Sunday, 23 January 2011

Analysis of Hearts

1) Introduction
I'm not a bridge expert, but heart is one of my favourite... Hearts, being a popular game and an elementary practice for bridge players, someone just tell me to write something on hearts.
Poker games, are most likely linked with "luck". Without luck we can't win even one game.
Is that true?
Yes, but luck don't determine 100% of the win and lose.
Unlike Big2 game, Hearts and Bridge's win and loses are mostly likely to be determined by your skills.

Moreover, let's assume that the % of result are determined by luck (we don't have experimental result of this %, just a hypothetical data)
One claims that the % are as following: Big2 60%, Tenhou 40%, Hearts 10%, Bridge 2%
Without cheating, the "luck" is totally, independently, fairly random. It is in normal distribution.
"luck" can't be quantitized since each hand's "luck" is determined by the distribution of whole system instead of your identical hands. Make it easy that the standard deviation = "luck %"/6 and the mean is "luck %"/2, i.e., 99.7% data is included.
Consider Big2, the skill determines 40% of the results.
In order to give a fairer review we can look at 100 games.
When the skill is perfect, 100(0.4^100) = 1.60694* 10^-38 games can be won without anyluck, but this happens with the probability of 0.15% ^ 100 = 4.06 * 10^-283 !
Integrate the bad luck part (mean luck < theoratical mean)
It wons 2.24 * 10^-16 games, while the prob. is 0.5^100 = 7.88 * 10^-31
That is, even your luck is extremely bad, your skill helps you to win about square root of # of games played.

How about in a good luck?
The expected games won will be 100(0.4+x)^100 dx, by integrate from 0.3 to 0.6, the answer is 0.99!
That is, you must win at least one of the 100 games.

Note that when you're in a good luck, since the good combinations are limited, others will have relatively lower luck, and your chane to win is much higher. Then you will win more than the mean among the games in reality.

When you have no skill, even you are in very good luck (every game's "luck" is above mean, you chance would be 3.88 * 10^-23, much less than those with skills.

Even in such a gambling game we can see that skill determines a lot, then it's far more important in Hearts and Bridges. What makes Hearts intersesting is the passing round which determines most of the game.

Here's some rules that is important for the discussion of stragities:
-S Q worth 13 pts, each H worths 1 pt
-left pass, right pass, vertical pass, no pass
-shooting the moon (one eats all)
-Broken heart (H can't be played actively until one plays heart passively, unless you have no other suits)
-No pt should be given out in the first round
-Club 2 goes first in the first round.

3)Determination of your hand in the first glance - Passing skill
Passing determines the results a lot so it's quite important for you to plan what to pass to give you a safer hand, But before that we should know how to determine what is a good hand:
Good hand: Hands that tends to have no or low pts (0~4 pts), hands that surely shoot the moon (26 pts)
e.g. 1: S 9873 H J83 D 9832 C A4
This is quite safe as you don't have the risk to eat SQ, while the C A can be discarded in the first round.
Expected pts: 0
e.g. 2: S AJ9873 H QT2 D 93 C 32
This is quite safe as well since S A won't be forced out under the protection of 6 card S suit, the only potential points comes from the H suits.
Expected pts: 0~4
e.g. 3: S AQ8 D AJ H AKQJ10 C AKQ
With the 5 top cards in H suits, it's possible that you can pull out all hearts after pulling out your S Q and get yourself. The potential hazard comes from losing the active role while playing D J.
Expected pts: 26

Normal hand: Hands that may worth a number of pts (5~9 pts), Hands that may eat S Q with a few Hearts (13~15 pts)

e.g. 4: S JT92 H AT103 D J65 C AK
You can't discard both C A, K in the first turn, discard later is quite danger; if others play H, potentially this hand eats once.
Expected pts: 4~8
e.g. 5: S Q H A8732 D AK7653 C 8
Most probably you'll have to collect 13 pts from yourself, but you don't need to worry on the H and D suit under well-protection.
Expected pts: 13

Dangerous hand: Hands that eats a lot of Hearts, and potentially eats S Q (10~13, 16+ pts)

e.g. 6: S AQ3 H KJ D AQJT75 C 2
When you are required to play one of S, H or D suits, you can't throw away the active role easily. That potentially gives a large amount of points.
Expected pts: 8~9, or 16~20
e.g. 7: S AK97 H ---- D KQJ C JT7653
Cs are played in the first round, so it's also quite hard to throw away the active role.
Expected pts: Very high or very low

Gambling hand: Hands that may eat all OR a very high pts without eating all (24~26 pts)

e.g. 8: S AK765 H AQT987 D AK C -----
If H K does not appear, you can't eat all.
Expected pts: 24~26
e.g. 9: S AQ9 H A3 D AKJ6 C AQJ10
It's hard to collect Hs by such a small H card.
Expected pts: 23

Let's look at the use of different suits:
S - long suit is very good - to protect yourself from getting S Q without other choice.
H - long suit in your hand also prevents getting mroe pts
D - better shorten it if you only have 4 or less, a void suit would be useful in discarding other big cards.
C - one top C is safe which can be used in the first round.

It's quite important that we should not discard S suit.

Let's take the previous examples.
Good Hand - discard D, C
Normal Hand - discard D C top cards, then shorten D suit
Dangerous Hand - upgrade to gambling hand or passing the A,Ks away
Gambling Hand - discard the smallest cards
 e.g. 1: S 9873 H J83 D 9832 C A4
discard D 98, C4 or H J D9 C4 - discarding one C gives you a voided suit, discarding D make you safe in D suit.
e.g. 2: S AJ9873 H QT2 D 93 C 32
discard H QT C 3 - the H suit is quite dangerous.
e.g. 3: S AQ8 D AJ H AKQJ10 C AKQ
Nothing should be discarded lol, I suggest S8, DJ and C Q.
e.g. 4: S JT92 H AT103 D J65 C AK
discard H A C AK, this gives you a chance to discard one unwanted card in the first round.
e.g. 5: S Q H A8732 D AK7653 C 8
discard H A8 C8, it'll be more dangerous if you recieve S AK without S Q.
e.g. 6: S AQ3 H KJ D AQJT75 C 2
discard H KJ C2 to produce two viod suit, as to avoid being the active role, and if you are in active, it has a chance to shoot the moon.
e.g. 7: S AK97 H ---- D KQJ C JT7653
discard D KQJ and hope you opponents send you some S.
 e.g. 8: S AK765 H AQT987 D AK C -----
discard S 765.
e.g. 9: S AQ9 H A3 D AKJ6 C AQJ10

discard H3 D J6.

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