(This article was first published on

**Econometrics by Simulation**, and kindly contributed to R-bloggers)I often want to simulate outcomes varying across a set of parameters.In order to accomplish this in an efficient manner I have coded up a little function that takes parameter vectors and produces results. First I will show how to set it up with some dummy examples and next I will show how it can be used to select the optimal blackjack strategy.

SimpleSim <- function(..., fun, pairwise=F) {

# SimpleSim allows for the calling of a function varying

# multiple parameters entered as vectors. In pairwise form

# it acts much like apply. In non-paiwise form it makes a

` # combination of each possible parameter mix`

# in a manner identical to block of nested loops.

returner <- NULL

L <- list(...)

# Construct a vector that holds the lengths of each object

vlength <- unlist(lapply(L, length))

npar <- length(vlength)

CL <- lapply(L, "[", 1) # Current list is equal to the first element

# Pairwise looping

if (pairwise) {

# If pairwise is selected than all elements greater than 1 must be equal.

# Checks if all of the elements of a vector are equal

if (!(function(x) all(x[1]==x))(vlength[vlength>1])) {

print(unlist(lapply(L, length)))

stop("Pairwise: all input vectors must be of equal length", call. =F)

}

for (i in 1:max(vlength)) { # Loop through calling the function

CL[vlength>1] <- lapply(L, "[", i)[vlength>1] # Current list

returner <- rbind(returner,c(do.call(fun, CL),pars="", CL))

}

} # End Pairwise

# Non-pairwise looping

if (!pairwise) {

ncomb <- prod(vlength) # Calculate the number of combinations

print(paste(ncomb, "combinations to loop through"))

comb <- matrix(NA, nrow=prod(vlength), ncol=npar+1)

comb[,1] <- 1:prod(vlength) # Create an index value

comb <- as.data.frame(comb) # Converto to data.frame

colnames(comb) <- c("ID", names(CL))

for (i in (npar:1)) { # Construct a matrix of parameter combinations

comb[,i+1] <- L[[i]] # Replace one column with values

comb<-comb[order(comb[,(i+1)]),] # Reorder rows

}

comb<-comb[order(comb[,1]),]

for (i in 1:ncomb) {

for (ii in 1:npar) CL[ii] <- comb[i,ii+1]

returner <- rbind(returner,c(do.call(fun, CL),pars="", CL))

}

} # End Non-Pairwise

return(returner)

} # END FUNCTION DEFINITION

# Let's first define a simple function for demonstration

minmax <- function(...) c(min=min(...),max=max(...))

# Pairwise acts similar to that of a multidimensional apply across columns

SimpleSim(a=1:20,b=-1:-20,c=21:40, pairwise=T, fun="minmax")

# The first set of columns are those of returns from the function "fun" called.

# The second set divided by "par" are the parameters fed into the function.

SimpleSim(a=1:20,b=-1:-20,c=10, pairwise=T, fun="minmax")

# Non-pairwise creates combinations of parameter sets.

# This form is much more resource demanding.

SimpleSim(a=1:5,b=-1:-5,c=1:2, pairwise=F, fun="minmax")

# Let's try something a little more interesting.

# Let's simulate a game of black jack strategies assuming no card counting is possible.

blackjack <- function(points=18, points.h=NULL, points.ace=NULL,

cards=10, cards.h=NULL, cards.ace=NULL,

sims=100, cutoff=10) {

# This function simulates a blackjack table in which the player

# has a strategy of standing (not asking for any more cards)

# once he has either recieved a specific number of points or

# a specific number of cards. This function repeates itself sims # of times.

# This function allows for up to three different strategies to be played.

# 1. If the dealer's hole card is less than the cuttoff

# 2. If the dealer's hole card is greater than or equal to the cuttoff

# 3. If the dealer's hole card is an ace

# In order to use 3 level strategies input parameters as .h and .ace

# It returns # of wins, # of losses, # of pushes (both player and dealer gets 21)

# and the number of blackjacks.

# This simulation assumes the number of decks used is large thus

# the game is like drawing with replacement.

if (is.null(points.h)) points.h <- points

if (is.null(points.ace)) points.ace <- points.h

if (is.null(cards.h)) cards.h <- cards

if (is.null(cards.ace)) cards.ace <- cards.h

bdeck <- c(11,2:9,10,10,10,10) # 11 is the ace

bdresult <- c(ppoints=NULL, pcards=NULL, dpoints=NULL, dcards=NULL)

for (s in 1:sims) {

dhand <- sample(bdeck,1) # First draw the deal's revealed card

phand <- sample(bdeck,2, replace=T)

# Specify target's based on dealer's card

if (dhand) {

pcuttoff <- points

ccuttoff <- cards

}

if (dhand>=cutoff) {

pcuttoff <- points.h

ccuttoff <- cards.h

}

if (dhand==11) {

pcuttoff <- points.ace

ccuttoff <- cards.ace

}

# player draws until getting above points or card count

while ((sum(phand))&(length(phand) )){

phand <- c(phand, sample(bdeck,1))

# If player goes over then player may change aces to 1s

if (sum(phand)>21) phand[phand==11] <- 1

}

# Dealer must always hit 17 so hand is predetermined

while (sum(dhand)<17) {

dhand <- c(dhand, sample(bdeck,1))

# If dealer goes over then dearler may change aces to 1s

if (sum(dhand)>21) dhand[dhand==11] <- 1

}

bdresult <- rbind(bdresult,

c(ppoints=sum(phand), pcards=length(phand),

dpoints=sum(dhand), dcards=length(dhand)))

}

# Calculate the times that the player wins, pushes (ties), and loses

pbj <- (bdresult[,1]==21) & (bdresult[,2]==2)

dbj <- (bdresult[,3]==21) & (bdresult[,4]==2)

pwins <- ((bdresult[,1] > bdresult[,3]) & (bdresult[,1] < 22)) | (pbj & !dbj)

push <- (bdresult[,1] == bdresult[,3]) | (pbj & dbj)

dwins <- !(pwins | push)

# Specify the return.

c(odds=sum(pwins)/sum(dwins),

pwins=sum(pwins),

dwins=sum(dwins),

push=sum(push),

pcards=mean(bdresult[,2]),

dcards=mean(bdresult[,4]),

pblackjack=sum(pbj),

dblackjack=sum(dbj))

}

blackjack(points=18, sims=4000)

# We can see unsurprisingly, that the player is not doing well.

blackjack(points=18, points.h=19, sims=4000)

# We can see that by adopting a more aggressive strategy for when

# the dealer has a 10 point card or higher, we can do slightly better.

# But overall, the dealer is still winning about 3x more than us.

# We could search through different parameter combinations manually to

# find the best option. Or we could use our new command SimpleSim!

MCresults <- SimpleSim(fun=blackjack, points=15:21, points.h=18:21,

points.ace=18:21, cutoff=9:10, cards=10, sims=100)

# Let's now order our results from the most promising.

MCresults[order(-unlist(MCresults[,1])),]

# By the simulation it looks like we have as high as a 50% ratio of loses to wins.

# Which means for every win there are 2 loses.

# However, I don't trust it since we only drew 100 simulations.

# In addition, this is the best random draw from all 224 combinations which each

# have different probabilities.

# Let's do the same simulation but with 2000 draws per.

# This might take a little while.

MCresults <- SimpleSim(fun=blackjack, points=15:21, points.h=18:21,

points.ace=18:21, cutoff=9:10, cards=10, sims=5000)

# Let's now order our results from the most promising.

MCresults[order(-unlist(MCresults[,1])),]

hist(unlist(MCresults[,1]), main="Across all combinations\nN(Win)/N(Loss)",

xlab = "Ratio", ylab = "Frequency")

# The best case scenario 38% win to loss ratio appears around were we started,

# playing to hit 18 always and doing almost as well when the dealer is high

# (having a 10 or ace) then playing for 19.

# Overall, the odds are not in our favor. For every win we expect 1/.38 (2.63) loses.

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