# How to maximize expectation with pocket aces and an Aces Cracked bonus running

I was playing $3-$6 Limit live Texas Hold'em and there was a $200 Aces Cracked bonus running - meaning if you have pocket Aces and lose, you get$200. The house rake is pretty steep - $3 per hand +$2 drop for the bonus/jackpot fund. The table was typically loose for $3-6 - about 4 or 5 players average seeing a flop. Post-flop they would sometimes fold rather than call two or more bets, but most would chase any kind of draw or weak pair for one bet. I was dealt pocket Aces and after the showdown, was berated for my play. I understand that with the bonus, you give up some expected value in the$200 for every player that folds (that isn't drawing dead), but you give up expected value by playing passive, too. I don't think I played the hand perfectly, but I had very different ideas about where my mistakes were than my critic did.

## Preflop:

Hero in big blind with A:clubs:A:spades: Four players limp and I raise. All four players call.

## Flop ($25 after the house took their piece): 9:clubs:A:diamonds:5:clubs: Hero bets and four players call. ## Turn ($40):

9:clubs:A:diamonds:5:clubs:3:hearts:

Hero bets and 3 players fold. Button calls.

## River ($52): 9:clubs:A:diamonds:5:clubs:3:hearts:4:diamonds: Hero bets and button raises. Hero calls. ## Final Pot Size:$76

• Is it correct to raise pre-flop? I'm not sure.
• Is it correct to bet out on the flop? I think so because I think anyone who has a chance of drawing out will call, so I'm not worried about chasing bonus money away.
• Is it correct to bet out on the turn? Again, I think so because if someone has two clubs or four to a straight, they are going to call. The only thing I might chase away is an inside straight draw and the pot is already pretty big. This is main point my table-mate had a problem with, though. He was quite adamant that I should have checked the turn.
• Is it correct to bet out on the river? I think this was a mistake. If I have the best hand, the Villain is going to fold anyway. If my aces are cracked, the Villain will likely raise and I'll have to spend another bet to showdown. I don't know what the critic thought of this move because he couldn't get past his problems with my play on the turn.

Of course it is correct to call the raise - I can't lose. Either I win the pot or the bonus. But did I maximize my expectation?

• Is this live play or online ? If you can still remember (probably just approximately), it would be great to post the bet/raise/call amounts. You say bet, but not how much you bet. You say raise, but not how much you raise etc. – Radu Murzea Jun 30 '12 at 10:12
• Your line looks fine to me but, without extra details like bet sizing or stack sizes, Its difficult to say more. Also, is this Limit (LHE) or No-Limit (NLHE)? – Toby Booth Jun 30 '12 at 11:11
• Sorry, I said it was 3-6, which means $3/$6 LHE, so the raises are fixed size. Also, this was live. I have edited the question to hopefully make that more clear. – jhericks Jul 1 '12 at 2:07

Since this is limit holdem, the maximum pot for this scenario is $355, of which you can invest a maximum of$69 (after a forced blind post). Hence you will never win less than $200 -$69 = $131 if you always bet/raise and are shown a better hand. However, it is still possible to play this hand suboptimally even if it's impossible to lose at showdown. Here's the optimal line: 1. If you are a pot-odds dog vs. your opponent's range, always check/call. 2. If you are a pot-odds favorite vs. your opponent's range, always bet/raise. As you are favored on every street in the hand history you posted, your line up to the river is definitely optimal. On the river, your expectation for calling is (1 - x) *$70 + x($200 -$6) = $124x +$70, where x is how often you lose. Your expectation for raising is (1 - y) * $70 + y[(1 - z) *$76 + z($200 -$12)] = y($112z +$6) + $70, where y is how often your opponent calls and z is how often you lose when he calls (we've assumed he never reraises for simplicity). Hence your line is correct if $124x + $70 > y($112z + $6) +$70. If your opponent always calls and wins, the inequality becomes $124x >$118, making calling the best play if you lose the hand more than about 95% of the time when you do call.

This is as simple as it gets without analyzing bluffing ranges etc. and is quite a bit more complicated and probably not worth doing for no limit games and limit games for which the pot frequently exceeds the bonus. (And for significantly larger jackpots, utility considerations come into play.)

To summarize, if your opponent can fold a better hand to your reraise, which seems impossible here, reraising would be a clear mistake. As played out the dollar difference either way doesn't warrant being berated, and in general the optimal line for this game with the given bonus is given by the two rules above.

Interestingly, the better the Aces Cracked bonus is, the more that skilled players draw from it; and the worse it is, the less that skilled players draw from it. In this case, then, the value of the promotional drop is worse than breakeven for the worst player at the table. (Think seven-deuce side bet in no limit.)

The justification for #2 has an analog in no limit through fold equity on a draw, though it presents itself in the reverse manner of putting in money. That is to say, it's only correct to raise all-in on a draw as a pot-equity underdog if you have enough fold equity to cover the loss. In your case, you always have enough "backup equity" to cover the case when you are behind your opponent's range, but when you are behind with a greater frequency than your pot odds, you lose money on every additional dollar that goes in the pot even though it's deducted from a net gain.

This is essentially a pot odds argument. You're obviously ahead on the turn since your opponent has more draws in his range than made hands better than yours. Without dissecting entirely, let's say you're a 2:1 favorite. You win roughly that portion of the pot and are compensated $200 when you lose. It's always going to be good to bet since your bet lays you almost 8:1. The key addition is that when you bet/raise as the favorite, you never bet out a hand which beats you - the strategy changes if this were not the case. Do whatever it takes to get to showdown, but do so passively as the pot underdog. There may be special cases where aggression as the favorite folds out a better hand, but this is mostly against overly cautious players or particularly scary boards. EDIT: Correction for multiway pots... My apologies, the above analysis is only good when the hand goes heads-up. The multiway case depends on the stakes relative to the bonus, which here I think "trying to lose" is a good basic strategy. Your preflop play is correct as you are unlikely to fold out players who stand to release your bonus, at the same time value raising if you happen to win the hand anyway. If you were not last to act, raising may be suboptimal. From the flop forward, I like playing the hand passively until the hand goes heads-up, after which you adopt the strategy I outline above (playing normal, good poker). I think Villain is correct about your mistake on the turn. I like a check on the river, planning on check-calling. Your reasoning is sound, and there is an off-chance you can gain a bet from a bluff. The "trying to lose" strategy has an analog in tournament poker bubble situations in which you can maximize your money equity by cooperatively checking down. To be a bit more thorough, on the turn your hand is 70% to win against Villains with assumed ranges of {any two clubs}, {pair plus straight draw}, {any nine}, and {a random hand}, of which the draws are about 26% chance to catch up. If we fold out straight draws half the time and all non-flush-draw hands, our bet wins us$6 * 70% + $200 * 23% =$50, whereas checking would win us $200 * 30% =$60.

• I was wrong on my pot sizes (since corrected) but you are also wrong. The bet size on the flop is $3, so 5 bets =$15, not $30. – jhericks Jul 13 '12 at 8:33 • Sorry, halve all my bets. I play no limit, so the big bet and the big blind are the same. In that case, the$200 bonus is a really good deal when you have aces, but a really crappy deal if a drop is taken for it. I'll update my post later :). – Loc Nguyen Jul 13 '12 at 10:14
• There is a $3 drop for the house and a$2 drop for the progressive jackpot fund, which funds more than the aces cracked bonus. Theoretically, the house pays out all of that $2 drop eventually. – jhericks Jul 13 '12 at 15:01 • Also, the river analysis is exactly the kind of analysis I was looking for, but can you justify your optimal line for turn? I was playing that way (bet if I'm favored), but I don't know how to justify it. – jhericks Jul 13 '12 at 15:03 • You'll find the river analysis applied to the turn is consistent with the notion of pot odds since pot odds is just shorthand thinking for expectation, i.e. pot * odds_win - bet * (100% - odds_win) > 0 simplifies to odds_win > bet / (pot + bet). The bonus value is largely irrelevant unless your strategy prevents a losing showdown. – Loc Nguyen Jul 13 '12 at 17:16 Your line was correct upto flop. Your bet on flop as a continuation bet is correct and that's why everybody called. You already know that some players are now chasing a flush, straight, set or playing with an A-x. Now to extract maximum money with an Aces cracked pot, you should slow down after the flop. Now with everyone calling the flop, a check on Turn is the correct play because you want them to improve on river or if someone bets just call after taking your time. A raise/bet on the river should be the next logical move to improve the odds of maximizing your profit. Regardless when you bet on the turn, you chased away 3 fishes who could have raised (bluffing or hitting) (or more likely to call if they have an overpair or an A-x) on the river. I don't understand why you just called on the river, you should take your time and re-raise him to extract maximum money upto 200. • The player had been passive during the rest of the hand (and during most of the night). The only hand that beats me on the river is a 2, but his raise on the end makes me believe he has it, so I think I'm beat. I win the pot if my set of aces holds up, but I don't get the$200 unless they are beat. So rather than trying to extract the most money, I am trying to spend the least amount of money to win the $200 bonus. – jhericks Jul 1 '12 at 10:52 • The nuts is 2 6 and so if you check on the turn and re-raise on the river would make more sense and if they re-raise you then you can call but I don't think 12 bucks is too much to lose when you just called irrespective of your previous plays. – Subs Jul 1 '12 at 20:00 • So, I should have checked the turn and re-raised on the river to get the most if my set of aces hold up. I can agree with the check on the turn, but I have a hard time with the re-raise on the river. If he doesn't have the nuts - say just a two or a set of 4's he may fold. If he folds a loser, I don't get any more money. If he calls with a loser I get an extra$6. But if he folds a winner I lose ~$130 ($200 - pot size). – jhericks Jul 2 '12 at 16:26

I've never played with this "Aces Cracked" bonus, so take my answer with that fact in mind.