Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^m_{1,2}$ $\ldots$ $B^m_{1,f(1)}$
$B^m_{2,1}$ $B^m_{2,2}$ $\ldots$ $B^m_{2,f(2)}$
$\ldots$
Variables in the same row are iid, with $B^m_{n,i} = (1+2m)^K(1-m)^{n-K}$, where $K$ ~ Bin($n$, $\frac{1}{2}$).
Intuition. $B^m_{n,i}$ is Agent $i$'s bankroll after betting proportion $m$ of her bankroll $n$ times in succession, from initial bankroll 1, on a bet with win probability $\frac{1}{2}$ and payoff odds 2:1. On Day 1, $f(1)$ agents have each bet once; on Day 2, $f(2)$ agents have each bet twice; and so on.
Question. Let $S^m_n$ be the sum of row $n$. I'm interested in comparing $S^m_n$ for different values of $m$ as $n \to \infty$. Say that $m$ does better than $m'$ if $\mathbb{P}(S^m_n > S^{m'}_n) \to 1$ as $n \to \infty$. Say that $m$ does best if $m$ does better than $m'$ for all $m' \neq m$. Does there exist $f$ such that some $m$ other than $\frac{1}{4}$ or 1 does best?
Background and motivation. In the regime above, on Day $n$, $f(n)$ agents have each bet $n$ times. Looking at simpler regimes sheds some light on the complicated regime, and shows how the question came up.
First, suppose instead that on Day $n$, $n$ agents have each bet once. That regime may be represented by a sequence of iid random variables $X^m_n$, where $X^m_i$ is Agent $i$'s bankroll after betting, so is $(1+2m)$ or $(1-m)$ with equal probability. Let $S^m_n = \Sigma_{i=1}^n X^m_i$. Clearly, $m=1$ does best in this regime, since the Law of Large Numbers tells us that the bettors' average bankroll tends to the expected bankroll, and $m=1$ maximizes the expected bankroll.
Next, suppose that on Day $n$, one agent has bet $n$ times. That regime fits the form above, with $f(n) = 1$ for all $n$. By the Kelly Criterion, $m=\frac{1}{4}$ does best. Here's the idea behind the Kelly Criterion. The expected growth rate of betting $m$ is $(1-m)^{\frac{1}{2}}(1+2m)^{\frac{1}{2}}$. Each time the agent wins, she multiplies her bankroll by $(1+2m)$ and each time she loses she multiplies it by $(1-m)$. The agent's actual growth rate on Day $n$ is the $n$-th root of her bankroll on Day $n$. So, by the Law of Large Numbers, for large $n$, her actual growth rate tends to be close to her expected growth rate, and the Kelly proportion $\frac{1}{4}$ maximizes that quantity.
Summarizing so far: Betting proportion 1, which maximizes expected bankroll, does best when many agents each bet once. Betting proportion $\frac{1}{4}$, which maximizes expected growth rate, does best when one agent bets many times. Both results are applications of LLN. I hope to find a regime in which some proportion other than $\frac{1}{4}$ or 1 does best. My idea was to look at an $n$ by $f(n)$ structure (on Day $n$, $f(n)$ have each bet $n$ times), for some $f$, in the hope that a 'weighted average' of the many-agents-bet-once and one-agent-bets-many-times regimes will lead to some proportion between $\frac{1}{4}$ and 1 doing best.
Suppose, for example, that on Day $n$, $n$ agents have each bet $n$ times in succession. That regime fits the form above with $f(n) = n$. Will that do? No. An argument I got from Ewain Gwynne shows that, again, $m=\frac{1}{4}$ does best. Indeed, by applying Hoeffding's Inequality to $K$, we have that with probability at least $1-2e^{-2\epsilon^2n}$:
\begin{equation} (1-m)^{n(\frac{1}{2} + \epsilon)}(1+2m)^{n(\frac{1}{2} - \epsilon)} \leq B^m_{n,i} \leq (1-m)^{n(\frac{1}{2} - \epsilon)} (1+2m)^{n(\frac{1}{2} + \epsilon)} \end{equation}
for any $\epsilon$, $n$, and each $i=1,\ldots,n$. By the union bound, we have that with probability at least $1-2ne^{-2\epsilon^2n}$, all the $B^m_{n,i}$ lie within these bounds simultaneously. So with probability at least $1-2ne^{-2\epsilon^2n}$:
\begin{equation} n(1-m)^{n(\frac{1}{2} + \epsilon)}(1+2m)^{n(\frac{1}{2} - \epsilon)} \leq S_n^m \leq n (1-m)^{n(\frac{1}{2} - \epsilon)} (1+2m)^{n(\frac{1}{2} + \epsilon)} \end{equation}
for any $\epsilon,n$. Furthermore, if $(1-m)(1+2m) > (1-m')(1+2m')$ then for sufficiently small $\epsilon$ (depending on $m, m'$) and any $n$:
\begin{equation} n \cdot (1-m)^{n(\frac{1}{2} + \epsilon)}(1+2m)^{n(\frac{1}{2} - \epsilon)} > n \cdot (1-m')^{n(\frac{1}{2} - \epsilon)} (1+2m')^{n(\frac{1}{2} + \epsilon)} \end{equation}
That is, the lower bound for $S^m_n$ is higher than the higher bound for $S^ {m'}_n$. So with probability at least $1-2ne^{-2\epsilon^2n}$, we have $S^m_n > S^{m'}_n$. Finally, noting that for any $\epsilon$, $1-2ne^{-2\epsilon^2n} \to 1$ as $n \to \infty$ and $\frac{1}{4}$ maximizes $(1-m)(1+2m)$ over $m \in [0,1]$, we have the result.
Note that Gwynne's form of argument shows, more generally, what proportion does best in this regime when we vary the win probability and payoff odds. And it shows that in order for some $m \neq \frac{1}{4}$ to do best, $f$ will need to grow at least exponentially.
I've focused on a bet with win probability $\frac{1}{2}$ and payoff odds 2:1. But that's just for concreteness. The more general aim is: given a regime, find which $m$ does best as a function of the win probability and payoff odds.
Any suggestions much appreciated! Thank you!
