Satoshi Dice -- Statistical Analysis

[etotheipi]

Yes, but I use the Armory code to load the blockchain to produce the reports.  It loads the blockchain that the bitcoin-qt client writes to disk.

Dooglus, I should apologize that you got stuck with the original analysis script I wrote.  It was intended half as "sample armoryengine.py code" and half "Let's see how much information we can deduce about SD."  As such, I implemented the search on the python side which was much simpler but painfully slow.  If I'd done it "right", it could've been on the C++ side, where it's literally about 100x faster.  Oh well.  If it's a problem, I might be able to outsource some of the search code to C++ for you... this thread has become important enough...

Even though I started the thread, I have been watching with great interest, and appreciate that you took over when I had no more time.  I just sent you a donation for the value you have provided me (regardless of whether you think it's weird that I'd be donating to you for this ).

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[dooglus]

Dooglus, I should apologize that you got stuck with the original analysis script I wrote.  It was intended half as "sample armoryengine.py code" and half "Let's see how much information we can deduce about SD."  As such, I implemented the search on the python side which was much simpler but painfully slow.  If I'd done it "right", it could've been on the C++ side, where it's literally about 100x faster.  Oh well.  If it's a problem, I might be able to outsource some of the search code to C++ for you... this thread has become important enough...

Even though I started the thread, I have been watching with great interest, and appreciate that you took over when I had no more time.  I just sent you a donation for the value you have provided me (regardless of whether you think it's weird that I'd be donating to you for this ).

Thanks a lot for the donation.

I modified the script so that it saves its state between runs, so it only needs to search the new blocks each time I run it.

The vast majority of the time it takes it loading the blockchain files, which is all done in C++ anyway.

I guess if there was some way of just loading the newest bits of the blockchain that would speed it up a lot, but it's really not an issue at the moment to load the blockchain into your script once a day.

All I do at the moment to load the blockchain is:

   TheBDM.setBlocking(True)
   TheBDM.setOnlineMode(True)

[Syke]

Is it possible that an SD conspirator is running a bot to inflate the numbers? Would such a graph reveal it if it were true?

It would be like a casino handing a stack of chips to an employee to go play at the casino so that the casino doesn't appear empty. Whether they win or lose, the employee hands whatever chips he has left back to the casino. The only cost to inflate the activity graphs would be the transaction fee, a minimal .0005 BTC at this time.

There is no way to tell if there is an insider placing SD bets.

[etotheipi]

Thanks a lot for the donation.

I modified the script so that it saves its state between runs, so it only needs to search the new blocks each time I run it.

The vast majority of the time it takes it loading the blockchain files, which is all done in C++ anyway.

I guess if there was some way of just loading the newest bits of the blockchain that would speed it up a lot, but it's really not an issue at the moment to load the blockchain into your script once a day.

All I do at the moment to load the blockchain is:

   TheBDM.setBlocking(True)
   TheBDM.setOnlineMode(True)

Yeah, and there's no reason it can't collect the SD statistics as it does that first blockchain load.  Then it'd be done before you run any code after "setOnlineMode".  Unfortunately, it's not as simple as just adding the addresses to the initial scan wallet, because you need to ignore non-bet transactions.  It would require altering the C++ scan loops, but I bet it wouldn't be too terrible...

But other priorities call.  If it is running okay, then why fix it if it ain't broken!

[StarenseN]

Is it possible that an SD conspirator is running a bot to inflate the numbers? Would such a graph reveal it if it were true?

It would be like a casino handing a stack of chips to an employee to go play at the casino so that the casino doesn't appear empty. Whether they win or lose, the employee hands whatever chips he has left back to the casino. The only cost to inflate the activity graphs would be the transaction fee, a minimal .0005 BTC at this time.

There is no way to tell if there is an insider placing SD bets.

+ 13% of 1.9% (0.25%) of each bets, theorically, that goes to the shareholders.

[dooglus]

+ 13% of 1.9% (0.25%) of each bets, theorically, that goes to the shareholders.

- (3 million shares) * (temporary price increase due to 'whale' business)...

This guy uses address "1Lucky..." and lived up to his name:

Mar  9 05:57:31 2013 <32000 85bae6cd4926ebd8:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET   128.00000000 LOSE  -127.36000000
Mar  9 06:25:18 2013 <32000 300695f8664aa659:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET   256.00000000 WIN    256.98508800

Mar  9 06:46:52 2013 <32000 334e52c5be795dbd:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET    12.80000000 LOSE   -12.73600000
Mar  9 07:37:30 2013 <32000 373c5a4ea42e2a01:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET    25.60000000 LOSE   -25.47200000
Mar  9 08:02:35 2013 <32000 fb7d684cf4a9ce7a:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET    51.20000000 LOSE   -50.94400000
Mar  9 08:48:26 2013 <32000 812c0388ea7dd045:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET   102.40000000 LOSE  -101.88800000
Mar  9 09:57:08 2013 <32000 31ff8650d4c9489b:0  1LuckyWwuK7CMF9PveFT4ZXbBcDz6xripS BET   250.00000000 WIN    250.96200000

Total of 86 bets unaccounted for.

Results: 2013-Mar-09 10:58am (up to block 225040)

   Address  Target   Should Win |    #Bets |        Win        |   Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP 
----------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    97152 |       1 (0.00001) |   95634 |    1517 |    1119.83 |    1280.08 |   120.41 |   -160.24 | 114.310
 1dice1Qf4       2      0.00003 |     5121 |       0 (0.00000) |    4603 |     518 |      85.83 |       0.01 |    18.66 |     85.81 |   0.022
 1dice2pxm       4      0.00006 |     7574 |       1 (0.00014) |    7122 |     451 |     114.30 |     159.99 |    11.90 |    -45.68 | 139.970
 1dice2vQo       8      0.00012 |    11653 |       5 (0.00045) |   11192 |     456 |     213.64 |     431.92 |    10.07 |   -218.28 | 202.173
 1dice2WmR      16      0.00024 |    13291 |       1 (0.00008) |   12855 |     435 |     445.51 |       4.80 |    22.06 |    440.71 |   1.079
 1dice2xkj      32      0.00049 |    16523 |       6 (0.00037) |   16118 |     399 |     912.64 |     813.76 |     1.38 |     98.87 |  89.166
 1dice2zdo      64      0.00098 |    18233 |      19 (0.00107) |   17753 |     461 |    1335.44 |     451.88 |    55.81 |    883.55 |  33.838
 1dice37Ee     128      0.00195 |    19313 |      36 (0.00191) |   18823 |     454 |    2320.36 |    1547.60 |    48.37 |    772.75 |  66.697
 1dice3jkp     256      0.00391 |    24086 |     105 (0.00443) |   23597 |     384 |    4568.04 |    7357.22 |    13.20 |  -2789.17 | 161.058
 1dice4J1m     512      0.00781 |    33043 |     266 (0.00822) |   32080 |     697 |    5672.91 |    5648.39 |    10.01 |     24.51 |  99.568
 1dice5wwE    1000      0.01526 |   115117 |    1753 (0.01529) |  112860 |     504 |   35590.18 |   30846.98 |     2.04 |   4743.20 |  86.673
 1dice61SN    1500      0.02289 |    24365 |     562 (0.02341) |   23448 |     355 |    7631.70 |    8099.77 |    15.08 |   -468.06 | 106.133
 1dice6DPt    2000      0.03052 |    72279 |    2237 (0.03112) |   69646 |     396 |   36543.12 |   33222.67 |     9.36 |   3320.44 |  90.914
 1dice6gJg    3000      0.04578 |    28518 |    1291 (0.04596) |   26796 |     431 |    9179.81 |    9984.92 |    25.17 |   -805.11 | 108.771
 1dice6GV5    4000      0.06104 |    34345 |    2093 (0.06164) |   31861 |     391 |    7598.04 |    7417.35 |    31.35 |    180.68 |  97.622
 1dice6wBx    6000      0.09155 |    45842 |    4196 (0.09241) |   41208 |     438 |   15866.18 |   17651.54 |     7.26 |  -1785.36 | 111.253
 1dice6YgE    8000      0.12207 |   178249 |   21829 (0.12281) |  155921 |     499 |   90199.82 |   88782.54 |   100.44 |   1417.27 |  98.429
 1dice7EYz   12000      0.18311 |    90638 |   16512 (0.18324) |   73600 |     526 |  169197.82 |  170921.55 |  3314.87 |  -1723.72 | 101.019
 1dice7fUk   16000      0.24414 |   237109 |   57692 (0.24388) |  178865 |     552 |  364087.04 |  350390.40 |  2322.40 |  13696.63 |  96.238
 1dice7W2A   24000      0.36621 |   237963 |   87363 (0.36802) |  150022 |     578 |  581768.79 |  573520.67 |  1013.21 |   8248.12 |  98.582
 1dice8EMZ   32000      0.48828 |  1099758 |  535923 (0.48802) |  562229 |    1606 |  860965.50 |  843529.06 |  2924.81 |  17436.44 |  97.975
 1dice97EC   32768      0.50000 |   583613 |  291108 (0.49973) |  291418 |    1087 |  655894.15 |  640298.75 |  6521.08 |  15595.40 |  97.622
 1dice9wcM   48000      0.73242 |   328447 |  240980 (0.73509) |   86843 |     624 |  296658.23 |  289324.90 |  5455.24 |   7333.32 |  97.528
 1dicec9k7   52000      0.79346 |    72889 |   57448 (0.79373) |   14929 |     512 |   60651.44 |   59310.90 |  1187.54 |   1340.53 |  97.790
 1dicegEAr   56000      0.85449 |    60565 |   51370 (0.85611) |    8634 |     561 |   76590.30 |   75963.79 |   400.50 |    626.51 |  99.182
 1diceDCd2   60000      0.91553 |    92530 |   84292 (0.91632) |    7698 |     540 |   69985.13 |   69198.68 |     0.48 |    786.45 |  98.876
 1dice9wVt   64000      0.97656 |    16214 |   14842 (0.97980) |     306 |    1066 |   23943.27 |   23532.23 |   240.15 |    411.03 |  98.283
----------------------------------------------------------------------------------------------------------------------------------------------
           small (bets < 4 BTC) |  3451151 | 1419961           | 2015044 |   16146 |  755615.95 |  741416.20 |   254.28 |  14199.75 |  98.121
            big (bets >= 4 BTC) |   113279 |   51970           |   61017 |     292 | 2623523.18 | 2568276.27 | 23628.70 |  55246.91 |  97.894
----------------------------------------------------------------------------------------------------------------------------------------------
                                |  3564430 | 1471931           | 2076061 |   16438 | 3379139.13 | 3309692.47 | 23882.98 |  69446.66 |  97.945
----------------------------------------------------------------------------------------------------------------------------------------------

SD Profit before fees:      69446.66155124 BTC (2.055%)
Cumulative Fees Paid:        2765.15607500 BTC
SD Profit after fees:       66681.50547624 BTC (1.973%)
Pending Liabilities:           -6.70932707 BTC
Final SD Profit:            66688.21480331 BTC (1.974%)
----
Since Satoshi Dice started, there have been:
Blockchain Tx: 11233556  :  SatoshiDice Tx:  6566436  (58.5%)
Blockchain MB:   4818.7  :  SatoshiDice MB:   2696.8  (56.0%)

[misterbigg]

+ 13% of 1.9% (0.25%) of each bets, theorically, that goes to the shareholders.

If the cost of running a collusion bot (increased transaction fees plus payouts to shareholders) is less than the expected gain in value from share price appreciation, then running the bot would be profitable.

How do we interpret this graph? It looks like SatoshiDICE is consistently beating expectations. Over time shouldn't the actual revenue be very close to the red line (house edge)? Why is it so much higher? And why is it that in two particular months there was a large jump in the activity (i.e. change in slope)? Is this an indicator of botting?

[dooglus]

How do we interpret this graph? It looks like SatoshiDICE is consistently beating expectations. Over time shouldn't the actual revenue be very close to the red line (house edge)? Why is it so much higher? And why is it that in two particular months there was a large jump in the activity (i.e. change in slope)? Is this an indicator of botting?

The red and black lines are pretty close together (61k vs 66k).  Notice that zero is a few miles off the bottom of the graph, and the graph is auto-scaled to make the best use of space.  House profits are about 5k higher than they should be, or less than 10% higher than they should be.  A few big losses will correct for that.  If you look at the graph for January you'll see the red and black lines cross each other at least 17 times, meaning the house was doing exactly as expected.  In general the slope of the two lines should roughly match each other.  The law of large numbers says that the two lines will get arbitrarily close to each other in the long run, but we're not at the "long run" yet, when a single bet can change the total house profits by more than 1%.

The whale lost big in the beginning of February, which put the black line well above the red line, where it has stayed since.

December and January saw a lot of betting activity from one user, which made the profit graph jump up and down erratically and made the red line go steeper than usual.  It was almost certainly a bot, just because I doubt if a person would have the stamina to keep up such a level of play for so long.  Whether it was a "house bot" or a real player it's impossible to prove.  Erik says it's a real player and I believe him.

[wisard]

I thought the house edge was based on the number of bets. And not on the amount of betting.

[BitcoinHoarder]

All bets * (1-.981).  That is roughly how much the house is expected to rake.

Think of SatoshiDice as a laundrer, it returns you 98% (roughly) of your money over time.  Now some people play wierdly trying to "beat the numbers" and they might actually lose 100% of their money.  That sweetens Satoshi's profits.  But as Dooglus has shown, pretty much it's just laundering and taking a 1.9% fee.

[dooglus]

I thought the house edge was based on the number of bets. And not on the amount of betting.

The house keeps 1.9% of all bets, on average.

If a million people bet 1 BTC each, or 100k people bet 10 BTC each, the total bet volume is 1 million BTC either way, and the house will keep 19k BTC on average in both cases.

Does that make sense?

[wisard]

I thought the house edge was based on the number of bets. And not on the amount of betting.

The house keeps 1.9% of all bets, on average.

If a million people bet 1 BTC each, or 100k people bet 10 BTC each, the total bet volume is 1 million BTC either way, and the house will keep 19k BTC on average in both cases.

Does that make sense?

Thanks.

But this assumes that the average bet will remain in the same range.

My contention is that the average bet will keep on reducing as the BTC / USD rate keeps on increasing. The amount of total BTC bet on the first 100,000 bets will be higher than the amount of BTC made on the next 100,000. (I have not looked at the numbers and I don't know if I'm right or wrong, it just feels that way to me intuitively.)

If I'm right, then the gap between the black line and the red line won't go away as more time goes by.

[dooglus]

If a million people bet 1 BTC each, or 100k people bet 10 BTC each, the total bet volume is 1 million BTC either way, and the house will keep 19k BTC on average in both cases.

But this assumes that the average bet will remain in the same range.

My contention is that the average bet will keep on reducing as the BTC / USD rate keeps on increasing. The amount of total BTC bet on the first 100,000 bets will be higher than the amount of BTC made on the next 100,000. (I have not looked at the numbers and I don't know if I'm right or wrong, it just feels that way to me intuitively.)

I don't think anything I said assumes anything about the average bet size.

I don't have data readily available about the average bet size, but I do have the total amount bet over time.  Here's a graph showing the total bet volume over time since September last year when the price was down around $10/BTC.  Notice how the slope of the line (indicating bet volume per unit time) is about the same now as it was then.  There are a few very steep parts from when the whale was playing, but the overall volume doesn't seem to be affected by the dollar value of Bitcoins, odd as that seems.  Maybe the price goes up because more people are using it, so more people are betting, which offsets the small size per bet, but that's just a guess.

[dooglus]

I don't have data readily available about the average bet size

... but I do now.

This graph shows the moving average bet size (in BTC) over the previous 7 days:



I also tried using a 1-day window, but it's very spiky:



and here's the average over the previous 28 days:

[organofcorti]

I don't have data readily available about the average bet size

... but I do now.

This is where boxplots or violin plots might be more useful. If you're not sure what I mean (and tbh I haven't exactly explained myself well) if you post the data (or pm me) I'll post some examples.

[dooglus]

This is where boxplots or violin plots might be more useful. If you're not sure what I mean (and tbh I haven't exactly explained myself well) if you post the data (or pm me) I'll post some examples.

http://dooglus.com/sd-data.txt.gz has 2 numbers per line.  Time (seconds since 1970) and average bet size in the block with that timestamp:

1362850998 0.0800365152562
1362851859 0.44658473297
1362852003 5.26315630659e-05

[hashman]

All bets * (1-.981).  That is roughly how much the house is expected to rake.

Think of SatoshiDice as a laundrer, it returns you 98% (roughly) of your money over time.  Now some people play wierdly trying to "beat the numbers" and they might actually lose 100% of their money.  That sweetens Satoshi's profits.  But as Dooglus has shown, pretty much it's just laundering and taking a 1.9% fee.

Thanks for the stats folks
It also might be interesting to see what fraction of users are selecting to have winnings routed to another address. 
If the money goes back to the same address, it's hardly a very useful launderer.

[dooglus]

It also might be interesting to see what fraction of users are selecting to have winnings routed to another address. 
If the money goes back to the same address, it's hardly a very useful launderer.

Well, if I send my money to satoshidice with a public note saying "send my winnings back to 1MyOtherSecretAddress so they don't know it's me" then that's not a very useful laundry either.

I've seen this "SDICE launders money" argument before, but I think it's bogus.  Everything they do is public, so is absolutely not laundering money.

[organofcorti]

This is where boxplots or violin plots might be more useful. If you're not sure what I mean (and tbh I haven't exactly explained myself well) if you post the data (or pm me) I'll post some examples.

http://dooglus.com/sd-data.txt.gz has 2 numbers per line.  Time (seconds since 1970) and average bet size in the block with that timestamp:

1362850998 0.0800365152562
1362851859 0.44658473297
1362852003 5.26315630659e-05

Is it possible paste a list of all bets, rather than the average size per time period (I'm guessing it's per block)?

[dooglus]

Is it possible paste a list of all bets, rather than the average size per time period (I'm guessing it's per block)?

I don't have a list of all bets but can generate it.  I'm a little busy at the moment getting ready to give back $82k that I've become kind of attached to tomorrow.  But I'll prepare the bet list when I get a chance.

Total of 24 bets unaccounted for.

Results: 2013-Mar-10 01:58pm (up to block 225242)

   Address  Target   Should Win |    #Bets |        Win        |   Lose  | Refunds |   BTC In   |  BTC Out   |  Refund  |   Profit  |   RTP  
----------------------------------------------------------------------------------------------------------------------------------------------
 1dice1e6p       1      0.00002 |    97797 |       1 (0.00001) |   96258 |    1538 |    1128.50 |    1280.12 |   122.46 |   -151.61 | 113.435
 1dice1Qf4       2      0.00003 |     5152 |       0 (0.00000) |    4632 |     520 |      87.00 |       0.02 |    19.91 |     86.98 |   0.028
 1dice2pxm       4      0.00006 |     7607 |       1 (0.00014) |    7153 |     453 |     114.74 |     159.99 |    12.83 |    -45.24 | 139.433
 1dice2vQo       8      0.00012 |    11922 |       5 (0.00044) |   11460 |     457 |     223.84 |     431.97 |    10.07 |   -208.13 | 192.980
 1dice2WmR      16      0.00024 |    13552 |       1 (0.00008) |   13113 |     438 |     454.34 |       4.85 |    22.06 |    449.49 |   1.068
 1dice2xkj      32      0.00049 |    17221 |       6 (0.00036) |   16815 |     400 |     940.35 |     813.90 |     1.38 |    126.44 |  86.553
 1dice2zdo      64      0.00098 |    18504 |      19 (0.00105) |   18022 |     463 |    1351.72 |     451.96 |    55.81 |    899.76 |  33.436
 1dice37Ee     128      0.00195 |    19441 |      36 (0.00190) |   18949 |     456 |    2329.67 |    1547.65 |    48.38 |    782.02 |  66.432
 1dice3jkp     256      0.00391 |    24171 |     105 (0.00441) |   23682 |     384 |    4574.58 |    7357.25 |    13.20 |  -2782.67 | 160.829
 1dice4J1m     512      0.00781 |    33135 |     266 (0.00820) |   32172 |     697 |    5681.90 |    5648.44 |    10.01 |     33.46 |  99.411
 1dice5wwE    1000      0.01526 |   116106 |    1765 (0.01527) |  113829 |     512 |   35674.60 |   30946.84 |     2.05 |   4727.76 |  86.748
 1dice61SN    1500      0.02289 |    24440 |     564 (0.02342) |   23520 |     356 |    7642.67 |    8116.88 |    15.09 |   -474.20 | 106.205
 1dice6DPt    2000      0.03052 |    73050 |    2269 (0.03123) |   70385 |     396 |   36599.03 |   33294.69 |     9.36 |   3304.34 |  90.972
 1dice6gJg    3000      0.04578 |    28583 |    1295 (0.04601) |   26853 |     435 |    9185.74 |    9992.64 |    25.17 |   -806.90 | 108.784
 1dice6GV5    4000      0.06104 |    34384 |    2095 (0.06163) |   31898 |     391 |    7601.72 |    7417.69 |    31.35 |    184.03 |  97.579
 1dice6wBx    6000      0.09155 |    46102 |    4217 (0.09235) |   41447 |     438 |   15893.24 |   17665.29 |     7.26 |  -1772.04 | 111.150
 1dice6YgE    8000      0.12207 |   179501 |   21980 (0.12280) |  157017 |     504 |   90316.60 |   88902.10 |   100.45 |   1414.49 |  98.434
 1dice7EYz   12000      0.18311 |    91901 |   16749 (0.18331) |   74622 |     530 |  169299.83 |  171039.94 |  3314.87 |  -1740.10 | 101.028
 1dice7fUk   16000      0.24414 |   238889 |   58142 (0.24395) |  180193 |     554 |  364448.62 |  350823.74 |  2322.40 |  13624.88 |  96.262
 1dice7W2A   24000      0.36621 |   240217 |   88169 (0.36793) |  151469 |     579 |  582144.14 |  573880.85 |  1013.21 |   8263.28 |  98.581
 1dice8EMZ   32000      0.48828 |  1104941 |  538454 (0.48803) |  564867 |    1620 |  865098.75 |  847579.77 |  2924.87 |  17518.97 |  97.975
 1dice97EC   32768      0.50000 |   590929 |  294737 (0.49970) |  295093 |    1099 |  656935.31 |  641231.25 |  6521.11 |  15704.05 |  97.609
 1dice9wcM   48000      0.73242 |   332680 |  244096 (0.73511) |   87959 |     625 |  297266.69 |  289904.63 |  5455.25 |   7362.06 |  97.523
 1dicec9k7   52000      0.79346 |    73546 |   57965 (0.79370) |   15066 |     515 |   60823.13 |   59474.12 |  1187.54 |   1349.00 |  97.782
 1dicegEAr   56000      0.85449 |    61170 |   51901 (0.85640) |    8703 |     566 |   76787.21 |   76164.40 |   400.52 |    622.81 |  99.189
 1diceDCd2   60000      0.91553 |    94091 |   85718 (0.91627) |    7833 |     540 |   70164.87 |   69369.23 |     0.48 |    795.64 |  98.866
 1dice9wVt   64000      0.97656 |    16365 |   14974 (0.97972) |     310 |    1081 |   23964.48 |   23553.02 |   240.17 |    411.45 |  98.283
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           small (bets < 4 BTC) |  3481872 | 1433439           | 2032178 |   16255 |  760085.53 |  745835.80 |   258.71 |  14249.73 |  98.125
            big (bets >= 4 BTC) |   113525 |   52091           |   61142 |     292 | 2626647.87 | 2571217.53 | 23628.70 |  55430.33 |  97.890
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                                |  3595397 | 1485530           | 2093320 |   16547 | 3386733.40 | 3317053.33 | 23887.41 |  69680.07 |  97.943
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SD Profit before fees:      69680.07288378 BTC (2.057%)
Cumulative Fees Paid:        2796.14317500 BTC
SD Profit after fees:       66883.92970878 BTC (1.975%)
Pending Liabilities:           -5.93485875 BTC
Final SD Profit:            66889.86456753 BTC (1.975%)
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Since Satoshi Dice started, there have been:
Blockchain Tx: 11312047  :  SatoshiDice Tx:  6621333  (58.5%)
Blockchain MB:   4853.5  :  SatoshiDice MB:   2720.6  (56.1%)