Introduction:

I was talking to a friend of mine before econ class, and somehow the topic turned to incels. So we're talking about them, then he says these words:

"Incels are just dudes with no game."

Writing it out mathematically, the statement becomes, "If you are an incel, you are a dude with no game", so the contrapositive implies that if you could somehow give "game" to an incel, the incel would cease to be an incel. But I wager that if even if you somehow found a large number of Hitches to give them "game", it would be completely ineffective, as the incel mindset is, at its core, anti-woman. At the basis of the ideology is a fundamental objectification of women, simply turning them into objects of use, and hence an incel would likely use that "game" to commit harm against women.

So we can't use "game", but still, how can we help incels, or at least prevent other people from believing their ideology and becoming incels themselves?

Teach all people reverse game theory, also known as mechanism design.

Background:

Immediately, the question is why teaching people mechanism design will help with the incel problem. This is because incel theory has a huge mechanism design flaw.

See, according to incel theory, women constantly lie to men, putting on a show that they're good people, when they actually just want to use men for various things. If a man is "trapped" in a relationship with them, women reveal that they're actually hideous beasts simply out to use and harm men.

That is bad economics, specifically bad mechanism design. Using the aforementioned assumptions, we will prove that incel theory is internally inconsistent.

The Model:

Let's create the setting. We have one lady, whom we will denote as W. There are N guys who are attracted to her, and want to become her boyfriend. She will only have one boyfriend.

The key thing to note is that each guy does not know how much he actually values a relationship a relationship with W. This is because W may seem great in public, but in private and in a relationship she may be completely different. Maybe she's just putting on an act, like incel theory states.

So the guys don't know their exact values (V) of a relationship with W, but they have expected values (E(V)), which are dependent on a signal (S) they each receive about W. Maybe the signal is a rumor they heard about her, an Instagram post, whatnot.

We will assume that signals are positively correlated with values. In other words, a high/low signal means it's more likely that a guy's true value of a relationship with W will be high/low. Furthermore, we will assume that each guy's signals and values are "affiliated" with the others'. In other words, the higher other guys' signals/values are, the more likely you will have a high signal/value. We can say that in this model, values are interdependent and not independent. Furthermore, signals are drawn from the same distribution for each guy. Finally, every guy is risk-neutral.

Let's take the point of view of one guy who likes W, and there are N-1 other guys. We will let S denote the signal of this guy and Yi denote the i-th highest signal of the other N-1 guys. So Y_1 is the highest signal of the others, Y_2 is the 2nd-highest, and so on, until Y{N-1} is the N-1 highest of the others, i.e. the lowest.

Now let's use the incel assumptions to construct the mechanism that decides who becomes W's boyfriend. Accordingly, W is purely materialistic, shallow, and in general a bad person. She only cares about what the guys can do for her, be it buying her food, clothes, and whatnot. Hence, she will only enter a relationship with the guy who is willing to spend the most money on her (maybe there are other things she wants too, but if so let's just take the Euclidean norm) to combine those things into one value). This means that each guy will keep trying to do the others in terms of how much they spend on her (assuming that budget constraints do not bind). As each guy declares how much money he will spend on W, this "raises the cost" of a relationship with W. Eventually, there will come a point where the expected utility from being in a relationship with W will be negative; this occurs for a guy when the cost is too high. And so he drops out. More and more guys drop out, until there are only 2 guys left in this spending game. Finally, one of the guys will raise the cost just too high for the other guy, and so the other guy drops out, and thus the guy who raised the cost becomes W's boyfriend. The cost to W's boyfriend is thus the point at which the second-to-last guy dropped out.

You may notice that this mechanism sounds almost exactly like W setting up an English auction. And since this is an English auction, we know from Vickrey (1961) that this is equivalent to a second-price sealed bid auction in equilibrium for independent private values; specifically, the equilibrium cost at which a winner wins is the same across both auctions. However, values here are interdependent, so equilibrium is not exactly the same. However, I will examine the second-price case, because the qualitative result is the same, and it's also easier. Just for robustness though, we'll discuss how the final qualitative result holds when analyzing the English auction perspective.

Now, return to the guy whose POV we are taking. Let this be his expected value of a relationship, condition on his signal equaling some $x$, and all the other guys' signals being whatever they are.

However, remember the setting! If this guy wins, he pays the cost at which the last person among the others dropped out. Furthermore, signals and values are correlated, and due to the symmetry assumption from above, we have that the highest signal equals the highest expected value of a relationship. This means that our specific guy only cares about the signal of the person with the highest signal among the others, because the cost at which that guy drops out is the cost our guy pays. So we can rewrite the condition expected value in this manner, for some signal y.

Finally, we will let b^*(x) denote the cost one pays if he becomes W's boyfriend, given that his signal is x. So in the second-price sealed bid auction setting, since we are assuming risk neutrality here, expected utility is like so. Let's draw a graph!

Cost is on the vertical axis, and the highest signal among the other guys is on the horizontal axis.

Let's suppose our guy shares that signal. Then his cost function will look something like this.

Next, we'll graph the expected value of a relationship, fixing our guy's signal at some positive value x. Notice that this doesn't start at 0, since one's own signal is positive, so even if others' signals are 0, one will have positive expected value. Right now we won't prove why the slope is less than that of the b function; we'll shiow that later. But the intuition, I think, is clear: your own signal has a higher effect on you than others' signals, and while your own signal is constant in the expected value function, it changes with y in the b function. So the b function has a greater slope.

Now, remember that all the guys are risk neutral, meaning our guy only wants to win when expected value is greater than or equal to cost, or lose when expected value is less than cost. So we add these labels to our graph, for convenience.

Now observe that b(y) is another way of writing one's expected value given that your signal, and the highest signal among the others, is y. This shows that the slope of the b function is greater than that of the expected value function when S = x.

Anyways, because of the previous observation, we can characterize the reservation cost, i.e. the cost at which one is indifferent between winning and losing, as b^*(x).

And this occurs when one's signal is x. So, what is the ramification?

Well, because of everything we have shown in the graph's thus far, we can define b^*(x) in terms of condition expected value: it is the expected value of a relationship, given that highest signal among the others is the same as yours. In other words, for each guy, it is an optimal strategy to act as though everyone else has the same signal as he does, so as to guarantee that he drops out when the cost goes too high, but stays in when it hasn't reached that point yet!

Proving the Internal Inconsistency of Incel Theory:

Supposedly, it is in W's interest to lie and hide her true nature from the guys. But does that actually hold, assuming W is rational and intelligent in the Myersonian, game-theoretic sense?

Let's say that W will also get a signal about herself. She can commit to one of two plans: Plan A is to reveal the signal, and Plan B is to not reveal the signal. Remember, her goal is to extract the largest cost out of the guys as possible. So given this objective, which plan is optimal?

Well, let's define new variables. Let X_i define the i-th highest signal overall. So for example, if we take the perspective of the guy with the highest signal, then his signal is X_1, but Y_1 is the next-highest signal, i.e. Y_1 = X_2 in this case.

Let us also define S_W to be the signal that W can choose to revealing or not revealing. Suppose that S_W = w. Then if she chooses to reveal it, a guy with signal x will act according to the above, while also taking into account W's signal, in this manner.

Incel theory dictates that W's optimal strategy is to commit to not reveal her signal. Now, you may know of the Linkage Principle, as proven by Milgrom and Weber (1982), which will tell you whether or not incel theory is correct. But we will do a direct proof here. spoiler: incel theory is wrong, and it is actually W's optimal strategy to commit to revealing her signal.

Now we know that W's benefit comes from the cost at which the guy with the second-highest signal drops out. So let us take the perspective of the guy with the second-highest signal.

We begin with the identity between cost and expected value that was proved earlier.

Next, we will use the Law of Iterated Expectation. Using the LIE, we get this next line here.

Next, since the outside conditions have your signal and the highest among the others equal to x, we will fix those conditions for the inside as well.

This next line follows from the identity between the b function and conditional expected value.

Next, we know that we are looking from the perspective of the person with the second-highest signal, and as such S = X_2.

Now we shake things up. Suppose that the highest signal of the others is greater than or equal to x. Since all the guys' signals and values are affiliated, it means that now, the expected cost is greater than or equal to what it was before.

Next, we can simplify the expression like so, since S = X_2 = x already implies that the highest signal among the others' is greater than or equal to x.

Finally, we know we're looking at the guy with the second-highest signal, so we can get rid of the S.

What did we just prove? Remember that the cost W gets from her boyfriend is the cost at which the guy with the second-highest signal dropped out, i.e. his reservation cost. Hence, this proof shows that the reservation cost of the guy with the second-highest signal is less than or equal to his expected reservation cost, given that W chose to reveal S_W! QED.

In other words, it is optimal for W to commit to revealing her signal, which proves that incel theory is internally inconsistent with its assumptions!

Discussion:

The intuition is that the second-highest reservation cost underestimates the true value to that guy of a relationship with W, since the guy with the second-highest signal operates under the assumption that the highest signal among the others equals his. This is false, though, as the highest of the other signals is greater than his, and if he knew that, then his expected value would increase. Hence, by committing to revealing S_W, W corrects for this underestimate and raises, on average, the second-highest reservation cost. This is because her signal is affiliated with the highest other signal.

Now let's loop back to the English perspective. Using the Linkage Principle, it can be proved that English case has an expected cost greater than or equal to the second-price case. Furthermore, W revealing her signal yields the same qualitative result: this is more beneficial to her than hiding her signal.

Hence, teaching people mechanism design will showcase this internal inconsistency in incel theory.

Incel theory is bad, and now we know that not only is it bad morally, but economically as well.