Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good (“cheating”). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or “tax” on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.
Original languageEnglish (US)
Article number052309
JournalPhysical Review E
Volume95
DOIs
StatePublished - May 11 2017

Fingerprint

Taxation
games
penalties
Coexistence
Penalty
optimal control
Game
Population Growth
Fixed point
Tax
Interior Point
Replicator Dynamics
Quasilinearization
Second-order Ordinary Differential Equations
differential equations
costs
Optimal Control Problem
Optimal Control
Interior
Unstable

Cite this

@article{9493f0d344d3460682e8c535ad740057,
title = "Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation",
abstract = "We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good (“cheating”). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or “tax” on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.",
author = "Christopher Griffin",
year = "2017",
month = "5",
day = "11",
doi = "https://doi.org/10.1103/PhysRevE.95.052309",
language = "English (US)",
volume = "95",
journal = "Physical Review E",
issn = "2470-0045",
publisher = "American Physical Society",

}

TY - JOUR

T1 - Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation

AU - Griffin, Christopher

PY - 2017/5/11

Y1 - 2017/5/11

N2 - We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good (“cheating”). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or “tax” on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.

AB - We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good (“cheating”). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or “tax” on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.

U2 - https://doi.org/10.1103/PhysRevE.95.052309

DO - https://doi.org/10.1103/PhysRevE.95.052309

M3 - Article

VL - 95

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

M1 - 052309

ER -