Estimating arbitrator's hidden judgement in Final Offer Arbitration

In Final Offer Arbitration the arbitrator has to choose between two prices - salary offer a and demand b, presented by two negotiating sides. This situation is common e.g., in Major League Baseball since 1974, for players with at least three years experience. We model the arbitrator's decision process as follows. First, the arbitrator chooses an appropriate sum Z. Then he selects the one among a and b which is closer to the 'proper sum' Z. The final amount selected is Y. Based on the history of n cases, the information available is the triplets {(Y _i, a _i, b _i): i = 1, 2, ... n} (the {Z _i} are hidden). It is assumed that the {Z _i} are random variables with a distribution which depends on an unknown parameter θ and the challenge is to estimate θ. Furthermore, since each case had different merits and characteristics, the resulting distribution is case-specific. Thus, our model allows the inclusion of explanatory variables. The statistical algorithm which we shall use is the Expectation-Maximization (EM) algorithm. In the paper, the statistical model is introduced in detail and the application of the EM algorithm to the available data is explained. Two numerical examples illustrate the use of the EM algorithm in estimating the arbitrator's hidden judgements.