Estimating Buyer Willingness-to-Pay and Seller Reserve Prices from Negotiation Data and the Implications for Pricing

We consider a company selling heterogeneous products where the final price is set by negotiation between sales agents and customers. Negotiation is common in many industries including retail automotive sales, consumer lending, insurance and many business-to-business markets. We assume that each sales agent has a minimum reserve price that he/she will accept and each customer has a maximum willingness-to-pay; however, these values are not directly observable. If the customer’s willingness-to-pay is greater than the sales agent’s reserve price, then a sale takes places at the Nash bargaining equilibrium price, otherwise no trade takes place. Given a record of the outcomes of a series of such negotiations, we would like to estimate the underlying joint distribution of willingness-to-pay and reserve prices. To do so, we propose and analyze a structural estimation method based on the expectation-maximization (EM) algorithm that allows for the willingness-to-pay and reserve price distributions to depend on an arbitrary set of covariates (e.g. characteristics of the customer, product or sales agent) and be potentially correlated. Using a realworld data set, we show, via out-of-sample tests, that our estimation method generates accurate estimates of customer-price response and the prices of lost deals. We also show how our estimated distributions can be used to optimize controls on negotiated prices that can significantly increase revenues relative to both unconstrained negotiations and centrally-optimized fixed prices. These results provide a justification for the policy commonly observed in practice of providing sales staff limited discretion to change prices relative to a centrally-established price list.