Main     Study 9X35 by Richard Pavlicek    

Dealer Advantage

Long sought mystery finally solved

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 PositionRaw Score AnalysisPar Score AnalysisIMPs to Par AnalysisSpecific 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.

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Opening Bids by Position

Before 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.

DescriptionNone VulBoth VulDealer NVDealer Vul Overall
Number of results1916719530197871892777411
1st seat opening percent50.0847.0953.7244.7848.96
2nd seat opening percent28.5129.2424.1732.7628.62
3rd seat opening percent17.4717.7418.6116.3817.56
4th seat opening percent3.825.623.325.674.60
Passed out percent0.110.310.180.410.25

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?

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Raw Score Analysis

The 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 StatsNone VulBoth VulDealer NVDealer Vul Overall
Number of results1916719530197871892777411
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 total280003040221203085650361181013519610
Minus score total265826041885803610030294273013399600
Net score+141770-166460-524380+669080+120010
Average score+7.3966-8.5233-26.5012+35.3506+1.5503
Standard deviation379558458469471
Plus results965796389796941538506
Minus results948898319955943538709
Tied results (passouts)22613677196
Won-lost percent50.4449.5149.6049.9549.87

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.

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Par Score Analysis

Perhaps 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 S (nonvul) and 6 S 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 StatsNone VulBoth VulDealer NVDealer Vul Overall
Number of results1916719530197871892777411
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 total230498033248002718850280161011150240
Minus-to-par score total225232034270902810820263999011130220
Net score+52660-102290-91970+161620+20020
Average score+2.7474-5.2376-4.6480+8.5391+0.2586
Standard deviation341499408420421
Plus results871686778823848334699
Minus results864688038949851934917
Tied results (= par)18052050201519257795
Won-lost percent50.1849.6849.6849.9049.86

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IMPs to Par Analysis

Now let’s see the effect of converting the par-score difference to IMPs. For example, if the dealing side stops in 3 H (vul) when 4 H 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 StatsNone VulBoth VulDealer NVDealer Vul Overall
Number of results1916719530197871892777411
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 total50589627755692755427225718
Minus-to-par IMPs total49549643925667554836225452
Net IMPs+1040-1617+252+591+266
Average IMPs+0.0543-0.0828+0.0127+0.0312+0.0034
Standard deviation6.648.227.307.407.41
Plus results838483378569815533445
Minus results835485038601827033728
Tied results (par +/-10)242926902617250210238
Won-lost percent50.0849.5849.9249.7049.82

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Specific Hand Types

Finally, let’s see how various hand types fare according to whether holder is dealer or second hand. Columns show net raw score, number of cases, and average raw score per case; first when held by the dealer and then when held by second hand. The last column shows the dealer advantage (plus) or disadvantage (minus) in raw score per case. Vulnerability breakdown is ignored; i.e., only the overall picture is shown. Existence of a particular hand type for dealer does not preclude second hand having the same, or vice versa.

This table brings out the most significant aspect 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, in general show a disadvantage to being dealer. The latter seems counterintuitive, as one would expect that being the first to call would favor any hand, but apparently not. 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 anomaly is the huge spike toward dealer advantage with 20-21 balanced. Evidently this is because it is a common 2 NT opening range, and being able to do this unhampered (as dealer) promotes greater scoring versus having to cope with right-hand opponent’s drivel.

Another anomaly is the dealer disadvantage with a 5 card major in an unbalanced hand. This peculiarity prompted me to verify my settings and rerun the tests, but it proved to be correct. Out of curiosity I also checked it by strength, which might explain: If the hand has opening values (defined as 13+ revalued points) being dealer has a clear advantage (+33). Only with weaker hands is being dealer a disadvantage (-26). Perhaps this reflects the ability to make a one-level overcall with hands that would not qualify to open.

Hand TypeDealerSecond HandDealer
Net ScoreCasesAverageNet ScoreCasesAverageAdvantage
Any balanced-107963037974-28-83791038258-22-6
10-11 balanced+179206732+3+189006938+30
12-14 balanced+7833107816+100+8266207816+106-6
15-17 balanced+8662203636+238+9924403934+252-14
18-19 balanced+3702501116+332+3677801084+339-7
20-21 balanced+196250424+463+145370382+381+82
22+ balanced+116950278+421+68390164+417+4
Any unbalanced+122911042164+29+68843041880+16+13
5 card major (unbal)+42962012526+34+52786012662+42-8
5 card minor (unbal)+1621010554+2-14645010510-14+16
6 card major+5380206800+79+3411506704+51+28
6 card minor-416906606-6-2043206484-32+26
7+ card major+2179101624+134+1959201566+125+9
7+ card minor+576401692+34-140201594-9+43

What does it all mean?

The data sample seems sufficiently large enough to conclude in general that dealer advantage exists for unbalanced hands (especially 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. Order your advance copy here!

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© 2015 Richard Pavlicek