Partial Lying and the Poisson Binomial Distribution

Abstract

In a one-step trinary lying experiment, subjects privately observe a random device that indicates a low payoff, an intermediate payoff or a high payoff. Subjects are paid whatever they report, inducing some subjects to lie in order to receive a higher payoff. This paper presents a methodology for analyzing the experimental results based on the Poisson binomial distribution. I derive closed-form expressions for the conditional probability that a subject will lie given the number of low and intermediate payoff reports. Given these reports, in addition to the conditional probability of lying, I use the binomial and Poisson binomial distributions to calculate the probability that a subject did lie and the expected number of liars. I use these to calculate a Bayesian update of the binomial priors of observing each type of payoff. All of these are then combined to create the most likely scenario explaining the results.