# Offers with Asymmetric Information

#### Base Case

Suppose a buyer is bidding on an asset of unknown value \(V\). The buyer knows that \(V\) falls between 0 and 1 (in particular, it is uniformly distributed between 0 and 1). The seller of the object does know the value of \(V\). How much should the buyer bid on the asset?

Let’s first think about how the seller will behave as a function of the buyer’s bid amount \(b\). The seller doesn’t want to lose money, so she’ll only complete the trade if \(b \ge V\). This means the buyer faces the following problem:

\[ \underset{b\in{[0,1]}}{\text{max}} E \big[(V-b) \mathbf{1}[b \ge V]\big] \,.\]

\(\mathbf{1}[b \ge V]\) is an indicator function that equals one when the bid is at least as big as the realized value \(V\) and is zero otherwise.

We can calculate the expected value in the problem above. \[\begin{align*} E \big[(V-b) \mathbf{1}[b \ge V]\big] =& \int_0^1 (V-b) \mathbf{1}[b \ge V] dV \\ =& \int_0^b (V-b) dV \\ =& \frac{V^2}{2} - bV \Big|_{V=0}^{V=b} \\ =& \frac{-b^2}{2}\\ \end{align*}\]

The bid that maximizes this expected value is 0! With asymmetric information here, the buyer should bid nothing to avoid being adversely selected by the superior information of the seller.

#### Case with Known Liquidity Shock for Seller

Now let’s assume the buyer knows that the seller has a liquidity need to sell the asset. In particular, the seller will accept any offer greater than half of the asset’s actual value \(V\). How much should the buyer bid on the asset in this case?

The buyer now solves:

\[ \underset{b\in{[0,1]}}{\text{max}} E \big[(V-b) \mathbf{1}[b \ge 0.5 V]\big] \,.\]

The expected value in this problem is: \[\begin{align*} E \big[(V-b) \mathbf{1}[b \ge 0.5 V]\big] =& \int_0^{\text{min}(2b,1)} (V-b) dV \\ =& \frac{V^2}{2} - bV \Big|_{V=0}^{V=\text{min}(2b,1)} \\ =& \mathbf{1}[b \le 0.5] \left( \frac{4b^2}{2} - 2b^2 \right) + \mathbf{1}[b > 0.5] \left( \frac{1}{2} - b \right) \\ =& \mathbf{1}[b \le 0.5] \cdot 0 + \mathbf{1}[b > 0.5] \left( \frac{1}{2} - b \right) \,. \end{align*}\]

Thus, an optimal bid is any bid between 0 and 0.5. In expectation, the bidder will lose money anytime they bid over 0.5, but their expected value is zero for any bid between 0 and 0.5.

#### Case with Uncertain Liquidity Shock for Seller

Now suppose that the buyer does not know for sure that the seller has a liquidity need. In particular, the buyer only knows that there is a 50% chance that the seller has a liquidity need. What is the optimal bid in this circumstance?

The buyer now solves:

\[ \underset{b\in{[0,1]}}{\text{max}} \frac{1}{2} E \big[(V-b) \mathbf{1}[b \ge V]\big] + \frac{1}{2} E \big[(V-b) \mathbf{1}[b \ge 0.5 V]\big] \,.\]

For bids below 0.5, the expected value is

\[ \frac{1}{2} \cdot \frac{-b^2}{2} + \frac{1}{2} \cdot 0 \,,\]

which is negative for all nonzero bids.

For bids greater than or equal 0.5, the expected value is

\[ \frac{1}{2} \cdot \frac{-b^2}{2} + \frac{1}{2} \cdot \left(\frac{1}{2} - b\right)\,, \]

which is negative for all bids greater than or equal to 0.5.

Therefore, the optimal bid is again 0, and there is no trade.