Robust screening under ambiguity

Abstract

We consider the problem of screening where a seller puts up for sale an indivisible good, and a buyer with a valuation unknown to the seller wishes to acquire the good. We assume that the buyer valuations are represented as discrete types drawn from some distribution, which is also unknown to the seller. The seller is averse to possible mis-specification of types distribution, and considers the unknown type density as member of an ambiguity set and seeks an optimal pricing mechanism in a worst case sense. We specify four choices for the ambiguity set and derive the optimal mechanism in each case.

Appendix

1.1 The revelation principle

Here we provide for the reader’s convenience a version of the Revelation Principle (due to Myerson [20, 21]) useful for our purposes. It is adapted from the forthcoming book by T. Börgers [11]. We restrict attention to deterministic mechanisms.

In selling an object to a potential buyer with some valuation for the object (unknown to the seller), the seller can engage into an extensive game with the buyer, where the terminal history is associated with a probability distribution over $[0,1] \times \mathbb {R}$. This distribution is interpreted as the probability of the buyer getting the object, and the payment made by the buyer to the seller. The seller will also commit to a strategy during the game. Naturally, the buyer will also commit to a strategy so as to maximize his expected utility. In our setting, as we assumed that the distribution of buyer valuations is not known with certainty by the seller, we want to determine the extensive game and the strategy of the seller who maximizes his/her worst-case expected revenue where the worst-case is minimum with respect to a set of probability distributions over the valuations of the buyer.

More formally, a mechanism $\Gamma$ is a triplet $(\Theta ,q,\mu )$ where $\Theta$ is a measurable space of messages that the buyer can send to the seller, q is a function mapping each message to the probability that the buyer wins the object, and $\mu$ is a function mapping each message to the payment the buyer makes to the seller (regardless of whether he wins the object or not). Therefore, in our setting the seller wants to determine the mechanism $\Gamma$ that will maximize his/her worst-case expected revenue from the sale where the worst-case is the minimum with respect to the set of distributions of valuations for the buyer.

While the space of messages in an arbitrary mechanism may be very complex and lead to an extensive game, there is a class of mechanisms called direct mechanisms with a very simple message structure: a message is simply the declared valuation (type) of the buyer.

Since the buyer only declares a type (valuation), a “direct mechanism” in our case can be reduced to allocation and payment vectors: i.e., $\{A_i\}_{i=1}^m$ and $\{p_i\}_{i=1}^m$, respectively. A buyer’s optimal strategy is a mapping $\sigma$ from the the first m integers to the first m integers (or, equivalently from $\{t(1),\ldots ,t(m)\}$ to itself) that indicates for every true type of a buyer the declared type of the buyer.

Proposition 4

For every mechanism $\Gamma$ and every optimal buyer strategy $\sigma$ in $\Gamma$ there is a direct mechanism $\Gamma '$ and an optimal buyer strategy $\sigma '$ in $\Gamma '$ such that

• (i.) The strategy $\sigma '$ satisfies

  $$\begin{aligned} \sigma '(j) = j \;\text{ for } \text{ every } \;j \in \{1,\ldots ,m\}, \end{aligned}$$

  i.e., $\sigma '$ is a truth-telling strategy.

• (ii.) For every $j \in \{1,\ldots ,m\}$ the allocation $A_j$ and the payment $p_j$ equal the allocation probability and the payment that result in $\Gamma$ if the buyer plays her optimal strategy $\sigma$.

Proof

For every $j \in \{1,\ldots ,m\}$ we define $A_j$ and $p_j$ as required by (ii) in the statement of the proposition. We show that for this direct mechanism the strategy $\sigma '(j) = j$, i.e., a truth-telling strategy, is optimal for the buyer. Under this strategy, for every j, the buyer with type j (and valuation t(j)) obtains in the mechanism $\Gamma '$ the same worst-case expected utility as in the mechanism $\Gamma$ when choosing strategy $\sigma$. Moreover, when declaring some other type $j' \ne j$, the buyer obtains the same worst-case expected utility that she would have obtained had she played the strategy of type $j'$, $\sigma (j')$ in $\Gamma$. The optimality of truthful declaration for type j in $\Gamma '$ is then a consequence of the optimality of $\sigma (j)$ in $\Gamma$. $\square$