ABSTRACT The FIFA Men’s World Cup (FWC) is the most important football (soccer) competition, attracting worldwide attention. A popular practice among football fans in Brazil is to organize contests in which each participant informs guesses on the final score of each match. The participants are then ranked according to some scoring rule. Inspired by these contests, we created a website to hold an online contest, in which participants were asked for their probabilities on the outcomes of upcoming matches of the FWC. After each round of the tournament, the ranking of all participants based on a proper scoring rule was published. In this article we estimate, by means of simulations, the ability of the best forecasters of our contest, considering that their good performances could be due to randomness. We also study the performance of some methods to aggregate individual forecasts, in order to study if some sort of wisdom of crowds (WOC) phenomenon was verified in the contest.

RESUMO Apresentamos duas demonstrações, por métodos diferentes, de que polinômios positivos em caixas fechadas no ℝ 2 podem ser escritos como polinômios de Bernstein bivariados com coeficientes estritamente positivos. Ambas as estratégias de demonstração podem ser estendidas para provar o resultados análogo para polinômios que são positivos em caixas fechadas no ℝ n, n > 2.

ABSTRACT We present two different proofs that positive polynomials on closed boxes of ℝ 2 can be written as bivariate Bernstein polynomials with strictly positive coefficients. Both strategies can be extended to prove the analogous result for polynomials that are positive on closed boxes of ℝ n, n > 2.