384 



Fishery Bulletin 89(3), 1991 



10 20 30 40 



FAVORITISM COEFFICIENT (a) 



Figure 4 



Catch-per-effort for North Pacific albacore predicted by model 

 as function of favoritism coefficient, a, which reflects the 

 ability of fishermen to locate fish concentrations. The expected 

 values of CPE new (solid line) and CPE , d (dashed line) are 

 compared with the true mean population density in CPE 

 equivalents (dotted line). The maximum error in the approx- 

 imation for E[CPE n( , w ] is less than 0.7 CPE units in all cases. 



Figure 5 



Favoritism index of vessels fishing for North Pacific albacore, 

 calculated using Equation 3. The index measures the degree 

 to which fishermen concentrate on strata with high fish 

 populations. It is the proportion of effort in each year exerted 

 in the top 20th percentile of strata where strata are ranked 

 by catch-per-effort. 



P [stratum i missing] = 



(1 - pi) e '« if i in low-density area 

 (1 - Ph) e t»i if i in high-density area. 



(7) 



In each area the expected number of strata with at least one visit is the total minus the expected number of miss- 

 ing strata. Thus, 



E[v,] = n, - n^l-p,)^.; E[v,] = n h - n h (l - p h ) e .ot. 

 The new CPE is the average over the visited strata of the CPE values observed therein, that is, 



v,d| + v h d h 



(8) 



CPE r 



v, + v h 



(9) 



The functional form of Equation 9 does not allow substitution of expected values of Vj and v h to get the expected 

 value of CPE strat exactly. However, it has been shown for a similar situation that it is a close approximation as 

 long as the expected number of visited strata is not too small (Deriso and Parma 1988). The expected value of 

 the new CPE is approximated by 



E[CPE r 



E[ V| ]di + E[v h ]d h 

 E[v,] + E[v h ] 



(10) 



with an error less than (d, + d h )/(E[v,] + E[V]]) based on Deriso and Parma's formulation. 



With some straightforward algebra it can be shown that when a = 1 (no favoritism), E[CPE o]d ] and E[CPE r 

 are both equal to the true mean population density (in CPE equivalents), that is, 



E[CPE old ] = E[CPE new ] = 



nidi + n h d h 

 n,-rn h 



(11) 



