Probability, Minimax Approximation and Nash-Equilibrium. Estimating the Parameter of a Biased Coin

TL;DR

This paper applies approximation theory and game theory to estimate the bias of a coin, characterizing optimal estimators, their asymptotics, and Nash-equilibria, including a complete solution for two tosses.

Contribution

It introduces a novel combination of approximation and game theory techniques to solve the biased coin parameter estimation problem.

Findings

Characterization of the optimal estimator for biased coin

Asymptotic behavior of estimators as number of tosses increases

Nash-equilibrium established for the estimation game

Abstract

This paper deals with the application of Approximation Theory type techniques to study a classical problem in Probability: estimating the parameter of a biased coin. For this purpose, a Minimax Estimation problem is considered and the characterization of the optimal estimator is shown, together with the weak asymptotics of such optimal choices as the number of coin tosses approaches infinity; in addition, a number of numerical examples and graphs are displayed. At the same time, the problem is also discussed from the Game Theory viewpoint, as a non-cooperative, two-player game, and a Nash-equilibrium is established. The particular case of n=2 tosses is completely solved.