J Morgan: Brain Teaser: Betting Strategy

[J Morgan]

I ask, again, for Don's indulgence in this Three Card Poker brain teaser. There is possibly some relevance to other games, including blackjack.

You have $65 in chips. You cannot buy in again, you cannot borrow, no markers, no friends. $65. That's it. You're playing a full-pay (1/4/6/30/40 and 1/4/5) Three Card Poker game ($5-$500 limits) where you know two of the dealer's cards, giving you roughly a 17.2% edge. There will be two more rounds before the dealer goes home (and then you also go home).

To simplify this problem, I require you to play the 2-card poker strategy given in _Beyond Counting_, giving you the 17.2% edge on each of the last two rounds. Also, all bets must be in increments of $5 (in fact, this is a rule at some Vegas joints). Of course, you must be prepared to match your Ante bet with an equal Play bet to remain in the hand.

If your goal is to maximize your total expected win for the two remaining rounds, what would you bet on each round?

(Notice, we are not risk averse. Forget about Kelly and all that. We're going for maximum expectation.)

[Mr. X]

I'd guess bet the most you can, half your bankroll, or, if an odd multiple of $5, half minus $2.50.
So, 1st hand bet $30. Win both, bet $60, etc.
High variance, but I would guess that maximizes the expectation.
I know I'm missing a trick, probably has to do with that odd $5
PS- Thanx for a very interesting chat.

[One_More_Shoe]

OK, here is an attempt:

Since you say we are NOT risk averse and we want to MAXIMIZE
EV, then the hell with trying to save a bet for round 2. SINCE
we have a large edge, go for it on round one.

Therefore, bet $30 on the ANTE for round 1, saving $30 for
the potential play bet. Save the $5 for later, if needed.

IF Fold then:
Bet $15 on the ANTE for last round, saving $15 for a potential
play bet. I thought about throwing the final $5 out on the
pair plus bet, but that being a negative EV bet, Save the money.

IF WIN ANTE ONLY then:
Bet $45 on the ANTE for last round, saving $45 for the Play bet.
Screw the $5 and don't make a neg ev bet on pairs plus.

IF WIN ANTE AND PLAY then:
Bet $60 on ANTE for last round, saving $60 for the pairs plus.
Screw the $5 for the pairs plus. Just save it.

IF WIN any better hand bonus, split total bank in 1/2 and
bet 1/2 on ANTE saving the rest for Play. Any odd $5 amount
just hold onto.

OF course, For $5 I'd screw it and play the PAIRS PLUS bet with
any last remaining odd $5 on the last round.

OK, now how badly did I muck this up?
Curious to know

Best regards,

-- oms --

[ET Fan]

I have never sat down to play a single hand of TCP, and don't have the vaguest notion what an equal Play bet is or a 1/4/ ... pay off schedule. But in chat you said all must respond, so here is my lame contribution:

1) One thing often glossed over is the difference between maximizing EV expressed as a percentage, and the "total expected win" required in your problem. They are often different, especially with finite capital.
2) If the rounds are independant (again, pleading ignorance), then without the betting limits you'd surely want to maximize E(W) on the first round. But the $500 max might create a situation where you'd settle for a little less, since any winnings over $500 represent a wasted opportunity to grab that 17.2% edge "the second time around."

ETF

[J Morgan]

With only $65 to work with, the way to maximize total expected profit for the two remaining rounds is:

Right now: $30 on Ante (saving $30 to back it up), and $5 on Pair Plus.

Last hand: Half of chips (if necessary, rounded down to nearest $5 increment) on Ante, and $0 on Pair Plus.

[Mr. X]

Doh! Of course!! It seems so obvious now, sacrifice a little E on the pair plus for a chance at greater E the next round on the ante.
I'm embarassed, I shouda gotten that.
Question- I had some trouble last trip with snoopy dealers looking at my 3CP discards. Tossing a K and a QJ caused some comments. Are there any tricks you use to minimize that from happening? I thought of intentionally tossing a pair, then retracting as the dealer picks it up, to establish that I am sloppy or sleepy, but I figured that would cause the dealer to check my discards more often, to protect me from myself, so to speak.