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Thanks for the more concrete example—the pizza situation in the article seemed really contrived and unintuitive to me! One question this leaves me with and I don’t think the article addresses is what happens when losses are sustained when collaborating. Your stock example provides a good basis for an exploration of this question.

Say A puts $4000 in and B puts $2000 in with the expectation that they will double their money, but instead they lose it all. By the procedure outlined in article, the “pie” worth -$6000 would be evenly distributed, meaning both A and B would have to pay $3000. So A ends up with $1000 and B ends up with -$1000. This seems especially punitive to B, and they end up in a worse place than they would have with what the article calls the “power” arrangement, under which they would have $0 after losing only $2000 instead of $3000.

I’m curious about how many weaker parties continue to favor the “pie” method when losses are divided the same way as gains—-my intuition is that a “pie”-style deal is typically reached only when there is an almost-certain probability of gain, but I’m curious about how negotiators actually behave!

There is a whole subcategory of behavioral economics that can be summed up as "people treat potential losses and potential gains very differently". But what you're saying isn't the example that is being presented. You have A and B being equal partners with a loan from A to B of $1000. That's different from the kind of ultimatum style (with 100% certainty) game that is proposed.

I would say a better example is that A owns 66% of a startup and B owns 33%. They decide to sell the tech for X dollars. However, they are offered a 2X buyout if both of them agree to work for 3 years at the acquiring company. There's no "partial" offer though, it's both partners or neither. How is the 2X split up?

The contrived example fails because of the 33% and 66% wouldn't be assigned randomly.

Without knowing much about the field, intuitively this seems reasonable. More potential return on your money generally means taking a higher risk (more volatile stock = more potential returns). The higher risk here means loosing a higher percentage of your investment (in fact >100%).

I actually think the method proposed in the article works perfectly in this case as well.

If you found a magic investment that was guaranteed to return 200%, but were some money short so couldn't invest at all, if I made up the difference and said we had to split the gains evenly it would still be in your best interests to say yes.

Of course the obvious response is that investments aren't guaranteed to return money. Going by this method, the losses would be "split" evenly as well rather than in proportion. So if the company could only return 50% of the initial investment, you would claim extra money towards your principal, thus reducing your overall risk at the expense of the person who invested less money.

This is a good line of questioning. I believe that the difference between the two situations is that the pizza deal is implied to be just for Alice and Bob. If either walks away then the deal is totally sunk, so they both have equal "leverage" over the deal.

When you frame it as an investment question, the implication is that if either person walks away then they can be replaced by a third party who is willing to stake the cash. Since it is easier to replace someone when their stake is lower, the person putting up the $2k has less leverage in this case.

If you explicitly limited the investment to two specific people (imagine a crazy term in a will or something), then I think you'd end up back with a $7k/$5k split.

Definitely agree, a big part of this is also how badly the person putting more into it needs that other partner.

This example system starts falling apart as the amounts shift. If one party has 5999, and the other 1 dollar, is that type of split still fair? It would never benefit anyone to be the higher of the contributors, a system of everyone looking for suckers.

If there really were a contrived scenario where exactly two people are allowed to contribute, the full 6k does need to be provided, and one person has no means to exceed 5999 then the split is still fair in some sense (and is the only "fair" allocation in the sense described by Shapley -- efficient, symmetric, and linear).

What you're seeing in this hyperbolic "counter-example" is that the assumptions of that model don't hold; the person providing 1 rather than 5999 is probably very replaceable and would be willing to settle for much less profit due to market forces, the person providing 5999 probably has the means to at least temporarily acquire another 1, the deal probably wouldn't fall through in the first place if it were for slightly less than 6k, and so on.

No, of course not. That's why I pointed out that at the end of the day it comes down to what seems fair, and what the two parties are willing to negotiate.

If I were the $4000 person, I would reject the 7/5 split as unfair, because, conventionally, the expectation is that the return is proportional to the participation.

Consider an extreme sceanrio - $5990 and $1. As per the article the return should be $8999 and $3001. That's not going to fly in the real world.

> let's say the value is guaranteed to increase, in this example by 200% to $12,000.

Still kind of contrived:-(