Introduction to Scoring

Let us agree that we have a fair coin and a fair way to flip that coin. Neither of us, then, should be very surprised should a flip come up heads. Nor should either of us be very surprised should a flip come up tails.

Suppose I am surprised by an amount $s_H$ if the outcome is heads, and an amount $s_T$ if the outcome is tails. Then on average I will be surprised by an amount $$ S = \frac{1}{2}\times s_H+\frac{1}{2}\times s_T $$ Very reasonably, we should each be equally surprised regardless of the outcome, so $S=s_H=s_T$. Surprise is a weird thing to quantify, so let us define the surprise caused by a fair coin toss to be 1.

The surprise of two coin tosses, then should be 2, right? There are four equally likely outcomes: heads followed by heads, heads followed by tails, tails-tails, and tails-heads, so $$ 2=S=\frac{1}{4}\times s_{HH}+\frac{1}{4}\times s_{HT}+\frac{1}{4}\times s_{TT}+\frac{1}{4}\times s_{TH} $$ and therefore equal surprise from each: $$ S=s_{HH}=s_{HT}=s_{TT}=s_{TH} $$ With three tosses, $$ 3=S=8\times\frac{1}{8}\times s_{ttt} $$ and so on. The important takeaway here is that $S$ units of surprise result from realizing one of $2^S$ equally likely outcomes.