Question: You start with just the number 1 written on a slip of paper in a hat. Initially, there are no other slips of paper in the hat. You will draw from the hat $100$ times, and each time you draw, you have a choice: If the number on the slip of paper you draw is $k,$ then you can either receive $k$ dollars or add $k$ higher numbers to the hat.For example, if the hat were to contain slips with the numbers $1$ through $6$ and you drew a $4,$ you could either receive $\$4$ or receive no money but add four more slips numbered $7, 8, 9$ and $10$ into the hat. In either case, the slip with the number $4$ would then be returned to the hat.

If you play this game perfectly — that is, to maximize the total amount of money you’ll receive after all $100$ rounds — how much money would you expect to receive on average?

## Solution

Each time we draw a number $k,$ we have to figure out if it’s worth more to cash it in or to hedge to add $(h+k)$ numbers in the bowl to start the next round, taking into account how many rounds we have left, $\ell.$

### Approximate argument

Suppose we’re playing the game with current highest number $h.$

Crudely, if we decide to cash in for $n$ rounds in a row, the expected value of play is

\[n\langle k \rangle = \frac12 nh.\]If we decide to hedge for the first $(n-1)$ turns, and cash in on the last one, then the expected value of play is

\[\frac12 \left(\frac{h+\frac12h}{h}\right)^{n-1} h = \frac12 \left(\frac32\right)^{n-1}h,\]since we expect to raise the highest number $h$ by $\frac12h$ on each draw, and the expected value of the cash in is half the highest number at the time of the draw.

So, the hedging strategy wins out at $n=5,$ and we expect the best outcome to scale like $\approx \left(3/2\right)^\ell.$

In the real game, this $n$ should be lower since the first turn doubles $h$ from $1$ to $2.$

### Rigorous argument

Let’s call the expected value of our position $\Omega(h, \ell, k)$ where $h$ is the biggest number currently in the bowl, $\ell$ is the number of turns we have left, and $k$ is the number we just drew from the bowl.

If we cash in, then we immediately get $k$ and can expect to add the average value of all the games we can immediately move it, i.e.

\[\langle\text{cash in}\rangle_{h,\ell,k} = k + \frac{1}{h}\sum\limits_{j=1}^h \Omega(h, \ell-1, j).\]If we hedge then we get nothing immediately but raise the value of $h$ to $(h+k)$ in the next game:

\[\langle\text{hedge}\rangle_{h,\ell,k} = \frac{1}{h+k}\sum\limits_{j=1}^{h+k} \Omega(h+k, \ell-1, j).\]So, our strategy

\[\Omega(h,\ell,k) = \max\{ \langle\text{cash in}\rangle_{h,\ell,k}, \langle\text{hedge}\rangle_{h,\ell,k} \}.\]which can be evaluated analytically.

When $\ell=1,$ we have just one turn left and there is nothing we can do but take the money on the table

\[\Omega(h, 1, k) = k.\]Plugging this in to the decision, we get

\[\begin{align} \Omega(h, 2, k) &= \max\{ \langle\text{cash in}\rangle_{h,2,k}, \langle\text{hedge}\rangle_{h,2,k} \} \\ &= \max\{ k + \frac{1}{h}\sum\limits_{j=1}^h \Omega(h, 2-1, j), \frac{1}{h+k}\sum\limits_{j=1}^{h+k} \Omega(h+k, 2-1, j)\} \\ &= \max\{ k + \frac{1}{h}\sum\limits_{j=1}^h j, \frac{1}{h+k}\sum\limits_{j=1}^{h+k} j\} \\ &= \max\{k + \frac{1}{h}\frac{h(h+1)}{2}, \frac{1}{h+k}\frac12(h+k+1)(h+k)\} \\ &= \max\{k + \frac12(h+1), \frac12(h+k+1)\} \\ &= k + \frac12(h+1), \end{align}\]so, with $2$ turns left, we should take the cash out.

Going again, the decision is

\[\begin{align} \Omega(h, 3, k) &= \max\{ \langle\text{cash in}\rangle_{h,3,k}, \langle\text{hedge}\rangle_{h,3,k} \} \\ &= \max \{k+h+1, \frac12 k + h+1\} \\ &= k+h+1, \end{align}\]and, with $3$ turns left, we should cash out.

With $4$ turns left, the decision is

\[\begin{align} \Omega(h, 4, k) &= \max\{ \langle\text{cash in}\rangle_{h,4,k}, \langle\text{hedge}\rangle_{h,4,k} \} \\ &= \max \{\frac12 k + \frac32(h+1), \frac32(k+h+1)\} \end{align}\]and, at last, we should hedge.

From here on out, the $(k+h+1)$ picks up one factor of $\frac32$ on each further iteration, giving

\[\boxed{\Omega(1,\ell,1) = 2\times\left(\frac32\right)^{\ell-2}},\]for $\ell \geq 2$ and $\Omega(1,1,1) = 2.$

For a $100$ round game, we expect to win

\[2\times\left(\frac32\right)^{98} \approx \\$361,387,713,364,635,766.58\]### Update: typical performance

From Laurent Lessard, the distribution of scores is log-normal, i.e. the typical outcome is significantly less than the mean.

Knowing the strategy from above, we can model this using a random walk. For $t$ hedging steps, we expect $h$ to rise by a factor of

\[\begin{align} h_t &= h_0 (1+x_1)(1+x_2)\cdots(1+x_t) \\ &= e^{\log(1+x_1) + \log(1+x_2) + \cdots + \log(1+x_t)}\\ &= e^{t\langle \log(1+x)\rangle} \end{align}\]where $x$ is the random extent of $k$ relative to $h$ in a given run (which has average $\frac12$).

The average of $\log(1+x)$ is

\[\begin{align} \lambda &= \langle\log(1+x)\rangle \\ &= \int\limits_0^1dx\,\log(1+x) \\ &= \int\limits_1^2 dz\, \log z \\ &= 2\log 2 -1. \end{align}\]According to the strategy, we should hedge $97$ times and then cash in three times.

This will lead to an underestimate since $\lambda$ is the asymptotic limit of the growth rate of $h.$

The first jump is a doubling, the second jump is the factor $7/4,$ and so on, settling to $\lambda$ as $h$ grows. Replacing the first two factors of $e^\lambda$ with these more exact figures, we get an estimate of

\[\begin{align} \Omega(1,100,1)_\text{median} &\approx \frac32 \cdot 2\cdot \frac74 e^{95\lambda} \\ &= 4.54\times10^{16}. \end{align}\]