The Toll and the Bridge
Every platform eventually asks whether to cut out its middleman. The answer turns on a single number — and the platform gets to choose that number itself.
Here is a thesis in one sentence: the power to move first is usually described as a weapon, but when the party you are moving against both helps you and charges you, moving first makes you more restrained rather than less — and which of those two worlds you are in is something you decide yourself.
That sentence took two rejections to arrive at, and the rejections are the more useful part of the story, so I will start there.
A paper that was wrong in an interesting way
The original paper had three actors. A rule-setter chooses some feature of a system. A second party — call it the intermediary — sits between that system and the people who use it, and takes a cut. Then investors decide how much to put in, having seen both decisions.
The economics was about displacement. If the rule-setter builds enough capability directly into the system, users can go around the intermediary, and the intermediary’s cut shrinks. So the rule-setter has a reason to build more than it otherwise would: not because the extra capability is worth having on its own, but because it weakens someone else. That is a commitment effect, and the paper’s headline was that it requires sequential timing. If both parties moved at once, the distortion vanished.
The Journal of Institutional and Theoretical Economics rejected it with two referee reports. It was rebuilt and sent to Games and Economic Behavior, which desk-rejected it without review. The editor’s letter was short and it was right:
The idea that a first mover in an industry may overinvest to deter entry was formalized in the literature fifty years ago in papers by Spence and Dixit... These results are not surprising, given that the surplus function decreases in player B’s investment... But that begs the question as to the need for such an intermediary in the supply chain.
Two objections, and the second is the fatal one. Michael Spence in 1977 and Avinash Dixit in 1980 showed that an incumbent can overinvest in capacity to deter entry, and Drew Fudenberg and Jean Tirole organised the whole family of such effects in 1984 into a taxonomy that economists have taught ever since. A first mover distorting its own choice to weaken a rival is not news.
But look at what the model had assumed. The intermediary in that paper was pure extraction. The socially optimal amount of it was zero. So the paper had built a parasite, and then proved that the rule-setter wants to kill the parasite and overdoes it. Of course it does. The result was smuggled in through the setup.
Why the fix is not obvious
The repair is to give the intermediary a reason to exist. Let its action do two things at once: raise the productivity of the whole system and levy a charge on it.
This is not an artificial construction. It describes most real intermediaries. A payment processor genuinely settles transactions and genuinely takes a percentage. A distributor genuinely reaches customers you could not reach and genuinely takes a margin. An app store genuinely provides discovery, trust and billing, and genuinely takes thirty per cent. A certifier genuinely supplies assurance and genuinely prices it. In every case the same activity is the service and the toll.
Now the question stops being rhetorical. If the middleman contributes something real, does the rule-setter still try to crush it? And if it does, is that good or bad?
The naive answer is that you protect a complement. Something that makes your system more valuable is an asset, and you would not want to destroy it. The naive answer is wrong roughly half the time, and the interesting part is working out which half.
Why game theory rather than argument
This is the point at which prose runs out. You can argue either side plausibly. The platform will crush the middleman because it wants the margin. Also: the platform will protect the middleman because users value what it does. Both are stories. Neither tells you when.
What a model does is force the two effects onto the same page and make them compete quantitatively. And the answer that falls out is not one that either story would have suggested.
Write the users’ investment decision. They put in more when the system is more productive, and less when they are taxed. The middleman does both. So its net effect on users is its contribution minus its charge — and whether that is positive or negative is not a fixed property of the middleman.
It depends on the rule-setter’s own choice. That is what “displacement” means: a higher rule makes the middleman easier to bypass, which lowers the toll it can levy while leaving what it contributes intact. Set the rule low and the middleman charges more than it gives. Set it high and it gives more than it charges. The same middleman, with nothing about it changed, is a drain in one region and an asset in the other.
The result
Here is the whole thing, and it is three lines of calculus.
Compare two worlds. In the first, the rule-setter moves before the middleman and can commit. In the second, they move at the same time. The rule-setter’s optimality condition in the two worlds differs by exactly one term — the middleman’s net marginal contribution, multiplied by how strongly the middleman responds to the rule.
That term is what commitment buys. And it changes sign precisely where the middleman’s contribution equals the toll it can still charge.
Below the switch, when the middleman takes more than it gives, commitment is a weapon: the rule-setter that can commit pushes harder to bypass it than one that cannot. That is the classical case — Spence and Dixit’s world, and the original paper’s world, which turns out to be one edge of this rather than the general rule.
Above the switch, commitment reverses. The rule-setter that moves first knows that pushing harder does not only shrink the toll; it also shrinks the service. Moving first forces it to internalise that. Commitment stops being a weapon and starts being a discipline.
Show Image
The solid line is a platform that moves first and can commit. The dashed line is one that moves simultaneously. They cross. To the left, commitment makes the platform more aggressive; to the right, more restrained.
The part that surprised me
There is a standard way economists classify commitment problems, due to Fudenberg and Tirole. You ask two questions: does the leader want the follower to do more or less, and how does the follower respond to the leader? The combination tells you the posture — the famous “top dog”, “puppy dog”, “fat cat”, “lean and hungry look”.
Both of those are normally fixed features of the situation. Here the second one is fixed. The way the middleman responds to the rule never changes character; measured properly, that quantity keeps one sign across the entire range.
What changes is the first. Whether the leader wants more or less of the follower’s activity flips at an interior point. So the game sits in a single cell of the textbook taxonomy by its response structure, while the leader’s own choice decides which posture that cell calls for.
I did not expect that, and I initially wrote it up wrongly — as a change in strategic complementarity, which is the thing the taxonomy tracks. It is not. Checking the relevant second derivative showed it never switches sign at all. The switch is in a first derivative: a contribution, not a complementarity. Those are different objects and they need not move together.
What this is for
The structure recurs wherever someone who sets the rules of a system also has to decide how much of the system to build themselves.
Platform governance. An operating system vendor deciding how much payment, identity or discovery functionality to build natively, when third parties currently supply it and charge for it. The model says the answer depends on whether, at the current level of native functionality, those third parties are net contributors — and that the vendor’s own past choices determine the answer.
Payment rails. A network deciding how much to disintermediate acquirers and processors.
Standards bodies and certification. Where the certifier both reduces genuine uncertainty and prices that reduction.
Retail and distribution. The oldest version of the problem, and the one where “cut out the middleman” became a slogan long before anyone checked when it was right.
And a regulatory reading, which is the one I find most useful. When a competition authority looks at a platform absorbing an intermediary’s function, the instinct is to ask whether the platform is foreclosing a rival. This says: ask instead whether, at the platform’s chosen configuration, the intermediary was giving more than it took. That is a measurable question, and its answer determines whether the platform’s aggression was excessive or disciplined.
What I am not claiming
Two things, because the temptation to overclaim is exactly what got the earlier version rejected.
First, I have not shown that the welfare ranking of the two timing structures flips at the switch. It does in the calibrations I ran, and there is a clean intuition for why it should. But there is at least one parameter region where the leader’s action reverses and welfare does not, and until that is characterised properly the welfare claim is not a theorem. It appears in the paper as a property of an example and nothing more.
Second, the model compares two fixed timing structures rather than deriving which one arises. The natural next step is to let the parties choose when to move and see whether they select the socially preferable structure. That is a different paper.
The short version: whether the ability to commit is good or bad has no general answer. It depends on what you are committing against — and if you are the one setting the rules, you are also the one deciding which case you face.