FISHERY BULLETIN: VOL. 78, NO. 1 



inclusion of an absorbing state would seem to be a 

 more realistic assumption. It is included in what 

 follows. 



A coarser grid implies, in a sense, less informa- 

 tion about the state of the system. As the interval 

 (0, Xj) becomes large, our information has de- 

 creased about the true state of the population and 

 this increased uncertainty is reflected in increased 

 risk of absorbtion. Similarly, a finer grid implies 

 more exact information — a grid should not be used 

 which is finer than the precision of the estimate of 

 the population size. 



Optimal policies for grids of 16, 26, 51, 101, and 

 501 equally spaced points (including zero) for both 

 rivers are shown in Table 1. The optimal equilib- 

 rium population for the equivalent deterministic 

 models are shown also. All numbers are in units of 

 millions offish. 



The optimal policies are all of the base stock 

 variety, i.e. it is optimal to harvest to a fixed 

 number of spawners, or else not to harvest at all. If 

 the 501-point grid is taken as the standard, it can 

 be seen that each coarser grid has as its base stock 

 size the grid point closest to the base stock size for 

 the 501-point grid. 



Figure 1 gives the long-run (ergodic) cumula- 

 tive distribution of being in any state when follow- 

 ing an optimal policy on grids of 16, 26, 51, and 101 

 points. Grid size can be seen to play a crucial part 

 in estimating the probabilistic behavior of the 

 population. For the Wood River, extinction with 

 probability one is predicted on grids of 16 and 26 

 points, while the probability is zero on grids of 51 

 and 101 points, so long as zero is not the initial 

 state. Similar but not identical results are valid 

 for the Branch River. It should be emphasized that 

 for a = 1, i.e., when the objective is given by Equa- 

 tion (1.2a), the estimated average per period har- 

 vest of any policy depends entirely on the ergodic 

 distribution that arises from that policy. There- 

 fore, this variation in estimated long-run behavior 

 due to changes in grid size is nontrivial. 



Probability one of extinction occurs because for 

 a finite state, irreducible Markov chain with an 

 absorbing state, the absorbing state is reached in 



Table l. — Optimal policies for the different grid sizes. 



Wood River 



Grid size 



Branch River 



Vf = min {x,. 0.9333) 

 Vj = min (xj, 0.840) 

 /( = min (x^, 0.700) 

 Yf = min (Xj, 0.770) 

 /f = min (Xf, 0.742) 



Equilibrium stock 0.735 



16 



26 



51 



101 



501 



Deterministic 



y, = min (x^, 0.3333) 

 y = min (x^, 0.4000) 

 y = min (x,, 0.3000) 

 y = min (x,, 0.3500) 

 y = min (Xj, 0.3500) 



Equilitxium stock 0.345 



STATE (WOOD RIVER) 



2 3 



STATE (BRANCH RIVER) 



Figure l. — Ergodic cumulative distributions for optimal har- 

 vesting strategies on grid sizes of 16, 26, 51, and 101 points for 

 the Wood River and the Branch River. 



finite time with probability one. However, for the 

 larger grids, there exist policies that are reducible, 

 in the sense that if the chain does not start in the 

 interval (0, x^), it will never enter that interval. 

 Since P[jC;e(0,Xj )] = 0, and a fraction of this proba- 

 bility has been assigned to the zero state, then P(jc, 

 = 0) = 0. When using the smaller grids that induce 

 Markov chains that are irreducible, the estimated 

 time till absorption varies greatly also. For exam- 

 ple, for the Branch River, ifP(Xi = 0) = andP(Xj 

 = w) = 1/{N-1), where w is a grid point and A^ is 

 the number of states, then a 16-point grid predicts 

 absorption with probability one after 2,000 itera- 

 tions, the 26-point grid predicts only a 76% chance 

 of absorbtion after 2,000 iterations, and the 51- 

 point grid predicts only a 17% chance of absorp- 

 tion. 



When maximizing total expected discounted re- 

 turn, the discounted mean return depends on the 

 values of these intermediate probability distribu- 

 tions, so that coarser grids can be expected to un- 

 derestimate the long-run value of the harvest. 



Finally, for the Wood River, note that the 51- 

 and 101-point grids have similar long-run be- 

 havior. These results suggest that in order to find 



38 



