A bidding strategy using multivariate distribution

Since Friedman published his seminal paper in 1956, many models for competitive bidding have been proposed. Two major criticisms of most of these bidding models are the use of a single distribution with fixed parameters to model the bidding patterns of all bidders and the assumptions of independence of bids among competitors. However, many attempts to develop multivariate distributions for the bidding patterns fail due to insufficient data. This paper proposed to develop multivariate distributions for groups of contractors. Contractors are clustered into groups according to the official classification that are based on size and experience of the contractors. Distributions and covariance matrix for the groups can then be determined and realistic multivariate distributions to represent different bidding patterns and dependence among bidders can be developed. This paper also proposes a method that uses the EM algorithm to estimate the covariance matrix of bids from different contractor groups. A large data set consisting of 2,885 bids from 267 projects submitted during the period 1990 to 1996 in Hong Kong was used to develop the multivariate model. This data set also includes the cost estimates of the 267 projects supplied by a collaborating contractor who determined these estimates during the bidding processes of the projects. In this paper, a bid to cost ratio is first used to measure the level of competitiveness of a contractor. Sample means and sample standard deviations of the log of the bid to cost ratio for the contractor groups are obtained and outliers removed from the data set. A multivariate log-normal distribution is used successfully to represent the bidding patterns and correlations among bidders. The paper demonstrates, with examples, the application of the multivariate log-normal model in developing bidding strategy by calculating the probabilities of winning and the corresponding mark-ups for contract bidding.