Package 'EvolutionaryGames' reference manual

Title:	Important Concepts of Evolutionary Game Theory
Description:	Evolutionary game theory applies game theory to evolving populations in biology, see e.g. one of the books by Weibull (1994, ISBN:978-0262731218) or by Sandholm (2010, ISBN:978-0262195874) for more details. A comprehensive set of tools to illustrate the core concepts of evolutionary game theory, such as evolutionary stability or various evolutionary dynamics, for teaching and academic research is provided.
Authors:	Daniel Gebele [aut, cph], Jochen Staudacher [aut, cre, cph]
Maintainer:	Jochen Staudacher <[email protected]>
License:	GPL-2
Version:	0.1.2
Built:	2025-02-20 04:00:01 UTC
Source:	https://github.com/cran/EvolutionaryGames

Brown-von Neumann-Nash dynamic

Description

Brown-von Neumann-Nash replicator dynamic as a type of evolutionary dynamics.

Usage

BNN(time, state, parameters)
BNN(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Brown, G. W. and von Neumann, J. (1950) "Solutions of games by differential equations", In: Kuhn, Harold William and Tucker, Albert William (Eds.) "Contributions to the Theory of Games I", Princeton University Press, pp. 73–79.

Examples

dynamic <- BNN
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)
dynamic <- BNN
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)

BR dynamic

Description

Best response dynamic as a type of evolutionary dynamics.

Usage

BR(time, state, parameters)
BR(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Gilboa, I. and Matsui, A. (1991) "Social Stability and Equilibrium", Econometrica 59, pp. 859–867.

Examples

dynamic <- BR
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)
dynamic <- BR
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)

ESS for two-player games with a maximum of three strategies

Description

Computes Evolutionary Stable Strategies of a game with two players and a maximum of three strategies.

Usage

ESS(A, strategies = c(), floats = TRUE)
ESS(A, strategies = c(), floats = TRUE)

Arguments

`A`	Numeric matrix of size 2x2 or 3x3 representing the number of strategies of a symmetric matrix game.
`strategies`	String vector of length n that names all strategies whereas n represents the number of strategies.
`floats`	Logical value that handles number representation. If set to `TRUE`, floating-point arithmetic will be used, otherwise fractions. Default is `TRUE`.

Value

Numeric matrix. Each row represents an ESS.

Author(s)

Daniel Gebele [email protected]

References

Smith, J. M. and Price, G. R. (1973) "The logic of animal conflict", Nature 246, pp. 15–18.

Examples

ESS(matrix(c(-1, 4, 0, 2), 2, byrow=TRUE), c("Hawk", "Dove"), FALSE)
ESS(matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=TRUE))
ESS(matrix(c(-1, 4, 0, 2), 2, byrow=TRUE), c("Hawk", "Dove"), FALSE)
ESS(matrix(c(1, 2, 0, 0, 1, 2, 2, 0, 1), 3, byrow=TRUE))

Evolutionarily stable set for two-player games with three strategies

Description

Computes evolutionarily stable sets of a game with two players and three strategies.

Usage

ESset(A, strategies = c("1", "2", "3"), floats = TRUE)
ESset(A, strategies = c("1", "2", "3"), floats = TRUE)

Arguments

`A`	Numeric matrix of size 3x3 representing the number of strategies of a symmetric matrix game.
`strategies`	String vector of length 3 that names all strategies.
`floats`	Logical value that handles number representation. If set to `TRUE`, floating-point arithmetic will be used, otherwise fractions. Default is `TRUE`.

Value

Numeric matrix. Each row represents the start and end point of a line (ESset). In addition, a plot of the ESset in the game will be created.

Author(s)

Daniel Gebele [email protected]

References

Thomas, B. (1985) "On evolutionarily stable sets", Journal of Mathematical Biology 22, pp. 105–115.

Examples

# Please note that the computation of evolutionarily stable sets 
# is rather time-consuming. 
# Depending on your machine you might need to wait more 
# than 10 seconds in order to run the following example.
## Not run: 
A <- matrix(c(-2, 5, 10/9, 0, 5/2, 10/9, -10/9, 35/9, 10/9), 3, byrow=TRUE)
strategies <- c("Hawk", "Dove", "Mixed ESS")
ESset(A, strategies)

## End(Not run)
# Please note that the computation of evolutionarily stable sets 
# is rather time-consuming. 
# Depending on your machine you might need to wait more 
# than 10 seconds in order to run the following example.
## Not run: 
A <- matrix(c(-2, 5, 10/9, 0, 5/2, 10/9, -10/9, 35/9, 10/9), 3, byrow=TRUE)
strategies <- c("Hawk", "Dove", "Mixed ESS")
ESset(A, strategies)

## End(Not run)

ILogit dynamic

Description

Imitative Logit dynamic as a type of evolutionary dynamics.

Usage

ILogit(time, state, parameters)
ILogit(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Jochen Staudacher [email protected]

References

Weibull, J. W. (1997) "Evolutionary Game Theory", MIT Press.

Examples

dynamic <- ILogit
A <- matrix(c(-1, 0, 0, 0, -1, 0, 0, 0, -1), 3, byrow=TRUE)
state <- matrix(c(0.1, 0.2, 0.7, 0.2, 0.7, 0.1, 0.9, 0.05, 0.05), 3, 3, byrow=TRUE)
eta <- 0.7
phaseDiagram3S(A, dynamic, eta, state, TRUE, FALSE)
dynamic <- ILogit
A <- matrix(c(-1, 0, 0, 0, -1, 0, 0, 0, -1), 3, byrow=TRUE)
state <- matrix(c(0.1, 0.2, 0.7, 0.2, 0.7, 0.1, 0.9, 0.05, 0.05), 3, 3, byrow=TRUE)
eta <- 0.7
phaseDiagram3S(A, dynamic, eta, state, TRUE, FALSE)

Logit dynamic

Description

Logit dynamic as a type of evolutionary dynamics.

Usage

Logit(time, state, parameters)
Logit(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Fudenberg, D. and Levine, D. K. (1998) "The Theory of Learning in Games", MIT Press.

Examples

dynamic <- Logit
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
eta <- 0.1
phaseDiagram3S(A, dynamic, eta, state, FALSE, FALSE)
dynamic <- Logit
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
eta <- 0.1
phaseDiagram3S(A, dynamic, eta, state, FALSE, FALSE)

Maynard Smith replicator dynamic

Description

Maynard Smith replicator dynamic as a type of evolutionary dynamics.

Usage

MSReplicator(time, state, parameters)
MSReplicator(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Smith, J. M. (1982) "Evolution and the Theory of Games", Cambridge University Press.

Examples

dynamic <- MSReplicator
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)
dynamic <- MSReplicator
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)

Phase Diagram for two-player games with two strategies

Description

Plots phase diagram of a game with two players and two strategies.

Usage

phaseDiagram2S(
  A,
  dynamic,
  params = NULL,
  vectorField = TRUE,
  strategies = c("1", "2")
)
phaseDiagram2S(
  A,
  dynamic,
  params = NULL,
  vectorField = TRUE,
  strategies = c("1", "2")
)

Arguments

`A`	Numeric matrix of size 2x2 representing the number of strategies of a symmetric matrix game.
`dynamic`	Function representing an evolutionary dynamic.
`params`	Numeric vector representing additional parameters for the evolutionary dynamic.
`vectorField`	Logical value that handles vector field presentation. If set to `TRUE`, vector field will be shown, otherwise not. Default is `TRUE`.
`strategies`	String vector of length 2 that names all strategies.

Value

None.

Author(s)

Daniel Gebele [email protected]

Examples

A <- matrix(c(-1, 4, 0, 2), 2, 2, byrow=TRUE)
phaseDiagram2S(A, Replicator, strategies = c("Hawk", "Dove"))
A <- matrix(c(-1, 4, 0, 2), 2, 2, byrow=TRUE)
phaseDiagram2S(A, Replicator, strategies = c("Hawk", "Dove"))

Phase Diagram for two-player games with three strategies

Description

Plots phase diagram of a game with two players and three strategies.

Usage

phaseDiagram3S(
  A,
  dynamic,
  params = NULL,
  trajectories = NULL,
  contour = FALSE,
  vectorField = FALSE,
  strategies = c("1", "2", "3")
)
phaseDiagram3S(
  A,
  dynamic,
  params = NULL,
  trajectories = NULL,
  contour = FALSE,
  vectorField = FALSE,
  strategies = c("1", "2", "3")
)

Arguments

`A`	Numeric matrix of size 3x3 representing the number of strategies of a symmetric matrix game.
`dynamic`	Function representing an evolutionary dynamic.
`params`	Numeric vector with additional parameters for the evolutionary dynamic.
`trajectories`	Numeric matrix of size mx3. Each row represents the initial values for the trajectory to be examined.
`contour`	Logical value that handles contour diagram presentation. If set to `TRUE`, contour diagram will be shown, otherwise not. Default is `FALSE`.
`vectorField`	Logical value that handles vector field presentation. If set to `TRUE`, vector field will be shown, otherwise not. Default is `FALSE`.
`strategies`	String vector of length 3 that names all strategies.

Value

None.

Author(s)

Daniel Gebele [email protected]

Examples

A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)

phaseDiagram3S(A, Replicator, NULL, state, FALSE, FALSE)
phaseDiagram3S(A, Replicator, NULL, state, TRUE, TRUE)

# Plot two trajectories rather than only one:
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3, 0.6, 0.2, 0.2), 2, 3, byrow=TRUE)
phaseDiagram3S(A, Replicator, NULL, state, FALSE, FALSE)
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)

phaseDiagram3S(A, Replicator, NULL, state, FALSE, FALSE)
phaseDiagram3S(A, Replicator, NULL, state, TRUE, TRUE)

# Plot two trajectories rather than only one:
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3, 0.6, 0.2, 0.2), 2, 3, byrow=TRUE)
phaseDiagram3S(A, Replicator, NULL, state, FALSE, FALSE)

Phase Diagram for two-player games with four strategies

Description

Plots phase diagram of a game with two players and four strategies.

Usage

phaseDiagram4S(
  A,
  dynamic,
  params = NULL,
  trajectory = NULL,
  strategies = c("1", "2", "3", "4"),
  noRGL = TRUE
)
phaseDiagram4S(
  A,
  dynamic,
  params = NULL,
  trajectory = NULL,
  strategies = c("1", "2", "3", "4"),
  noRGL = TRUE
)

Arguments

`A`	Numeric matrix of size 4x4 representing the number of strategies of a symmetric matrix game.
`dynamic`	Function representing an evolutionary dynamic.
`params`	Numeric vector with additional parameters for the evolutionary dynamic.
`trajectory`	Numeric vector of size 4 representing the initial value for the trajectory to be examined.
`strategies`	String vector of length 4 that names all strategies.
`noRGL`	Logical value that handles diagram rotation. If set to `FALSE`, diagram will be rotatable, otherwise not. Default is `TRUE`.

Value

None.

Author(s)

Daniel Gebele [email protected]

Examples

A <- matrix(c(5, -9, 6, 8, 20, 1, 2, -18, -14, 0, 2, 20, 13, 0, 4, -13),
 4, 4, byrow=TRUE)
state <- c(0.3, 0.2, 0.1, 0.4)
phaseDiagram4S(A, Replicator, NULL, state)
A <- matrix(c(5, -9, 6, 8, 20, 1, 2, -18, -14, 0, 2, 20, 13, 0, 4, -13),
 4, 4, byrow=TRUE)
state <- c(0.3, 0.2, 0.1, 0.4)
phaseDiagram4S(A, Replicator, NULL, state)

Replicator dynamic

Description

Replicator dynamic as a type of evolutionary dynamics.

Usage

Replicator(time, state, parameters)
Replicator(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Taylor, P. D. and Jonker, L. B. (1978) "Evolutionary stable strategies and game dynamics", Mathematical Biosciences 40 (1-2), pp. 145–156.

Examples

dynamic <- Replicator
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)
dynamic <- Replicator
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)

Smith dynamic

Description

Smith dynamic as a type of evolutionary dynamics.

Usage

Smith(time, state, parameters)
Smith(time, state, parameters)

Arguments

`time`	Regular sequence that represents the time sequence under which simulation takes place.
`state`	Numeric vector that represents the initial state.
`parameters`	Numeric vector that represents parameters needed by the dynamic.

Value

Numeric list. Each component represents the rate of change depending on the dynamic.

Author(s)

Daniel Gebele [email protected]

References

Smith, M. J. (1984) "The Stability of a Dynamic Model of Traffic Assignment – An Application of a Method of Lyapunov", Transportation Science 18, pp. 245–252.

Examples

dynamic <- Smith
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)
dynamic <- Smith
A <- matrix(c(0, -2, 1, 1, 0, -2, -2, 1, 0), 3, byrow=TRUE)
state <- matrix(c(0.4, 0.3, 0.3), 1, 3, byrow=TRUE)
phaseDiagram3S(A, dynamic, NULL, state, FALSE, FALSE)

Triangle for 2-simplex operations

Description

Generates a triangle representing the 2-simplex.

Usage

triangle(labels = c("1", "2", "3"))
triangle(labels = c("1", "2", "3"))

Arguments

labels

String vector of length 3 that names the edges of the triangle.

Value

List of size 2 with members coords and canvas. coords holds edge coordinates of the 2-simplex, canvas a ggplot2 plot object of the 2-simplex.

Author(s)

Daniel Gebele [email protected]

Examples

triangle()
triangle()

Package 'EvolutionaryGames'

Help Index

Brown-von Neumann-Nash dynamic

Description

Usage

Arguments

Value

Author(s)

References

Examples

BR dynamic

Description

Usage

Arguments

Value

Author(s)

References

Examples

ESS for two-player games with a maximum of three strategies

Description

Usage

Arguments

Value

Author(s)

References

Examples

Evolutionarily stable set for two-player games with three strategies

Description

Usage

Arguments

Value

Author(s)

References

Examples

ILogit dynamic

Description

Usage

Arguments

Value

Author(s)

References

Examples

Logit dynamic

Description

Usage

Arguments

Value

Author(s)

References

Examples

Maynard Smith replicator dynamic

Description

Usage

Arguments

Value

Author(s)

References

Examples

Phase Diagram for two-player games with two strategies

Description

Usage

Arguments

Value

Author(s)

Examples

Phase Diagram for two-player games with three strategies

Description

Usage

Arguments

Value

Author(s)

Examples

Phase Diagram for two-player games with four strategies

Description

Usage

Arguments

Value

Author(s)

Examples

Replicator dynamic