Rick,Thanks for your in-depth analysis of my play.Right now for me,the big drop in variance is worth the .22 drop in EV.But I'm going to be on the hunt for good Mult-D machines.
If I can't find M.D. 9/6 in AC,I wonder if M.D.8/5 Bonus would be a good alternative(which I have never played).Probably,my only other possible choice.
Marc
-----Original Message-----
From: Rick E. Percy <ricke.percy@yahoo.com>
To: vpFREE <vpFREE@yahoogroups.com>
Sent: Wed, Nov 9, 2011 5:49 pm
Subject: Re: [vpFREE] Re: Always Play 5 Coins
If you observe Marc's play in retrospect only, you will find (using a negative game like 9/6 Jacks or Better as an example) that he lost 4 x (n - 1) coins fewer by playing his strategy on each streak of n losses. You will also find that he gained 80% fewer coins on the first win only, except once per 51,481 hands of 1-coin play (or once per 94,361 hands overall) when he had that 1-coin royal and won considerably less than he would have with 5-coin play.
With the switching strategy, his EV declines from the target 99.54% to 99.32%, presuming that he plays both 5-coin and 1-coin strategies perfectly, but his variance drops from 487.87 coins squared (25 x 19.51468 per single coin that is given in WinPoker) to 224.39 coins squared. The portion of the variances that is downside variance (attributed to hands that lose coins) drops from 13.51 coins squared to 6.44 coins squared.
His net expected losses are reduced from 0.0228 coins per hand with normal 5-coin play to 0.0192 coins per hand with his mixed play. Of course, the difference in loss and the fact that it is lower is due to the fact that he is betting on average 2.8177 coins per play rather than 5. And I presume with the continual switching bet sizes that he plays slower so his average losses per hour with the switching strategy are probably lower still.
Of course, there is nothing magic about switching when he does. He would experience these same numbers if he played the first 54.56% of his session at 1 coin with the remaining 45.44% at 5 coins except that he might play quicker and possibly make fewer hold mistakes because of the difference in the two playing strategies. Any similar trigger would produce similar results (like play 1 coin when the minute hand of the clock is from 0 to 32 and play 5 coins when it is from 33 to 59, if he played complete hours for each session).
So, it is more than luck that he would be experiencing lower loss levels in his losing sessions with this type of play. His losses in losing sessions are EXPECTED to be lower The above analysis is based on averages so sessions with longer losing streaks would likely have still lower loss results in comparison to 5-coin play. Certainly he would expect a lower expected net result in gain/loss over the lifetime of his play, but his ride should be more stable.
I would guess that there might be differences in probability of winning during a 4-hour session, but I have not analyzed that enough to comment meaningfully.
If he wants to continue with this betting strategy, it would be better if he could move up and down in denomination, always betting 5 coins so he doesn't miss out on the full-coin royals, but depending upon where he is, he may not be able to find the same pay schedule at the lower denomination.
It isn't always about EV. Some may be willing to trade a little expected value to get less variance; that's usually a bit more subjective of a decision. In this case, the implied trade-off is 0.22% less EV for a big drop in variance. People don't put all their money in the stock market even if it has greater expected value than T-bills, cds, savings, or checking accounts.
Certainly Marc could get an
even bigger drop in variance by playing even a higher percentage of hands at 1-coin,
or a lower dropoff in EV by using some other trigger, so it is not clear that he has consciously made an optimal informed EV-Variance tradeoff; but his decision may also not have been as horrific as has been intimated.
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