BONUS! Maximizing Surprise

TL;DR

The paper tackles how to maximize audience surprise in a two-player, $n$-round contest by tuning the final-round bonus $x$ under a Beta prior on per-round win probability $p$. It introduces a main technical lemma that makes the analysis tractable by showing a fixed ratio between consecutive rounds’ expected surprise, reducing the problem to a trade-off between the final and penultimate rounds and yielding an $O(n)$ algorithm for general Beta priors. The authors derive closed-form or near-closed-form solutions for the finite case in symmetric, uniform, and certain priors, and provide asymptotic characterizations linking the optimal bonus to the prior’s skewness and uncertainty (e.g., the “expected lead” in the certain case and a unique maximizer in the symmetric/asymptotic regimes). These results illuminate how prior beliefs shape optimal scoring rules and offer practical guidance for designing suspenseful competitions, with potential field experiments and extensions to account for time effects and more complex score structures.

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.

Figures (13)

• Figure 1: Optimal bonus in asymptotic case: for all $\alpha,\beta\geq 1$, we use $\frac{\alpha-\beta}{\alpha+\beta}$ to measure the skewness of the prior and $\frac{1}{\alpha+\beta}$ to measure the uncertainty of the prior. The figure shows that when $n$ is sufficiently large (we use $n=10000$ here), the relationship between the optimal bonus size and skewness/uncertainty.
• Figure 2: Belief curves with low/high overall surprise
• Figure 3: Overview of our method
• Figure 4: A single round
• Figure 5: Two rounds

Theorems & Definitions (30)

• Definition 2.1: Surprise
• Claim 2.2: Beta's posterior is Beta
• Claim 2.3: Order Independence
• Corollary 2.4: State Dependence
• Lemma 3.1: Main Technical Lemma
• Claim 3.2
• proof : Proof of Claim \ref{['cla:roundsurp']}
• proof : Proof of Lemma \ref{['lem:ratio']}
• Theorem 4.1
• Lemma 4.2