Robustness of optimal mixed strategies

Patsy Haccou, Yoh Iwasa

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in 'ideal' and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small.

Original languageEnglish
Pages (from-to)485-496
Number of pages12
JournalJournal of Mathematical Biology
Volume36
Issue number5
DOIs
Publication statusPublished - Jan 1 1998

Fingerprint

Mixed Strategy
Optimal Strategy
Robustness
Phenotype
phenotype
Perturbed System
Converge
Exact Distribution
Continuous Distributions
Zero
Small Perturbations
Fitness
Perturbation
Strategy

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

Cite this

Robustness of optimal mixed strategies. / Haccou, Patsy; Iwasa, Yoh.

In: Journal of Mathematical Biology, Vol. 36, No. 5, 01.01.1998, p. 485-496.

Research output: Contribution to journalArticle

Haccou, Patsy ; Iwasa, Yoh. / Robustness of optimal mixed strategies. In: Journal of Mathematical Biology. 1998 ; Vol. 36, No. 5. pp. 485-496.
@article{9add38f2a55c48e191ce252cd2a231a9,
title = "Robustness of optimal mixed strategies",
abstract = "Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in 'ideal' and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small.",
author = "Patsy Haccou and Yoh Iwasa",
year = "1998",
month = "1",
day = "1",
doi = "10.1007/s002850050110",
language = "English",
volume = "36",
pages = "485--496",
journal = "Journal of Mathematical Biology",
issn = "0303-6812",
publisher = "Springer Verlag",
number = "5",

}

TY - JOUR

T1 - Robustness of optimal mixed strategies

AU - Haccou, Patsy

AU - Iwasa, Yoh

PY - 1998/1/1

Y1 - 1998/1/1

N2 - Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in 'ideal' and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small.

AB - Mixed strategies, or variable phenotypes, can evolve in fluctuating environments when at the time that a strategy is chosen the consequences of that decision are relatively uncertain. In a previous paper, we have shown several examples of explicit forms of optimal mixed strategies when an environmental distribution and payoff function are given. In many of these examples, the mixed strategy has a continuous distribution. In a recent study, however, Sasaki and Ellner proved that, if the distribution of the environmental parameter is modified in certain ways, the exact ESS distribution becomes discrete rather than continuous. This forces us to take a closer look at the robustness of optimal mixed strategies. In the current paper we prove that such strategies are indeed robust against small perturbations of the environmental distribution and/or the payoff function, in the sense that the optimal strategy distribution for the perturbed system, converges weakly to the optimal strategy distribution for the unperturbed system as the magnitude of the perturbation goes to zero. Furthermore, we show that the fitness difference between the two strategies converges to zero. Thus, although optimal strategies in 'ideal' and perturbed systems can be qualitatively different, the difference between the distributions (in a measure theoretic sense) is small.

UR - http://www.scopus.com/inward/record.url?scp=0003211436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003211436&partnerID=8YFLogxK

U2 - 10.1007/s002850050110

DO - 10.1007/s002850050110

M3 - Article

AN - SCOPUS:0003211436

VL - 36

SP - 485

EP - 496

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 5

ER -