# R/Smith.R In EvolutionaryGames: Important Concepts of Evolutionary Game Theory

#### Documented in Smith

```#' @name Smith
#' @title Smith dynamic
#' @description Smith dynamic as a type of evolutionary dynamics.
#' @aliases Smith
#' @export Smith
#' @author Daniel Gebele \email{dngebele@@gmail.com}
#' @param time Regular sequence that represents the time sequence under which
#'  simulation takes place.
#' @param state Numeric vector that represents the initial state.
#' @param parameters Numeric vector that represents parameters needed by the
#'  dynamic.
#' @return Numeric list. Each component represents the rate of change depending on
#'  the dynamic.
#' @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)

Smith <- function(time, state, parameters) {
a <- parameters
states <- sqrt(length(a))
A <- matrix(a, states, byrow = TRUE)
A <- t(A)

dX <- c()
val <- c()

for(i in 1:states) {
dX[i] <- sum(state * A[i, ])
}

fMax <- 0
sMax <- 0

for(i in 1:states) {
for(j in 1:states) {
if(i == j) next

fMax <- fMax + state[j] * max(0, dX[i] - dX[j])
sMax <- sMax + max(0, dX[j] - dX[i])
}

val[i] <- fMax - state[i] * sMax
fMax <- 0
sMax <- 0
}

return(list(val))
}
```

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EvolutionaryGames documentation built on April 11, 2022, 5:07 p.m.