# Simulation of Blackjack: the odds are not with you

July 19, 2013
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`  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==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 demonstrationminmax <- 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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