The thesis addresses decision-making under uncertainty, where agents do not know the true probabilities of events but instead use their own estimates. We model these estimates as the sum of the true probability and a random error with expected value of zero. We present two introductory models: the Prediction Tournament and the Gentleman's bet and show that it is possible to quantify the relative performance of agents in estimating probabilities without knowing the true probabilities. We then analyze three more complex applications: In the Bookmaker's Dilemma, we identify the optimal price range that the bookmaker sets for agents and show that it depends on the inaccuracy of the bookmaker's own estimate, leading to more conservative pricing. In the Kelly rules, we examine the impact of error on the long-term logarithmic growth of capital. Finally, we provide a continuous generalization and numerically illustrate the behavior of the average error cost in the best-item selection problem and in an auction. In all the models considered, the principle of quadratic error consistently holds: for small errors, the error cost in probability estimation scales as the variance of the error.
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