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Profitably dropping in Tonk Profitably dropping in Tonk

10-06-2008 , 04:40 PM
This is for the math guys out there. A friend and I were reliving some middle school days the other night by playing tonk, and I started to wonder something.
If you don't know how to play, you really don't need to know much for this question other than this: (5 card game with rules slightly similar to rummy) if it is your turn you can put your hand face up on the table if you think that the total point value of the cards in your hand is lower than the total point value of the cards in your opponents hand (Ace=1, 2-10=that number, face cards=10, eg. AA23K=17). If your total is lower you get one unit, if theirs is lower or ties you lose two units (or they win two units).
My question is this: Assuming you are first to act after a deal and no one has had a chance to discard yet, at what point value is it profitable to perform this showdown? I know you have to win 2 out of 3 to break even, but I don't know what this translates to for card values.
Thanks guys. Any clarification just reply with your question.
10-07-2008 , 11:23 AM
Odd game. Assuming you draw before you discard, the best hands shouldn't table, since they make twice as much when villain eventually tables and they beat him. Tabling early with middling strength or sandbagging with a pat monster are about the only two ways to play a good hand, and a bad hand obviously just keeps drawing.

The math tool would be to find the point value for all possible hands and table anything that beats 67 percent of them immediately, but if dealt anything significantly better, continue drawing and sandbag. Finding that break point would be a lot more interesting exercise.

If dealt AAAA2, for example, you would never table, and once it became obvious your hand was too good to table, villain would never table either, and you'd both keep drawing until they shut out the lights and turn a hose on you.
10-07-2008 , 01:21 PM
good reply. I have to apologize as I may have made it seem like tabling was the only way to win. Say villain catches 3 of a kind or a 3 straight flush, hey can lay that down(like rummi) after drawing, then discards leaving him with two cards. His next turn he can table and probably beat us if we don't make anything(he can also wait and see if he can draw and lay down the remaining 3 in the same manner as above, this will win him 2 units, so there's a lot of ways he can win if we let him).
I guess my question was: I want to end the game as early as possible, so if I wanted to drop first turn, at what point value would it be profitable?
Thanks
10-07-2008 , 03:19 PM
Quote:
Originally Posted by bobalobagous Profitably dropping in Tonk
My question is this: Assuming you are first to act after a deal and no one has had a chance to discard yet, at what point value is it profitable to perform this showdown? I know you have to win 2 out of 3 to break even, but I don't know what this translates to for card values.
Thanks guys. Any clarification just reply with your question.
Check the archives here or on the probability board. All the Tonk math was posted a year or so ago. For the situation you describe, I believe one sigma is 23 points, but don't hold me to that.
10-07-2008 , 03:45 PM
Quote:
Originally Posted by electrical Profitably dropping in Tonk
If dealt AAAA2, for example, you would never table, and once it became obvious your hand was too good to table, villain would never table either, and you'd both keep drawing until they shut out the lights and turn a hose on you.
In many games, this would be a 'little tonk' and could be tabled before the start of play for two units.

If I had an AAAA2 later in the play, I might lay AAA knowing that I couldn't be beat by two cards, and trying to precipitate anyone with a meld to lay down. But that would be FPS; somebody could still beat me by laying six cards, so I would lay the hand and go on to the next deal.
10-07-2008 , 04:52 PM
Is this Tonk, the Tonk that John Scarne describes in Scarne On Cards?
10-08-2008 , 03:32 AM
Thanks Mack, 23 sounds like a pretty good number as there are only 4 ways to make aaaa2 but 16*15*14*12= a lot of ways to make fifty so the distribution is a little skewed. After just playing for a while I assumed it was a little less(somewhere around ~17-18) but I appreciate having more that just a "guesstimate." Good to know I can lower my tabling requirements with raggedy draws :-)
10-08-2008 , 11:02 AM
Quote:
Originally Posted by Al Mirpuri Profitably dropping in Tonk
Is this Tonk, the Tonk that John Scarne describes in Scarne On Cards?
Can't remember the reference by Mr. Scarne, but it is this Tonk: Pagat page.
10-11-2008 , 01:05 AM
I hacked up a quick calculator.

Without any dead cards, the distribution of hands looks like this:

Code:
 i    count cumulative
--  ------- ----------
 6        4 0.0000
 7       28 0.0000
 8       92 0.0000
 9      240 0.0001
10      484 0.0003
11      920 0.0007
12     1552 0.0013
13     2492 0.0022
14     3784 0.0037
15     5724 0.0059
16     8344 0.0091
17    11988 0.0137
18    16520 0.0201
19    22144 0.0286
20    28948 0.0397
21    36708 0.0539
22    45584 0.0714
23    55712 0.0928
24    67600 0.1188
25    79416 0.1494
26    92416 0.1850
27   103808 0.2249
28   115520 0.2693
29   125188 0.3175
30   134052 0.3691
31   140224 0.4231
32   146936 0.4796
33   149268 0.5370
34   147784 0.5939
35   143676 0.6492
36   136344 0.7016
37   127484 0.7507
38   116832 0.7956
39   105176 0.8361
40    92548 0.8717
41    82176 0.9033
42    65532 0.9285
43    52556 0.9488
44    40436 0.9643
45    31216 0.9763
46    22496 0.9850
47    16720 0.9914
48    10640 0.9955
49     7280 0.9983
50     4368 1.0000
So, if this is correct, a hand worth 30 points will be beaten only 32% of the time (and tied an additional 5%, what happens then?) Unfortunately you win only 63% so this is a slight loser. 29 points should always be good.

I tried a bunch of 30 point hands and they look profitable assuming ties are EV=0, so perhaps dead cards are enough to swing 30 points to a win always.

If you hold AA8TT (30 points) then you lose 29.59% of the time, win 65.10% of the time, and tie the remaining 5.31%.

If you hold 5555K (30 points) then you lose 30.53%, win 64.58%, and tie 4.89% .

2288J: lose 30.11%, win 64.77%, tie 5.12%.

Last edited by MarkGritter; 10-11-2008 at 01:19 AM. Reason: Fixed incorrect result.
10-11-2008 , 04:11 PM
Nice work, as always, by Mr. Gritter.

I would like to point out to the casual reader that these lay-down values are for one opponent.

Another value that I believe is calculable:

Let's say you are playing heads-up. Your opponent is first to act. He draws, then discards a face card. What is your break-even laydown value?

Re: Skewing. The fact that no one has 'Big" tonked may slighltly skew the results, but many games play 'Little' tonks that allow immediate laydowns of 15, 14, or 13 points, which could skew the values in the opposite direction. After playing for +40 years, I can say I've never played two games w/ the same rules.
10-12-2008 , 11:15 PM
There is also a strategy element to this question because your alternative to dropping is to keep playing, which is not a zero-EV outcome. If you have a really bad playing hand you should be willing to take slightly the worse of the drop just to end the hand. The opposite applies if you have strong prospects for either laying down cards or discarding some tens.
10-11-2011 , 08:13 PM
Can anyone figure out the probability of having the best hand after 1 and 2 draws assuming the highest point value card is discarded in a heads up format?

      
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