nash: Computes harvesting rates at the Nash Equilibrium (NE)

In ThomasDelSantoONeill/nash: An Efficient Algorithm for Nash Equilibria when Harvesting Interacting Living Resources

Computes harvesting rates at the Nash Equilibrium (NE)

Description

Takes a numeric vector of initial harvesting rates to be optimised ('par') by evaluating the objective function that is to be maximised ('fn'). Its structure in terms of arguments and outputs is similar to that of optim.

Usage

nash(
  par,
  fn,
  ...,
  method = "LV",
  yield.curves = FALSE,
  conv.criterion = 0.001,
  Bcons = 0,
  F.increase = 0.1,
  progress = TRUE,
  track = FALSE
)

Arguments

par	Double type numeric vector of harvesting rates of length equal to the number of harvested species for which the NE is desired.
fn	Function that runs the multispecies/ecosystem model with par as input and returns simulated yields at equilibrium.
...	Further arguments to be passed to fn.
method	Method utilised to compute the Nash Equilibrium: (i) 'LV' or (ii) 'round-robin' method (see 'Details' section for specifics).
yield.curves	Logical TRUE/FALSE if equilibrium yield curves for each of the optimised species are to be computed.
conv.criterion	Absolute convergence tolerance set by default to < 0.001.
Bcons	Constraints for biodiversity conservation. Double type numeric vector set to 0 by default.
F.increase	Double type numeric vector indicating the step size used to compute the effective interaction matrix M.
progress	Logical that if TRUE information on the progress of the optimisation is produced.
track	Logical that if TRUE will return all fishing mortality rates that where iteratively computed during the search. \loadmathjax

Details

For ecosystem models where there is some interest in keeping some or all harvested species above a certain biomass state limit, Bcons should be populated with non-zero biomass values. The length of this vector must be the same as par and set to non-zero where relevant. In the literature, is common practice to fixed such constraints of biodiversity conservation \insertCiteMatsuda2006nash as a fraction (e.g. \mjeqn0.1-0.25ascii) of the unfished biomass \mjeqnB_0ascii; biomass threshold at which a stock is considered collapsed \insertCiteWorm2009nash.

Equilibrium yield curves are obtained for each i species by applying different harvesting values to i whilst keeping the other species j at the optimised par levels (i.e. at \mjeqn\mathbfF_Nashascii). The harvesting values applied to i run from 0 to \mjeqnF_Nash,i\times 2ascii with a desired sequence length of length.out=30 (see seq for details). As raised by \insertCiteThorpe2017;textualnash, this is one of the advantages of using the NE as the multispecies extension of the Maximum Sustainable Yield concept.

To compute the interaction matrix a second order central difference quotient is used to approximate derivatives. F.increase is employed during this calculation as a step-size set by default to 0.1 to avoid truncation and/or rounding errors \insertCitePope2019nash.

The 'LV' method is set by default given its performance advantage \insertCiteOneill2023nash over the 'round-robin' method and is based on the protocol devised by \insertCiteFarcas2016nash. For each species i in turn, round-robin iteratively maximises the yield by adjusting the harvesting rates whereas LV does the same simultaneously for all species per iteration.

Value

The function nash returns a list with the following components:

par	Harvesting rates at the NE.
value	Yield values of fn corresponding to the optimised par.
counts	Number of fn evaluations until NE.
convergence	Statement indicating the number of iterations for conv.criterion to be reached.

References

\insertRef

Matsuda2006nash

\insertRef

Worm2009nash

\insertRef

Thorpe2017nash

\insertRef

Pope2019nash

\insertRef

Farcas2016nash

\insertRef

Lucey2020nash

ThomasDelSantoONeill/nash documentation built on Aug. 10, 2024, 1:49 a.m.