Risk-Sensitive Competitive Bidding Model and Impact of Risk Aversion and Cost Uncertainty on Optimum Bid

Abstract

A decision analysis model is presented that can be used by a contractor to determine the optimum lump-sum bid that maximizes expected utility for an upcoming job. Required inputs include the contractor’s utility curve, the distribution of the opponents’ low bid-to-cost ratio, and the distribution of the project cost. Closed-form expressions for expected utility are developed for an exponential utility curve and normally distributed project cost that can also be used as approximations for other models. Any appropriate bidding model can be used to estimate the probability to win. Important concepts introduced include the certain equivalent and the risk premium for a project profit lottery, corporate indifferent buying price (CIBP), and risk-adjusted cost (RAC). Parametric curves from the maximization of expected utility over a total of 21,462 combinations of the contractor’s risk tolerance and the standard deviation of project cost illustrate the considerable impact of risk aversion and cost uncertainty on the optimal bid-to-cost ratio, on the probability of losing money on the job, and on the certain equivalent and risk premium for profit. This sensitivity analysis shows that maximizing expected utility can result in an optimal bid-to-cost ratio that can be significantly different (smaller or larger) than the bid-to-cost ratio found by only maximizing expected profit and ignoring the significant impact of uncertainty in project cost σ. The presented procedure allows contractors to use any appropriate model for the probability of winning (such as those proposed by Friedman, Gates, Carr, or Ioannou), assess their risk tolerance and the standard deviation of project cost, and select an optimal bid price that maximizes their expected utility for profit.