##### https://github.com/cran/GPGame
Tip revision: cbe720d
crit_PNash.R
## ' Probability for a strategy of being a Nash Equilibrium by simulation
## ' @title Simulated Pnash probability
## ' @param s strategy considered
## ' @param integcontrol is a list containing: \code{integ.pts}, a [npts x dim] matrix defining the grid,
## ' \code{expanded.indices} a matrix containing the indices of the integ.pts on the grid and \code{n.s},
## ' a nobj vector containting the number of strategies per player
## ' @param model list of km
## ' @param nsim number of conditional simulations for computation
## ' @param eps numerical jitter
## ' @export
## ' @examples
## ' \dontrun{
## ' Pnashs <- apply(expanded.indices, 1, Pnash, expanded.indices = expanded.indices,
## '                 integ.pts = integ.pts, model = model, nsim = 100)
## ' }
Pnash <- function(s, integcontrol, model, nsim, eps = 1e-6){
if(is.null(eps))
eps <- 1e-6

expanded.indices <- integcontrol$expanded.indices integ.pts <- integcontrol$integ.pts

## Loop over objectives
p <- 1
for(i in 1:length(s)){
idx <- get_alternatives(s, i, expanded.indices)
preds <- predict(model[[i]], newdata = integ.pts[idx,], type = 'UK', checkNames = F, cov.compute = TRUE, se.compute = FALSE)
sims <- mvrnorm(n = nsim, mu = preds$mean, Sigma = preds$cov + diag(eps, length(preds$mean))) sisBest <- apply(sims, 1, function(x) min(x) == x[s[i]]) p <- p * sum(sisBest)/nsim } return(p) } ## ' Probability for a stragy of being a Nash Equilibrium with mnormt ## ' @title Exact Pnash probability ## ' @param s strategy considered ## ' @param integcontrol is a list containing: \code{integ.pts}, a [npts x dim] matrix defining the grid, ## ' \code{expanded.indices} a matrix containing the indices of the integ.pts on the grid and \code{n.s}, ## ' a nobj vector containting the number of strategies per player ## ' @param model list of km ## ' @param eps numerical jitter ## ' @examples ## ' \dontrun{ ## ' Pnashs2 <- apply(expanded.indices, 1, Pnash_exact, expanded.indices = expanded.indices, ## ' integ.pts = integ.pts, model = model) ## ' } ## ' @importFrom mnormt pmnorm ## ' @importFrom mvtnorm pmvnorm ## ' @details inspired from DiceOptim qEI function ## ' @export Pnash_exact <- function(s, integcontrol, model, eps = 1e-8){ p <- 1 expanded.indices <- integcontrol$expanded.indices
integ.pts <- integcontrol$integ.pts ## Loop over objectives for(i in 1:length(s)){ idx <- get_alternatives(s, i, expanded.indices) preds <- predict(model[[i]], newdata = integ.pts[idx,], type = 'UK', checkNames = F, cov.compute = TRUE, se.compute = FALSE) mu_k <- preds$mean[s[i]] - preds$mean[-s[i]] Sigma_k <- covZk(preds$cov, s[i])
Sigma_k <- Sigma_k[-s[i], -s[i]]

if(length(idx) < 20){
pNashsi <- pmnorm(x = -mu_k, varcov = Sigma_k + diag(eps, length(preds$mean) - 1)) # Error if more than 20 dimensions }else{ pNashsi <- pmvnorm(upper = -mu_k, sigma = Sigma_k + diag(eps, length(preds$mean) - 1))
}
p <- p * pNashsi[1]
}
return(p)
}

## Just add argument i for compatibility with mclapply
Pnash_wrap <- function(i, s, integcontrol, type = 'simu', model, control = list(nsim = 100, eps = 1e-6)){
if(is.null(nrow(s)))
s <- matrix(s, nrow = 1)
if(type == 'simu')
return(Pnash(s = s[i,], integcontrol=integcontrol, model = model, nsim = control$nsim, eps = control$eps))
if(type == 'exact')
return(Pnash_exact(s = s[i,], integcontrol=integcontrol, model = model, eps = control$eps)) } #' Acquisition function for solving game problems based on the probability for a strategy of being a Nash Equilibrium. #' The probability can be computed exactly using the mutivariate Gaussian CDF (\code{mnormt}, \code{pmvnorm}) or by Monte Carlo. #' @title Probability for a strategy of being a Nash Equilibrium #' @param idx is the index on the grid of the strategy evaluated #' @param integcontrol is a list containing: \code{integ.pts}, a [\code{npts x dim}] matrix defining the grid, #' \code{expanded.indices} a matrix containing the indices of the \code{integ.pts} on the grid and \code{n.s}, #' a \code{nobj} vector containting the number of strategies per player #' @param type '\code{exact}' or '\code{simu}' #' @param model is a list of nobj \code{\link[DiceKriging]{km}} models #' @param control list with slots \code{nsim} (number of conditional simulations for computation) and \code{eps} #' @param ncores \code{\link[parallel]{mclapply}} is used if \code{> 1} for parallel evaluation #' @param eps numerical jitter for stability #' @return Probability of being a Nash equibrium corrsponding to \code{idx}. #' @export #' @seealso \code{\link[GPGame]{crit_SUR_Eq}} for an alternative infill criterion #' @references #' V. Picheny, M. Binois, A. Habbal (2016+), A Bayesian optimization approach to find Nash equilibria, #' \emph{https://arxiv.org/abs/1611.02440}. #' @examples #' \donttest{ #' ############################################## #' # Example 1: 2 variables, 2 players, no filter #' ############################################## #' library(DiceKriging) #' set.seed(42) #' #' # Define objective function (R^2 -> R^2) #' fun <- function (x) #' { #' if (is.null(dim(x))) x <- matrix(x, nrow = 1) #' b1 <- 15 * x[, 1] - 5 #' b2 <- 15 * x[, 2] #' return(cbind((b2 - 5.1*(b1/(2*pi))^2 + 5/pi*b1 - 6)^2 + 10*((1 - 1/(8*pi)) * cos(b1) + 1), #' -sqrt((10.5 - b1)*(b1 + 5.5)*(b2 + 0.5)) - 1/30*(b2 - 5.1*(b1/(2*pi))^2 - 6)^2- #' 1/3 * ((1 - 1/(8 * pi)) * cos(b1) + 1))) #' } #' #' # Grid definition #' n.s <- rep(11, 2) #' x.to.obj <- c(1,2) #' gridtype <- 'cartesian' #' integcontrol <- generate_integ_pts(n.s=n.s, d=2, nobj=2, x.to.obj = x.to.obj, gridtype=gridtype) #' #' test.grid <- integcontrol$integ.pts
#' expanded.indices <- integcontrol$expanded.indices #' n.init <- 11 #' design <- test.grid[sample.int(n=nrow(test.grid), size=n.init, replace=FALSE),] #' response <- t(apply(design, 1, fun)) #' mf1 <- km(~., design = design, response = response[,1], lower=c(.1,.1)) #' mf2 <- km(~., design = design, response = response[,2], lower=c(.1,.1)) #' model <- list(mf1, mf2) #' #' crit_sim <- crit_PNash(idx=1:nrow(test.grid), integcontrol=integcontrol, #' type = "simu", model=model, control = list(nsim = 100)) #' crit_ex <- crit_PNash(idx=1:nrow(test.grid), integcontrol=integcontrol, type = "exact", model=model) #' #' filled.contour(seq(0, 1, length.out = n.s[1]), seq(0, 1, length.out = n.s[2]), zlim = c(0, 0.7), #' matrix(pmax(0, crit_sim), n.s[1], n.s[2]), main = "Pnash criterion (MC)", #' xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors, #' plot.axes = {axis(1); axis(2); #' points(design[,1], design[,2], pch = 21, bg = "white") #' } #' ) #' #' filled.contour(seq(0, 1, length.out = n.s[1]), seq(0, 1, length.out = n.s[2]), zlim = c(0, 0.7), #' matrix(pmax(0, crit_ex), n.s[1], n.s[2]), main = "Pnash criterion (exact)", #' xlab = expression(x[1]), ylab = expression(x[2]), color = terrain.colors, #' plot.axes = {axis(1); axis(2); #' points(design[,1], design[,2], pch = 21, bg = "white") #' } #' ) #' } #' @importFrom mnormt pmnorm #' @importFrom mvtnorm pmvnorm #' crit_PNash <- function(idx, integcontrol, type = 'simu', model, ncores = 1, control = list(nsim = 100, eps = 1e-6)){ if (is.null(control$nsim)) control$nsim <- 100 if (is.null(control$eps)) control$eps <- 1e-6 # if(is.null(nrow(x))) # x <- matrix(x, nrow = 1) # # if (is.null(idx) && is.null(x)) { # cat("At least one of the inputs idx and x should be provided \n") # return(NA) # } s <- integcontrol$expanded.indices[idx,,drop=FALSE]

if(ncores > 1)
return(unlist(mclapply(X = 1:nrow(s), s = s, FUN = Pnash_wrap, model = model, integcontrol=integcontrol,
mc.cores = ncores, type = type, control = control)))

return(apply(matrix(1:nrow(s), ncol = 1), 1, Pnash_wrap, s = s, integcontrol=integcontrol,
type = type, model = model, control = control))
}

## ' Return the designs s[-i]
## ' @param s .
## ' @param i .
## ' @param expanded.indices .
## ' @export
get_alternatives <- function(s, i, expanded.indices){
return(which(apply(expanded.indices[,-i, drop = FALSE], 1, function(x) all(x == s[-i]))))
}

covZk <- function(sigma,index){
result <- sigma
q <- nrow(sigma)
I <- (1:q)[-index]
result[I,index] <- result[index,I] <- sigma[index,index] - sigma[index,I]
result[index,I] <- sigma[index,index] - sigma[I,index]
M <- crossprod(sigma[index,I,drop=FALSE], rep(1,q-1))
result[I,I] <- sigma[I,I] + sigma[index,index] - M - t(M)
return(result)
}