Under what circumstances are we fair in this classical dorm situation?

Question

So I am going to college this fall. I had two roommates sharing the same room. I will bring the fridge, $\$150$, jack will bring a TV, and Kyle will bring a printer. The TV and the printer are of the same value, let us call its dollar value $x$. So all three of us are using the fridge, so each is paying $\$50$. However, I will not be using the TV Jack has brought, but I will be using the printer Kyle brought. But Kyle and Jack will use all three things.

My question is how much will I have to pay for the printer and pay to whom for it will make it fair?

First attempt: Suppose I do not use anything else except the fridge which I have already paid for $\$50$. Say now I want to use the printer too, I will pay $\frac{x}{3}$ dollars to Kyle, so we will be fair because kyle had the printer.

Second attempt: Suppose I do not use anything else except the fridge which I have already paid for $\$50$. Say now I want to use the printer too, I will pay $\frac{x}{6}$ dollars to Kyle and $\frac{x}{6}$ to jack, so we will be fair. Because in this case I view the (printer and the TV) belongs to both jack and kyle, so jack and kyle both have shares in the printer now.

Any thoughts? I am really confused.

Answer 1

Your first attempt is closer. Usage of the TV should cost $x/2$ because there are only two people using it. You and Jack should pay $x/3$ to Kyle for the fridge, but Kyle should pay $x/2$ to Jack for the TV. They should each pay you 50 for the fridge. If you want, you can net things out, but it may be easier just to make these payments.

Your second attempt works, too. In the first, you are buying into the printer at the start, when Kyle owns it. He will also get $x/3$ from Jack. In the second, Jack has paid Kyle $x/2$ because you initially did not want to use the printer. You now need to buy a one-sixth share from each of them, which you are doing.

Answer 2

Your second attempt is fair. Let's say the printer has a cost of 300. If you initially don't want to use the printer, Jack and Kyle pay 150 each. If you want to use the printer now, it is only fair that everyone has paid the same price, which ist 100. Therefore half of your 100 go to Kyle and the other half to Jack, so all three of you have paid 100.

Answer 3

In either case you will have paid $\$50$ for the fridge and $\$\frac x3$ for the printer, so the question of fairness is completely a matter of what funds pass between Jack and Kyle.

In your second scenario, in which you pay Jack and Kyle $\$\frac x6$ each for the printer because they both "own" it, is fair if Jack has already made (or will make) a sufficient payment to Kyle so that Jack would "own" half of the printer. Namely, it requires Jack to pay Kyle $\$\frac x2$ for half the printer. If Jack only pays $\$\frac x3$, he has paid for only $\frac13$ of the printer; then Kyle still "owns" two of the remaining $\frac13$ shares of the printer and you should buy one of them directly from him.

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In more detail:

Which of the scenarios is fair—the first, the second, both, or neither—is entirely a matter of what transpires between Jack and Kyle. You have already paid your fair share for what you use, but perhaps Jack gives no money to Kyle, so Jack is getting free use of a printer at Kyle's expense (and a windfall of $\$\frac x6$ on top of that, in one case, but in either case Jack is taking advantage of Kyle).

Let's assume that in either scenario, Jack and Kyle each pay you $\$50$ for their use of the fridge, and Kyle pays Jack $\$\frac x2$ for his share of the TV, in addition to any other payments you make to each other.

Your first scenario ($\$\frac x3$ to Kyle) is fair under the following conditions: Jack pays Kyle $\$\frac x3$ for his share of the printer, and you pay Kyle $\$\frac x3$ for your share of the printer.

Your second scenario is fair under the following conditions: Jack pays Kyle $\$\frac x2$ so that they now have equal shares in the printer. Now, in order for you to get a $\frac13$ share in the printer so that you can use it, you buy $\frac13$ of Kyle's share and $\frac13$ of Jack's share for $\$\frac x6$ each. Now you all have equal shares in the printer.

In either of the fair scenarios, you originally paid $\$150$ to obtain the fridge you brought, but you got back $\$100$, and everyone has made net payments (money paid out, minus money received back) of $\$50$ for the fridge. Kyle paid $\$x$ to obtain the printer, but received back $\$\frac23x$, Jack either paid $\$\frac x2$ and got back $\$\frac x6$ or simply paid $\$\frac x3$ with no money back, and you paid $\$\frac x3$, so everyone made net payments of $\$\frac x3$ for an equal share in the printer. Finally, Jack paid $\$x$ for the TV but received $\$\frac x2$ from Kyle, so Jack and Kyle each made net payments of $\$\frac x2$ for the TV.