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Assume nonnegative wealth and rewards, and initially $0<p_i\le 1$. Set $F_0(x)=x$, and let $R_i$ take the values $0,2a_i$ with equal probability conditional on not being caught. For the retirement threshold with multiplicative capture risk, the dynamic programming equation is the displayed maximum; capture contributes zero. Write $t={\max }_i{a}_i{q}_i/p_i$.

Inductively, if $x\ge t$, all future successful wealth levels also lie above $t$, so $F_{s-1}(x+R_i)=x+R_i$. Continuing in town $i$ is then worth $q_i(x+a_i)\le x$, and retirement is optimal. If $x<t$, choose $i$ with $a_i{q}_i/p_i>x$. Continuing for just one night and then retiring is worth $q_i(x+a_i)>x$, so immediate retirement cannot be optimal for any $s\ge 1$. This proves an optimal policy continues exactly when $\boxed{x<t}$, with retirement chosen at the threshold.

Now suppose $q_1>q_2$ and $q_1{a}_1>q_2{a}_2$. Since also $p_1<p_2$, town $1$ has the larger threshold, so $t=a_1{q}_1/p_1$. The functions $F_s$ are nondecreasing in wealth and in $s$, and $F_s(x)\ge x$; these properties follow by induction from the dynamic programming equation.

Prove by induction on the remaining horizon that town $2$ is never optimal when continuation is worthwhile. For $s=1$, the difference between the town values is $(q_1-q_2)x+q_1{a}_1-q_2{a}_2>0$. If $a_2\le a_1$, monotonicity giveswith strictness because the first bracket is positive under the hypotheses.

For $a_2>a_1$, first suppose $x+2a_2\ge t$. The second town's successful large-reward branch retires, so its value is $q_2[F_{s-1}(x)+x+2a_2]/2$. Town $1$ is worth at least $q_1[F_{s-1}(x)+x+2a_1]/2$. Their difference is at leastIf instead $x+2a_2<t$, all states after either first-night reward remain below the threshold. By induction a policy starting in town $2$ then chooses town $1$ on its second night. Its value isInterchanging the first two towns gives exactly the same value, because capture probabilities multiply and independent rewards add. After starting in town $1$, however, the induction hypothesis says town $2$ is strictly inferior on the second night at every possible surviving wealth level. Replacing that second choice by the optimal one strictly improves the value. This proves town $1$ dominates throughout every finite horizon.

If a town has $p_i=0$ and $a_i>0$, its threshold is interpreted as infinity and retirement is not optimal before the horizon ends. A town with $q_i{a}_i=0$ offers no gain; the natural threshold is zero. These conventions handle the degenerate capture probabilities rather than dividing by zero.