In so called red and black, a player starts with a given fortune and wants to reach a given target. The reader my want to have a look at the exposition How to Gamble If You Must by Kyle Siegrist located at https://www.maa.org/press/periodicals/loci/how-to-gamble-if-you-must for further reference.

For concreteness, let us call the player Milan, and for simplicity let us say that the original fortune is $x$, $0<x<1$, and the target is $1$. In each round, Milan bets some part of his fortune and wins with some fixed probability $p$ and looses with probability $1-p$. In case of a win, Milan gets his bet back and additionally the same amount of money as the bet was. In case of a loss, Milan looses the amount that he bet. The game is over if Milan reaches the goal or if he has no money left.

One of the strategies is called bold play: if the fortune is less than one half, Milan bets everything. Otherwise the bet is exactly such that in case of a win, the target is reached. Thus the probability of winning is in case of $0\leq x<\frac{1}{2}$ \begin{equation} \varphi(x)=p\varphi(2x), \end{equation} and \begin{equation} \varphi(x)=p+(1-p)\varphi(2x-1) \end{equation} if $\frac{1}{2}\leq x\leq 1$ (here we also allow the cases $x=0$ and $x=1$). Equivalently stated \begin{align} \varphi\left(\frac{x}{2}\right)&=p\varphi(x),\\ \varphi\left(\frac{x+1}{2}\right)&=(1-p)\varphi(x)+p \end{align} for all $0\leq x\leq 1$.

If the probability $p$ is not one half, then $\varphi$ is singular.

In our case, the success function of a strategy is the probability that Milan reaches his target starting with $x$. A strategy $S$ is optimal if any other strategy's success function is bounded from above by $S$'s success function for every admissible bet. It turns out that if $p$ is less or equal than one half, bold play is an optimal strategy (but not the only optimal one, see the section about Unfair Trials in Siegrist's paper).

As far as I can tell, George de Rham was the first to study such kind of systems (in a different context allowing $p$ to be a complex number with absolute value less than 1) in the paper Sur quelques courbes definies par des equations fonctionnelles (Univ. e Politec. Torino. Rend. Sem. Mat. 16 1956/1957 101--113). He shows that the unique bounded solution is a continuous function and the derivative, if it exists, can only be $0$. Further, he points out that the continuous function that solves the system has been studied before. The reader may be interested in the paper Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions by E. de Amo, M. Díaz Carrillo, and J. Fernández-Sánchez, (Acta Appl. Math. 119 (2012), 129--148), especially Proposition 2, where the here relevant properties of $\varphi$ are listed.

In the so-called red and black, a player starts with a given fortune and wants to reach a given target. The reader may want to have a look at the exposition How to Gamble If You Must by Kyle Siegrist for the further reference.

For concreteness, let us call the player Milan, and for simplicity let us say that the original fortune is $x$, $0<x<1$, and the target is $1$. In each round, Milan bets some part of his fortune and wins with some fixed probability $p$ and looses with probability $1-p$. In case of a win, Milan gets his bet back and additionally the same amount of money as the bet was. In case of a loss, Milan looses the amount that he bet. The game is over if Milan reaches the goal or if he has no money left.

One of the strategies is called bold play: if the fortune is less than one half, Milan bets everything. Otherwise the bet is exactly such that in case of a win, the target is reached. Thus the probability of winning is in case of $0\leq x<\frac{1}{2}$ \begin{equation} \varphi(x)=p\varphi(2x), \end{equation} and \begin{equation} \varphi(x)=p+(1-p)\varphi(2x-1) \end{equation} if $\frac{1}{2}\leq x\leq 1$ (here we also allow the cases $x=0$ and $x=1$). Equivalently stated \begin{align} \varphi\left(\frac{x}{2}\right)&=p\varphi(x),\\ \varphi\left(\frac{x+1}{2}\right)&=(1-p)\varphi(x)+p \end{align} for all $0\leq x\leq 1$.

If the probability $p$ is not one half, then $\varphi$ is singular.

In our case, the success function of a strategy is the probability that Milan reaches his target starting with $x$. A strategy $S$ is optimal if any other strategy's success function is bounded from above by $S$'s success function for every admissible bet. It turns out that if $p$ is less or equal than one half, bold play is an optimal strategy (but not the only optimal one, see the section about Unfair Trials in Siegrist's paper).

As far as I can tell, George de Rham was the first to study such kind of systems (in a different context allowing $p$ to be a complex number with absolute value less than 1) in the paper Sur quelques courbes definies par des equations fonctionnelles (Univ. e Politec. Torino. Rend. Sem. Mat. 16 1956/1957 101--113). He shows that the unique bounded solution is a continuous function and the derivative, if it exists, can only be $0$. Further, he points out that the continuous function that solves the system has been studied before. The reader may be interested in the paper Singular Functions with Applications to Fractal Dimensions and Generalized Takagi Functions by E. de Amo, M. Díaz Carrillo, and J. Fernández-Sánchez, (Acta Appl. Math. 119 (2012), 129--148), especially Proposition 2, where the here relevant properties of $\varphi$ are listed.