[Stephen H]

Alright, I've been thinking about this a while now, and reading the previous posts, as well as re-reading the fabulous [0,1] game posts by Chen/Ankenman. Almost all the math I've lifted directly from the [0,1] posts, so please correct me where I've missapplied it. Here's what I see, starting with analysis of the fixed limit game:

If the game allows no folding and no checkraises, this is exactly game #3, with the result that you should raise with the top sqrt(2)-1 percent hands (Labeled the "Golden Mean of Poker" in the [0,1] posts) that you put your opponent on. Working out a chart, I get that with the top 2 cards in the game, you should be willing to put in the 15th bet; for the 16th bet you need the top card. This means that, as stated, you would bet and then put in 7 raises on top of that for a total of 16 bets back to you after your opponent re-raises your last raise. At this point you would just call.

First question: Since the cards are discrete values, should I be rounding in this table at each step? Or, since a strategy could be, say, when comtemplating making it 10 bets, to raise it with the top 148 cards, and also raise the 149th 68% of the time, is rounding simply adding error? I suspect there's no need to round, and as was said before, he who has the most accurate calculator (and randomization device, I guess!) has the +EV.

Once you allow folding, you introduce the bluff. The bluff rate needs to be 1 bluff raise for every p value raises, where p is the size of the pot after you raise. However, the range of value raising hands does not change; you still raise those the same amount. So, for each bet, you can add in a # of bluffs equal to (# of raising hands)/(value of pot after your raise). As the # of hands is going down quickly, and the value of the pot is going up, this becomes less than 1 around the 14th bet.
2 interesting points I've found.
First, since your opponent may be bluffing, his hand range is a little larger than before..and this actually adds up to enough to allow you to put in the 16th bet with the top 2 cards, one more bet than before (although it doesn't change the answer to the question as stated, since it's an even bet #). I also calculate that you can value raise about 14% of the time for the 17th bet.
Second, while you stop value raising with the 999,999 after 17 bets, the bluff rates go on forever (but are remarkably small), so you might be bluffing with the 999,999. I'm not entirely sure how the bluffing works once you reach the degenerative case of only value betting the 1,000,000, but I'm fairly sure that you'll be bluffing a fairly low percentage of the time past this point.

Questions: How often SHOULD you be bluffing once you reach the degenerative case? And, again, should I be doing rounding here? When I calculate the range of hands you put your opponent on for use with the "Golden Mean of Poker", should I be including the amount of bluffs? For example:
When the first bet goes in, you should have at least the top 414,213 cards (I can see why the [0,1] game was written as lowball - so you can say "top X cards" and "X or better" without confusion). You should also be bluffing 25% of the time, so you bet out with effectively 552,284.75 hands out of 1,000,000. Do you re-raise for value with the top 171,572 hands (.414 times 414,213) or the top 228,763 hands (.414 times 552,284)? I've assumed for my numbers above that you do include the bluffing range.

For pot limit, the first thing to note is that pot size/bet size considerations only affect bluff/fold percentages, and not value raise percentages. If you're ahead, you're ahead, and your raise will only gain you 1:1 on the value of your raise no matter how much is in the pot. Now, the bluff rate should be constant, because you're always offering the same pot odds to call; 2:1. Because of this, the hand range stays large a little longer and takes longer to converge to the top 1 card. According to my calculations, you're now value betting 999,999 up to 26 bets, value betting the 27th bet 75% of the time, and value betting the 28th bet 8.8% of the time. After that, the only raises going in with 999,999 should be bluffs. The bluff factor seems more important here, but again, I'm not sure just how we're supposed to bluff once we reach the degenerative case.

I haven't bothered to figure out any of the fold/call percentages, mainly because they would be a lot more work (for me, anyways..maybe it's trivial!) and aren't really required for this part of the problem. I'd like a little confirmation that I'm on the right track before I bothered to work on those parts.

To construct the tables, I used the following formulas (rather than put the whole tables in here)
Fixed Limit:
Nth bet
Pot size after raise = Pot size of (n-1)th bet + 2
Raising range = (sqrt(2)-1)*(Opponent's Hand Range)
Bluffing range = (Raising Range) / (Pot size after raise)
Opponent's Hand Range = (n-1)th Raising range + (n-1)th Bluffing range

1st bet: pot size = 3, Opponent hand range = 1,000,000

Pot Limit:
Nth bet:
Raising range = (sqrt(2)-1)*(Opponent's Hand Range)
Bluffing range = (Raising Range) / 2
Opponent's Hand Range = (n-1)th Raising range + (n-1)th Bluffing range

1st bet: Opponent's Hand Range = 1,000,000

Comments/Criticisms?