A new dualism

Despite the pressure, evolutionary game theory persisted and, beset on all sides, thrived. Yet, forming connections between itself and its natural ally, non-cooperative game theory, was delayed. Due to evolutionary game theory operating on its own and needing a new identity, a new dualism was introduced.

When Maynard Smith outlined how game theory can be used in the new application, he had used the words ‘classical game theory’ to describe models of (sort of) rational humans. This was to be juxtaposed with ‘evolutionary games’, describing models of selection of traits within biological populations. Being (to put it mildly) light on formalism, he made no attempt to define either classical or evolutionary games in mathematical terms. Instead, the distinction was only made in terms of what we (informally) assume about the subjects of the games:

“Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest.”

This is the only time Maynard Smith talks about ‘classical game theory’. I don’t think it’s a worthy exercise to debate whether this is accurate either for the state of non-cooperative game theory of the 1980s (when Maynard Smith wrote this) or the 1940s (the state of game theory literature he chose to adapt).

What I will say though is that the dualism that Maynard Smith had introduced, almost unwittingly, turned out to be almost as consequential for game theory as Nash’s deliberate one. Evolutionary game theorists, for whom the jumping-off point was Maynard Smith’s work rather than that of Neumann or Nash, and who defined their identity in opposition to the game theory “of the economists”, ended up building up on this distinction. Instead of bridging the gap that exists much more in interpretation and domain of application rather than mathematics, future generations of evolutionary game theorists widened it. The chasm between evolutionary and non-cooperative is now wider than non-cooperative and cooperative game theory! Whose doing this was initially, mainstream game theorists’ for failing to take note of evo game theory, or evo game theorists’ for failing to conform is pointless to speculate on. Eventually, evolutionary game theorists made the separation a part of their identity. Pick up an evo game theory paper, go to a presentation, or read a twitter bio and you’ll see evidence of the forced dualism between “evolutionary” and “classical” all around. The closer the evo game theorist is to biology and further from economics, the stronger they tend to feel about this distinction.

The merits and faults of the distinction

The dualism between non-cooperative and cooperative exists because of two different languages trying to answer two different questions. The difference is already apparent in the simplest frameworks of the two fields and is fundamental. The definitions for “game” use two different mathematical objects:

On the left we have the payoff function of a normal-form non-cooperative game, u, the simplest and most studied object in non-cooperative game theory. It is a function that maps some strategy space X to an n-vector. Each component of the vector is a payoff value ("utility") for one of the n players. On the right we have the payoff function (a.k.a. characteristic function) of a cooperative (TU-)game v, again, the simplest and most studied object in cooperative game theory. It maps the set of subsets (coalitions) of the player set N to one real number for each coalition that represents its worth, or “value”.

The difference is clear also in what is considered a “solution” to a game. In non-cooperative world, a solution is an element or a subset of the strategy space X, specifying “what strategies the players might pick”. In cooperative world, a solution is an n-vector (or a set of them) specifying “how much of the pie each player ought to receive”.

The mathematical difference between evolutionary and non-cooperative frameworks is nonexistent: a (static) evolutionary game is the same object u with the restriction that it must be a symmetric game -- something that non-cooperative game theory has studied extensively both before and after evolutionary game theory arrived to the scene. A solution of the game is, again, an element or subset of X. You may add all sorts of dynamics to both the non-cooperative and the evolutionary frameworks if you want, but as long as the game is like the above, mathematically, evo game theory is a special case of non-cooperative game theory.

I’ve heard evo game theorists say that the difference between “classical” and “evolutionary” game theory is the solution concept used. The former types of games are solved by the Nash equilibrium, while the latter by the ESS. But even though the Nash equilibrium is now the "default" solution concept, it was never used to define non-cooperative game theory. In fact, a lot of non-cooperative game theory is coming up with refinements, generalizations, or alternatives to the Nash equilibrium, of which the ESS is one. Hell, you can define any sort of solution concept to any sort of game. If I say let the ‘Unicorn equilibrium’ be defined by all players picking the first element of their strategy set (ordered and finite, leave me alone), then I can solve whatever non-cooperative game by this powerful and novel equilibrium concept! Yet, even though my approach is groundbreaking, it doesn’t mean I’ve found a unique approach to game theory that deserves its own field.

The differences are solely in the fields of application, which, I will readily admit, make using different solution concepts make a lot of sense. Non-cooperative games applied to (sort of) rational behavior has implications on the types of mathematical assumptions we can make that you wouldn't make in a biological application. For instance, if we study (sort of) rational choice, the labels on the strategies are not so important. To show an example, a coordination game such as

becomes an anti-coordination game if I switch the labels of strategies of Player 2 but not for Player 1.

Both games accommodate two pure Nash equilibria. (Eenie, Eenie) and (Meenie, Meenie) in the first game and (Eenie, Meenie) and (Meenie, Eenie) in the second game, as well as a mixed one each where both players play Eenie and Meenie with probability 0.5. It is clear that, except for the labels, both games describe the same behavioral situation: the challenge for the players is to coordinate their actions in some way. In a behavioral application, these games might even fall into the same equivalence class.

If I interpret it as an evolutionary game (more precisely, if I make it explicit that I interpret this interaction as typical pairwise interactions occurring in a single biological population with many individuals), which is typically represented by depicting the game as a single matrix instead of the bi-matrix, the story is much different. The ESS structure of my coordination game

is super sensitive to switching the labels asymmetrically. There are two ESSs, pure Eenie or pure Meenie. Selection in my biological population is characterized by bi-stability, either everyone’s an Eenie or everyone’s a Meenie. Whereas in the anti-coordination game that I would get by switching the labels for Player 2

there is only one ESS, given by my population being exactly 50% Eenie and 50% Meenie. Instead of bi-stability, I have stable coexistence. Whereas the Nash equilibrium structures map to each other 1-to-1, the ESSs don't (given that they are a subset of symmetric Nash equilibria which care about the labels of the strategies). In the evolutionary application, these two games can’t be considered equivalent.

Why I feel strongly about the name

The evolutionary mindset has great value and certainly stands on its own legs despite evolutionary game theory being a subset of non-cooperative, mathematically speaking. Why does it matter to me then what it is called? And finally, I get to speak to the point of the whole matter.

First, defining “classical game theory” as “the kind of game theory that deals with (sort of) rational decision making” and then setting it equal to non-cooperative game theory completely forgets about cooperative game theory. Cooperative game theory was there right from the beginning, right from Neumann. In His wisdom, Nash saw it fit to separate the two and He was pleased. 3 Nobel Prizes and many exciting and literally life-changing applications (e.g., in kidney exchange or school choice) say that it is not just a niche math exercise but a powerful tool on its own right that anyone who claims to be a game theorist should be aware of. It also happens to inform policy in ways that evo game theory has not yet managed to. It is a vital component of (sort of) rational decision making and something whose potential contributions to evo game theory will remain undiscovered until evo game theorists at least recognize its existence. A very low-cost and important first step would be to call non-cooperative game theory by its name, letting the mind wander onto what the complement of ‘non-cooperative’ might be.

Second, the evolutionary vs classical distinction is often framed as two conceptually different subfields of game theory. In strict mathematical terms, this is simply not right. The distinction exists in the application and there it is important. Yet, turning the distinction into a separation allows the not-crème-de-la-crème of game theorists who have a habit of reinventing the wheel to say that “oh my research is only on evolutionary game theory and I’m not interested in classical game theory” (a sentence I’ve heard way too many times). Have you just rediscovered correlated equilibrium for the 10th time? Not a problem. Just publish it where economists aren’t invited to review and you’ll have a prestigious biology or interdisciplinary publication. Picking the right nomenclature motivated by methodology and not hiding behind the borders of scientific discipline not only builds bridges between related fields of science, it is actually a pretty cheap way to introduce quality control.

Third, not knowing the name is objectively bad science. Even if you never tried to repackage existing results for a new audience (or at least not consciously) and claiming it to be novel, failing to recognize non-cooperative game theory is IMHO a bit of a stain on someone’s resume (or paper abstract or presentation slide or twitter bio).

‘Non-cooperative games’ isn’t a name that I came up with and keep using to annoy others, it is literally the title of Nash’s 1951 paper where he defined the Nash equilibrium!

The paper that is mentioned and cited over and over again by hundreds if not thousands of evo game theory papers. Yet, modern evo game theorists keep telling me they haven't heard of this 'non-cooperative' game theory! Can it be that some of my evo game colleagues, despite claiming to build on game theory as laid out by Neumann and Nash, never looked at the fundaments they're supposedly standing on?

Fourth and finally. Forget the other three points. Forget that it is bad science, that it creates an unnecessary gap between related fields, and that it ignores a more fundamental and truer dualism within game theory. Forget all of that. What you’re left with is that non-cooperative game theory is an endonym. It is what the authors of the field itself chose to call it and keep calling it to this day. And whether or not you agree with that choice being a correct one, it is simply not right to call it anything else. If you don't agree, then from this moment on, your name is now Hermann because I thus decided.

John Maynard Smith is excused from this. His efforts and results, given his pioneering status and the constraints he found himself in are nothing short of laudable. He had to establish a point of departure from the existing frame of reference and the word ‘classical’ serves this purpose fine. He only used the word the one time and didn’t assert any sort of fundamental mathematical difference between evolutionary game theory and what had come before. But none of these excuses are valid to the field of evolutionary game theory as it exists today.

50 years have passed since Maynard Smith and Price had created evolutionary game theory. 50 years the field has spent defining itself in opposition to “classical game theory”. Trenches were dug between two heavily related fields that held back both, but IMHO, it held back evo game theory more. Therefore, for the good of evolutionary game theory, I think it’s time to begin filling the trenches. A first step would be to call a spade a 'spade' and non-cooperative game theory 'non-cooperative game theory'.

To conclude, please let me indulge myself in citing Attila József, perhaps the greatest literary figure of Hungary. In the first paragraph of his autobiographical short note entitled ‘Curriculum Vitae’ he writes:

“[...]my foster-parents had always called me Pista [Steve]. They consulted the neighbours about the name Attila and I heard them come to the conclusion that there was no such name. This astounded me, I felt they were casting doubt on my very existence.”

Attila József had a short, difficult, and ultimately tragic life that was beset by difficulties of all kinds. Dysfunctional family, absent and abusive parents, more abusive foster parents, poverty, romantic disappointment, sexual frustration, you name it. Yet, none of his traumas stopped him from producing some of the most beautiful lines ever written in the Hungarian language. There’s hardly a soul in Hungary or elsewhere in the world, who knows the names of his foster parents. I think there’s a lesson there.

March 12, 2023

Footnotes:

• Neumann had kind of made up the ‘von’ part of his name, like I, Bayer Péter (again, Hungarian name order), would sometimes call myself Peter von Bayer when I felt like cringe, except I grew out of it. The ‘von’ implies a noble rank that Neumann’s father did indeed get granted from Franz Joseph, the Emperor of Austria/King of Hungary. His dad was thus called Margittai Neumann Miksa, or Max Neumann von Margitta, with Margitta being some random town that the family had nothing to do with: when you're a noble you have to be "of somewhere" and the family picked Margitta because they thought it sounded cool. János inherited the noble rank and hence the name. Nobility fell out of fashion for a bit during the Hungarian Soviet Republic in 1919, whereas Margitta got annexed by Romania at the end of World War 1, so the 'von Margitta' was dropped. When he emigrated to Germany, he took the name Johann von Neumann, then in the US he changed it to John von Neumann. It is funny to me that of all the ‘vons’ and ‘vans’ in the world, like Bismarck or Beethoven, his is the only one that I see remembered. Ever the prankster, I imagine Neumann in the beyond with a grin on his face every time some says ‘von Neumann utility function’.

• Attila József was the absolute cringe lord, in the best possible sense. Just try this:

Nincsen apám, se anyám, I have no father, nor mother, Without father without mother,

Se istenem, se hazám, No god, no homeland, Without God or homeland either

Se bölcsőm, se szemfedőm, No crib, no coffin, Without crib or coffin-cover

Se csókom, se szeretőm. No kiss, no lover. Without kisses or a lover.

Original Lit. translation Poetic translation (by Thomas Kabdebo)

It’s no wonder that Hungarian teenagers of all generations love him.

• Interestingly, Attila József also produced some of the ugliest lines ever written in the Hungarian language too. While he was under psychological care he was asked to just write whatever came to mind. And he did. Oh boy, he did. Unfortunately for him, the document was preserved, and it is the meanest, most disjointed, foulmouthed, and just… “yeesh” series of thoughts ever put to paper. I won’t quote it here but whatever you’re imagining now, it’s a lot worse.

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