BONUS! Maximizing Surprise

TL;DR

This paper derives the optimal bonus size in multi-round competitions to maximize audience surprise, considering prior beliefs about players' abilities, and finds it depends on prior skewness and uncertainty.

Contribution

It introduces a novel analysis method to compute the optimal bonus, linking it to prior belief characteristics and providing solutions for finite and infinite rounds.

Findings

Optimal bonus depends on prior belief skewness and uncertainty.

In a special case, the optimal bonus approximates the expected lead.

Higher prior skewness and uncertainty lead to larger optimal bonuses.

Abstract

Multi-round competitions often double or triple the points awarded in the final round, calling it a bonus, to maximize spectators' excitement. In a two-player competition with $n$ rounds, we aim to derive the optimal bonus size to maximize the audience's overall expected surprise (as defined in [7]). We model the audience's prior belief over the two players' ability levels as a beta distribution. Using a novel analysis that clarifies and simplifies the computation, we find that the optimal bonus depends greatly upon the prior belief and obtain solutions of various forms for both the case of a finite number of rounds and the asymptotic case. In an interesting special case, we show that the optimal bonus approximately and asymptotically equals to the "expected lead", the number of points the weaker player will need to come back in expectation. Moreover, we observe that priors with a higher skewness lead to a higher optimal bonus size, and in the symmetric case, priors with a higher uncertainty also lead to a higher optimal bonus size. This matches our intuition since a highly asymmetric prior leads to a high "expected lead", and a highly uncertain symmetric prior often leads to a lopsided game, which again benefits from a larger bonus.