The Fishing Problem

Question

Suppose you want to go fishing at the nearby lake from 8AM-8PM. Due to overfishing, a law has been instated that says you may only catch one fish per day. When you catch a fish, you can choose to either keep it (and thus go home with that fish), or throw it back into the lake and continue fishing (but risk later settling with a smaller fish, or no fish at all). You want to catch as big a fish as possible; specifically, you want to maximize the expected mass of fish you bring home.

Formally, we might set up this problem as follows: fish are caught at a certain rate (so, the time it takes to catch your next fish follows a known exponential distribution), and the size of caught fish follows some (also known) distribution. We want some decision process which, given the current time and the size of a fish you just caught, decides whether to keep the fish or throw it back.

So the question is: how should this decision be made? Is there some simple (or complicated) way of deciding when to stop fishing? I think the problem is equivalent to determining, for a given time t, what expected mass of fish an optimal fisher would take home if they started at time t; the optimal decision process would keep a fish if and only if the fish is heavier than that expected mass. But that seems sort of self-referential; we're defining the optimal fishing strategy in terms of an optimal fisher, and I'm not quite sure how to proceed.

Answer 1

Let $\lambda$ denote the rate of the Poisson process and let $S(x)=1-F(x)$ where $F(x)$ is the cumulative distribution function of the fish size distribution.

Let $t=0$ denote the end of the day and let $g(t)$, $t\le 0$, denote the expected catch in the interval $(t,0)$ we obtain if using the optimal strategy. Clearly $g(0)=0$. Also, if we catch a fish of size $x$ at time $t$ we should keep it and stop fishing if it is larger then $g(t)$. So this is our decision rule. Thus, a realisation of the process and the realised decision (green point) may look as follows:

Working in continuous time, using ideas from stochastic dynamic programming, the change in $g(t)$ backwards in time is described by a simple differential equation. Consider an infinitesimal time interval $(t-dt,t)$. The probability that we catch a fish of size $X>g(t)$ in this time interval is $$ \lambda dt S(g(t)), $$ otherwise our expected catch will be $g(t)$.

Using a formula for mean residual life, the expected size of a fish larger than $g(t)$ as $$ E(X|X>g(t))=g(t)+\frac1{S(g(t))}\int_{g(t)}^\infty S(x)dx. $$

Hence, using the law of total expectation, the expected catch in the interval $(t-dt,0)$ becomes $$ g(t-dt) =[\lambda dt S(g(t))][g(t)+\frac1{S(g(t))}\int_{g(t)}^\infty S(x)dx] + [1-\lambda dt S(g(t)] g(t). $$

Rearranging, we find that $g(t)$ satisfies $$ \frac{dg}{dt}=-\lambda \int_{g(t)}^\infty S(x) dx. \tag{1} $$ Note how $g(t)$ towards the end of the day decline at a rate equal to the product of the Poisson rate $\lambda$ and the mean fish size $\int_0^\infty S(x)dx$ reflecting that we at that point will be best off keeping any fish we might catch.

Example 1: Suppose that the fish sizes $X\sim \exp(\alpha)$ such that $S(x)=e^{-\alpha x}$. Equation (1) then simplifies to $$ \frac{dg}{dt}=-\frac\lambda\alpha e^{-\alpha g(t)} $$ which is a separable differential equation. Using the above boundary condition, the solution is $$ g(t) = \frac1\alpha\ln(1-\lambda t), $$ for $t\le 0$ shown in the above Figure for $\alpha=\lambda=1$. The following code compares the mean catch using this strategy computed based on simulations with the theoretical mean $g(-12)$.

g <- function(t,lambda, rate) {
  1/rate*log(1-lambda*t)
}
catch <- function(daylength=12, lambda=1, rfn=runif, gfn=g, ...) {
  n <- rpois(1,daylength*lambda)
  starttime <- -daylength
  arrivaltimes <- sort(runif(n,starttime,0))
  X <- rfn(n,...)
  j <- match(TRUE, X > gfn(arrivaltimes,lambda,...))
  if (is.na(j))
    0
  else
    X[j]
}
nsim <- 1e+5
catches <- rep(0,nsim)
for (i in 1:nsim)
  catches[i] <- catch(gfn=g,rfn=rexp,rate=1,lambda=1)
> mean(catches)
[1] 2.55802
> g(-12,1,1)
[1] 2.564949

Example 2: If $X \sim U(0,1)$ a similar derivation leads to $$ g(t) = 1 - \frac1{1-\lambda t/2} $$ as the solution of (1). Note how $g(t)$ tends to the maximum fish size as $t\rightarrow -\infty$.