$200 Blog Post

August 11, 2008

WSOP Texas Hold ‘Em Bonus Poker

I suppose every experience has to have at least one positive aspect, and that’s that no matter how bad it is, you can blog it. Well, I hope someone values this blog post at $200, because that’s what it cost me. Over the weekend, I lost $200 playing a table game called World Series of Poker Texas Hold ‘Em Bonus Poker (say it quickly for added fun). The main lesson was one that I already should have known: table games are there because they are profitable for the casino. In light of the general principle that extraordinary claims require extraordinary evidence, you would need strong evidence in order to believe that a table game can actually be “beaten.” Perhaps I can say I’ve learned that my brother’s declaration that “This game has to be beatable,” does not constitute strong evidence.

This past week has actually been pretty eventful. Starting on August 1st, my parents and I took a trip up the East Coast to tour the houses of my siblings. The trip culminated in a successful wedding of my sister, leaving me the only unmarried person in my immediate family.

The wedding was at a hotel/golf resort 12 miles outside of Atlantic City, NJ, so one of the major nighttime activities was going to the casinos. I went expecting to play $2/$4 Limit Hold ‘Em, which is also not beatable for the average player, but at least you aren’t playing against the house, so I figured it was worth a shot. Since the tables were full, however, I was forced to wait, and as I waited, my brother encouraged me to play the table game.

In the Hold ‘Em table game, the object is to beat the dealer’s hand. You only play against the dealer, not the other players at your table. The way the play goes is, you pay a $15 ante to see two hole cards. After seeing the cards, you can pay $30 to see the flop, or fold. Once you have made this bet, you can go all the way to the end without folding for no additional payment. If you pay to see the flop and win, you get 1:1 on your $30, but no payoff on your ante unless you win with a flush or better. After you see the flop, you have two more chances to bet: after the flop you can bet an additional $15, then another $15 after the turn. You can’t bet after the river. So, the maximum amount of money you can lose in any hand is $75, the maximum you can win is $60.

The limited analysis I was able to make of this game indicates that it’s a lot harder to win than my brother thought. The optimal strategy and analysis, however, are fairly interesting.

Having paid to see your hole cards, you have some information about your chances of beating the dealer’s (random) hand. Any two cards have odds against a random hand which can be calculated with one of the online poker calculators. Since, pre-flop, you are betting $30 in order to win $45, you want to fold any hand that has a worse than 40% equity against a random hand. Unfortunately, this only describes about 18% of all possible hands, so you are going to play some pretty bad hands.

For the bets on the turn and river, you are betting $15 to win $15 (1:1 payout), so you want to bet hands that have at least 50% equity against a random hand, given the cards you see on the board. That’s pretty straightforward.

So your expected payout is going to depend on two things: how many hands are “playable,” (i.e. have a positive expectation from betting vs. not betting), and the average likelihood of winning with your playable hands. Here, the pre-flop analysis is a lot easier than post-flop.

The basic problem is that your hole cards just aren’t informative enough. Say you play every single hand. Your chance of winning any given hand is 50%, and since you win $45 every time you win, and lose $30 every time you lose, you expect to make $7.50 per hand by betting. Problem is, of course, you pay $15 per hand to play, so you are at -$7.50 net. Now, if you play only hands with an equity above 40%, your expected payout is still only a bit less than 55% on that set of hands (vs. 50% if you played every hand). When we play this way, our pre-flop expectation is:

-$15 + .82 [($45* .55)-($30*.45)] = -$5.78

If the game were only the $15 ante and the $30 pre-flop bet, we’d expect to lose $5.78 every time we play.

There’s one more thing I can do to improve my expectation. Pre-flop, the average equity on hands that have an equity of at least 50% against a random hand is .58, so I can make a little more money by betting on these hands, giving me

-$5.78 + .5 * ($30 * [.58 – .42] ) = -$3.38

If I commit to betting all the way on hands with at least 50% equity pre-flop, my expected loss is $3.38 per hand. I’m still not doing very well, but it’s better than losing $5.78.

That’s a lot of ground to make up in our flop and turn betting, and unfortunately the analysis kind of ends here because to go further would require a lot of simulations that I don’t know how to do efficiently. Of the hands I play, I would expect to win almost 55%. Before seeing the flop, I was prepared to make bets “in the dark” on roughly 62% ( 5/8 ) of the hands I played. Thus it seems to me I should be betting somewhere between 62% and 55% of the hands I play, but I don’t know this number precisely. I also don’t know the expected equity on the hands that I do bet. When I was betting 62% of my played hands, my equity was .58. If I somehow had perfect information, I would bet 55% of the time with equity of 1. So, as I know more and more, the equity on my bets increases, but I don’t know at what rate. Certainly, higher than .58, since that’s what I could get without any board information, but I don’t know how much higher.

-$5.78 + (.82 * x * [p – (1-p )] * $15) + (.82 * y * (q [1-q ])* $15 ) = ???

x – percentage of hands I can bet post-flop (between 62.5% and 55% )

y – percentage of hands I can bet post-turn (between 62.5% and 55%, y<x )

p – average equity on post-flop bets (p > .58 )

q – average equity on post-turn bets ( q > p > .58 )

One equation, five unknowns. That’s hard to solve. But the loss ought to be less than $3.38 if my strategy is to be an improvement.

Well, without knowing the precise figures, it would appear that the intuitions of my brother and myself were wrong, and this game is not an attempt by the casino to give away money. Had I figured this out two days ago, I would be a richer man.

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