’s way of defining and characterizing expected utilities. This does not use strategies. Instead, this approach lists a small number of desiderata, the so-called “axioms” (which word is used differently here than in most of mathematics, yes I notice the irony), that solution concepts should be expected to have. This approach is silent on what actions players take to arrive at the solution. The desiderate may be formal notions of fairness and efficiency, or some sort of consistency between games. In some of the most beautiful results of the field, these “axioms” are shown to be necessary and sufficient conditions to define a solution concept of cooperative game theory and only that solution concepts. Most famously, a cooperative game’s Shapley value is the solution that (1) always gives null players zero, (2) distributes the value of the grand coalition exactly, (3) treats symmetric players equally, and (4) satisfies additivity between games, and it is the only solution concept that can do all of these. Even more elegantly, some of these “axioms” define exactly the same outcome as the (subgame perfect) Nash equilibria of the non-cooperative bargaining game would prescribe, a beautiful union of the cooperative and the non-cooperative approach.

• When I imply that evo game theorists’ nomenclature is not perfect, I do not mean to imply that non-coop nomenclature is perfect, or that non-coop has nothing to learn from evo game theory or anyone else. Quite the opposite, in fact. And if people believe the way economists are thinking of game theory can be improved, our notation, and our nomenclature can be improved, I hope they’ll tell us how. But in doing so, I hope they note that it is a complex system of theorems, notations, and conventions, and you can’t just alter things “willy-nilly”. ‘Classical game theory’, while it is indeed a household terminology of evo game theory, is not a part of any system because it’s exclusively used as an exonym for non-cooperative game theory. As such, changing it is costless to the precision of the theory.

• All other arguments for the use of ‘classical game theory’ that I’ve seen and are not mentioned in my posts were appeals to authority. “I’ve read it in [The Famous Book] by [Renowned Figure]”, “I’ve learned it in the class of [Brilliant Young Figure] who learned it from [Brilliant Senior Figure]”, “See? Even [Accomplished Person Outside the Field] recognizes the term!”. The fact that such appeals to authority exist in academia is not surprising, the fact that they exist in a mathematical field is a bit more so. In any case, this is one of the most well-known logical fallacies and I don’t find much persuasive value in it. Unless all the brilliant people using the phrase have some brilliant reason to do so, these brilliant people should probably stop using it too to raise their levels of brilliance and contribute to the common good.

Comments should be addressed to peter.bayer7@gmail.com.