# Poker 51: how much an initial decoded stamp pattern tells us

**Research note for the v3.5 Standard game.** This extends the initial, pre-bet posterior in the supplied v3.5 mathematical delta. It is an exact information calculation under a uniform initial deal, not a measurement of visitor recall, game returns, or jbits. The playable Poker 51 rules and current odds panel are unchanged.

## Observation and posterior

Condition on the player's original five identities. Let $U$ be the other $u=46$ identities; the House initially holds a uniformly chosen five-subset $H\subset U$. Let $E\subset U$ be the $e$ earned identities the player can decode without error. The observed stamped set is $K=H\cap E$, with $k=|K|$. Every earned identity not seen is known absent from the House. Put $B=U\setminus E$, $b=u-e$, and $h=5-k$. Conditional on the complete observed $K$, the unresolved House identities are a uniform $h$-subset of $B$.

For an unordered $r$-card draw set $D\subset U\setminus K$, let $j=|D\cap B|$. The exact posterior is

\[
\Pr(D\mid K)=\frac{\binom{b-j}{h}}{\binom{b}{h}\binom{u-5}{r}}.
\]

The numerator counts compatible House hands avoiding $D$; the last denominator counts draws from any resulting $(u-5)$-card pool. A known House card has probability zero of appearing next. An earned-but-absent identity has next-card probability $1/(u-5)$; each unearned identity has probability $(b-h)/(b(u-5))$. Simply deleting recognized House cards and treating everything else as uniform misses the negative evidence from earned stamps that are absent.

## Information in bits, with no payoff conversion

Before seeing $K$, the House hand has $\binom{u}{5}$ equally likely possibilities. A **specific** observed set $K$ of size $k$ leaves $\binom{b}{5-k}$ possibilities, each equally likely. Its information gain about the initial House hand is

\[
I_k=\log_2\frac{\binom{u}{5}}{\binom{b}{5-k}}\quad\text{bits}.
\]

The number $k$ is hypergeometric:

\[
\Pr(|K|=k)=\frac{\binom e k\binom b{5-k}}{\binom u5}.
\]

Therefore the mean information gain is the exact finite sum $\sum_k\Pr(|K|=k)I_k$. It equals the Shannon entropy of the observed stamp pattern because that pattern is a deterministic function of the House hand. For fixed $u=46$, it is nondecreasing as the earned set grows: if $E_1\subseteq E_2$, the $E_1$ observation can be read from the $E_2$ observation.

At $e=8$, the mean is approximately **3.95616 bits**. The chance of seeing none of the eight earned identities is exactly $\binom{38}{5}/\binom{46}{5}\approx0.36618$. Even then, the observation removes eight possible House identities and yields about 1.44938 bits. The next-card probability of each earned-but-absent identity is $1/41\approx2.4390\%$, versus $33/(38\cdot41)\approx2.1181\%$ for each unearned identity: a ratio of $38/33\approx1.1515$. These statements describe the uniform initial deal only.

| Effective earned identities outside the player's hand | Mean information about initial House hand |
|---:|---:|
| 0 | 0 bits |
| 4 | 1.98156 bits |
| 8 | 3.95616 bits |
| 12 | 5.92217 bits |
| 24 | 11.73841 bits |
| 46 | 20.38654 bits, the entire House identity set |

## What the information can and cannot establish

If observing stamps costs nothing, the legal actions and payoff rule stay the same, and the opponent follows the same policy, the **optimal expected** utility with stamps is weakly greater than without them. The informed player can always ignore the stamps. This policy-class inclusion does not promise an improvement on every deal, a positive betting return, a value per symbol, or effective human recall. Acquisition costs, uncertain decoding, and House behavior after observed bets require separate models.

The current Poker 51 odds panel intentionally ignores stamps and reports categories rather than win probabilities. A useful next experiment is to freeze deals and the House policy, compare baseline and correctly stamp-aware player choices on identical states, and report paired utility by $e$ and $k$. After a House bet or draw, recompute the posterior with that action likelihood; the initial formula above no longer suffices.

## Replay

Run `python verify_stamp_information.py`. The verifier uses exact fractions and exhaustively compares the posterior with direct House-hand enumeration in three small models; it also checks the 46-card single-draw marginals. [Receipt](stamp_information_receipt.json) records the stated model, probabilities, and approximate information values. It imports no game evaluator.