# How to auction shared item with a roommate?

My roommate and I split the cost of a game console a year ago (~$400) but he is now moving out. We were wondering how to fairly auction the console given the situation: • We both paid an equal price of$200+
• Right now half the market price of the console is \$175 • We both would prefer having the console than being paid for it • He will be splitting the price with his new roommate whereas I will be paying by myself With regular bidding, as my roommate representing both him and his next roommate and me myself, is he at an advantage because he has double the market power? If yes, how can we fix that problem? Outcome: Thank you for your answers. While the chosen answer seemed the best way to do this, we both felt it involved too much strategy and would have resulted in envy from one of us. We instead went with the Divide and Choose method, which worked out fairly for both of us. ## 2 Answers If you think that the distribution of your value and your roommate's value are the same then I would suggest that you consider the following bidding protocol: • each bidder submits a sealed bid,$b_i$. • the highest bidder,$i$, received the console and makes a payment of$\frac{b_i+b_j}{2}$to the loser, where$b_j$is the loser's bid. Cramton, Gibbons, and Klemperer (Econometrica, 1987; Section 4) show that this is an efficient way to dissolve a partnership with joint ownership stakes in an asset. If you think that the two of you have differing value distributions (e.g. because your roommate has a new friend) then Cramton et al. show in Theorem 2 how to adjust the pricing rule to restore efficiency. I highly recommend taking a look at the Cramton et al. paper. It's surprisingly readable for an Econometrica. Let us know how you get on if you decide to try it out! • I'm very confused by this. Person A submits a bid of \$175, and Person B submits a bid of \$200. By this protocol, Person B gets the console for only \$25? Person A is getting compensated far less than what either person thinks the item is worth, and Person B gets it at a great bargain. – Nuclear Hoagie Apr 16 '18 at 15:51
• @NuclearWang yeah, I made a mistake in the answer. What I meant is that the payment splits the difference between the reported bids. So in your example, B would have to pay \$187.50. – Ubiquitous Apr 16 '18 at 15:55 When you state that you would prefer having the console than being paid for it, I imagine you mean you'd prefer having it than being paid your share of the current market price. Despite what game data is on the console, there is surely some price at which you are happy to part ways with it. Both of you seem like you would be happy to buy each other's half for$175, but not sell it for that much.

Suppose you and your former roommate with his new roommate both bid on your respective halves of the console. Each of you have some sort of reservation price, the highest price you'd be willing to pay to get the other person's half of the console, which will be above \$175. (Likewise, it'd be the minimum you'd need to receive to part with your half, if both "halves" are homogeneous.) It's true that your former roommate will have someone else helping him pay, but that should not be a problem for you as long as you are not constrained by your budget and have to bid strategically. That is, if your budget is greater than your reservation price, you'll be fine.

There are two straightforward mechanisms I could suggest.

• Both parties enter a sealed bid (your former roommate and his friend can determine their own cost sharing rule). Whoever bids highest wins, but pays the amount the other person wrote down. This is effectively a second price auction, which normally prevents strategic bidding, but as denesp notes, payoffs do depend on your own bid.

This method's split is not equitable. One person will end up getting exactly what they would need to part with their half (again, assuming both "halves" of the console are identical).

• Both parties enter a sealed bid. Highest bid wins, and you split the difference between the two bids and pay that. So if you bid 250 and your friend bids 220, then you win, and end up paying your friend 235. This has a little more room for finagling around with strategic bidding, but the max-min strategy is to also bid your reservation price. If you underbid and lose, then you might get less money than you would want for having to part with your half. So there is some protection here.

This method is equitable, but again, a little more prone to strategy.

Take those suggestions for what they are worth. I think they are relatively easy to implement, so maybe there is something more efficient, but eh.

• The second price auction does not prevent strategic bidding in this setting because bidders' payoff depends on their bid even if they do not win. Consider a case where your reservation price is 200 and mine is 100. By bidding 150 instead of 100 I increase my payoff since I get 150 instead of 100. – Giskard Jun 4 '16 at 8:30
• Envy-freeness does not hold either. If I value the item at 200 and bid 200 while you bid 150, then my surplus is 50 = 200 - 150. But you get 150, which I would prefer. Envy-freeness clearly does not hold in the first priced sealed bid case either. – Giskard Jun 4 '16 at 8:35
• You are correct about the streategy, but for envy freeness, it's not just about 50 vs 150, it's 50 vs 150 and not having the console. I'll amend the answer. – Kitsune Cavalry Jun 4 '16 at 16:16
• The console is also included in my calculaton as I attributed 200 dollars value to it. – Giskard Jun 4 '16 at 17:22
• Ah, you are correct. – Kitsune Cavalry Jun 4 '16 at 17:39