Three Card Whist

I've just found out about a game of winning tricks at Whist that's been in several Grosvenor casinos.

Players "Ante", receive three cards and the trump suit is revealed. Players look at their hands and can "Play" or "Fold". Then, one by one, the Dealer reveals his three cards; the aim is for the Player to win tricks. You have to follow suit or trump if you can. Players need to win 2 or 3 tricks to be paid out.

Quote: http://www.ukcasinotablegames.info/3cardwhist.html

Three Card Whist is a casino table game played with one standard pack of playing cards without jokers.

The Player competes against the Dealer and not against other Players. One to six Players can play each round.

Each Player and the Dealer are dealt three cards per round. An additional card is dealt face up and its suit becomes the trump suit. Any card that is the same suit as the trump suit is a trump card

The cards are ordered by rank, with Deuce low and Ace high. A card of the same suit as another card but of higher rank, beats that card.

A trump card always beats a card that is not of the same suit as the trump suit.

The Dealer reveals his three cards sequentially.

If the Player plays a card that beats the current Dealer card, the Player wins a trick. The Player tries to win as many tricks as possible against the three Dealer cards.

I'm looking into a possible strategy for this and get that you should fold just under 41% of hands.

Approximately (there are a few close exceptions)
(i) 3 or 2 trumps - play
(ii) 1 trump - usually play but fold if your other two cards are very small
(as a rough rule fold with non-trump 6x or worse, your actual trump card doesn't seem to matter much).
(iii) 0 trumps

• 3 cards in same suit - fold
• Two different suits
  - fold if your two card suit is 10-high or lower.
  - fold if your one card suit is 7 or lower.
  - play J2+ with A, J5+ with K, Q4+ with Q, Q8+ with J, K7+ with 10, A7+ with 9, AT+ with 8.
  - exception fold Q3-2 Q, K3-2 J, A3-2 T.
  
• Three different suits - you need 21 outs (i.e. there are 21 cards left in the pack lower than your cards, so you play 999 but fold 998; except play QJ3, KJ2 and fold A76).

I got the stated House Edge but there are a few quirks about discarding.

First Round
In the first round, where you can't trump or follow suit, you normally discard your lowest card. The only case you don't is with two cards in one suit (and one in another) when it is sometimes better to discard one of these than the odd card. The logic is that you have more chances of winning the second trick by holding two different suits which can overcome the disadvantage of possibly having a lower card left for the third trick. e.g. AhKh8d discard the Kh.

Second Round
There's also a small chance of discarding the wrong card on the second round. Consider a hand such as Ah (low)h Kd, where you won the first trick with (low)h, but now have to discard on dealer's As. Kd now has a better chance to win than Ah (11 to 10 outs), so you discard Ah. In essence this only happens where the diamond is only one rank lower than the higher heart, so this is fairly rare and has a very minor effect on the House Edge.
Edited this after considering the discard quirks.

Only in England could they turn freakin' Whist into a table game. What's next? King's Corner?

I'm surprised a whist variant got off the ground. I don't think I'm going to fuss with this unless it gets to a dozen placements or so. Are you interested in doing a full analysis? I'd be happy to write it up, giving you full credit. We could even pay you a little bit, but that shouldn't be your primary motive.

At last I managed to play the game for real this weekend and apparently it is in at least two casinos. Most people were playing the minimum but some were also playing the sidebets (which are similar ideas to those in 3-card poker). I sensed many had little idea of a proper strategy and were playing too many hands. I even saw one person play A 2 3, but at least they won the straight bet!

The game itself ran fairly slowly as each game consists of three tricks. For my part, playing £2, I worked through £20 before a win. On average you lose about 70% of hands (so £2 or £4) but stand to win £6 or £8 for winning three tricks. So a few wins can bring you back into the game.

I've come up with a simple strategy but it still requires knowledge of poker "outs".
(i) 3 Trumps : Play
(ii) 2 Trumps : Play
(iii) 1 Trump : Play with any non-trump 7 or higher.
(iv) 0 Trumps, 3 different suits : Play with 21+ outs (e.g. 999 as each card has seven outs: 8765432)
(v) 0 Trumps, 2 different suits : Ignoring the lower card in the doubleton, only play AT+ KJ+ QQ.
(vi) 0 trumps, flush (i.e. all in one non-trump suit) : Fold
The cost in House Edge for playing this way is about 0.36%

In what order are the dealer's cards played? Random?

So the entire HE depends on 1 thing:

You must decide to play or fold (forfeit) before knowing any dealer cards.

Is that correct?

And dealer randomly turns cards, or looks at the hand and plays a strategy?

I think I would like this game very much, but would want to play it.

Quote: charliepatrick

<snip>Second Round
There's also a small chance of discarding the wrong card on the second round. Consider a hand such as Ah (low)h Kd, where you won the first trick with (low)h, but now have to discard on dealer's As. Kd now has a better chance to win than Ah (11 to 10 outs), so you discard Ah. In essence this only happens where the diamond is only one rank lower than the higher heart, so this is fairly rare and has a very minor effect on the House Edge.
Edited this after considering the discard quirks.

charliepatrick,

This is true for heads-up play, but if the table has other players, then their discards would influence your decision: if on the first trick other players discarded diamonds, the Kd would be less valuable than the Ah on the third trick.

For that matter, your seat position would also be important if you get to see other players' discards on a given trick before you play your own.

Since you've played this game, perhaps you would answer three questions. First, how many players can play simultaneously? Second, on a given trick do all the players reveal their chosen cards simultaneously, or sequentially? Third, what is the penalty for reneging: a one-way trip to the Tower of London? ;-)

Dog Hand

Quote: DogHand

...Since you've played this game, perhaps you would answer three questions. First, how many players can play simultaneously?

Per the initial post of the rules: 6

Quote: DogHand

Second, on a given trick do all the players reveal their chosen cards simultaneously, or sequentially?

I wonder about this too, but I suspect the procedure is:
The dealer turns their card.
Players place a card face down into a "play" area, and cannot change it once set.
The dealer then reveals each player's card, collects losers, and pays winners.
The procedure is repeated for each round.

I wonder what the house way chart looks like.

Firstly I did enjoy the game and it seemed others did as well.

The dealer's cards are dealt face down and turned over one at a time. Players make their decisions to each trick by playing their cards face up - there's a nice method on the box with the words "WIN" nearer the dealer and "LOSE" nearer the player and you move your card to the top or bottom of the box to show whether you won or lost.

On the first trick technically there might be some help if you can see other people's cards, but bear in mind they can only discard if they haven't got a trump nor the suit led. Strangely, assuming it isn't a trump, that meant you both started with two suits (doubleton and singleton) and your correct decision is still to throw your lower card of the doubleton. The other occasion is if the dealer starts with a trump and you have three suits, then it might be worth looking at other people's cards.

On the second trick I can see you might have more information from others player's cards.

There is a penalty for revoking (although I'm not sure what the official rules are). The only time I saw it was when the dealer's first two cards were Clubs (Q then 5); another player put a Spade out for the first trick and then the 8 clubs for the second. He wasn't allowed to rectify his mistake and had to play the 8 on the Q (as "the card had already been played") and the King Of Clubs on the 5. He lost the last trick so would have won if he'd played correctly.

There is no House Way, the Dealer's cards are turned up one at a time.

Quote: charliepatrick

Firstly I did enjoy the game and it seemed others did as well.

The dealer's cards are dealt face down and turned over one at a time. Players make their decisions to each trick by playing their cards face up - there's a nice method on the box with the words "WIN" nearer the dealer and "LOSE" nearer the player and you move your card to the top or bottom of the box to show whether you won or lost.

On the first trick technically there might be some help if you can see other people's cards, but bear in mind they can only discard if they haven't got a trump nor the suit led. Strangely, assuming it isn't a trump, that meant you both started with two suits (doubleton and singleton) and your correct decision is still to throw your lower card of the doubleton. The other occasion is if the dealer starts with a trump and you have three suits, then it might be worth looking at other people's cards.

On the second trick I can see you might have more information from others player's cards.

There is a penalty for revoking (although I'm not sure what the official rules are). The only time I saw it was when the dealer's first two cards were Clubs (Q then 5); another player put a Spade out for the first trick and then the 8 clubs for the second. He wasn't allowed to rectify his mistake and had to play the 8 on the Q (as "the card had already been played") and the King Of Clubs on the 5. He lost the last trick so would have won if he'd played correctly.

There is no House Way, the Dealer's cards are turned up one at a time.

Sounds like a game ripe for shot-takers. Not a good thing. I still say I would enjoy it, but I think that might kill it in US casinos. The dealer would be constantly auditing/policing the hands.

Quote: charliepatrick

(iii) 0 trumps

Three different suits - you need 21 outs (i.e. there are 21 cards left in the pack lower than your cards, so you play 999 but fold 998; except play QJ3, KJ2 and fold A76).

I have been verifying Charlie's excellent strategy. I have found one additional exception to the above rule -you should also play a QQ2, which is a 20 outs hand.

Here is how often a player will get each kind of hand:

1. 3 Trumps: 0.010564
2. 2 Trumps: 0.123601
3. 1 Trump, 2 different off-suits: 0.292149
4. 1 Trump + 2 in one off-suit: 0.134838
5. 0 Trumps; 3 different off-suit: 0.105498
6. 0 Trumps; 2 different off-suits: 0.134838
7. 0 Trumps; 3 in one off-suit: 0.0412

In the above categories, you will always play #1 and #2 and always fold #7. The other 82.5% of the time you will need to make a FOLD/PLAY decision based on the ranks of your cards.

#7 is by far the most catastrophic for the player, even with high cards. If you are dealt a suited A-K-Q of a non-trump suit, PLAY has an EV= -1.46404. Because FOLD is EV= -1, you must always fold 3 suited cards if they are not in the trump suit.

For #5 (0 trumps 3 different offsuits), you will have a positive EV with 30 outs or more. The hand AAA (all offsuit) has 36 outs and an EV =0.660037. As Charlie has noted, when you hand has 21 or more outs you have EV>-1. As a rule of thumb, an "out" is worth roughly 0.09-0.11 in EV.

The hands x22, where x=any other card including a trump, have an EV=-2 for Play, because with two 2s there is no way to win 2 twice. The same is true of X32 in which the 3-2 are in the same off-suit. The hand 332 (3 off-suits) has EV= -1.9898821.

Thanks for your insight and spot of QQ2. Interestingly when I ran a simulation of 100m hands it would have played QQ2 KJ2 but folded QJ3 based on results; they're quite close which is why using 21-outs is a reasonable approach. The other fact you show is the ratio of hands with 1 trump and so how important it is to get this part of the strategy right.

I've also just thought of a different way than outs and that is to add up the value of the cards where A=14,K=13,Q=12,J=11 10-2 asis and usually you need 27 or more with no trumps and three suits.

Thanks, I am a geek who likes to calculate this kind of stuff even when it is marginally useful.

With one trump and 2 off-suit cards in the same suit (I label this 1-200), I propose this as a good approximate strategy:

PLAY if the ranks of the suited (off-suit) cards are at least this high

Trump Card	Minimum Same-Suit (off-suit) Cards
T-A	7-2
3-9	8-2
2	8-5

Otherwise FOLD

By analogy, I imagine that a more detailed strategy for when the hand is one trump and 2 unsuited cards (1-110) would be

PLAY if the ranks of the unsuited (off-suit) cards are at least this high

Trump Card	Minimum (unsuited) Off-suit Cards
T-A	6-2
3-9	7-2
2	7-5

Otherwise FOLD

I have also developed for these hands a "perfect play" strategy with higher granularity which also factors in whether the "trump indicator" card is higher or lower than your trump card. But I suspect that no one really wants all that detail.

Does the side bet pay, even if the player folds? A non-trump suited staight flush would still be a paying hand.

I have just realized a much easier way to state a strategy:

In order to bet PLAY (not fold) with hands with 1 Trump card, your highest unsuited card must have at least 4 (or more) clear "outs" depending upon the rank of your trump card:

Trump Card	Minimum Outs for Highest Non-trump Card
T-A	4
3-9	5
2	6

This strategy applies, regardless of whether your off-suit cards are suited or unsuited.

Basically when you have a 1-Trump hand such as Qh-7d6d you are trying to win two tricks by
a) winning your trump card, Qh, and
b) winning with your 7d by hoping the dealer has a 2d, 3d, 4d, or 5d.

That's a pretty slender chance -you need the dealer to have one of those 4 cards!

I always assume the upcard for trumps is a 2, but you can still define a Hi trump as the best 6 and a Lo trump as the lowest 6. So A-9 will always be high and 7-2 will always be lo, an 8 depends.
This is an internediate strategy I had created.

Usually PLAY if either non-trump is 7 or higher, however :-
(a) If your trump is Hi (A-9) then you can play one rank lower (i.e. 6 or higher)
(b) If your two non-trumps are in same suit then you need one rank higher.
Optional extra: It is worth learning how to play these ideally as it saves about 0.12%

Quote: charliepatrick

This is an internediate strategy I had created.

Usually PLAY if either non-trump is 7 or higher, however :-
(a) If your trump is Hi (A-9) then you can play one rank lower (i.e. 6 or higher)
(b) If your two non-trumps are in same suit then you need one rank higher.
Optional extra: It is worth learning how to play these ideally as it saves about 0.12%

Yes, that is almost exactly what my last post said, but in a different way.

Similarly, if the "trump indicator card" is lower than the trump card in your hand, then its as if your trump card is one rank lower.

If the "trump indicator card" is 2h, then your 10h plays identically to a 9h with a trump indicator card >9h. Even in trumps, it is all about the number of suited "outs" below the rank of your card.

Well, it sounds as if you are in the process of writing that WOO webpage for the Wizard on the game. Good luck with that and let me know if there is anything I can do to help.

^ Thanks - it's taken a while but I've finally sent a draft to WoO. For the intermediate strategy It seemed easier to ignore the actual up-card...

Trump card	Number of side suits		Intermediate strategy
A K Q J	2	K♠ 5♥ 4♦	Play if the total of the two non-trump cards is … 9 or more
10 9 8 7 6	2	6♠ 6♥ 4♦	… 10 or more
5 4 3 2	2	3♠ 9♥ 2♦	… 11 or more
A K Q J 10 9	1	K♠ 7♥ 2♥	Play with any 7 or higher
8 7 6 5 4 3 2	1	7♠ 8♥ 2♥	Play with any 8 or higher

Quote: charliepatrick

Thanks - it's taken a while but I've finally sent a draft to WoO.

I finally put up the material Charlie sent over onto the site: Three Card Whist. There may have been some mistakes in the conversion, which are on me, not Charlie. Although I'm listed as an author, I'm not, Charlie is. We need to add his name to the list of official site authors.

A huge thanks to Charlie for this outstanding effort! This is not only Wizard worthy, but better than what I would have done with it.

Any updates on this game? Is it still played in London? Is it spreading? I played a lot of whist back in the day and would love to try this. I suppose it won't catch on in the US but hey, War is the simplest game in the Whist Family and look how it did in the casinos. Maybe I'll make some player mats to at least try it.

Question: In the WoO picture I can see the "Bonus Bet" spot, but what is the side bet to the right? It looks like "Hn5". I did some online searching and got shown the WoO "21+3" page which kind of makes sense but I dunno...

Sadly the game isn't in my local casino any more; I'm not sure about other casinos. The side bet and basic game is described here - http://www.ukcasinotablegames.info/3cardwhist.html . The other side bet is a variant on Three-Card prime paying if colours are the same. (3/1 Player; 4/1 Player and Dealer, 5/1 Player, Dealer and Trumps).

That is sad. It's such a clever idea, one of those rare casino games where you really have to think about every card you play. I was already memorizing the strategy you devised. Oh well. Maybe it'll get picked up by online casinos.