See
http://gronbog.org/results/blackjack...litAces.6.html which is the same strategy generated independently by me.
lds means to double for any amount but for a little as possible, otherwise stand
dls means to double for any amount but for as much as possible, otherwise stand
ls means to double for less only of you can do so for less than the critical fraction of your initial bet.
If you hover your mouse over the data in a given cell, you will see the expected value of each available action. In the case of A,3 vs A, you will see
Double for zero (theoretical) has an EV of -0.31 or -31% of your original bet
Double for the full amount has an EV of -0.62 or -62% of your original bet
Double for other amounts will have an EV between these two extremes
Stand has an EV of -0.66 or -66% of your original bet
This is why the strategy for this hand is to double for as little as possible. It is a defensive play. Basic strategy would be to hit, but you don't have that option. The next best option is to pay as little as possible to take a card. The less you can double for, the closer this becomes to hitting.
Of special note are A,6 vs T and A and A,7 vs 9 and T. For these hands you double for less only if you can do so for less than the critical fraction of your original bet, otherwise stand. Hovering your mouse over A,7 vs T you will see that the critical fraction is 4.86%. That is, if you had bet $100, you would need be able to double for $4 or less.
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