R/normal_form.R
In yukiyanai/rgamer: rgamer: Learn Game Theory Using R
Defines functions
normal_form
Documented in
normal_form
#' @title Define a normal-form (or strategic-form) game
#' @description \code{normal_form()} defines a normal-form game and
#' creates an object of "normal_form" class.
#' @details Creates an object of "normal_form" class, which can be passed to
#' functions in order to find solutions of the game.
#' @param players A character vector of the name (label) for the players.
#' @param s1 A character vector of pure strategies for Player 1 (row player).
#' Required only when the player has discrete-choice strategies.
#' @param s2 A character vector of pure strategies for Player 2 (column player).
#' Required only when the player has discrete-choice strategies.
#' @param payoffs1 The payoff of Player1. This argument can be specified in
#' three different ways. First, it can be a numeric vector of payoffs.
#' Second, it can be a character string of the payoff function
#' (e.g., payoffs1 = "x^2 - y"). Third, it can be an R function of payoff.
#' @param payoffs2 The payoff of Player 2. See the explanation of
#' \code{payoffs1} for detail.
#' @param cells A list of vectors to specify the payoff for each cell. Each
#' element of the list should be a numeric vector of two players' payoffs.
#' @param discretize A logical value. Set this \code{TRUE} to evaluate payoff
#' functions at some discrete values of strategies \code{s1} and \code{s2}.
#' Default is \code{FALSE}.
#' @param discrete_points A numeric vector of length 2 to set how many discrete
#' points should be used to discretize the game defined by payoff functions.
#' Default is \code{c(6, 6)}, which shows equally spaced 6 values from the
#' range of the strategies \code{par1_lim} and \code{par2_lim}. Instead of
#' setting this parameter, you can specify the vectors of arbitrary
#' strategies by setting \code{s1} and \code{s2}.
#' @param symmetric A logical value. Set this \code{TRUE} when the payoffs for
#' two players are symmetric as in the prisoners' dilemma. Then, payoffs1 is
#' recycled for payoffs2. Default is \code{FALSE}.
#' @param byrow A logical value. If \code{TRUE}, payoffs will be lined up by
#' row. Default is \code{FALSE}. Only used when both \code{s1} and \code{s2}
#' are provided.
#' @param pars A character vector of parameters that are selected by players 1
#' and 2, respectively. Only used when \code{payoffs1} and \code{payoffs2}
#' are specified by payoff functions (either as character strings or R
#' functions).
#' @param par1_lim A numeric vector of length 2, which defines the range of
#' parameters from which Player 1 chooses her strategy.
#' @param par2_lim A numeric vector of length 2, which defines the range of
#' parameters from which Player 2 chooses his strategy.
#' @param cons1 A named list of parameters contained in \code{payoffs1} that
#' should be treated as constants, if any.
#' @param cons2 A named list of parameters contained in \code{payoffs2} that
#' should be treated as constants, if any.
#' @param cons_common A named list of parameters contained in \code{payoffs1}
#' and \code{payoffs2} that should be treated as constants, if any. If
#' \code{cons1} and \code{cons2} are exactly same, you can specify
#' \code{cons_common} instead of both \code{cons1} and \code{cons2}.
#' @return An object of "normal_form" class, which defines a normal-form (or
#' strategic-form) game.
#' @author Yoshio Kamijo and Yuki Yanai <yanai.yuki@@kochi-tech.ac.jp>
#' @export
#' @examples
#' game1 <- normal_form(
#' s1 = c("T", "B"),
#' s2 = c("L", "R"),
#' payoffs1 = c(4, 2, 3, 1),
#' payoffs2 = c(4, 3, 2, 1),
#' players = c("Row Player", "Column Player"))
#'
#' game1b <- normal_form(
#' s1 = c("T", "B"),
#' s2 = c("L", "R"),
#' cells = list(c(4, 4),
#' c(2, 3),
#' c(3, 2),
#' c(1, 1)),
#' players = c("Row Player", "Column Player"))
#'
#' game2 <- normal_form(
#' players = c("A", "B"),
#' payoffs1 = "-x1^2 + (28 - x2) * x1",
#' payoffs2 = "-x2^2 + (28 - x1) * x2",
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30),
#' pars = c("x1", "x2"))
#'
#' fx <- function(x, y, a, b) -x^a + (b - y) * x
#' fy <- function(x, y, s, t) -y^s + (t - x) * y
#' game3 <- normal_form(
#' payoffs1 = fx,
#' payoffs2 = fy,
#' pars = c('x', 'y'),
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30))
#'
#' \dontrun{
#' ## This throws an error because payoffs1 and payoffs2 are in different forms.
#' game4 <- normal_form(
#' payoffs1 = fx,
#' payoffs2 = "-y^2 + (28 - x) * y",
#' pars = c('x', 'y'),
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30)
#' )
#' }
normal_form <- function(
players = NULL,
s1 = NULL,
s2 = NULL,
payoffs1 = NULL,
payoffs2 = NULL,
cells = NULL,
discretize = FALSE,
discrete_points = c(6, 6),
symmetric = FALSE,
byrow = FALSE,
pars = NULL,
par1_lim = NULL,
par2_lim = NULL,
cons1 = NULL,
cons2 = NULL,
cons_common = NULL) {
# If players are not named, give them default names.
if (is.null(players)) players <- c("Player 1", "Player 2")
# If payoffs are provided by cell, create payoffs1 and payoffs2 from cells
if (!is.null(cells)) {
n_cells <- length(cells)
payoffs1 <- payoffs2 <- rep(NA, n_cells)
for (i in 1:n_cells) {
payoffs1[i] <- cells[[i]][1]
payoffs2[i] <- cells[[i]][2]
}
}
# error message used for a few times below
stop_message <- "For a game with discrete strategies, please specify both s1 and s2.\nFor a game with continuous strategies, please specify all of payoffs1, payoffs2, pars, par1_lim, and par2_lim. When dicretize = TRUE, par1_lim and par2_lim are not necessary."
if (is.null(pars)) {
## Game with discrete-choice strategies
if (is.null(s1) | is.null(s2)) {
# error if strategies are not provided for both players
stop(stop_message)
}
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
if (symmetric) payoffs2 <- payoffs1
# error if the length of payoffs differs from the number of cells
if (length(payoffs1) != n_cells) stop("the length of payoffs1 must equal the number of cells.")
if (length(payoffs2) != n_cells) stop("the length of payoffs2 must equal the number of cells.")
# Player 1's payoff matrix
mat1 <- matrix(payoffs1, nrow = n_rows, byrow = byrow)
# Player 2's payoff matrix
byrow2 <- ifelse(symmetric, !byrow, byrow)
mat2 <- matrix(payoffs2, nrow = n_rows, byrow = byrow2)
# Gather game elements in a data frame
if (byrow) {
row <- rep(1:n_rows, each = n_cols)
s1_vec <- rep(s1, each = n_cols)
column <- rep(1:n_cols, times = n_rows)
s2_vec <- rep(s2, times = n_rows)
} else {
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_rows)
s2_vec <- rep(s2, each = n_rows)
}
df <- data.frame(row,
column,
s1 = s1_vec,
s2 = s2_vec,
payoff1 = payoffs1,
payoff2 = payoffs2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else if (is.character(payoffs1) & is.character(payoffs2)) {
## Game whose payoffs are defined by character strings describing functions
if (is.null(par1_lim) | is.null(par2_lim)) {
# error if domains of parameters are not provided
stop(stop_message)
} else if (length(par1_lim) != 2 | length(par2_lim) != 2) {
# error if parameter domains are not specified correctly
stop("Each of par1_lim and par2_lim must be the numeric vector of length 2.")
} else {
if (discretize) {
# transform char functions to R functions
new_fs <- char2function(f1 = payoffs1,
f2 = payoffs2,
pars = pars)
# pick out discrete strategies
if (is.null(s1)) {
s1 <- seq(from = par1_lim[1],
to = par1_lim[2],
length.out = discrete_points[1])
}
if (is.null(s2)) {
s2 <- seq(from = par2_lim[1],
to = par2_lim[2],
length.out = discrete_points[2])
}
s_set <- expand.grid(s1, s2)
names(s_set) <- c("x", "y")
payoff1 <- purrr::pmap(s_set, new_fs[[1]]) |> unlist()
payoff2 <- purrr::pmap(s_set, new_fs[[2]]) |> unlist()
s1 <- as.character(s1)
s2 <- as.character(s2)
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
# payoff matrix for Players 1 and 2
mat1 <- matrix(payoff1, nrow = n_rows, byrow = FALSE)
mat2 <- matrix(payoff2, nrow = n_rows, byrow = FALSE)
# gather game elements in a data frame
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_cols)
s2_vec <- rep(s2, each = n_rows)
df <- data.frame(row,
column,
s1,
s2,
payoff1,
payoff2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoff1, payoffs2 = payoff2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else {
value <- list(player = players,
strategy = list(s1 = par1_lim, s2 = par2_lim),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
pars = pars,
type = "char_function")
}
}
} else if (is.function(payoffs1) & is.function(payoffs2)) {
## Game whose payoffs are defined by function objects
if (discretize) {
# pick out discrete strategies
if (is.null(s1)) {
s1 <- seq(from = par1_lim[1],
to = par1_lim[2],
length.out = discrete_points[1])
}
if (is.null(s2)) {
s2 <- seq(from = par2_lim[1],
to = par2_lim[2],
length.out = discrete_points[2])
}
s_set <- expand.grid(s1, s2)
names(s_set) <- pars
payoff1 <- purrr::pmap(s_set, payoffs1) |> unlist()
payoff2 <- purrr::pmap(s_set, payoffs2) |> unlist()
s1 <- as.character(s1)
s2 <- as.character(s2)
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
# payoff matrix for Players 1 and 2
mat1 <- matrix(payoff1, nrow = n_rows, byrow = FALSE)
mat2 <- matrix(payoff2, nrow = n_rows, byrow = FALSE)
# gather game elements in a data frame
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_cols)
s2_vec <- rep(s2, each = n_rows)
df <- data.frame(row,
column,
s1,
s2,
payoff1,
payoff2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoff1, payoffs2 = payoff2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else {
if (is.null(par1_lim) | is.null(par2_lim)) {
# error if domains of parameters are not provided
stop(stop_message)
} else if (length(par1_lim) != 2 | length(par2_lim) != 2) {
# error if parameter domains are not specified correctly
stop("Each of par1_lim and par2_lim must be the numeric vector of length 2.")
} else {
if (!is.null(cons_common)) {
constants <- list(cons_common = cons_common)
} else {
constants <- list(cons1 = cons1, cons2 = cons2)
}
# list for return
value <- list(player = players,
strategy = list(s1 = par1_lim, s2 = par2_lim),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
pars = pars,
constants = constants,
type = "function")
}
}
} else {
stop("Please specify strategies (if payoffs are not functions) and payoffs in a proper way.")
}
# return a "normal_form" class object
structure(value, class = "normal_form")
}
yukiyanai/rgamer documentation built on June 14, 2024, 7:38 p.m.
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Defines functions normal_form
Documented in normal_form
#' @title Define a normal-form (or strategic-form) game
#' @description \code{normal_form()} defines a normal-form game and
#' creates an object of "normal_form" class.
#' @details Creates an object of "normal_form" class, which can be passed to
#' functions in order to find solutions of the game.
#' @param players A character vector of the name (label) for the players.
#' @param s1 A character vector of pure strategies for Player 1 (row player).
#' Required only when the player has discrete-choice strategies.
#' @param s2 A character vector of pure strategies for Player 2 (column player).
#' Required only when the player has discrete-choice strategies.
#' @param payoffs1 The payoff of Player1. This argument can be specified in
#' three different ways. First, it can be a numeric vector of payoffs.
#' Second, it can be a character string of the payoff function
#' (e.g., payoffs1 = "x^2 - y"). Third, it can be an R function of payoff.
#' @param payoffs2 The payoff of Player 2. See the explanation of
#' \code{payoffs1} for detail.
#' @param cells A list of vectors to specify the payoff for each cell. Each
#' element of the list should be a numeric vector of two players' payoffs.
#' @param discretize A logical value. Set this \code{TRUE} to evaluate payoff
#' functions at some discrete values of strategies \code{s1} and \code{s2}.
#' Default is \code{FALSE}.
#' @param discrete_points A numeric vector of length 2 to set how many discrete
#' points should be used to discretize the game defined by payoff functions.
#' Default is \code{c(6, 6)}, which shows equally spaced 6 values from the
#' range of the strategies \code{par1_lim} and \code{par2_lim}. Instead of
#' setting this parameter, you can specify the vectors of arbitrary
#' strategies by setting \code{s1} and \code{s2}.
#' @param symmetric A logical value. Set this \code{TRUE} when the payoffs for
#' two players are symmetric as in the prisoners' dilemma. Then, payoffs1 is
#' recycled for payoffs2. Default is \code{FALSE}.
#' @param byrow A logical value. If \code{TRUE}, payoffs will be lined up by
#' row. Default is \code{FALSE}. Only used when both \code{s1} and \code{s2}
#' are provided.
#' @param pars A character vector of parameters that are selected by players 1
#' and 2, respectively. Only used when \code{payoffs1} and \code{payoffs2}
#' are specified by payoff functions (either as character strings or R
#' functions).
#' @param par1_lim A numeric vector of length 2, which defines the range of
#' parameters from which Player 1 chooses her strategy.
#' @param par2_lim A numeric vector of length 2, which defines the range of
#' parameters from which Player 2 chooses his strategy.
#' @param cons1 A named list of parameters contained in \code{payoffs1} that
#' should be treated as constants, if any.
#' @param cons2 A named list of parameters contained in \code{payoffs2} that
#' should be treated as constants, if any.
#' @param cons_common A named list of parameters contained in \code{payoffs1}
#' and \code{payoffs2} that should be treated as constants, if any. If
#' \code{cons1} and \code{cons2} are exactly same, you can specify
#' \code{cons_common} instead of both \code{cons1} and \code{cons2}.
#' @return An object of "normal_form" class, which defines a normal-form (or
#' strategic-form) game.
#' @author Yoshio Kamijo and Yuki Yanai <yanai.yuki@@kochi-tech.ac.jp>
#' @export
#' @examples
#' game1 <- normal_form(
#' s1 = c("T", "B"),
#' s2 = c("L", "R"),
#' payoffs1 = c(4, 2, 3, 1),
#' payoffs2 = c(4, 3, 2, 1),
#' players = c("Row Player", "Column Player"))
#'
#' game1b <- normal_form(
#' s1 = c("T", "B"),
#' s2 = c("L", "R"),
#' cells = list(c(4, 4),
#' c(2, 3),
#' c(3, 2),
#' c(1, 1)),
#' players = c("Row Player", "Column Player"))
#'
#' game2 <- normal_form(
#' players = c("A", "B"),
#' payoffs1 = "-x1^2 + (28 - x2) * x1",
#' payoffs2 = "-x2^2 + (28 - x1) * x2",
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30),
#' pars = c("x1", "x2"))
#'
#' fx <- function(x, y, a, b) -x^a + (b - y) * x
#' fy <- function(x, y, s, t) -y^s + (t - x) * y
#' game3 <- normal_form(
#' payoffs1 = fx,
#' payoffs2 = fy,
#' pars = c('x', 'y'),
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30))
#'
#' \dontrun{
#' ## This throws an error because payoffs1 and payoffs2 are in different forms.
#' game4 <- normal_form(
#' payoffs1 = fx,
#' payoffs2 = "-y^2 + (28 - x) * y",
#' pars = c('x', 'y'),
#' par1_lim = c(0, 30),
#' par2_lim = c(0, 30)
#' )
#' }
normal_form <- function(
players = NULL,
s1 = NULL,
s2 = NULL,
payoffs1 = NULL,
payoffs2 = NULL,
cells = NULL,
discretize = FALSE,
discrete_points = c(6, 6),
symmetric = FALSE,
byrow = FALSE,
pars = NULL,
par1_lim = NULL,
par2_lim = NULL,
cons1 = NULL,
cons2 = NULL,
cons_common = NULL) {
# If players are not named, give them default names.
if (is.null(players)) players <- c("Player 1", "Player 2")
# If payoffs are provided by cell, create payoffs1 and payoffs2 from cells
if (!is.null(cells)) {
n_cells <- length(cells)
payoffs1 <- payoffs2 <- rep(NA, n_cells)
for (i in 1:n_cells) {
payoffs1[i] <- cells[[i]][1]
payoffs2[i] <- cells[[i]][2]
}
}
# error message used for a few times below
stop_message <- "For a game with discrete strategies, please specify both s1 and s2.\nFor a game with continuous strategies, please specify all of payoffs1, payoffs2, pars, par1_lim, and par2_lim. When dicretize = TRUE, par1_lim and par2_lim are not necessary."
if (is.null(pars)) {
## Game with discrete-choice strategies
if (is.null(s1) | is.null(s2)) {
# error if strategies are not provided for both players
stop(stop_message)
}
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
if (symmetric) payoffs2 <- payoffs1
# error if the length of payoffs differs from the number of cells
if (length(payoffs1) != n_cells) stop("the length of payoffs1 must equal the number of cells.")
if (length(payoffs2) != n_cells) stop("the length of payoffs2 must equal the number of cells.")
# Player 1's payoff matrix
mat1 <- matrix(payoffs1, nrow = n_rows, byrow = byrow)
# Player 2's payoff matrix
byrow2 <- ifelse(symmetric, !byrow, byrow)
mat2 <- matrix(payoffs2, nrow = n_rows, byrow = byrow2)
# Gather game elements in a data frame
if (byrow) {
row <- rep(1:n_rows, each = n_cols)
s1_vec <- rep(s1, each = n_cols)
column <- rep(1:n_cols, times = n_rows)
s2_vec <- rep(s2, times = n_rows)
} else {
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_rows)
s2_vec <- rep(s2, each = n_rows)
}
df <- data.frame(row,
column,
s1 = s1_vec,
s2 = s2_vec,
payoff1 = payoffs1,
payoff2 = payoffs2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else if (is.character(payoffs1) & is.character(payoffs2)) {
## Game whose payoffs are defined by character strings describing functions
if (is.null(par1_lim) | is.null(par2_lim)) {
# error if domains of parameters are not provided
stop(stop_message)
} else if (length(par1_lim) != 2 | length(par2_lim) != 2) {
# error if parameter domains are not specified correctly
stop("Each of par1_lim and par2_lim must be the numeric vector of length 2.")
} else {
if (discretize) {
# transform char functions to R functions
new_fs <- char2function(f1 = payoffs1,
f2 = payoffs2,
pars = pars)
# pick out discrete strategies
if (is.null(s1)) {
s1 <- seq(from = par1_lim[1],
to = par1_lim[2],
length.out = discrete_points[1])
}
if (is.null(s2)) {
s2 <- seq(from = par2_lim[1],
to = par2_lim[2],
length.out = discrete_points[2])
}
s_set <- expand.grid(s1, s2)
names(s_set) <- c("x", "y")
payoff1 <- purrr::pmap(s_set, new_fs[[1]]) |> unlist()
payoff2 <- purrr::pmap(s_set, new_fs[[2]]) |> unlist()
s1 <- as.character(s1)
s2 <- as.character(s2)
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
# payoff matrix for Players 1 and 2
mat1 <- matrix(payoff1, nrow = n_rows, byrow = FALSE)
mat2 <- matrix(payoff2, nrow = n_rows, byrow = FALSE)
# gather game elements in a data frame
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_cols)
s2_vec <- rep(s2, each = n_rows)
df <- data.frame(row,
column,
s1,
s2,
payoff1,
payoff2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoff1, payoffs2 = payoff2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else {
value <- list(player = players,
strategy = list(s1 = par1_lim, s2 = par2_lim),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
pars = pars,
type = "char_function")
}
}
} else if (is.function(payoffs1) & is.function(payoffs2)) {
## Game whose payoffs are defined by function objects
if (discretize) {
# pick out discrete strategies
if (is.null(s1)) {
s1 <- seq(from = par1_lim[1],
to = par1_lim[2],
length.out = discrete_points[1])
}
if (is.null(s2)) {
s2 <- seq(from = par2_lim[1],
to = par2_lim[2],
length.out = discrete_points[2])
}
s_set <- expand.grid(s1, s2)
names(s_set) <- pars
payoff1 <- purrr::pmap(s_set, payoffs1) |> unlist()
payoff2 <- purrr::pmap(s_set, payoffs2) |> unlist()
s1 <- as.character(s1)
s2 <- as.character(s2)
n_rows <- length(s1)
n_cols <- length(s2)
n_cells <- n_rows * n_cols
# payoff matrix for Players 1 and 2
mat1 <- matrix(payoff1, nrow = n_rows, byrow = FALSE)
mat2 <- matrix(payoff2, nrow = n_rows, byrow = FALSE)
# gather game elements in a data frame
row <- rep(1:n_rows, times = n_cols)
s1_vec <- rep(s1, times = n_cols)
column <- rep(1:n_cols, each = n_cols)
s2_vec <- rep(s2, each = n_rows)
df <- data.frame(row,
column,
s1,
s2,
payoff1,
payoff2)
# list for return
value <- list(player = players,
strategy = list(s1 = s1, s2 = s2),
payoff = list(payoffs1 = payoff1, payoffs2 = payoff2),
df = df,
mat = list(matrix1 = mat1, matrix2 = mat2),
type = "matrix")
} else {
if (is.null(par1_lim) | is.null(par2_lim)) {
# error if domains of parameters are not provided
stop(stop_message)
} else if (length(par1_lim) != 2 | length(par2_lim) != 2) {
# error if parameter domains are not specified correctly
stop("Each of par1_lim and par2_lim must be the numeric vector of length 2.")
} else {
if (!is.null(cons_common)) {
constants <- list(cons_common = cons_common)
} else {
constants <- list(cons1 = cons1, cons2 = cons2)
}
# list for return
value <- list(player = players,
strategy = list(s1 = par1_lim, s2 = par2_lim),
payoff = list(payoffs1 = payoffs1, payoffs2 = payoffs2),
pars = pars,
constants = constants,
type = "function")
}
}
} else {
stop("Please specify strategies (if payoffs are not functions) and payoffs in a proper way.")
}
# return a "normal_form" class object
structure(value, class = "normal_form")
}
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