  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.
