Tarot at 5 players

Evaluating the hand

First problem

Here’s a hand reported by Benoit Lessard from Montreal :

5-player tarot – friendly game at 5 pennies per point (a serious game !)

17-11-10-9-8-7-6-5-4-3-2
-
-
x-x
x-x

You are first to speak. What is your bid? (Unpleasant, isn’t it?) Before giving a final answer, I simulated the play one hundred times to estimate the average result. My intuition was that a simple garde would be preferable, but I wanted to check.

Suppose you bid garde and someone overcalls with garde without the dog behind you. Should you show the Poignée ? Here the answer is fairly clear. Declarer probably holds a hand with 2 oudlers, 5 or 6 trumps, a king, and possibly a void. Or perhaps two oudlers on a short trump holding with some scattered points. If the attackers’ honors are in and , they are quite likely to make the contract despite the 11 trumps in your hand.

Moreover, we do not want to alert declarer’s partner to the urgency of discarding points, nor discourage declarer from taking a finesse or ducking a king. The priority is to defeat the contract, and showing the double Poignée is more likely to help the attackers. I say “more likely” because revealing it might occasionally cause them to panic and misplay—for example, by playing K-Q or K-Q together on the same trick when they could have taken two separate tricks. That would be the only small advantage of showing the double Poignée.

How did the actual game end? A player overcalled with garde without the dog over the garde, holding two oudlers (21 and the Excuse) and many honors. Unfortunately for him, the called king was in the dog, and he went down by only 2.

Evaluation

I ran a series of simulations with this hand, playing it 100 times. I recorded a clear win or loss only when the outcome was reasonably certain. For example, if the defense held a hand guaranteed to take four trump tricks including the three oudlers, plus two side-suit tricks, I automatically counted the deal as a loss for declarer. In more delicate cases—where the result depended on whether a defender discarded a specific card or not—I counted only a half win.

I instructed the defenders to play as follows: give up the Petit at the first opportunity to any player, and discard points blindly, especially when the defense knows that declarer bid without oudlers and therefore almost certainly has a long trump holding. Declarer, on the other hand, follows a fixed plan: draw trumps as soon as he gains the lead. He improves his chances by exhausting trumps before putting partner in with the called king. The partner can then cash any established winners. Sometimes he can even run a long suit, or at least cash several good cards. That is the line of play I gave declarer whenever possible.

The result for garde against the dog was 49 wins out of 100 deals. This figure should probably be weighted upward for two reasons. First, some players do not discard points blindly, and in this deal defenders will often have to discard without knowing who the partners are. Second, declarer’s line of play is fairly straightforward, while defensive errors are more likely. Another source of error is distribution variance: it is possible that declarer was unlucky in these 100 deals—for example, calling a partner without an oudler more often than average.

The opposite could also be true—he may have been lucky. That’s why margins of error always go both ways. One hundred deals is a very small sample, and variance can be high. Even after adjusting generously in declarer’s favor, we might reach at best 60% success (and that is optimistic), which is still well below the usual profitability threshold for a garde against the dog. That threshold is around 80%. However, that figure assumes you are comparing against a garde that would win almost 100% of the time. If you are considering garde against the dog, your hand would normally be an almost certain winning garde. But here, the garde is far from automatic, as shown by the 100 simulated deals played as garde :

Results over 100 deals

58 wins

37 losses

5 overbids

The 5 overbid contracts went down twice and succeeded three times.

The success rate of the garde does not increase dramatically compared to garde against the dog. This is because when declarer turns over a weak dog, he is no better off than in garde against. He benefits only when he finds a key card (a king or an oudler), or additional trumps that allow him to discard losers.

During the simulations, I found very few oudlers in the dog. I therefore expect the success rate of the garde to rise with a more typical distribution of oudlers. Still, based on these results, garde appears to be the better bid compared with garde against the dog. If I round the garde-against success rate to 55% and the garde rate to 65%, we can estimate the following equity :

Note : since the called king is found in the dog about 4.9% of the time, I first calculated the score from those 5 deals before evaluating the remaining 95. I split the score two-thirds to declarer and one-third to partner.

Garde against the dog:
5 king in the dog × (-900) = -4000
53 wins × (+440) = +23320
42 losses × (-380) = -15960
Total over 100 deals = +3360

Garde:
4.5 king in the dog × (-260) = -1170
0.5 king in the dog × (+240) = +120
63 wins × (+160) = +10080
32 losses × (-140) = -4480
Total over 100 deals = +4550

Note: the 0.5 “king in the dog” value reflects the possibility that declarer makes the contract when he finds the called king in the dog along with a trump or an oudler (this may be slightly optimistic). I also assigned higher average scores to successful contracts, since a favorable dog may allow declarer to show a Poignée that would otherwise remain hidden. Likewise, if partner holds the 21, he has good chances to capture the Petit in defense.

It is interesting to note that kings in the dog tilt the balance in favor of the garde. The king in the dog represents the initial handicap of garde against and garde without, which must be offset by superior equity in the remaining deals.

Probabilities

Some interesting probabilities for this hand : what are the chances of finding 0, 1, 2, or 3 oudlers in the dog when declarer holds none? What about the chance of finding a particular card, such as the called king? These probabilities apply to 5-player tarot only :

Probabilities with
no oudlers in hand

Find...	Chances (in %)
0 oudler	86%
1 oudler	13,4%
2 oudlers	0,5%
3 oudlers	0,0025%
A specific card	4,9%

What about garde without the dog ?

Simulating deals is a long and tedious task, which is why I did not calculate the exact equity of garde without the dog. However, by comparing garde and garde against, we can reasonably suppose that garde without has an equity close to that of garde, assuming the initial estimates are correct.

This hand lacks controls but has the right shape for an overcall. The two contracts are probably very close in value. I will try to simulate them one day when I have time, but for now we can already say that choosing one or the other is not a serious mistake. Only garde against seems slightly optimistic.

Second problem

Your hand :
21-20-19-17-13-12-7-2
-
-
10-7
K-9-6-4-1

You are first to speak. What is your bid? Do you show the Poignée? What is your opening lead ?

This one is a little easier: we bid garde against the dog and call the king of diamonds. We show the Poignée since our plan is to hunt the Petit by cashing the top trumps, openly displaying our strength. There is little reason to hide the Poignée here: it is hard to go down with this hand, and we do not intend to initiate the hunt by leading the 2 of trumps (which would be a good reason to conceal the Poignée).

Why hunt the Petit by cashing high trumps rather than leading the 2? Because two favorable things can happen : not only do you have a good chance to capture the Petit, you also have a good chance to catch the 18, which would set up your 17. These combined chances are too good to risk losing the Petit by leading the 2.

Another reasonable line is to draw two master trumps and, if nothing special happens, put partner in with a diamond toward his king. If partner has a third trump to return, we give ourselves a chance to score the 17 via a finesse or to capture the 18—especially if partner leading towards all defenders.

Finally, we could simply play on and let the defense manage the Petit position. Control of hearts and spades almost guarantees reaching 51 points even if the Petit escapes.

So, is it better to hunt the Petit or play clubs? Surprisingly, playing clubs will win more often. However, the gains from this line do not compensate for the huge swings when, for example, a defender ruffs the called king. Playing clubs is disastrous whenever partner is long in the suit. I ran some simulations and stopped after 36 deals, during which the defense ruffed the king of   8 times !

The best line of play is to show the Poignée and fire off the master trumps. No finesse needed here—the bulldozer works best !

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