## Tarot at 5 players

### First problem

Here's a hand reported by Benoit Lessard of Montreal:

Tarot at 5 players - friendly game at 5 pennies/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 bidding? (unpleasant, not?). Before trying to give a final answer, I will simulate the play a hundred times to see what is the average result. My feeling is that a simple garde is preferable but I will check anyway.

By supposing that you take a garde and that someone overbids garde-without-the-dog behind you, will you show the Poignée? Here the answer is easy. The declarer probably has a hand like 2 oudlers with 5 or 6 trumps and a king, and perhaps a void. Or 2 short impregnable oudlers with some points. If the honors of the attackers are in and , they are likely to make the contract in spite of the 11 trumps in your hand.

Moreover, we do not want to alert the partner of the taker on the urgency to discard his points, nor to discourage the taker to try a finesse or duck a king. The priority is to make declarer goes down, to show the double Poignée is likely to help the attackers. I say "probably" because it is also possible, by showing the double Poignée, to make them panic wrongly, for example by making them twin K-Q or K-Q on the same trick whereas they could make 2 tricks. It would be the only small advantage to show the double Poignée.

How did this game finish? A player overbid garde-without behind the garde, with 2 impregnable oudler (impregnable = the 21 and the Excuse) and a lot of honors. Unfortunately, he found the called king in the dog but went down only by 2.

### Evaluation

I made a series of simulation with this hand by playing it 100 times. I allocated a loss or a win only when the result was reasonably clear. For example, when there was a hand in defense assured of 4 tricks of trump with the 3 oudlers + 2 tricks in any suits, I automatically allocated a loss to the taker. In the more difficult cases where the contract could be made or lost by the fact that a defender discards or not a particular card, I only allocated a half win.

I made play the defenders in the following way: gift of the Petit on the first occasion to any player, and blind discards of points, especially when the defense knows that the taker has bid without oudlers and that the only possible hand's configuration of the taker is a trump length. The taker, however, does not have a choice of line of play. He plays trump as soon as he is on lead. He can improve his chances by drawing all trumps before giving the lead to the partner through the called king. The partner can then cash any card that is set up in his hand. It sometimes happens that he can close, or at least cash several good cards. I thus made play the taker in this manner when it was possible.

The result for the Garde-against-the-dog was 49 wins over 100 games. But probably that a rising weighting must be made, for 2 reasons. First, some players tend to not charge the points blindly and in this deal the defenders will often have to discard without being sure of who's the partner. Secondly, the taker cannot easily be mistaken in his line in play. If errors must be made, they will generally be made by the defense. Another factor is the margin of error in the distribution. It is possible that the taker was very unlucky in these 100 deals, for example it's possible that he called a partner without an oudler more often than the normal average.

The reverse is also possible, he was maybe very lucky (this is why the margin of error is always preceded by a "more or less"). 100 deals is a really small sample and the margin of error can be very high. But, even while balancing in favor of declarer with these factors, we will reach at best 60 percent (and this is generous), which is still far from the rate of profitability of a garde-against-the-dog. This rate is 80 %. However, when I say that the rate of rentablity is 80 %, it's when compared to an almost 100% winning garde (because if you are thinking to bid garde-against-the-dog with a hand, this hand is probably a 100% winning garde). But here the garde is far from being a 100% win, as we can see in the results of the 100 deals played as garde:

Résults on 100 deals
58 wins
37 losses
5 overbids

The 5 overbid contracts went down 2 times and succeeded 3 times.

The rate of success of the garde does not go up a lot compared with garde-against-the-dog. That is explained by the fact that when the taker turns a blank dog, he is not more advanced than if he had taken guard-against. He is rewarded when he finds some KEYCARD (king or oudler), or some trumps (which make it possible to eliminate losing cards from its hand by discarding them in his discard).

When playing the 100 deals, I found very few oudlers in the dog. I thus expects that the rate of success of the garde increases according to a more normal presence of oudlers in the dog. But with these results, the contract of garde seems to be the good bid, at least when compared to the contract of garde-against-the-dog. If I round the rate of the garde-against to 55% and the rate of the garde to 65%, I can calculate the following equity:

Note: since we find the called king in the dog in 4,9% of cases, I must calculate the score from these 5 deals before evaluating the 95 others remaining from my sample of 100 deals. I separated the score 2/3 to the taker and 1/3 to the partner.

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

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

Note: the 0.5 king in the dog value represents the possibility for the taker to make the contract when he finds the called king in the dog + a trump or an oudler (they is perhaps a little generous). I also granted more points on average for the successful contracts. Indeed, when the taker finds a nice dog, he can show the Poignée which otherway he would have hidden. Another example, when the partner has the 21, he has good chances to collect or capture the Petit in defense.

It is interesting to note that the kings in the dog tip the scales in favour of the garde. The king in the dog represents the initial handicap of the garde-against and garde-without which must be overcome by a good equity of the remaining deals.

#### Probabilities

Some interesting probabilities concerning this hand: which are the chances to find 0, 1, 2 or 3 oudlers in the dog when the taker does not hold any of it? Chances to find a particular card, for example the called king? Do not forget that they are probabilities for tarot at 5 players 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 %
1 particular card 4,9 %

To simulate the play of a deal is a annoying and tiresome task, that's why I do not evaluate the equity of guard-without-the-dog. However, by comparing the garde and the guard-against, we can think that the garde-without should have an equity close to the garde, by supposing that my initial evaluation is correct.

The hand misses controls but has the necessary configuration to be played in a over-contract. The 2 contracts will be probably difficult to separate. I will try to make a simulation when I will find time, but I think that we can already affirm that it is not a serious mistake to choose one or the other. Only the contract of guard-against seems to be a little optimistic.

### Second problem

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

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

This one is a little easier: we bid guard-against-the-dog and call the king of diamond. We show the Poignée since our intention is to hunt the Petit by cashing the top trumps, thus revealing our force brutally. In fact, there are really no restrictions to show the Poignée because it's really hard to go down with this hand and because I do not think that we want to initiate the hunt for the Petit by leading the 2 of trump (this tactic would be a good reason to hide the Poignée).

Why hunt the Petit by cashing the top trumps rather than leading the 2? It's because of the 2 favorable things that can happen: not only you have a good chance to capture the Petit, you also have a good chance to capture the 18 (that will set up your 17). These chances combined together are too favorable to risk the loss of the Petit by leading the 2 of trump.

Another reasonable line of play consists to draw 2 master trumps and, if nothing interesting occurs, to give the lead to the partner by playing diamond for his king. If the partner has a third trump to play, we give ourself a chance to make the 17 with a finesse or to capture the 18 (especially if the partner is at the bottom)

Lastly, we can simply play on and let the defense manage an exit for the Petit. The control of the heart and spade suits practically guaranteeing to reach 51 points even when the Petit escapes.

So, is it better to hunt for the petit or play on clubs? Surprisingly, we will be more often rewarded by playing on clubs. On the other hand, the profits which we'll make with this line of play will not compensate for the tremendous swings that we'll suffer when, for example, a defender ruffs the called king. To play on clubs is indeed ridiculous each time our partner is fitted in the suit. I made some simulations of play which I stopped after 36 deals during which defense ruffed 8 times the king of  !

The good line of play consists to show the Poignée and bang the master trumps. No subtleties with this hand, the brutal method is better !

Updated: 07/02/2016