# Allocation of an Indivisible Good

Two kids at school are arguing over who owns a cool video game cartridge, and the schoolteacher, a ruthlessly practical social planner who knows that kids can only enforce so much AND that the game is worthless to either kid except in its full cartridge, does not believe in sharing; she demands that the one of them buy the other's share of the game off of the other.

She tells each kid that they will privately submit two values to her, $b_1$ and $s_1$ for the first kid and $b_2$ and $s_2$ for the second. Here $b_i$ is (supposed to be) the maximum willingness to pay for the other agent's share of the game, and $s_i$ denotes the minimum price each agent would be willing to sell their share of the game for.

She starts to propose a mechanism:

• If $b_1 > s_2$, and $b_2 < s_1$, then kid 1 buys kid 2's share of the cartridge for $\frac{1}{2}(b_1 + s_2)$.
• If $b_2 > s_1$, and $b_1 < s_2$, then kid 2 buys kid 1's share of the cartridge for $\frac{1}{2}(b_2 + s_1)$.
• If $b_1 > s_2$, and $b_2 > s_1$, then whoever has the higher valuation of the game $(b_i + s_i)$ buys using the proposals shown above.

• She then gets stuck. What if $b_1 < s_2$, and $b_2 < s_1$?

In the first three cases, can we ensure incentive compatibility (no lying is best), individual rationality, Pareto efficiency, and budget balancing (teacher does not have to subsidize to make a trade)? How would you go about showing these results?

Also, suppose that it is illegal for schoolteachers to give subsidies to ensure trades in the fourth case. The lack of Coasian bargaining is particularly bad for the teacher, whose disutility from listening to little tykes fight would be less than the disutility of just subsidizing the trade.

What alternative mechanism might you suggest for the fourth case that could satisfy the four conditions above (IC, IR, PE, BB) if possible? It does not have to involve $b_i, s_i$ necessarily.

The first three cases don't seem exceedingly hard to show intuitively, but I'm curious as to how it would be done formally (just do case by case?). As for the other question, I don't think there is an alternative mechanism, since it looks like neither player is willing to trade their share to the other. Maybe the teacher could be authoritarian and simply allocate the good based on chance, weighted based on each person's $n$ or $1-n$ share of the good?

• I am a little perplexed because it seems to me that the mechanism proposed for the first three cases is clearly not IC. Is this also what you expect to show? – Giskard Apr 23 '16 at 19:40
• @denesp When I said it seemed the first three cases satisfied the conditions, that was me making a guess. If they aren't, then yeah the question is asking to show if it's not. – Kitsune Cavalry Apr 23 '16 at 20:41
• My reasoning is the usual one: if $b_1>s_2$ and you have to pay $\frac{1}{2}(b_1+s_2)$ then you are incentivized to change $b_1$ to some $b_1'$ that statisfies $b_1> b_1'>s_2$. The more I think about the problem the more confused I am. What are the players actually trying to maximize? Do they have a reservation price we can calculate surplus from? – Giskard Apr 24 '16 at 7:00
• The kids have a reservation price, but it is unrevealed to us. You can pick some variable to represent them if you like. – Kitsune Cavalry Apr 24 '16 at 9:02
• Isn't this a setting a la Cramton, Gibbons & Klemperer: Econometrica 87? – Bayesian Apr 27 '19 at 13:47