getEquilibrium {GPGame}

Equilibrium computation of a discrete game for a given matrix with objectives values

Description

Computes the equilibrium of three types of games, given a matrix of objectives (or a set of matrices) and the structure of the strategy space.

Usage

getEquilibrium(
  Z,
  equilibrium = c("NE", "NKSE", "KSE", "CKSE"),
  nobj = 2,
  n.s,
  expanded.indices = NULL,
  return.design = FALSE,
  sorted = FALSE,
  cross = FALSE,
  kweights = NULL,
  Nadir = NULL,
  Shadow = NULL,
  calibcontrol = NULL
)

Arguments

Z

is a matrix of size [npts x nsim*nobj] of objective values, see details,

equilibrium

considered type, one of "NE", "NKSE", "KSE", "CKSE"

nobj

nb of objectives (or players)

n.s

scalar of vector. If scalar, total number of strategies (to be divided equally among players), otherwise number of strategies per player.

expanded.indices

is a matrix containing the indices of the integ.pts on the grid, see generate_integ_pts

return.design

Boolean; if TRUE, the index of the optimal strategy is returned (otherwise only the pay-off is returned)

sorted

Boolean; if TRUE, the last column of expanded.indices is assumed to be sorted in increasing order. This provides a substantial efficiency gain.

cross

Should the simulation be crossed? (For "NE" only - may be dropped in future versions)

kweights

kriging weights for CKS (TESTING)

Nadir, Shadow

optional vectors of size nobj. Replaces the nadir and/or shadow point for KSE. Some coordinates can be set to Inf (resp. -Inf).

calibcontrol

an optional list for calibration problems, containing target a vector of target values for the objectives, log a Boolean stating if a log transformation should be used or not and offset a (small) scalar so that each objective is log(offset + (y-T^2)).

Details

If nsim=1, each line of Z contains the pay-offs of the different players for a given strategy s: [obj1(s), obj2(s), ...]. The position of the strategy s in the grid is given by the corresponding line of expanded.indices. If nsim>1, (vectorized call) Z contains different trajectories for each pay-off: each line is [obj1_1(x), obj1_2(x), ... obj2_1(x), obj2_2(x), ...].

Value

A list with elements:

• NEPoff vector of pay-offs at the equilibrium or matrix of pay-offs at the equilibria;

• NE the corresponding design(s), if return.design is TRUE.

Examples

## Setup
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)))
}

d <- nobj <- 2

# Generate grid of strategies for Nash and Nash-Kalai-Smorodinsky
n.s <- c(11,11) # number of strategies per player
x.to.obj <- 1:2 # allocate objectives to players
integcontrol <- generate_integ_pts(n.s=n.s,d=d,nobj=nobj,x.to.obj=x.to.obj,gridtype="cartesian")
integ.pts <- integcontrol$integ.pts
expanded.indices <- integcontrol$expanded.indices

# Compute the pay-off on the grid
response.grid <- t(apply(integ.pts, 1, fun))

# Compute the Nash equilibrium (NE)
trueEq <- getEquilibrium(Z = response.grid, equilibrium = "NE", nobj = nobj, n.s = n.s,
                         return.design = TRUE, expanded.indices = expanded.indices,
                         sorted = !is.unsorted(expanded.indices[,2]))

# Pay-off at equilibrium
print(trueEq$NEPoff)

# Optimal strategy
print(integ.pts[trueEq$NE,])

# Index of the optimal strategy in the grid
print(expanded.indices[trueEq$NE,])

# Plots
oldpar <- par(mfrow = c(1,2))
plotGameGrid(fun = fun, n.grid = n.s, x.to.obj = x.to.obj, integcontrol=integcontrol,
             equilibrium = "NE")

# Compute KS equilibrium (KSE)
trueKSEq <- getEquilibrium(Z = response.grid, equilibrium = "KSE", nobj = nobj,
                         return.design = TRUE, sorted = !is.unsorted(expanded.indices[,2]))

# Pay-off at equilibrium
print(trueKSEq$NEPoff)

# Optimal strategy
print(integ.pts[trueKSEq$NE,])

plotGameGrid(fun = fun, n.grid = n.s, integcontrol=integcontrol,
             equilibrium = "KSE", fun.grid = response.grid)

# Compute the Nash equilibrium (NE)
trueNKSEq <- getEquilibrium(Z = response.grid, equilibrium = "NKSE", nobj = nobj, n.s = n.s,
                         return.design = TRUE, expanded.indices = expanded.indices,
                         sorted = !is.unsorted(expanded.indices[,2]))

# Pay-off at equilibrium
print(trueNKSEq$NEPoff)

# Optimal strategy
print(integ.pts[trueNKSEq$NE,])

# Index of the optimal strategy in the grid
print(expanded.indices[trueNKSEq$NE,])

# Plots
plotGameGrid(fun = fun, n.grid = n.s, x.to.obj = x.to.obj, integcontrol=integcontrol,
             equilibrium = "NKSE")
par(oldpar)