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How to Not Let Them Get Away With It: A Note on the Mathematics of Infinite Exploitation

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Published: 10 December 2025

Volume 48, pages 125–127 (2026)

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How to Not Let Them Get Away With It: A Note on the Mathematics of Infinite Exploitation

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Recently, a friend and their partner shared that they were looking into buying a house. Having been through the mortgage-securing process myself in the last couple of years, I took the liberty of offering some unsolicited advice about what to keep in mind, what to watch out for, et cetera. Soon thereafter, my partner joined in this unsolicited and altruistic doling out of mortgage wisdom. She reminded me of a particular episode from when we bought our house, and on the drive home, as I replayed that encounter in my mind, it suddenly came to me that there was something mathematically interesting about the situation. After we got home and I had worked through the details, I found myself wishing that I had responded differently.

The situation was the following: the sellers, having agreed to sell us the house, were having their agent work with our agent to complete all the required paperwork to get us to closing. At one step, that of securing a title insurance policy, the title insurance company requested a copy of the sellers’ policy. Since producing such a document would save us somewhere between \(\$400\) and $1,500, we reached out to the sellers to request a copy. This, we thought, was an innocuous request. To our surprise (and to that of our agent), the sellers responded, through their agent: “My clients are willing to share a copy of their title policy if they receive a credit for half of the savings that their policy would provide.”

If the nature of what they were asking for is not clear, here is an analogous situation: Suppose you are at your local big-box grocery store and realize during checkout that you have forgotten your member savings card, the one without which you would miss out on not-insignificant savings on your order. You notice that the person behind you in line has one, and you ask whether you could use their card, since you have forgotten yours. Now imagine that they respond, “I would be happy to let you use my card as long as you share half your savings with me.” Figure 1 depicts ChatGPT’s imagining of this exchange.

I found the response then, and continue to find it now, at the very least, bizarre. My partner, who happens to be a moral philosopher and is thus better qualified than I to find just the right word, called it exploitative. We thought for a couple of days about how to respond. Before I tell you how we did, let me tell you how I now wish we had.

To be more technical, let the total savings available by the sellers’ producing a copy of their policy be denoted by M (money!), and let me begin by considering the sellers’ demand for an equal division of the savings; then later, I will return to a more general case. Thus, the message received from the sellers, mathematically stated, was this:

Because of us, there is now an extra M dollars available that you were not aware of, and the only way for you to get some of this is to agree that we get half, M/2. And you can have the other half. Our mere existence will then give your bank account an extra M/2 dollars.

If this deal were accepted, each party would have M/2 dollars.<br>Figure 1. The alternative text for this image may have been generated using AI.Full size image

An AI depiction of the exploitative request.

What’s going on here? This is an example of what game theorists call, very appropriately, bargaining games. At this point, we have what is called an ultimatum game: take it or leave it. What I wish now I had done was to have turned the situation into a specific type of sequential bargaining game. More precisely, this is how I wish I had responded:

This initial split is fine. However, because of us, there is now an extra M/2 dollars in your life that you did not expect. I propose that we split the M/2 dollars that you were not expecting in two, since the only way for you to get M/4 in savings is to agree to our terms; our mere existence gives your bank account an extra M/4 dollars.

If the deal were accepted at this round, then the buyers would have 3M/4 (M/2 from the first round and then an extra M/4 that the sellers forfeited in the second round), whereas the sellers would have M/4 and a taste of their own medicine.

But the deal would not have been accepted in this round. If we know anything about the sellers, it is that they are a greedy and conniving bunch who want to have the last word and try to...