Is this Hi Lo game beatable

Hi,

Please read the game rules and description before answering the questions below (since I wouldn't be posting this if there wasn't at least some chance of this game having a 'long-term +EV' when played with a correct strategy).

1. If an online game had a single deck Hi-Lo game, could the initial house edge be turned into a game with a player edge*** in the long-term?

edge***: I already know that this games initial house edge can be reduced, (with the correct player choice) what i don't know is the chances of it happening and if a correct strategy could turn the game into a + EV game overall.

2. If yes to the first question, is the average player edge high enough to make it worth-while ?(i am looking for at least a 1% or greater average player edge for this game, when using the strategy mentioned below)

3, Is the strategy below, the 'most correct? if not post a better one, with proof shown whenever possible

(trying to tell me "to leave the game alone" could be a fine "strategy" as well, but I want to know the reasoning and proof you used)
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Game Rules etc

Game: Hi-Lo

Initial House Edge: 2.37 % (this would also be the overall house edge, if previous cards revealed in that round were reshuffled back into the deck, but they are not)

Ace is low, king is hi

Cards: single deck of 52 cards

Choices available: you MUST pick either hi or lo after the card has been dealt on the first deal, if your first pick doesn't lose, you can continue to play or 'cash out'.

nb: If you continue to play any previous cards seen are NOT returned to the deck, they are only returned to the deck and reshuffled when you lose or 'cash out'

Do the initial odds offered change at any time during play: no (see example)

example: drawing a 7 will pay $1.80 (hi or lo) on the first card drawn and if you reach 12 cards drawn in the same game a 7 will pay $1.80 ( hi or lo)

Pays are as follows:

For Lo:
A : not available
2: 12
3: 5
4: 4
5: 3
6: 2
7: 1.8
8: 1.5
9: 1.4
10: 1.3
J: 1.2
Q: 1.1
K: 1.0

For HI:
A : 1.0
2: 1.1
3: 1.2
4: 1.3
5: 1.4
6: 1.5
7: 1.8
8: 2
9: 3
10: 4
J: 5
Q: 12
K: not available

Maximum draws/(payout) til game is over: at least 12^ draws (maximum payout is not stated as far as I can see)

12^: i know that you can get to at least 12 draws in one game as the maximum I have made it to is 12 (11 hands won, 12th hand lost)

Best and (worse) RTP for each card before the first card has been dealt :
A: 100% (na)
2: 100.77%(100%)
3: 100% (84.62%)
4: 100% (97.69%)
5: 100% (93.85%)
6: 88.46% (84.62%)
7: 90.77% (90.77%)
8: 88.46% (84.62%)
9: 100% (93.85%)
10: 100% (97.69%)
J: 100% (84.62%)
Q: 100.77% (100%)
K: 100% (na)

Please note: ties push. eg first deal 7h and then a 7s on second deal is a push.
nb: remember, if you make it to the second round you can 'cashout' (this includes push rounds)

good cards and their best choice on the initial deal are listed below, in the first part of the basic strategy (good cards have an initial RTP of 97.63...% before the initial deal or better)

bad cards and their best choice on the initial deal are listed below in the first part of the basic strategy (bad cards have an initial RTP of less than 97.63...% before the initial deal)

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Basic Strategy (please improve if you think this strategy is not the best basic strategy)

Hi is the best choice for these good cards on the initial deal: A, 2 , 3, 9, 10

Hi is the best choice for this bad card on the initial deal: 6

Lo is the best choice for these good cards on the initial deal: K, Q, J, 5, 4

Lo is the best choice for this bad card on the initial deal: 8

7 is another bad card: choosing Hi or Lo has the same RTP

For the basic strategy below, assume $100 is the initial bet and -$2.37 is the EV figure we are trying to beat per bet.

First deal strategy is easy, just pick the one with the best EV (out of hi or lo)

My 'made it to the 2nd deal or further' strategy decisions are based on the following questions below (with scenarios):

For the best Hi, Lo or 'cash out' choice, is the EV to play better for the player than -$2.37?
(a) if yes then play the best Hi or Lo choice, or
(b) if not 'cash out'.

nb: you need to create a spreadsheet or similar program that shows the before and after card removal EV (or RTP ) figures

an example of where 'play on' is the best choice:
you are dealt a 2 on the first deal and would have chosen Hi at 100.78...% RTP (not the 100.77...% listed above in 'Game Rules etc'),
then you receive a 10 on the second deal, Hi is still the best choice 102% RTP (not the 100% listed above in 'Games Rules etc', see note below)

Note: The RTP's change from the 'before initial deal' RTP's because, those cards are not in the deck anymore (they only go back in the deck when you lose or 'cash out')

an example of where 'cash out' is the best choice:
you have passed the first deal on an A (100% RTP, only hi to choose) and receive a 6 on the second deal, best choice is to 'cash out', because the next best choices RTP is about 90.20...% for Hi (not 88.46...%, see note in previous example)

Note: there are no cards on the first deal that can overcome the 'bad' RTP of receiving a 6, 7 ot 8 on the second deal

an example of where 'I think' cash out is the best choice, but tell me if this wrong (if you know):
you have picked Hi on every deal so far(Hi was the best choice) after being dealt A(1st), 2(2nd), 3(3rd), and 4(on the 4th deal) after the first 4 deals, your $100 is now worth $171.6 ($100 bet x 1 x 1.1 x 1.2 x 1.3) you landed on a J, your spreadsheet says the two best choices are 'cash out' and Lo, (Lo has about an 98.3% RTP)

Would you 'cash out' or pick Lo for the above example?

I would pick 'cash out' (see reasoning below)

Reasoning - your bet size for the next deal (if you don't 'cash out) is $171.6 and the EV on that bet is
- $2.917... if you picked Lo (FYI: it is about -$14.6... if you picked Hi)

your initial bet is always $100, so for the next hand your EV on the initial deal is -$2.37

the EV (in $) to 'cash out' and play again is better by about $0.547, than if you continued on with the game by selecting Lo

Is the above reasoning correct, or did I stuff up? (I think it is correct, but would be happy for someone to point out if it is wrong, and explain why).

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NB: We were pretending that your are betting with $100 on the first deal in the strategy examples above just to make some of the math easier (for me), but your real bet can range from $1 to $100 (in $1 units)

Strategy written again below (since the examples may make it hard to read the strategy above properly)

1. On the first deal, pick the one with the best EV (out of hi or lo)

if the first bet results in a win or a 'push' go to step 2:

2. For the best Hi, Lo or 'cash out' choice, is the EV to play better for the player than -2.37%?
(a) if yes then play the best Hi or Lo choice, or
(b) if not 'cash out'.

nb: every time you lose or 'cash out' go back to step 1.
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again this took a while to write and is only first draft, so there may be word or grammatical errors
the math and reasoning should be correct for everything , except the 3rd example where I explained my reasoning, (I am only about 99% certain that part is correct)

thanks in advance

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update (at about 535 am EST):

I have never seen a bad card become a good card, though it is possible in theory, see example below
example: the first five cards out of the pack are all low cards and they came out of the pack as follows: A, 2, 3, 4, 5 and then the sixth card drawn is a 7 and selected Hi (choosing Hi on the 7 is worth about 100.43% RTP, in this example, but it is a bad card to draw most other cases)

Also, this is not a 'Wazdan' or 'Tom Horn' game (fyi)

Someone pointed out to me in a PM that the WoO has a HIgh Low game page (see below)

https://wizardofodds.com/games/high-low/

Some of the main differences are below:

1. The average starting house edge is 2.41%*** for my game and 7.11% ^^^ for the WoO game (per initial bet made).

***: The house edge for my game has been adjusted from 2.37% to 2.41%, since I should have made the starting house edge for '1 deck and AFTER seeing the 1st card' instead of 'infinite deck'.

^^^: This figure includes getting an Ace or a King.

2. There are more opportunities*^* for the card counter that make is worth playing on in my game.

*^*: In mine, if the game is not over by the 2nd card you have at least 39 or 40 'good' cards left, with the WoO example you only have 7 or 8 'good' cards left (assuming a single deck version of the WoO example).

Note: A card counter can improve the house edge on either game, using a strategy similar to the one in the original post, but the higher house edge and 'less opportunities' in the WoO example make it extremely unlikely to ever have a long-term player edge for a card-counter
(I still don't know if my game has a long-term player edge for a card-counter).

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This reply was written quickly, since I have to go to work soon, I have not checked the grammar etc, of what was just written.

I only did a quick skim of this. I'll just say that card counting would be very effective in this game. Online games likely shuffle a card back into the shoe immediately after being used. Maybe that's obvious. I think it would help you summarized what you're trying to say. Right now it is hard to follow and looks like it came out of Google Translate from another language.

Thanks Wiz

Writing isn’t my strong suit.
The cards don’t seem to be shuffled back into the deck during a single game.
I will test the’ cards are not shuffled theory’ properly soon and post the results.

I decided to look at the help file again during the 'cards seen' test, and I found that the game automatically finishes after you win $250 or more.

Because of this I have stopped the test early.
Note: I originally planned to test at least 1000 games.

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The summary of the test was as follows:

Games played: 67

Did a card repeat during a single game: no

Maximum number of cards seen in a single game: 15
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Just out of curiosity, I would still like to know if the game is 'beatable in the long-run', betting $1 as your initial bet?