A little game theory on discursive conflict

[Slightly updated version here.]

There’s a really widespread belief in the social sciences that discourse matters, that language and narrative isn’t just a transparent medium through which we view the world, but also something that creates winners and losers, and is therefore fought over. Michel Foucault is the very obvious name to check here, but I don’t think the belief is widespread just because it was espoused by Foucault. I think it’s the other way round: Foucault is influential because he was really on to something. Even if you don’t drink the postmodernism kool-aid, “discourse” is just too popular a concept to be completely mistaken.

Economists have not traditionally paid much attention to language and narrative, but that’s been changing. Some big names have written papers and books, either saying “hey, maybe narrative is important, we should think about it”, or actually trying to think about narrative in the framework of an economic model. This is a bit ironic, because the essence of formal modelling is that you throw out all the thick description and reduce everything to narrative-free mathematics. But OK, fine, economists have to bring their unique approach to bear.

So, I’ll sketch how one might think about discourse from a game-theoretic point of view. This is a “paper I’m too lazy to write”, or probably one that’s too trivial to write: I’m no economic theorist, it’s a simple idea, make of it what you will. I’ll focus on discursive conflict — the idea that people might have conflicting interests, and there might be resulting struggles, over how things are described.

Here’s a classic 2 x 2 game: the Battle of the Sexes. Like all such games it has a cute story attached.

The story is: Jim wants to go to the boxing, and Jane wants to go to the ballet. (Game theory developed in an age of gender stereotypes.) But both of them would like to go with the other person. Jim chooses an action in the rows; Jane chooses an action in the columns. The numbers are Jim’s payoff, then Jane’s payoff. So, if they both go to the boxing, Jim gets 2 and Jane gets 1. If they both go to the ballet, Jim gets 1 and Jane gets 2. But they both prefer either of these alternatives to the sad situation where they go to different events alone.

There are two equilibria here: both go to the ballet, or both go to the boxing. Those are equilibria because neither person unilaterally wants to change their action. Still, each equilibrium benefits one person especially. Jim would rather get to the BOX, BOX equilibrium. Jane prefers BALL, BALL.

This story roughly fits many real world situations:

There can be a range of prices at which a buyer and seller will be better off than if they don’t trade. But there’s conflict over which price to buy at: a higher price benefits the seller. Here, the two actions might be BUY HIGH, BUY LOW and SELL HIGH, SELL LOW; if the players can’t agree on a price then there’s no trade.
Society is a “cooperative venture for mutual advantage”: rules and laws are better for everyone than anarchy. But the rich and poor might prefer different kinds of society with different laws. Here the actions might be CAPITALISM, COMMUNISM; if the actors can’t agree then there’s civil war.
As for laws, so for rulers. If everyone agrees that someone is the legitimate power in the land, then it is very risky to try obeying anyone else. So there are as many equilibria as there are potential rulers. Of course, each equilibrium benefits different people — the relevant ruler and his allies.

So, let’s drop the boxing and ballet story, and just write two abstract labels A and B for each action, and two labels 1 and 2 for the players.

Suppose everyone in society is playing the A, A equilibrium, benefiting the player 1s.

Of course, in the real world, actions aren’t labelled A and B, and players aren’t labelled 1 and 2. They’ll be known by some real world labels. Action A might be “respect for private property”. Action B might be “redistribution”. Players 1 and 2 might be “men and women” or “the employer and the employee”.

Is there room for discursive conflict here? Probably not. Everybody knows who the “man” is. Everybody knows that carriages drive on the left, not the right. If you tried to redefine left as right, you probably wouldn’t persuade anyone else, and you might have a nasty accident. You can’t just call yourself the King; if you do, you may end up working in the scullery and think yourself lucky.

Lambert Simnel: it didn’t end well, but it could have ended worse.

But now suppose a new game arises — another Battle of the Sexes. For example:

Say we all know what it means to own land. And we agree that only “the landowner” should be allowed to use his land, graze cows on it, buy it, sell it, etc.
Now suppose an inventor claims that he owns his invention. Just like with the land, other people shouldn’t be allowed to use it! Arguments about patents were a big deal in the 17th and 18th centuries, and the development of patent law may have helped propel the industrial revolution. (Or maybe not! Those arguments are still going on.)
Suppose we know what a newspaper is, and what a private letter is. A newspaper gets sued if a writer in the paper libels someone, but the Post Office doesn’t get sued if someone writes a libellous letter.
Now suppose there’s a new thing called the internet. Are firms who host content on the internet like newspapers, or like the Post Office? Who is responsible for the content? There are at least two legal equilibria here, but they benefit different actors differently.

The stage is now set for different types of discursive conflict.

Type I conflict: relabel the actions

“Pearls before swine,” giggled the white mouse. “Tee, hee!”
“How uneducated you are!” snorted the pig, turning up his snout. “Ladies before gentlemen; swine before pearls!”
Hugh Lofting, Dr Doolittle’s Post Office

In the old game, everyone is playing action A – whatever they call it. It’s very likely that in the new game, they’ll also play the action with the same label. But which action should have that label? If you can persuade everyone that the action which benefits you is the “real action A”, then you win.

Is an invention private property? Then nobody should steal your property. Inventors win. Is an invention just an idea, which anybody is free to talk about? Then people can copy your idea. Copiers win.

Type II conflict: relabel the players

The beggars have changed places but the lash goes on.
Yeats

You might think that player 1 in the old game has an advantage here. It’s natural to think that if action A benefits player 1 in the old game, then the action which benefits the same person in the new game must also be “action A”, and so then everyone expects everyone else to play it.

But that assumes we directly know the identities of the players. In a small village, maybe everybody knows that Bob always gets his way, and that expectation works across many different arenas. But in a big society, players are known as members of categories. Lords and peasants. Men and women. Whites and blacks.

Some of these labels are very sticky! It’s hard to change your sex or your ethnic group. People do try, though. And some labels are more context-dependent. We must all obey the king, right? But when the old king dies, who is the new king?

Type III conflict: relabel the game

Now suppose that before the new game arrives, there are already two existing games in society, with settled equilibria. In one, everyone plays A; in the other, everyone plays B. Suppose there are also obvious, settled ways in which the new game’s actions and players map to each of the old games’ actions and players. So there’s no scope for fighting over those labels. But, which existing game is the new game more like? People who prefer the A equilibrium will want it to seem like the first game; those who prefer B will argue that it’s like the second game. Comparisons and analogies will have real payoffs.

We all know that you need to “protect minorities”. We also all agree that you need to “defend free speech”. And we know who the players are in these stereotyped stories — the insulted minority, the cancelled plain-speaker — and how the story should end — we all come together to defend minorities/free speech against angry white men/woke loons. But which story better fits the latest Twitter hullaballoo?

Doctor Doolittle makes his inauguration speech

Case-, nutcase- and basketcase-based reasoning

Under the hood of this argument lies a theory of how people reason about new strategic situations. For the logic to work, players must draw analogies with games they already know about. There are some game-theoretic papers taking this perspective, which indeed seems quite plausible.

Can a game-theory perspective offer any new insights? I think one potential hypothesis is that in discursive conflict, breadth beats depth.

That is, when you’re trying to relabel something so as to change the equilibrium, you don’t have to persuade people about reality. You just need to persuade them about what other people are thinking. Or even about what other people think other people will think. Or… and so on. In the world of the Battle of the Sexes, everybody is just trying to predict what other people will do, so as to go along with it, and they know that everyone else is doing the same thing. If you can make an unconvincing analogy between one player/action/game and another, but you can make it loudly enough, it won’t convince people; but it might convince them that other people will be convinced.

So, it’s more important to spread your message very broadly than to make it deeply persuasive. This might be why the Big Lie remains a powerful technique. Does anyone, apart from a few nutcases, really think the election was stolen? That’s not the point, and it’s almost impossible to measure anyway. What matters is that people, especially politicians, know to say they do. If enough of them do that, the equilibrium changes anyway, whatever people’s “real” beliefs. So, as Hannah Arendt put it, “propaganda is marked by its extreme contempt for facts as such”.

That’s all! There’s no deep maths here. I think it is a simple way of bringing game-theoretic reasoning to bear on the study of narrative. I’ll end with an old joke:

A mathematician is at a conference hotel, working late on his paper. Suddenly he notices that his cigarette ash has set the waste paper bin on fire. Swiftly, he puts the fire out with the fire extinguisher. Then he goes to bed.
The next night he’s working late again, when he spots that his cigarette has set a piece of paper on his desk smouldering. Quick as a flash, he puts the paper into the bin, which is soon cheerfully ablaze. “An already-solved problem”, thinks the mathematician, and he goes to bed.

The mathematician’s logic is that it’s good to reuse your tools. So perhaps it’s a nice approach to understand discursive conflict using simple games, which we already know a lot about.