A pretty common situation.

In this case P1 cuts P2's chance of prevailing from 0.6399 to 0.6350, an improvement of 0.49%. I have updated the online spreadsheet with all of the latest refinements.

 

First of all, there is a typo in the example above, but I can't seem to edit the post anymore. "instead of 100" should read "instead of 700". So, to be clear, the example above is:

P2: BR 2151, Bet 700
P1: BR 1800, Bet 900 vs 697

Responding with 697 decreased P1's chances by 1.22% vs betting 900.

Thanks to PlayHunter for asking me privately about an apparent discrepancy between this result and another similar example where bj vs win was also given up in order to gain surrender for the low but P1's chances improved instead of getting worse.

The other example was

P2: BR 2201, Bet 800
P1: BR 1800, Bet 900 vs 797

In that case P1's chances improved by 0.49% by betting 797 (giving up BJ vs win in order to obtain surrender for the low).

The apparent discrepancy comes from an additional difference between the two examples. Can anyone tell me what it is? (PlayHunter is disqualified from answering).

 

No One?

No one can tell me the additional difference between these examples? Based on his "thanks" (thanks for that!), I suspect that Monkeysystem knows the answer.

 

Finding the diference

The answer can be found without adding specific results of bet outcomes to the existing bankrolls.

Since in both cases P2 (BR1) makes bets that surrender covers P1’s (BR2) push and P1’s surrender covers P2 loss then this excludes any results with surrender involved.
Since in both cases all bets are bigger than the difference between bankrolls then this excuse any straight gains (W-P-L) and swings.
The difference between the opponents’ bets is smaller then their bankrolls’ differences, which eliminates win/win.
That leaves us with combinations of bj-dw, dw-bj, and dw-dw.
Once again, since the differences between the opponents’ bets is smaller than 2/3rds of bankrolls' differences that purges results with blackjacks.
The only thing left is winning doubled bets by both players.

Now, if we look at the scores in the second example, both bets 900 and 797 don’t advance P1 (BR2) if both players win doubled bets – but the reduced bet (797) gains half percent by giving up bj vs win in order to obtain surrender for the low.
In the first example, reducing bet from 900 to 697 gains the same half percent but gives up benefits of P1 winning the match if both players win doubled bets.

In general, the main difference between the two situations is that in the second example BR1 goes into the last hand having lead of more than 20% of BR2 bankroll.

S. Yama

 

Correct (as usual)! Thanks again to PlayHunter for asking me about this apparent ambiguity.