TejasviTomar
October 2026 · Preprint

A certified upper bound for the beat the average game

Tejasvi Singh Tomar

In plain words

Draw four nonnegative numbers independently from the same distribution. What is the largest possible chance that the first three add up to less than twice the fourth? Fritz asked this on MathOverflow in 2024; with Bellec he proved it lies between 0.4006953… and 0.41672 and conjectured that the lower value is the answer. Georgiev, Gómez-Serrano, Tao and Wagner list it as Problem 6.39 and call it the beat the average game. This paper proves it is at most 0.4006983, within three millionths of the conjectured answer, with a computer-checked certificate.

Abstract

Let C be the supremum of P[X_1 + X_2 + X_3 < 2X_4] over all probability measures on [0, ∞), where X_1, ..., X_4 are independent with the given law. This is Problem 6.39 in the list of Georgiev, Gómez-Serrano, Tao and Wagner. Bellec and Fritz proved 0.4006953 ≤ C ≤ 673/1615 < 0.41672 and conjectured that the lower bound is the exact value. We prove C ≤ 4006983/10000000 = 0.4006983, which is within 3·10^-6 of the conjectured value. The proof applies the flag algebra method of Razborov to samples of eight points. After sorting, such a sample determines which of the 280 inequalities 2x_d > x_a + x_b + x_c among its coordinates hold, and the probability in question is the expected fraction of these inequalities that hold. Independence of the sample imposes positive semidefinite constraints on the distribution of these sign patterns, and since they hold for every measure, no limit argument is needed. We combine them in a semidefinite program, round its solution to an exact sum of squares, and evaluate the bound in exact rational arithmetic on all 45,319,744 patterns of eight points whose restrictions to seven points are realizable. Gordan certificates exclude the two of these patterns that exceed the bound, and trees of Gordan certificates show that the list of realizable patterns of seven points is complete. A checker that uses only the Python standard library verifies the certificate.

A half-disc fan opening upward from a black dot at the bottom centre. Three thick black curves lead to large yellow dots that branch into thirty small yellow dots; from each small dot a grey fan of fine lines spreads to an outer arc. Beyond the arc, thin teal spikes of different lengths form thirty flame-shaped tufts.
Illustration of the sign patterns behind the proof. The pattern of k sorted positive numbers records which of the comparisons “twice one number exceeds the sum of three others” hold among them; the probability in question is the expected fraction that hold in a random sample from the distribution. Dropping the largest number turns a pattern of k numbers into one of k − 1, so the patterns that occur form a tree: one for three numbers (black dot), 3 for four (large yellow dots), 30 for five (small yellow dots) and 1,290 for six (outer ends of the grey lines). Each teal spike is proportional in length to the number of patterns of seven numbers that extend the pattern of six at its base, 136,800 in all; within each tuft the spikes are ordered with the longest in the middle. The proof checks that these lists are complete and joins the seven-number patterns into 45,319,744 candidates for eight.