# Buy-or-sell deal when there is cash shortage

A company is owned by two partners: partner $A$ holds a fraction $a$ and partner $B$ holds a fraction $b$ (with $a+b=1$). Partner $A$ wants to break the partnership. A common procedure for this is that $A$ should suggest a price $P$ for the entire company and let $B$ choose either of two options:

• Buy $A$'s share for $aP$, or -
• Sell his share to $A$ for $bP$.

This procedure incentivizes $A$ to suggest a fair price. Moreover, if $B$ thinks that $A$'s price is unfair, he can profit from this - if he thinks that the price is too low he can opt to buy and if he thinks that the price is too high he can opt to sell. The advantage of this procedure is that it guarantees fairness without requiring for an external arbitrator.

A possible problem with this procedure is that it requires $B$ to have enough cash for buying $A$'s share. If $B$ doesn't have enough cash, and cannot get credit, he might have to select the "sell" option even when the price is low. Moreover, if $A$ knows that $B$ is short in cash, he might exploit this fact and deliberately suggest a low price in order to force $B$ to sell his share.

MY QUESTION IS: is there an improved division mechanism that handles this problem?

Let's assume $A$ stated that the company is worth $P$.

I propose the following mechanism:

$B$ has to choose one of two alternatives:

1. Sell his share for $bP$.
2. Pay an amount of his choice, $x\leq aP$, and then with probability $\frac{bP+x}{P}$ get $A$'s share and with probability $1-\frac{bP+x}{P}=\frac{aP-x}{P}$ lose his own share (in addition to paying $x$).

If $B$ thinks that the company is worth less than $P$, he should go with the first option, just like in the original scheme.

Otherwise he should pick the second alternative, paying as much as he can to maximize his profit expectation. Should $B$ have enough money, he would pay $aP$ and get $A$'s share w.p. 1, if not, there is no way of allowing $B$ to get the company without ripping $A$, unless this is done randomly..

• This is an interesting use of randomness. Thanks! The issue that I see here is that it seems unfair for risk-averse agents. I.e., if B is forced to bet on his share with a certain probability p, his utility is lower than receiving a share of p with probability 1. – Erel Segal-Halevi Dec 11 '14 at 9:27