Question for Don: How much should you bet?

[Gronbog]

There has been a lot of discussion about the individual trees in this scenario but not a lot on the forest. The Amazon-sized forest here is that you should not find yourself in this situation often enough in your lifetime for it to matter in the long term. If it happens once, learn from it and make sure that your session bankroll is sufficient before playing or, at the very least, before beginning a new shoe, because making a habit out of running on fumes and busting out while the count is high will cost you more in the long term than what you do with that final bet. If it happens again, OK, sh*t happens. If it keeps happening enough to actually matter, then shame on you!

However, it can still be useful to analyze these things in order to improve ones understanding of the game and, in this case to understand why you should not have entered this forest in the first place. What if you do find yourself in the forest? What is your actual situation? I submit that you are actually at a fork in the road leading to two smaller sub-forests. There has been even more discussion about individual trees (hands) in these forests under the assumption that the forests themselves are comparable. It turns out that they are not.

OK. Enough of the forest analogy. The OP wondered what should be bet should find yourself in the position where the appropriate bet for your bankroll is the 1/2 of your session bankroll. The options he considered were 1) betting it all, 2) betting 1/2 or less. Let's look at this in context of a 4.5/6 S17 DAS game using Hi-Lo and the I18 indices.

Betting It All

There has been discussion about how doubling and splitting will not occur often enough to matter and that, if the bet is warranted, you should go ahead and make it. The key here is if the bet is warranted. If you bet it all, this may no longer be the case. Why? Because by doing so, you have changed the rules of the game. You are now playing a 4.5/6 S17 SPL0 game with no doubling allowed. I made a comment earlier about how the loss in pre-deal EV would be in the fractions of a percent. I was grossly wrong about this. Here are the EVs and variances per true count for this horrible situation:

True                        Optimal Bet
Count    EV    Variance     %Bankroll
   4   -0.46%  0.977        0.00%
   5    0.05%   0.976        0.05%
   6    0.46%   0.975        0.47%
   7    0.89%   0.975        0.91%
   8    1.35%   0.975        1.38%
   9    1.79%   0.975        1.83%
  10    2.25%   0.975        2.31%
  11    2.62%   0.975        2.68%
  12    2.97%   0.976        3.04%
  13    3.35%   0.975        3.44%
>=14    4.04%   0.975        4.15%

Not being able to double or split impairs the game so much that you don't even have an advantage until TC>=5 and even at TC=5 the advantage is tiny at 0.05%. So you shouldn't even consider going all-in unless the true count is >=+5 and the optimal entry point does not occur until +6 or +7, depending on your intended spread. The OPs original proposed bet of the entire $600 is no longer warranted at any reasonable count with a $10k bankroll (full Kelly). With a $20k bankroll (half Kelly), the bet can be justified only >=+12.

Betting Half

You are now playing 4.5/6 S17 SPL1 NDAS. Here are the EVs and variances:

True                       Optimal Bet
Count   EV    Variance     %Bankroll
  0   -0.40%    1.288        0.00%
  1    0.15%    1.286        0.12%
  2    0.67%    1.283        0.52%
  3    1.17%    1.285        0.91%
  4    1.74%    1.291        1.34%
  5    2.36%    1.297        1.82%
  6    2.83%    1.297        2.18%
  7    3.57%    1.291        2.76%
  8    3.93%    1.283        3.06%
  9    4.51%    1.272        3.55%
  10   4.93%    1.265        3.90%
  11   5.26%    1.250        4.21%
  12   6.11%    1.246        4.90%
  13   7.21%    1.233        5.85%
>=14   6.83%    1.218        5.61%

Here the game is significantly less impaired due to the ability to double or split once and in fact the EVs are indeed affected by only a fraction of a percent, meaning you still have a small advantage at TC=1. Notice that the IBAs at all advantage counts are more than twice than when betting it all, except for >=+14, and in some cases significantly higher than that.

Conclusions

1. The bet which was "justified" based on having enough chips to play all situations may no longer be justified at all with limited chips due to an effective degradation in the rules. In a SPL3 DAS game, this technically means needing 8.5 initial bets to be able to begin a round.
2. The degradation in conditions begins the moment you have less than this on hand, but for practical purposes is likely not really a factor until you are down to 2 or 3 initial bets.
3. You should only make a bet if you have the bankroll to support the bet under the degraded conditions.
4. You probably don't have the bankroll to bet it all ($600).
5. Even if you do have the bankroll to support betting $600 all-in, your IBA for betting 1/2 ($300) is more than twice as high at all advantage counts. So given a choice of betting it all or betting 1/2, betting 1/2 appears to be superior.
6. There is likely a sweet spot fraction of your remaining chips which maximizes EV.
7. Stay out of the forest. Don't let this happen to you in the first place!