Analysis on “Draws”: What to Do with Your Hand?

Pre-Word

The answer to the question posed in title is the KEY to playing particular hand. In order to win a player should evaluate chances to finish with particular pattern. (Top 10 wining patterns you can find here). And, picking-and-throwing tiles do best to keep alternatives open. At certain time the decision “Where to Go?” must be made. Consider, for instance, typical choices:

·        having 5 Pairs is worth to go for “All Pungs” or “Seven Pairs”?,

·        having 2/3 Pairs is it worth to go for “All Pungs” or any Chow-based hand?

·        how many tiles for Knitted hands is worth to have in order to win?, etc.

First choice is about to go for Pung-hunt or wait on self-draw. In the second example Chow-based hands are very flexible though you can pick only from 1 player plus self-draw. The 3rd strictly suggests self-drawing only until the last tile to pick from anybody.

Approach

Underlying idea to go through is to use cumulative PROBABILITIES to get certain “useful” tiles (we will call them “MT” = Matching Tiles). Extensive calculations have been carried out in order to get Number-of-Moves to get (at certain “Level-of-Certainty”) required tile depending on starting conditions: “Buying Method”, “Tiles-in-the-Wall” and number of “Matching Tiles”. Before those results would be provided let clarify ingredients J.

See here for more on “Draws Analysis Ingredients”:

Table and How to Use It

Here is table for Matching Tiles (Excel file for convenience is in filter mode). Please, select values for the fields:

·        “Method” (= “Buying Method”),

·        “Start” (= “Tiles-in-the-Wall”),

·        “MT” (= “Matching Tiles”),

·        “Prob” (= “Level-of-Certainty”)

to get expected number of single moves to get required tile. Let’s consider typical CHOICE problems and try to solve them using attached table.

Example of Choice Problem Solution: 3 Pairs for “All Pungs”

Let’s consider 3 Pairs dealt for the start. How easy is to convert that 2221111111 distribution for “All Pungs”? The answer depends on how sure we want to be to get it. First goes “Lucky Guy”, cumulative probability = 0.5, which means in equivalent to pick right tile out of two. (Click here for full example).

So What?

Analysis listed above is only a sample of what MAY happen at the table. Real chances highly depend on:

·        flexibility of player’s strategy,

·        portion of luck (“Luck Factor”),

·        how friendly or not opponents’ discards are (“Environment Factor”).

Transformation Chains Examples

Let’s track transformations of several distributions. For simplicity we assume start of the hand (83 tiles in the Wall) and couple different “Luck Factors”.

Knitted Hands

Seven Pairs

 Rule of Thumb

To estimate at-the-table when you get next tile toward your hand the following Simplified Method might be applied.

1.      Find "Tiles-in-the-Wall" number (for instance, 83 for the start of the game).

2.      Find "Matching Tiles" number.

3.      By dividing "Tiles-in-the-Wall" onto Matching Tiles" you get number of:

This simplified formula is valid ROUGHLY for the "Level-of-Certainty" = 70% which is slightly lower than "Sure" level and much higher than "Luck" level. To increases "Level-of-Certainty" simply multiply estimate by 1.1, 1.2 or 1.3.

Small Examples on “Rule of Thumb”

1.      9 Knitted for 10-th tile: MT=28, 83/28=3 TURNS (Method="Draw");

2.      22222111 for a Pair: MT=9, 83/9=9 TURNS;

3.      22222111 for Pung, MT=10, 83/10=8 MOVES ("Full") etc.

What’s Next?

It is necessary to estimate WHEN (meaning how much tiles “passed”) is good to finish a hand depending on the hand score. 8 pts. limit for going out increases slightly game (compared to “No Limit” rule). Playing level of opponents and their propensity to compete might give a hint for HOW LONG current hand would be played. It is crucial to “fit” within time gap when choosing among the alternatives. Distribution table of the scores (and Elements) vs. single moves passed would provide good basis for further analysis.

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