3 Card Poker Strategy
The wizard of odds tells us that best strategy for 3 card poker with 5/4/1 ante bonus is to play on Q|6|4 and higher. This strategy will give the house lowest "edge": 3.373%.
However, i've calculated that playing on Q|6|2 and higher gives a better return on all stakes played:
97.989% Q|6|2
97.985% Q|6|4
I'm playing my strategy in future.
Discuss.
Trevor
It's your money.
https://wizardofodds.com/games/three-card-poker/
"If you want to know why queen/6/4 is the borderline hand it is because if you raise on queen/6/3 you can expect to lose 1.00255 units, more than the 1 unit by folding. However if you raise on queen/6/4 the expected loss is .993378, less than the 1 unit by folding."
I think that the "element of risk" is more relevant to the player.
Not here -- you're not comparing like amounts. In other words, by folding less often, you're putting out 2 units more often than the Q64 strategy and therefore the average wager increases. Even if you lose slightly less as a percentage of total wager, you're losing more dollars. Over the course of an hour's worth of play, you'll have a greater theoretical loss in currency by playing Q62 than Q64.Quote: TrevorThere are no discrepancies between my results and the Wizards results displayed on that page. And when looking at the "house edge", Q|6|4 does indeed return the lowest value. However, when comparing the "element of risk" or RTP on all stakes played, Q|6|2 gives me a higher return.
I think that the "element of risk" is more relevant to the player.
The closest analogy would be craps and the passline odds. Making an odds bet decreases the total edge as a percentage of all action, but does not impact the edge on the passline -- that's constant at 1.41% of every dollar wagered, regardless of odds. The analogy to 3CP would be if the edge on the passline got slightly worse if you made odds. For example, suppose the edge on the passline went from 1.41% to 1.6% when you took odds (somehow). You could still argue that the combined pass+odds bet had a greater return, but the expected loss on each bet goes from 7 cents to 8 cents. Over the course of 100 hands, your average dollar loss would go from $7.07 to $8.00. That's essentially what happens if you switch strategies in 3CP.
In short, I think expected loss is more important than percentages. Q62 has a slightly higher expected loss per hand than Q64.
I too disagree here.Quote: Trevor...I think that the "element of risk" is more relevant to the player.
+1Quote: MathExtremistNot here -- you're not comparing like amounts. In other words, by folding less often, you're putting out 2 units more often than the Q64 strategy and therefore the average wager increases. Even if you lose slightly less as a percentage of total wager, you're losing more dollars. Over the course of an hour's worth of play, you'll have a greater theoretical loss in currency by playing Q62 than Q64...
(64 of each, minus 4 flushes. So, 120 hands out of 22100. At 30 hands an hour, that's 737 hours to cover that number, divided by 120 is 6.13. So, once every 6 hours or so.)
Now, playing something like 3CP vs. playing slots, that's a different story. Or playing UTH and only raising 3x instead of 4x. It won't take too long to know you've been doing one vs. the other.
Quote: MathExtremistNot here -- you're not comparing like amounts. In other words, by folding less often, you're putting out 2 units more often than the Q64 strategy and therefore the average wager increases. Even if you lose slightly less as a percentage of total wager, you're losing more dollars. Over the course of an hour's worth of play, you'll have a greater theoretical loss in currency by playing Q62 than Q64.
I see what you mean but to look at it another way. If i bet $100 with Q|6|4 and then $100 using Q|6|2 i'll have won more money using Q|6|2.
Quote: TrevorI see what you mean but to look at it another way. If i bet $100 with Q|6|4 and then $100 using Q|6|2 i'll have won more money using Q|6|2.
I don't think you will.
Quote: TrevorI see what you mean but to look at it another way. If i bet $100 with Q|6|4 and then $100 using Q|6|2 i'll have won more money using Q|6|2.
You will lose more.
Quote: MathExtremistSure, but that's a different analysis. It won't make any difference over a lifetime of play either, not statistically anyway -- and by that I mean the expected lifetime distributions from playing Q64 vs. playing Q62 will not be significantly different. For that matter, neither will a lifetime of playing pass vs. playing don't pass in craps. In both cases, the variance swamps the slight difference in house edge over the typical gambler's lifetime session. Edge isn't everything over a finite number of plays, but everyone plays a finite number of plays.
Now, playing something like 3CP vs. playing slots, that's a different story. Or playing UTH and only raising 3x instead of 4x. It won't take too long to know you've been doing one vs. the other.
Yep, we agree, that I'm approaching it from the human perspective rather than the mathematical. It's 3CP, after all. I've tried to explain to Mrs Mosca the even though each bet is $10, there are fewer bets per minute than 50 lines X 5 on a penny slot.
Suppose the ante bet was free (or just gone) and the player simply made the raise bet after seeing their cards. To maximize your winnings, would you still play Q62 or would you only play Q64?Quote: TrevorI see what you mean but to look at it another way. If i bet $100 with Q|6|4 and then $100 using Q|6|2 i'll have won more money using Q|6|2.
If your answer is Q64, which it should be because playing Q62 by itself favors the house, then why would that change based on a bet you made before you saw the cards?
The correct strategy for a game is usually considered as the way to play that will make more (or lose less) money in the long run. Hence the reference to Q64 (play) and A63 (fold).
What you have found is that if you had a choice of making a marginal play rather than using the money for a new game then sometimes it is better to play. In the UK (ENHC) there is a similar argument whether to (say) double 11 vs 10 at Blackjack - it's slightly better to hit.
Strategy Q|6|4
House Edge = 3.3729%
Avg Bet = 1.674
EOR = 3.3729/1.674 = 2.0147%
$100/1.674 = 59.72973 games
winnings = 59.72973 * (1.674-2.014%) = $97.985
Strategy Q|6|2
House Edge = 3.3772%
Avg Bet = 1.679
EOR = 3.3772/1.679 = 2.0107%
$100/1.679 = 59.53664 games
winnings = 59.53664 * (1.679-2.0107%) = $97.989
Thanks,
Trev
The House Edge is looking at the cost of playing 100 games.
The EOR is looking at the cost of making $100 worth of bets.
The HE is higher in your second trial, but the reason it costs less is because you get through $100 in fewer games, so the HE operates on fewer hands.
(a) $100 costs $2.23c (59.72973 x 3.3729%)
(b) $100 costs $2.01c (59.53664 x 3.3772%)
Consider this, fairly stupid, simple game - red or black. $1 bet will pay 90c if you're correct and lose otherwise, but it also allows you to "double" and make an extra $1 bet that pays 95c. Do you make the additional bet?
The House Edge, if never doubling, is 5% (since 100 $1 games payback 50 wins of 90c = $45 less 50 losses of $1 = $50; net loss=$50-$45=$5).
The House Edge, if always doubling, is 7.5% (since 100 games payback 50 wins of 90c+95c = $92.50c less 50 losses of $2 = $100; net lose=$100-$92.50c=$7.50c).
However the EOR is lower if always doubling. You lose $5 per $100 playing the basic game and $3.75 per $100 if always doubling. Note it will take 100 hands to lose $5 playing the basic game, while if doubling you will only last 66 2/3 games.
This is the dilemma with your summary. Playing 100 hands will cost less folding Q63, however playing $100 will cost less calling Q63 but you will get fewer games.
Quote: charliepatrickThis is the dilemma with your summary. Playing 100 hands will cost less folding Q63, however playing $100 will cost less calling Q63 but you will get fewer games.
Agreed, so does the player care about the number of hands they got for $100 dollars or just about maximising returns for that amount?
Trev
Neither -- the player almost invariable cares about the time they play. Whether that's measured by hands or hours doesn't matter, but I have never known anyone playing recreationally to track their handle and stop after reaching a defined total wager amount. Have you?Quote: TrevorAgreed, so does the player care about the number of hands they got for $100 dollars or just about maximising returns for that amount?
If you're worried about the best investment you can make with $100, you won't find that in a casino. If you're looking for an hour or two of casino entertainment, on the other hand, it's usually wise to minimize your expected hourly loss on the game you've chosen to play. 3CP isn't the best game in the casino, not by a longshot. But once you've decided to play it, you should probably make the best plays, all things being equal.
The problem with "deciding to play 3CP" is that the Ante bet has a strong house edge. That is offset by the option to make the Play bet, which is positive under optimal strategy. Of course, so is doubling down in blackjack -- you only double when the extra money is a positive investment. It appears that you're suggesting that a different evaluation would be to examine the theoretical cost of making the Play bet compared to making another Ante bet on the next hand, and make the Play bet not only when it is positive but when it is not as negative as the next bet. What do you get overall when you do that analysis?
Quote: IbeatyouracesI have NEVER seen any one player play proper basic strategy for this game. Not one!!
That is because we are all addicted to the Pairs Plus bet. OF COURSE you won't get a straight flush every 40 hands. But there's nothing like that kick in the heart when you get paid that 40 to 1!
Quote: MoscaThat is because we are all addicted to the Pairs Plus bet. OF COURSE you won't get a straight flush every 40 hands. But there's nothing like that kick in the heart when you get paid that 40 to 1!
I'm not including the PP or any other side bet. Just strictly playing strategy.
Is that because proper "basic strategy" involves not sitting at tables where the dealer isn't flashing?Quote: IbeatyouracesI have NEVER seen any one player play proper basic strategy for this game. Not one!!
Quote: IbeatyouracesI'm not including the PP or any other side bet. Just strictly playing strategy.
I think that is because the people that would know the strategy refuse to play a game with such a large house advantage.
Quote: MathExtremistIs that because proper "basic strategy" involves not sitting at tables where the dealer isn't flashing?
Haha no. I'd call that "information strategy"
I haven't played OCP in a few years.
"Civilian Strategy" is QTx
Quote: IbeatyouracesI have NEVER seen any one player play proper basic strategy for this game. Not one!!
I've seen it-- and I've seen dealers who coach new players to fold anything that doesn't have at least a queen high (which isn't optimal, but one could do worse). Of course, some of these same dealers encourage the players that the PP and 6-card bonuses are really "where the money is" in the game.
But I know what you're saying- the tables are still full of people playing blind because they really don't think it makes any difference.
Quote: MathExtremistNeither -- the player almost invariable cares about the time they play. Whether that's measured by hands or hours doesn't matter, but I have never known anyone playing recreationally to track their handle and stop after reaching a defined total wager amount. Have you?
I take your point.
Quote: MathExtremistIf you're worried about the best investment you can make with $100, you won't find that in a casino. If you're looking for an hour or two of casino entertainment, on the other hand, it's usually wise to minimize your expected hourly loss on the game you've chosen to play. 3CP isn't the best game in the casino, not by a longshot. But once you've decided to play it, you should probably make the best plays, all things being equal.
Ok, i've got $100, i need to make some quick cash, i've only got a certain amount of time to do it in and the only game in town is 3CP.
Q|6|4 or Q|6|2?
Quote: MathExtremistThe problem with "deciding to play 3CP" is that the Ante bet has a strong house edge. That is offset by the option to make the Play bet, which is positive under optimal strategy. Of course, so is doubling down in blackjack -- you only double when the extra money is a positive investment. It appears that you're suggesting that a different evaluation would be to examine the theoretical cost of making the Play bet compared to making another Ante bet on the next hand, and make the Play bet not only when it is positive but when it is not as negative as the next bet. What do you get overall when you do that analysis?
I'll get back to you.
Trev
It's not that simple. My original premise was "all things being equal," but now you need some quick cash so things are *not* equal. If you have $100 and you need to get to $150 in 10 minutes or you'll be stabbed to death (or, less dramatically, just miss your plane), the marginal utility of the extra $50 is much greater than the initial $100. In that case, throw typical "gambling as entertainment" concepts out the window and focus on maximizing your chances to live (or make your flight). It's very likely that neither Q64 nor Q62 are the right answer there. As a degenerate case, if you needed an extra $100 in the next 30 seconds, the obvious move would be bet $50 on one hand and make the Play bet regardless of what you had. Folding something awful like 953 would nevertheless guarantee failure in your situation, even though you'd never play that in a normal casino setting.Quote: TrevorOk, i've got $100, i need to make some quick cash, i've only got a certain amount of time to do it in and the only game in town is 3CP.
Q|6|4 or Q|6|2?
Quote: MathExtremistIt's not that simple. My original premise was "all things being equal," but now you need some quick cash so things are *not* equal. If you have $100 and you need to get to $150 in 10 minutes or you'll be stabbed to death (or, less dramatically, just miss your plane), the marginal utility of the extra $50 is much greater than the initial $100. In that case, throw typical "gambling as entertainment" concepts out the window and focus on maximizing your chances to live (or make your flight). It's very likely that neither Q64 nor Q62 are the right answer there. As a degenerate case, if you needed an extra $100 in the next 30 seconds, the obvious move would be bet $50 on one hand and make the Play bet regardless of what you had. Folding something awful like 953 would nevertheless guarantee failure in your situation, even though you'd never play that in a normal casino setting.
You're not going to let me away with this are you :)
