Is 16 vs 10 really a stand at true count 0? What exact composition analysis says about the famous in

Hi-Lo's best-known index says stand on 16 against a ten at a true count of 0 or more, hit below it. I wanted to see what that decision is worth at each count from the exact composition rather than an average, so I built a shoe at each true count and priced both plays.

Setup. Six decks, dealer stands on soft 17, no surrender, three decks left. For each true count the dealt cards are taken in proportion to the full shoe, then shifted between the Hi-Lo tags until the running count matches (a +2 shoe with three decks left has six more low cards gone than high). The true count is the shoe's before the hand is dealt. The hand is 10-6 against a dealer ten, and the EVs come from the same finite-shoe combinatorial analysis I use for basic strategy, which agrees with the Wizard of Odds house edge calculator within 0.005 points on the 22 rule sets I check it against.

Result. Hitting is better until +2. Standing takes over at +2, not 0, and at +3 it's worth about +0.007 per unit over hitting. The two curves are as flat as everyone says, but the crossing sits two points above the published index.

I ran the same thing for the rest of the Illustrious 18 and the Fab 4. Sixteen of the 22 cross within half a point of the published index (insurance at +3.5, 12 vs 3 at +2.5, 13 vs 2 at -1, 10 vs 10 at exactly +4). The ones that don't:

Hand	Dealer	Published index	Engine crossover
16 (10-6)	10	0	+2
15	10	+4	+5.5
12	2	+3	+4.5
12	4	0	+1.5
14 (surrender)	10	+3	+4.5
10,10	6	+4	+5
9	2	+1	0

Every straggler but one is a stiff hand against a high card, where the two EVs run nearly parallel, so a small change in composition moves the crossing a long way. The full table, with each play's gain three points past its index, is on:

casinogamessimulator com/blackjack/card-counting, under the drills.

Three questions for anyone who has generated indices:

1. What are the published indices averaged over: compositions, shoe depths, both?
2. Is a two-card 10-6 known to carry a higher index than the average 16 against a ten? The hand itself takes a ten and a six out of the shoe.
3. Is pinning the composition to the true count, as above, a fair way to get a composition-dependent index, or is there a standard method I should be matching?

If any of this is a bug I'd rather hear it here. The analysis is open source (MIT) at github com/casinogamessimulator/engines, under packages/blackjack (deviations.ts and the dealtAtCount function in counting.ts), and the same page has a free trainer that drills these plays with the exact EVs shown after each answer.

Quote “ The true count is the shoe's before the hand is dealt.”

Good research, but this part is problematic. When talking about true counts or running counts, they mean those after the hand is dealt.

Quote:

When talking about true counts or running counts, they mean those after the hand is dealt.

You're right, and thanks: that's the convention every index is published in, and my table isn't in it. The shoe is pinned to the true count before the three cards come out, so the count a player actually sees at the decision also includes the hand's own tags. For 10-6 against a ten that's -1 (two tens and a six) over the 2.94 decks left after the deal, so my "+2" is about +1.7 in the usual convention, and the last count where hitting was still better, +1.5, is about +1.2. So the crossing for 10-6 sits between +1 and +2 rather than at +2. Still above 0, but less far.

The same shift moves most of the other stragglers toward the published numbers. 9 vs 2 (2-7 against a 2) carries +2 in tags, so its 0 becomes about +0.7 against the published +1, and insurance with 10-6 showing goes from +3.5 to about +3.2. 12 vs 2 moves the other way, to about +4.9.

Rather than convert by hand, I'll recompute the whole table with the count taken after the cards on the table are out, which is a small change to the engine, and post the corrected numbers here. Once the convention matches, the 12s and the 14 vs 10 surrender will show whether anything is left to explain.

Earlier in another thread, we discussed the difference between a surrender game and a non-surrender game. I am more interested in the former.

In a surrender game, two-card 16 vs 10 decisions are either hit or surrender, so there is no hit/stand decision for these 16 vs 10 hands at all. What is left for you to consider are those three-or-more-card 16 vs 10 situations. How about you consider this problem?