Study 8J49 Main
Dealer Advantage
by Richard PavlicekStudy 8J49 Main |
| by Richard Pavlicek |
Being the first to act is an advantage in many competitions — from tic-tac-toe, to the white pieces in chess, to dueling with pistols. But what about bridge? Does the dealing side have an advantage? Common sense dictates yes, but just how much of an advantage? Until recently this was subject to wide debate, but I now have a definitive answer and can prove it.
As the simple formula above shows, if X = dealing side HCP differential, and S = saturation limit of finite freakness, the dealer’s advantage can be calculated to extreme precision. The relevance of pi has been heralded an amazing discovery, and if you’re wondering how imaginary numbers enter the equation, I refer you to my son’s preempts (imagine those numbers). So far the formula is a special theory, relative to my own partnerships, but I’m working on a general theory to be published soon. Enough said; the theoretical issue is simple. Now let’s look at some empirical evidence from actual data.
| Opening Bids by Position | Raw Score Analysis | Par Score Analysis | IMPs to Par Analysis | Specific Hand Types |
All data is from vugraph archives of major events (Vanderbilt, Spingold, U.S. and World Championships) from 1996 to present. This comprises 72 events and 80,138 results, but those without recorded auctions are excluded, leaving 77,411 results to analyze. Note that two results are obtained from the same deal (one at each table) and occasionally four (rarely eight) if a deal is duplicated between matches.
Opening Bids by PositionBefore examining dealer advantage specifically, let’s look at the frequency of opening bids by position. The following table breaks this down by vulnerability condition, and overall combined.
Especially note the percent of first-seat openings when only the dealing side is nonvulnerable, compared to vice versa — quite a difference. Obviously this is because favorable vulnerability spawns more preempts, as well as light one-bids and psychs.
| Description | None Vul | Both Vul | Dealer NV | Dealer Vul | Overall |
|---|---|---|---|---|---|
| Number of results | 19167 | 19530 | 19787 | 18927 | 77411 |
| 1st seat opening percent | 50.08 | 47.09 | 53.72 | 44.78 | 48.96 |
| 2nd seat opening percent | 28.51 | 29.24 | 24.17 | 32.76 | 28.62 |
| 3rd seat opening percent | 17.47 | 17.74 | 18.61 | 16.38 | 17.56 |
| 4th seat opening percent | 3.82 | 5.62 | 3.32 | 5.67 | 4.60 |
| Passed out percent | 0.11 | 0.31 | 0.18 | 0.41 | 0.25 |
The number of boards with each vulnerability condition should be equal (or approximately so) in fair contests. The large disparity between Dealer NV and Dealer Vul (860 results) is caused by inept organization of the U.S. Bridge Championship, repeatedly playing Boards 1-60 (omitting 61-64) which loses two dealer-vul cases each time. Obviously, the contributing culprit is the lunacy of 15-board segments. Will they ever learn?
| Study 8J49 Main |  | Top Dealer Advantage |
Raw Score AnalysisThe following table analyzes raw score totals for the dealing side, broken down by vulnerability condition, and overall combined. The top row shows the number of results in each group, and the next two rows show potential bias (average difference per result) for the group in HCP and freakness; plus indicates a bias favoring the dealing side; minus indicates a bias favoring the non-dealing side.
The essence of this study is summarized by the average score (blue tint) showing the dealer advantage (gold tint) or disadvantage (red tint). I was surprised by this, as I expected the overall dealer advantage to be higher. The paltry 1.55 points per result is almost negligible, considering that 50 points is the minimum possible score (aside from passouts). Nonetheless, the net score of plus 120,010 points is hard to refute as an advantage to the dealing side.
More surprising, if not shocking, are the stats with both sides vulnerable. How can the dealing side be at a disadvantage? This caused me to question my programming, but repeated checks (also by other software) verified the numbers. To some extent this may be due to the dealing side’s HCP deficit for the group, but this would hardly account for the disadvantage. Could it be because the dealing side has the first chance to get in trouble? Food for thought. Note that with none vulnerable, the stats are predictable, favoring the dealing side; and at unequal they are biased by the vulnerable side earning higher scores, so no surprises there.
For interest sake I included the number of results that were plus, minus or tied, and the effective won-lost percent. These stats, however, have no bearing on the study; all competitions were scored in IMPs, so no one was concerned about board-a-match outcome. Curiously, the non-dealing side was consistently favored (except with none vulnerable) which probably only emphasizes the strategic differences between the two forms of scoring.
| Dealer Side Stats | None Vul | Both Vul | Dealer NV | Dealer Vul | Overall |
|---|---|---|---|---|---|
| Number of results | 19167 | 19530 | 19787 | 18927 | 77411 |
| HCP bias | +0.0039 | -0.0191 | +0.0069 | +0.0007 | -0.0019 |
| Freakness bias | +0.0189 | +0.0066 | +0.0038 | -0.0028 | +0.0066 |
| Plus score total | 2800030 | 4022120 | 3085650 | 3611810 | 13519610 |
| Minus score total | 2658260 | 4188580 | 3610030 | 2942730 | 13399600 |
| Net score | +141770 | -166460 | -524380 | +669080 | +120010 |
| Average score | +7.3966 | -8.5233 | -26.5012 | +35.3506 | +1.5503 |
| Standard deviation | 379 | 558 | 458 | 469 | 471 |
| Plus results | 9657 | 9638 | 9796 | 9415 | 38506 |
| Minus results | 9488 | 9831 | 9955 | 9435 | 38709 |
| Tied results (passouts) | 22 | 61 | 36 | 77 | 196 |
| Won-lost percent | 50.44 | 49.51 | 49.60 | 49.95 | 49.87 |
Note that a comparison of actual IMPs won or lost would be meaningless in this study, because whatever the dealing side gained at one table is lost at the other; hence the net score is always zero. Raw scores, however, provide a valid indication of which side has an advantage.
| Study 8J49 Main | | Top Dealer Advantage |
Par Score AnalysisPerhaps a better method to explore dealer advantage is not by the actual raw score but by how it compares to the par score for that deal. For example, if the dealing side bids 4 
(nonvul) and 6 is cold, it doesn’t win 480 points but loses 500 points to par. The following table summarizes these stats in a similar manner. The first three rows are perforce identical but repeated for convenience.
Again note the average score. Tinting pattern is the same as before, but the gaps narrow. Indeed, the overall dealer advantage is only about one-fourth of a point, which statistically may be too close to call.
| Dealer Side Stats | None Vul | Both Vul | Dealer NV | Dealer Vul | Overall |
|---|---|---|---|---|---|
| Number of results | 19167 | 19530 | 19787 | 18927 | 77411 |
| HCP bias | +0.0039 | -0.0191 | +0.0069 | +0.0007 | -0.0019 |
| Freakness bias | +0.0189 | +0.0066 | +0.0038 | -0.0028 | +0.0066 |
| Plus-to-par score total | 2304980 | 3324800 | 2718850 | 2801610 | 11150240 |
| Minus-to-par score total | 2252320 | 3427090 | 2810820 | 2639990 | 11130220 |
| Net score | +52660 | -102290 | -91970 | +161620 | +20020 |
| Average score | +2.7474 | -5.2376 | -4.6480 | +8.5391 | +0.2586 |
| Standard deviation | 341 | 499 | 408 | 420 | 421 |
| Plus results | 8716 | 8677 | 8823 | 8483 | 34699 |
| Minus results | 8646 | 8803 | 8949 | 8519 | 34917 |
| Tied results (= par) | 1805 | 2050 | 2015 | 1925 | 7795 |
| Won-lost percent | 50.18 | 49.68 | 49.68 | 49.90 | 49.86 |
| Study 8J49 Main | | Top Dealer Advantage |
IMPs to Par AnalysisNow let’s see the effect of converting the par-score difference to IMPs. For example, if the dealing side stops in 3 
(vul) when 4 is cold, the loss of 450 points to par (170 minus 620) is converted to minus 10 IMPs. The following table summarizes these stats, with the first three rows unchanged as before.
Wow, a photo! In fact this is why I chose to show the averages to four places. The overall dealer advantage is a microscopic 0.0034 IMPs per result. Don’t hold your breath! When we look at this next time with more data, don’t be surprised to find that “Gore beats Bush.”
Note that the tinting pattern changed in one case: Dealer NV now shows a slight advantage to the dealer. Evidently the conversion to IMPs overcame the bias of the non-dealing side having higher attainable scores.
The breakdown of results plus, minus or tied is changed when converted to IMPs, because a 10-point difference now becomes a tie (0 IMPs) instead of the win or loss by score alone.
| Dealer Side Stats | None Vul | Both Vul | Dealer NV | Dealer Vul | Overall |
|---|---|---|---|---|---|
| Number of results | 19167 | 19530 | 19787 | 18927 | 77411 |
| HCP bias | +0.0039 | -0.0191 | +0.0069 | +0.0007 | -0.0019 |
| Freakness bias | +0.0189 | +0.0066 | +0.0038 | -0.0028 | +0.0066 |
| Plus-to-par IMPs total | 50589 | 62775 | 56927 | 55427 | 225718 |
| Minus-to-par IMPs total | 49549 | 64392 | 56675 | 54836 | 225452 |
| Net IMPs | +1040 | -1617 | +252 | +591 | +266 |
| Average IMPs | +0.0543 | -0.0828 | +0.0127 | +0.0312 | +0.0034 |
| Standard deviation | 6.64 | 8.22 | 7.30 | 7.40 | 7.41 |
| Plus results | 8384 | 8337 | 8569 | 8155 | 33445 |
| Minus results | 8354 | 8503 | 8601 | 8270 | 33728 |
| Tied results (par +/-10) | 2429 | 2690 | 2617 | 2502 | 10238 |
| Won-lost percent | 50.08 | 49.58 | 49.92 | 49.70 | 49.82 |
| Study 8J49 Main | | Top Dealer Advantage |
Specific Hand TypesFinally, let’s see how various hand types fare according to position. Columns 2-4 show the number of cases, net raw score and average raw score when the hand type is held by dealer. Columns 5-7 do likewise when the hand type is held by second hand. Vulnerability breakdown is ignored, i.e., just the overall picture is shown. Existence of a particular hand type for dealer does not preclude second hand having the same, or vice versa. As before, gold-tinted lines indicate an advantage for dealer, and red-tinted lines a disadvantage.
This table brings out the most significant part of the study, that dealer advantage depends on the nature of the hand. Unbalanced hands in general show a clear advantage to being dealer, as common sense would predict. Balanced hands, however, show a general disadvantage. The latter seems counterintuitive, as one would expect that being first to call would favor any hand type. Perhaps the crux is the ability to lie in wait, forcing the dealer to make the first move, either to reveal something useful or occasionally hang himself (been there, done that).
A noteworthy aspect is the spike in dealer advantage with 20-21 balanced. Evidently this is because it’s a common 2 NT opening range, and being able to do this unhampered allows better scoring than having to cope with right-hand opponent’s drivel.
A curious anomaly is dealer’s disadvantage with a 5-card major in an unbalanced hand. This prompted me to verify my settings and rerun the tests, but it proved to be correct. Out of curiosity I also checked this by strength, which sheds some light: With opening values (defined as 13+ revalued points) dealer has the advantage, but this is more than offset by the disadvantage with weaker hands. Perhaps this reflects the necessity to pass as dealer but having a one-level overcall available as second hand.
| Hand Type | Dealer | Net Score | Average | Second Hand | Net Score | Average |
|---|---|---|---|---|---|---|
| Any balanced | 37974 | -1079630 | -28 | 38258 | -837910 | -22 |
| 10-11 balanced | 6732 | +17920 | +3 | 6938 | +18900 | +3 |
| 12-14 balanced | 7816 | +783310 | +100 | 7816 | +826620 | +106 |
| 15-17 balanced | 3636 | +866220 | +238 | 3934 | +992440 | +252 |
| 18-19 balanced | 1116 | +370250 | +332 | 1084 | +367780 | +339 |
| 20-21 balanced | 424 | +196250 | +463 | 382 | +145370 | +381 |
| 22+ balanced | 278 | +116950 | +421 | 164 | +68390 | +417 |
| Any unbalanced | 42164 | +1229110 | +29 | 41880 | +688430 | +16 |
| 5 card major (unbal) | 12526 | +429620 | +34 | 12662 | +527860 | +42 |
| 5 card minor (unbal) | 10554 | +16210 | +2 | 10510 | -146450 | -14 |
| 6 card major | 6800 | +538020 | +79 | 6704 | +341150 | +51 |
| 6 card minor | 6606 | -41690 | -6 | 6484 | -204320 | -32 |
| 7+ card major | 1624 | +217910 | +134 | 1566 | +195920 | +125 |
| 7+ card minor | 1692 | +57640 | +34 | 1594 | -14020 | -9 |
The data sample seems large enough to conclude in general that dealer advantage exists for unbalanced hands, especially those with a long suit, but not for balanced hands, which incur a disadvantage. But wait! This may all change when my general theory hits the market, and you can view an advance copy today!
| Study 8J49 Main | | Top Dealer Advantage |
© 2014 Richard Pavlicek