Size Matters, so Does Duration: The Interplay between Offer Size and Offer Deadline

Abstract

This paper investigates the interplay between offer size and offer deadline in a Stackelberg game involving a proposer and responder. The proposer acts first by making an offer to the responder with a deadline, and the responder, concurrently following a continuous time finite-horizon search for alternative offers, has to respond to the proposer's offer by the deadline. Taking into account the responder's reaction, the proposer's optimal strategy can vary from an exploding offer---an offer that has to be accepted or rejected on the spot---to an offer with extended deadline under different market conditions, proxied by characteristics of the alternative offer distribution. In particular, the proposer should offer an exploding offer when the alternative offer market is unfavorable to the responder, and the harsher it is, the smaller the offer size. On the other hand, when the alternative offer market is favorable to the responder, the proposer can benefit from making a smaller (compared to the exploding offer) non-exploding offer, and the more favorable the market, the smaller the offer size and the longer the deadline. Our analysis is further extended to the case where the responder has private knowledge of the alternative offers' arrival rate, and we characterize the optimal strategy for the proposer when she either makes a single offer or a menu of offers that serves as a self-selection mechanism. In the latter case, the optimal menu of offers can be implemented as a sign-up bonus type of contract.

Keywords: search theory, ultimatum, offer size, offer deadline, mean residual life order, sign-up bonus