Quote:
Originally Posted by indianaV8
Well post your proposals. We've made available such db with about 10^9 hands. AFAIU spadebidder is going to run allin samples against it.
Here's an EXPERIMENT.
1. Take N hands where exactly two players went all-in pre-flop, and exactly one player won (so exclude ties).
2. For each hand calculate FLOP equity, conditioned on no tie, for both players (will show how to do conditioning if needed, but very easy)
3. For each hand, keep only the equity that is >= 0.50 (if 0.50 pick whatever)
4. For each hand, note whether the player with equity >= 0.50 was the ultimate winner or not.
5. Now, consider equity intervals
[0.505,0.515) , [0.515,0.525) , ... , [0.985,0.995)
6. For each equity interval:
a. count the number of hands that fall in the interval (denote the count by TRIALS).
b. count the number of instances where the player ultimately won (denote the count by SUCCESSES)
7. For each equity interval, let MID denote its midpoint.
8. For each equity interval, calculate the quantity
EXPECTED SUCCESSES := MID * TRIALS
9. For each equity interval, calculate the quantity
DIFFERENCE := SUCCESSES - EXPECTED SUCCESSES
10. For each equity interval, calculate the quantity
STANDARD DEVIATION := sqrt( TRIALS * MID * [1 - MID] )
11. For each equity interval, calculate the quantity
OBSERVED DEVIATION := DIFFERENCE / STANDARD DEVIATION
12. Construct the following table: TABLE 1 (some hypothetical numbers inserted to facilitate description)
MID | OBSERVED DEVIATION
----------------------------------
0.51 | -0.3
0.52 | 1.2
0.53 | 0.1
...... | .....
...... | .....
...... | .....
0.99 | 1.2
----------------------------------
13. Plot TABLE 1. Call the result PLOT 1.
Now, PLOT 1 will be interesting in and by itself. But i'd like to take this a step further. Specifically:
Repeat EXPERIMENT but with the five community cards permuted.
For definiteness, let's say, repeat EXPERIMENT but with community cards reading in reverse order. This results in a graph, call it PLOT 2.
Note PLOT 1 and PLOT 2 should look pretty much the same, provided the sample is large enough (i.e. N is large enough)
I can explain further if needed.
Here are note on/rationale behind EXPERIMENT:
1. EXPERIMENT a basic summary statistics of sorts; so I think it has intrinsic value regardless of intentions behind it.
2. PLOT 1 is supposed to show whether people win as often as expected.
3. In EXPERIMENT, I take a shortcut in that I approximate each equity interval by its midpoint MID. I expect noise due to this to be minimal (one can easily obtain upper bounds on how much noise such an approximation produces, but it should be small).
4. Due to the finiteness considerations, the spectrum of flop equities will be discrete. So, one has to be a little careful about how equity intervals are chosen. In general though, I don't expect this to be an obstacle.
5. Due to finiteness considerations (also if N is not large enough), it's possible some equity intervals may not contain enough sample points in them. Should such a situation arise, one may drop the interval entirely (and then make the obvious modifications to PLOT 1).
But in general, one should observe pretty good convergence to the underlying Bernoulli distribution with as little as 100 trials. So, if for ridiculous simplicity we assume hands are pretty uniformly distributed among equity bins, then something like N = 10^4 all-in hands would suffice.
6. Comparing PLOT 1 and PLOT 2 should help reveal whether there's some structure to the order of community cards. Of course, since we're dealing with all in situations, such a structure does not impact final results, but, should it exist, it rises to proof-level that the deal is manipulated.
7. In step 1 of EXPERIMENT, I asked for hands to be all-in preflop. However, since EXPERIMENT only makes use of FLOP equities, hands going all-in on flop could also be included. But for the sake of uniformity/homogeneity in dataset, and also to allow for the production of PLOT 2, I'd like to exclude such hands.
/of notes.
One comment, the experiment is actually much easier to carry out than what may be suggested by its lengthy description/notes. One can do many variations on it. For example, one may restrict to all in situations where pot size is at least blah blah big blinds. One can look at different permutations of community cards, other than that of reverse order....etc.