rgamer: rgamer: Learn Game Theory Using R

#' @title Play a normal-form game by simulation (softly best response)
#' @description \code{sim_game_sbr()} simulates plays expected in a normal-form
#'     game.
#' @details Simulate plays expected in a normal-form game defined by
#'     \code{normal_form()} when each player choose the action that is more
#'     likely to improve their payoff given the other player's previous action.
#' @param game An object of \code{normal_form} class defined by
#'     \code{normal_form()}.
#' @param n_periods A positive integer specifying how many times the game is
#'     played within each sample.
#' @param init1 Player 1's first strategy. If not specified, a strategy is
#'     randomly selected from the player's strategy set.
#' @param init2 Player 2's first strategy. If not specified, a strategy is
#'     randomly selected from the player's strategy set.
#' @param omega A numeric value in [0, 1] to control the degree of inertia in
#'     each player's behavior. If \code{omega = 1}, each player does not change
#'     their choices over time. If \code{omega = 0}, each player does not stick
#'     to their previous choice at all.
#' @param lambda A positive value controlling the weight of the best response to
#'      the previous move of the opponent.
#' @param cons1 A named list of parameters contained in
#'     \code{game$payoff$payoffs1} that should be treated as constants, if any.
#' @param cons2 A named list of parameters contained in
#'     \code{game$payoff$payoffs2} that should be treated as constants, if any.
#' @return data.frame containing the history of the game played.
#' @author Yoshio Kamijo and Yuki Yanai <yanai.yuki@@kochi-tech.ac.jp>
#' @noRd
sim_game_sbr <- function(game,
                         n_periods,
                         init1 = NULL,
                         init2 = NULL,
                         omega = 0,
                         lambda = 1,
                         cons1 = NULL,
                         cons2 = NULL) {

  payoff1 <- payoff2 <- NULL

  if (lambda <= 0) stop("lambda must be positive.")

  play1 <- rep(NA, n_periods)
  play2 <- rep(NA, n_periods)

  if (game$type == "matrix") {

    s1 <- game$strategy$s1
    s2 <- game$strategy$s2

    n1 <- length(s1)
    n2 <- length(s2)

    pi1 <- game$mat$matrix1
    pi2 <- game$mat$matrix2

    # for the first round
    if (is.null(init1)) {
      play1[1] <- sample(1:n1, size = 1)
    } else {
      play1[1] <- which(s1 == init1)
    }
    if (is.null(init2)) {
      play2[1] <- sample(1:n2, size = 1)
    } else {
      play2[1] <- which(s2 == init2)
    }

    for (t in 2:n_periods) {
      ## Player 1
      if (stats::runif(1) < omega) {
        play1[t] <- play1[t - 1]
      } else {
        p <- pi1[, play2[t - 1]]
        p <- exp(lambda * p)
        p <- p / sum(p)
        play1[t] <- sample(1:n1, size = 1, prob = p)
      }

      ## Player 2
      if (stats::runif(1) < omega) {
        play2[t] <- play2[t - 1]
      } else {
        p <- pi2[play2[t - 1], ]
        p <- exp(lambda * p)
        p <- p / sum(p)
        play2[t] <- sample(1:n2, size = 1, prob = p)
      }
    }

    play1 <- s1[play1]
    play2 <- s2[play2]

  } else if (game$type == "char_function") {

    # for the first round
    s1 <- game$strategy$s1
    s2 <- game$strategy$s2
    if (is.null(init1)) play1[1] <- stats::runif(1, min = s1[1], max = s1[2])
    else play1[1] <- init1
    if (is.null(init2)) play2[1] <- stats::runif(1, min = s2[1], max = s2[2])
    else play2[1] <- init2

    for (i in 2:n_periods) {
      ## Player 1
      if (stats::runif(1) < omega) {
        play1[i] <- play1[i - 1]
      } else {
        f1 <- game$payoff$payoffs1 |>
          stringr::str_replace(game$pars[2],
                               as.character(play2[i - 1])) |>
          stringr::str_replace_all(game$pars[1], "XXX") |>
          str2expression()
        fd1 <- function(x) {
          eval(stats::D(f1, name = "XXX"),
               envir = list(XXX = x))
        }
        br <- try(stats::uniroot(fd1, interval = s1)$root,
                        silent = TRUE)
        if (methods::is(br, "try-error")) {
          play1[i] <- play1[i - 1]
        } else {

          if (br <= s1[1] | br >= s1[2]) {
            s_vec <- seq(from = s1[1], to = s1[2], length.out = 100)
          } else {
            left_len <- round((br - s1[1]) / (s1[2] - s1[1]) * 100, 0)
            left <- seq(from = s1[1], to = br, length.out = left_len)
            right <- seq(from = br, to = s1[2], length.out = 101 - left_len)[-1]
            s_vec <- c(left, right)
          }

          p <- eval(f1, envir = list(XXX = s_vec))
          p <- exp(p * lambda)
          p <- p / sum(p)
          play1[i] <- sample(s_vec, size = 1, prob = p)
        }

        play1 <- as.numeric(play1)
      }

      ## Player 2
      if (stats::runif(1) < omega) {
        play2[i] <- play2[i - 1]
      } else {
        f2 <- game$payoff$payoffs2 |>
          stringr::str_replace(game$pars[1],
                               as.character(play1[i - 1])) |>
          stringr::str_replace_all(game$pars[2], "YYY") |>
          str2expression()
        fd2 <- function(y) {
          eval(stats::D(f2, name = "YYY"),
               envir = list(YYY = y))
        }
        br <- try(stats::uniroot(fd2, interval = s2)$root,
                        silent = TRUE)
        if (methods::is(br, "try-error")) {
          play2[i] <- play2[i - 1]
        } else {

          if (br <= s2[1] | br >= s2[2]) {
            s_vec <- seq(from = s2[1], to = s2[2], length.out = 100)
          } else {
            left_len <- round((br - s2[1]) / (s2[2] - s2[1]) * 100, 0)
            left <- seq(from = s2[1], to = br, length.out = left_len)
            right <- seq(from = br, to = s2[2], length.out = 101 - left_len)[-1]
            s_vec <- c(left, right)
          }

          p <- eval(f2, envir = list(YYY = s_vec))
          p <- exp(p * lambda)
          p <- p / sum(p)
          play2[i] <- sample(s_vec, size = 1, prob = p)
        }

        play2 <- as.numeric(play2)
      }
    }

  } else {

    # for the first round
    s1 <- game$strategy$s1
    s2 <- game$strategy$s2
    if (is.null(init1)) play1[1] <- stats::runif(1, min = s1[1], max = s1[2])
    else play1[1] <- init1
    if (is.null(init2)) play2[1] <- stats::runif(1, min = s2[1], max = s2[2])
    else play2[1] <- init2

    for (i in 2:n_periods) {

      ## Player 1
      if (stats::runif(1) < omega) {
        play1[i] <- play1[i - 1]
      } else {
        f1 <- function(XXX) {
          if (is.null(cons1)) {
            arg_list <- list(XXX, play2[i - 1])
            names(arg_list) <- game$pars
          } else {
            arg_list <- c(cons1, XXX, play2[i - 1])
            names(arg_list) <- c(names(cons1), game$pars)
          }
          purrr::pmap(.l = arg_list,
                      .f = game$payoff$payoffs1)
        }
        br <- try(
          stats::optim(par = stats::median(s1),
                       fn = f1,
                       method = "L-BFGS-B",
                       lower = s1[1],
                       upper = s1[2],
                       control = list(fnscale = -1))$par,
          silent = TRUE)
        if (methods::is(br, "try-error")) {
          play1[i] <- play1[i - 1]
        } else {

          if (br <= s1[1] | br >= s1[2]) {
            s_vec <- seq(from = s1[1], to = s1[2], length.out = 100)
          } else {
            left_len <- round((br - s1[1]) / (s1[2] - s1[1]) * 100, 0)
            left <- seq(from = s1[1], to = br, length.out = left_len)
            right <- seq(from = br, to = s1[2], length.out = 101 - left_len)[-1]
            s_vec <- c(left, right)
          }

          p <- unlist(sapply(s_vec, f1))
          p <- exp(p * lambda)
          p <- p / sum(p)
          play1[i] <- sample(s_vec, size = 1, prob = p)
        }
        play1 <- as.numeric(play1)
      }

      ## Player 2
      if (stats::runif(1) < omega) {
        play2[i] <- play2[i - 1]
      } else {
        f2 <- function(YYY) {
          if (is.null(cons2)) {
            arg_list <- list(play1[i - 1], YYY)
            names(arg_list) <- game$pars
          } else {
            arg_list <- c(cons2, play1[i - 1], YYY)
            names(arg_list) <- c(names(cons2), game$pars)
          }
          purrr::pmap(.l = arg_list,
                      .f = game$payoff$payoffs2)
        }
        br <- try(
          stats::optim(par = stats::median(s2),
                       fn = f2,
                       method = "L-BFGS-B",
                       lower = s2[1],
                       upper = s2[2],
                       control = list(fnscale = -1))$par,
          silent = TRUE)
        if (methods::is(br, "try-error")) {
          play2[i] <- play2[i - 1]
        } else {


          if (br <= s2[1] | br >= s2[2]) {
            s_vec <- seq(from = s2[1], to = s2[2], length.out = 100)
          } else {
            left_len <- round((br - s2[1]) / (s2[2] - s2[1]) * 100, 0)
            left <- seq(from = s2[1], to = br, length.out = left_len)
            right <- seq(from = br, to = s2[2], length.out = 101 - left_len)[-1]
            s_vec <- c(left, right)
          }

          p <- unlist(sapply(s_vec, f2))
          p <- exp(p * lambda)
          p <- p / sum(p)
          play2[i] <- sample(s_vec, size = 1, prob = p)
        }
        play2 <- as.numeric(play2)
      }
    }
  }

  return(data.frame(play1 = play1,
                    play2 = play2,
                    period = 1:n_periods))
}