Backgammon Match Equities

Match Equities

ME at 1-away/2-away (crawford)

From:   Ian Shaw
Address:   ian.shaw@iee.org
Date:   30 April 2003
Subject:   Crawford 2-away is only worth 67.7%
Forum:   GammOnLine

I've completed 66,240 2-ply untruncated rollouts with GnuBg, half with Leader winning the opening roll, and half with Trailer winning it. I get a figure of 67.725% for Leader's Match Equity. The 95% Confidence Interval is 0.047%, so we should be able to confidently say that 67.7% is accurate to 1 decimal place. Other METs quote: 71.25% (Zadeh) 70.0% (Woolsey&Heinrich, Ortega & Kleinman) 69.0% (Jacobs & Trice) 68.5% (Snowie 2) My rollout value is lower than any of these, and lower than GnuBg's own evaluations at 0, 1 and 2 ply, and lower than a 0-ply rollout (all about 68.4%). One feature I find strange about the rollouts is that Leader wins more gammons than Trailer, no matter who gets the first move. This does not show up in the evaluations, where the player with the first roll always gets more gammons. The rollouts also show both sides getting more gammons than the evaluations. I expected Trailer to get more gammons, since he has the incentive to play for them. Maybe it is easier for Leader to avoid gammons than for Trailer to achieve them. I think I've adequately demonstrated on this forum that I'm no expert, so I'd welcome other comments and opinions. Here are the results in full. The overall MEQ for Leader is calculated as Wins + ((Losses - Gammon Losses) / 2). Rollouts 0-ply trials/side 7776 1st Roll w wg wbg l lg lbg cubeless cubeful Trailer 0.5232 0.1481 0.0126 0.4768 0.1887 0.0202 33.784% 33.784% std err 0.290% 0.290% Leader 0.5308 0.2192 0.0250 0.4692 0.1254 0.0099 69.939% 69.939% std err 0.292% 0.292% Leader's MEQ Combined for Leader 0.5038 0.2040 0.0226 0.4962 0.1368 0.0113 68.353% Combined Std Err 0.206% 95%CI 0.403% 2-ply trials/side 33120 1st Roll w wg wbg l lg lbg cubeless cubeful Trailer 0.5320 0.1527 0.0184 0.4680 0.1873 0.0259 34.236% 34.236% std err 0.034% 0.034% Leader 0.5208 0.2160 0.0323 0.4792 0.1271 0.0138 68.689% 68.689% std err 0.034% 0.034% Leader's MEQ Combined for Leader 0.4944 0.2017 0.0291 0.5056 0.1399 0.0161 67.725% Combined Std Err 0.024% 95%CI 0.047%

Ian Shaw writes:

There was a bug. Gnubg variance reduction was not taking into account the initial position setting. I guess the effect of this is to overcompensate for the "bad" rolls of never getting a double on the first roll. I'm not sure of the implication for my rollouts. Is the whole rollout invalidated or just the reported standard error? I suspect the whole thing is duff (groan). I'll redo my 0-ply ones initially and see what the effect is. Excerpt from gnubg change log: Fri Jun 27 20:42:14 2003 Joern Thyssen * rollout.c (BasicCubefulRollout): fix bug in variance reduction when rolling out as initial position. (reported on GammOnline).

Neil Kazaross writes:

Since this plays very fast it was easy to do 259200 rollouts with each side on roll. With the leader on roll he wins 51.39% of the games and 13.07% G's. With the trailer on roll the trailer wins 48.61% games and 14.21% G's Calculating that out results in the trailer winning 49.895% of all games with 13.64% gammons and thus his ME is 31.7675%. I'd estimate that the std dev of that ME is less than .1% and am running a set of 648,000 overnight and will do 648,000 with the other side on roll tomorrow. Note that my rollouts have the side on roll winning substantially less often than Ian's rollouts did. My 0-ply rollouts are also consistant with my 0-ply rollouts using VR. I cannot believe that using 2-ply play makes a big difference and believe that we should take another look at the long 2 ply rollout. The best way to do this is as a team with each member doing a few of the 15 possible opening rolls for each side using 2-ply world class w/VR. ..neilkaz..

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