How to measure the robustness to outliers of some regression models - r

Question

I am adding the variances of model coefficients and then returning the means of the sums.
I simply want to check which Regression Method is more robust to the outliers. I will examine many scenarios.
However, my code is giving me that ordinary least square is the best, but this is not expected results since MM estimation and Huber is called robust regression methods.
Am I doing something wrong with my code?
#####################################
rmn <- function(n, mu) {
p <- length(mu)
matrix(rnorm(n*p, mean = mu), ncol = p)
}
#####################################
RI<-function(y,x,a,mu,R=30,t=1000){
x <- as.matrix(x)
dm <- dim(x)
n <- dm[1]
bias1 <- bias2 <- bias3 <- numeric(t)
b1 <- b2<- b3 <- numeric(R)
### Outliers in X ######
for (j in 1:t) {
for (i in 1:R) {
id <- sample(n, a * n)
z <- x
z[id, ] <- rmn(length(id), mu)
b1[i] <- var(coef(lm(y ~., data = data.frame(z))))
b2[i] <- var(coef(rlm(y ~ ., data = data.frame(z), maxit = 2000, method = "MM")))
b3[i] <- var(coef(rlm(y ~ ., data = data.frame(z), psi = psi.huber,maxit = 300)))
}
bias1[j] <- sum(b1); bias2[j] <- sum(b2); bias3[j] <- sum(b3)
}
bias <- cbind("lm" = bias1,"MM-rlm" = bias2, "H-rlm" = bias3)
colMeans(bias)
}
#####################################
p <- 5
n <- 300
x<- matrix(rnorm(n * p), ncol = p)
y<-rnorm(n)
a=0.2
mu <-colMeans(x)+10
#####################################
RI(y,x,a,mu)
#####################################
UPDATE
I changed the idea of measuring Robustness, due to the first provided answer.
I measured robustness by calculating the mean absolute difference between coefficients when data is uncontaminated and when they are contaminated. I introduce outliers first in y and then in x. I still have a problem.
############ R CODE ##############
rmn <- function(n, mu, seed = TRUE) {
 if (seed) set.seed(12345)
 p <- length(mu)
 matrix( rnorm(n * p, mean = mu), ncol = p)
}
##################################
out.cv <- function(y, x, a, mu, R = 500, seed = TRUE) {
 ## y: response variable
 ## x: independent variables
 ## a: percent of outliers
 ## mu: how far should the outliers be. A vector if outliers in x,
 ## or a single number if outliers in y
 ## R: how many times to repeat this process
 x <- as.matrix(x)
 dm <- dim(x)
 n <- dm[1] ; d <- dm[2] + 1
 b1 <- b2<- b3 <- numeric(R)
 be <- coef( lm(y ~., data = as.data.frame(x[,-1]) ) )
####################################
 ### Outliers in Y ######
 if ( length(mu) == 1 ) {
 for (i in 1:R) {
 if (seed) set.seed(12345)
 id <- sample(n, a * n)
 z <- y
 if (seed) set.seed(12345)
 z[id] <- rnorm(id, mu) ## mu has to be a single number here
 ## mean absolute difference between coefficients of clean data
 ## and coefficients with contaminated data
 b1[i] <- mean( abs( coef( lm(z ~., data = as.data.frame(x[,-1])) ) - be) )
 b2[i] <- mean( abs( coef( rlm(z ~ ., data = data.frame(x[,-1]), maxit = 2000, method = "MM") ) - be ) )
 b3[i] <- mean( abs( coef( rlm(z ~ ., data = data.frame(x[,-1]), psi = psi.huber,maxit = 300) ) - be ) )
 }
########################
##### Outliers in X #########
 } else {
 for (i in 1:R) {
 if (seed) set.seed(12345)
 id <- sample(n, a * n)
 z <- x
 z[id, ] <- rmn( length(id), mu, seed ) ## mu must be a vector
 b1[i] <- mean( abs( coef( lm(y ~., data = as.data.frame(z[,-1])) )- be) )
 b2[i] <- mean( abs( coef( rlm(y ~ ., data = data.frame(z[,-1]), maxit = 2000, method = "MM") ) - be ) )
 b3[i] <- mean( abs( coef( rlm(y ~ ., data = data.frame(z[,-1]), psi = psi.huber,maxit = 300) ) - be ) )
 }
 }
 bias1 <- mean(b1) ; bias2 <- mean(b2); bias3 <- mean(b3)
 bias <- c(bias1, bias2, bias3)
 names(bias) <- c("lm", "MM-rlm","Huber-rlm")
 bias
}
################################
p <- 5
n <- 200
##############################
# Independent X and Y ####
#set.seed(12345)
#x<- matrix( rnorm(n * p), ncol = p)
#y<-rnorm(n)
## Related X and Y ####
set.seed(12345)
x <- rmn(n, numeric(p))
ber <- rnorm(p)
m <- x %*% ber
y <- rnorm(n, m, 1)
############################
a <- 0.2 #outliers 10%
mu <- 15 ## outliers in y
out.cv(y, x, a, mu)
###########################
mu <-colMeans(x)+15 ## outliers in x
out.cv(y, x, a, mu)
###################

Answer 1

First of all I do not see that you generate a sample from long tailed distribution. Please, use rt(n, 3) t student with very small df to get such distribution or play with other ones like log-normal. Thus do not use rnorm for sure. I see that you use some injection produce which seems to be overcomplicated.
Another thing is that specification of the MASS::rlm is not as trivial.
In my opinion start with quantreg::rq which is a quantile regression and treat it as a robust benchmark method.
Additionally your sampling procedure looks to not be a valid one. You generating a new observations each iteration which are not know apriori. I would expect bootstrapping on train or test set.