Incremental Betting

Python 2, 72 68 62 bytes

def g(a,s,n=1):
 for c in s:
    if a>=n:a,n=((a+n,1),(a-n,2*n))[c<'W']
 return a

Call it like so: g(15,'LLLWLLLL').

This simply loops through the string, changing the value of the money that we have based on the character.

Here is a sample program that runs tests on this function:

import random
def g(a,s,n=1):
 for c in s:
    if a>=n:a,n=((a+n,1),(a-n,2*n))[c<'W']
 return a

for i in range(14):
    s=''.join(('L','W')[random.randint(0, 1)] for e in range(random.randint(10, 15)))
    print'g(%i,%s):'%(i,`s`),
    print g(i,s)

Sample output:

g(0,'LLWWWWWWLWWWWW'): 0
g(1,'WLLWWWWWWWW'): 1
g(2,'WWWLLLWLLW'): 2
g(3,'LLLLWLWLWWWWLL'): 0
g(4,'LWWWWLWLWWW'): 12
g(5,'WWLWWLLWWW'): 12
g(6,'LLLWWWLLLLWLLWL'): 3
g(7,'WWLLWWLWLWLWLLL'): 7
g(8,'WLLLWWWWWLLWLL'): 2
g(9,'WWWLLWLLLLLWL'): 6
g(10,'LWWWWLLLWL'): 7
g(11,'WLLLLWLWWWW'): 5
g(12,'WLLWWLWWWL'): 17
g(13,'LLLWLLWLWLWLWW'): 6

With a little change to the tester, we can get the average profit of many runs:

import random
def g(a,s,n=1):
 for c in s:
    if a>=n:a,n=((a+n,1),(a-n,2*n))[c<'W']
 return a

r=[]
for i in range(5000):
    for i in range(1000):
        s=''.join(('L','W')[random.randint(0, 1)] for e in range(random.randint(10, 15)))
        r+=[i-g(i,s)]
a=0
for n in r:
    a+=n
print float(a)/len(r)

Sample output (took quite a while, since we are calling the function 5000000 times):

-0.0156148

Edit: Thanks to Howard and Danny for further golfing.

EDIT: now the program checks for whether there is enough money to make the bet. This actually saves bytes.

GolfScript, 33 characters

{
1\{2$2$<!{1&{+1}{:b-b.+}if.}*;}/;
}:g;

Examples (online):

> 13 'LLLWLLLL'
6
> 4 'LWWLWLWWWL'
9
> 5 'LLLWWWLLWLWW'
2
> 2 'LW'
1

Annotated code:

1\            # prepare stack a b r
{             # for each char in r
  2$2$<!{     #   if a>=b  
    1&        #     take last bit of character (i.e. 0 for L and 1 for W)
    {         #     if W
      +       #       a <- a+b
      1       #       b <- 1
    }{        #     else
      :b-     #       a <- a-b
      b.+     #       b <- 2*b
    }if       #     end if
    .         #     create dummy value
  }*          #   end if
  ;           #   drop (i.e. either the dummy or the character)
}/            # end for
;             # discard current bet value

R, 95 characters

g=function(a,r){n=1;for(i in 1:nchar(r)){s=substr(r,i,i);if(s=='L'){a=a-n;n=n*2}else{a=a+n;n=1};if(n>a)break};a}

Indented:

g=function(a,r){
    n=1
    for(i in 1:nchar(r)){
        s=substr(r,i,i)
        if(s=='L'){
            a=a-n
            n=n*2
            }else{
                a=a+n
                n=1
                }
        if(n>a)break
        }
    a
    }

Usage:

> g(15,'LLWLLLL')
[1] 1
> g(20,'WLLW')
[1] 22
> g(13,'LLWLLLLWWLWWWLWLWW')
[1] 7