A more profitable approach

BASIS FOR ANALYSIS

Analyzing this game m terms of EV (expected value) of individual bets can be difficult, since the EV of each bet is variable, and can be influenced by the outcome of the previous bet (and so forth back through time). Unless one is making single bets (or a limited and well defined number of bets), then it probably makes more sense to analyze the EV of the overall strategy (I'll call this EVS = Expected Value of Strategy - %) rather than the EV of individual bets. This is the approach used below

YOUR PROPOSED STRATEGY

You correctly pointed out that to maximize EVS, you should make a single bet and then quit for the week. In this case your EVS is

EVS = (Proportion of Gains x Magnitude of Gains) - (Proportion of Losses x Magnitude of Losses)

Assuming a basic strategy EV of -0.6% and a rebate of 10%, then

Proportion of Gains = 0.5 - (0.006/2) = 0.497

Proportion of Losses = 0.5 + (0.006/2) = 0.503

Magnitude of Gains = 1

Magnitude of Losses = 1 - Rebate = 1 - 0.1 = 0.9

So the EVS equation becomes

EVS = (0.497 x 1) - (.503 x 0.9) = 0.0443

This is how I obtained EV = 4.4% in my previous post. Please note that I conveniently left out the complicated payouts for blackjacks, doubling, pair splitting etc. Also, I assumed no re-betting of pushes, although it probably would be appropriate to do so. Overall I believe the above equation is a good approximation.

A high EVS is great, but it is just a percentage, and you can't put a percentage in your wallet and spend it. So we need to convert EVS into dollars. This is sometimes called Expected Win (EW), where:

EW = EVS x Dollars Risked

Let's assume you bet at the table limit is $100, and you follow the above strategy. Then your weekly expected win would be:

EW = 0.0443 x $100 = $4.43

AN IMPROVED STRATEGY

If your bankroll relatively small, (i.e. you cannot bet above the table maximum), then the strategy above is optimal, since it maximizes both EVS and by extension maximizes EW. However, if your bankroll is larger, then there are strategies that result in a higher EW, even though they have a lower EVS. This is achieved by increasing the weekly dollars risked to a value greater than the table limit.

I'll walk through one example which is probably close to optimal, and which I described in my earlier post. Let's asssume you follow the following strategy: once a week you make repeated table limit bets until you are either up by 3x the table limit, or down by 3x the table limit. At that point you immediately quit for the week. If the table maximum is $100, then you are effectively betting $300 each week.... either you win $300 or you lose $300 (let's call this strategy "Stop at +3 or -3"). Based on my simulation (these are not exact numbers), the proportion of weekly wins and losses under this strategy would be approximately:

Proportion of wins = 0.493

Proportion of losses = 0.507

Therefore, going back to our original EVS equation, and using the above values:

EVS = (0.493 x 1) - (0.507 x 0.9) = 0.0367

So you only win about 3.7% under this strategy ("Stop at +3 or -3"), as compared to 4.4% under the single bet strategy ("Stop at +1 or -1). However you are now betting $300/week rather than $100/week. So your weekly expected win increases to:

EW = 0.0367 x $300 = $11.01

In other words, you are now winning $11.01/week with this improved strategy, rather than $4.43/week under the original strategy. This improved strategy ("Stop at +3 or -3") probably gives close to the optimal win rate. My simulations are showing that "Stop at +4 or -4" gives a slightly lower weekly expected win rate, while "Stop at +5 or -5" and "Stop at +2 or -2" both give you a significantly lower win rate. For reference, you will play an average of 9 hands per week under "Stop at +3 or -3", and an average of 16 hands per week using "Stop at +4 or -4"

It is possible you could get a higher win rate using an asymmetric strategy (i.e. Stop at +3 or -4), but I have not analyzed that. And of course all the assumptions described in my earlier post apply.