## John Manning - bridge player and mathematician

### Introduction

Dr John Manning was a mathematician and played contract bridge in the county of Sussex UK. In retirement he ran the county's Web site. John made a technical study of bridge movements, discovering that many of the "standards" that we have used for years were improperly balanced with respect to competition between pairs.

After clarifying the underlying theory John devised his own - often very elegant and sometimes startling. Britain's EBU invited him to draw up its manual on bridge movements which is now the authoritative work on this subject. John used to say that - if you needed a bridge movement for a situation not covered in the EBU Manual, you should email him!

In the last few years of his life he was an occasional contributer on the internet discussion group rec.games.bridge John Manning died suddenly of a brain haemorrhage on 3 September 1998. This page is minimally modified from that which he had developed on his own web site and is a public document.

### Movements

#### Competition, balance and arrow-switching

I am often asked about arrow-switching, why there should not be more (or less), so here is an attempt to explain the ideas without (much) recourse to algebra.

#### Competition in Pairs Movements

In a pairs event the total amount of competition can be defined as the total number of match-points awarded to NS on all the travellers. Or EW - the total is the same.

This total is the product of four factors:
• (i) The number of tables in play
• (ii) The number of rounds played.
• (iii) The number of boards per round
• (iv) The value of half a top.

Factor (i) is determined by the number of players who turn up. Factors (ii) and (iii) depend on the time allocated - in most clubs the product of Factors (ii) and (iii) is likely to be in the range 24-28.
Only factor (iv) is at the disposal of the Director, who should maximise it by choosing a complete movement, where all the players play all the boards, and half a top is one less than the number of tables in play.

An incomplete movement simply cuts down the amount of competition, and makes the result more of a lottery, less of a test of skill.

The total amount of competition ought to be as large as possible - otherwise we might as well play rubber bridge.

For example, in a 7-table Mitchell, (and suppose for simplicity that we are playing 1-board rounds), the total amount of competition is 7 x 7 x 1 x 6 = 294.

This figure of 294 can be split up amongst the various pairs of pairs:
• Pair 1 competes against each of the other 13 pairs
• Pair 2 competes against Pairs 3 to 14
• .. and so on, 91 pairs of pairs in all

If there are P pairs competing, the number of pairs of pairs is P x (P-1)/2

#### The concept of balance in pairs movements

A movement is balanced if each pair competes fairly and equally against each of the other participating pairs. If you want a single-winner 7-table Mitchell movement each pair of pairs ought to compete by 294/91 = 3.23

This cannot be arranged exactly, because all measures of competition are whole numbers, but the movement would be as balanced as possible if all the amounts of competition were 3 or 4. By judicious arrow-switching we can approach the ideal.

This is how to get a precise measure of the amount of competition between two pairs, say Pair 1 & Pair 2:

Consider each board (or board set) separately and allocate a competition score thus:
• If Pair 1 and Pair 2 play the board against each other, score half a top.
• If Pair 1 and Pair 2 play the board in the same direction, score 1.
• If Pair 1 and Pair 2 play the board in opposite directions, but not against each other, score -1.

We can calculate the competition scores between every possible pair of pairs, and for a balanced movement these scores should all be equal or approximately so. To see how this scheme works out, look at the 7-table Mitchell, switched on the last round.

 1 2 3 4 5 6 7 Round 1 1A 8 2B 9 3C10 4D11 5E12 6F13 7G14 Round 2 1B14 2C 8 3D 9 4E10 5F11 6G12 7A13 Round 3 1C13 2D14 3E 8 4F 9 5G10 6A11 7B12 Round 4 1D12 2E13 3F14 4G 8 5A 9 6B10 7C11 Round 5 1E11 2F12 3G13 4A14 5B 8 6C 9 7D10 Round 6 1F10 2G11 3A12 4B13 5C14 6D 8 7E 9 Round 7 9G 1 10A 2 11B 3 12C 4 13D 5 14E 6 8F 7

(In this matrix, a group of symbols such as 2D14 under Table 2 in Round 3 means that Pair 2 is NS, Pair 14 is EW, and they play the fourth set of boards (D being the fourth letter of the alphabet)).

```The competition score between Pair 1 & Pair 2 works out thus:
On boards B,C,D,E,F Pair 1 & Pair 2 both play NS:               Score 5
On boards A and G Pair 1 & Pair 2 play in opposite directions:  Score -2

Total competition score = 3.

A further example: Pair 3 and Pair 8
On board E they encounter one another:               Score 6
On boards B and F they play in the same direction:   Score 2
On boards A,C,D,G Pair 3 is NS and Pair 8 is EW:     Score -4

Total competition score = 4.

Continuing in this way for all the pairs of pairs, we arrive at
42 competition scores of 3
42 competition scores of 4
7 competition scores of 0
```

The total competition score is equal to 42 x 3 + 42 x 4 = 294, as we know it must be.

A measure of the spread of competition scores (the smaller the better) is given by the standard deviation of these numbers, which is calculated as the square root of (SS - S x S/N)/N

```SS means the sum of squares of the scores (in our case 42 x 9 + 42 x 16 = 1050)
S means the total competition            (in our case 294)
N means the number of pairs of pairs     (in our case 91)
```

The standard deviation works out at 1.05 and cannot be further reduced by switching more or fewer boards.

For more complicated movements, a computer program is needed to calculate the standard deviation, and to find the switching regime which minimises it.

In the case of the unswitched Mitchell, the competition score is 7 between any two NS pairs, 7 between any two EW pairs, and 0 between any NS and any EW pair. Hence the two groups of players are uncoupled from each other, but there is more intense competition within the NS field and within the EW field. The total amount of competition (294) is the same whether or not you apply arrow-switching, but arrow-switching spreads the competition more thinly and enables you to issue a single list of results.

### Statistics of Team Matches

When the European Bridge Championship was held in Brighton, I was able to analyse most of the results in the 231 32-board matches between the 22 competing teams. The distribution of swing turned out as follows:

 Swing (IMPs) boards Percentage Swing (IMPs) boards Percentage 0 2268 29.0 10 387 4.9 1 1095 14.0 11 414 5.3 2 571 7.3 12 347 4.4 3 385 4.9 13 234 3.0 4 278 3.5 14 93 1.2 5 551 7.0 15 29 0.4 6 481 6.1 16 16 0.2 7 313 4.0 17 23 0.3 8 168 2.1 18 6 0.1 9 170 2.2 19 2 0.0

Total: 7831 = 100.0%

Note that, even at a high standard of bridge, 19.8% of the boards resulted in a double-figure swing.

The mean square swing is 38.14; taking into account the match results, it is possible to analyse this mean square into two components:

• 37.56 as a random element
• 00.58 as a difference in skill between teams

The square root of 0.58 is 0.76, so we are justified in saying that the difference in skill between two of the European teams is typically 0.76 IMPs per board.

The mean square swing for the 32-board matches was 1797, which breaks down into

• 1202 attributable to random variation
• 595 attributable to differences in skill.

The random element is less dominant, but still exceeds the variation due to genuine skill differences.

Next time you play a match, calculate the mean square swing. It will probably be well over 40, because you (and/or your opponents) are probably more erratic players than those chosen to represent their country.

Suppose that you are playing in a team match, and you reckon to be 1 IMP per board better than your opponents. Then you will expect to win the match by a margin of 32 IMPs, but the standard error of the margin is 34.7 IMPs (the square root of 32*37.56), and so you have a 18% chance of losing the match.

The graph and table show your chance of winning according to the length of the match and the edge you have over your opponents.

#### Probability of winning a match

 Edge over opponents (IMPs per board) Length of match 0 .5 1 1.5 2 8 Boards .500 .591 .678 .756 .822 16 Boards .500 .628 .743 .836 .904 24 boards .500 .655 .785 .885 .945 32 boards .500 .678 .822 .917 .968 48 boards .500 .714 .871 .955 .988

Even if you are as much as 1.5 IMPs per board better than your opponents, you still have a 4.5% chance of losing a 48-board match. Hence the underdog is always in with a chance, and seeded teams do get knocked out of the first round of competitions.

(The fact that there is a strong possibility of winning a match against a better team is also the reason why it is not necessarily right for selectors to pick the winners of a trial automatically).

#### A Swiss teams simulation

In a Swiss teams tournament we assume that each team has a merit value M, and that when two teams play against each other, the edge that one team has over the other is given by the difference of the two merit values, expressed as IMPs per board. In a 9-board match the expectation of the result of the match between the ith and jth teams is 9(Mi - Mj) IMPs, with a standard deviation of 3s, where s is the standard deviation on a single board. I ran a simulation program for 270 teams, varying the distribution of the Ms and the value of s, in an attempt to match the results obtained in an actual EBU Brighton Tournament. The best match was obtained when the Ms followed a Gaussian distribution with a variance of 1.5. The best value of s was 6.0 (better than 5.7 or 6.3).

The variation on each board is about the same as in the European championship, but the spread of abilities amongst the participating teams was considerably greater.

According to the simulations, the correlation between merit value and Victory Point score after each round was as follows:

 Round 1 0.474 Round 6 0.775 Round 11 0.844 Round 2 0.605 Round 7 0.793 Round 12 0.853 Round 3 0.669 Round 8 0.81 Round 13 0.861 Round 4 0.708 Round 9 0.833 Round 14 0.867 Round 5 0.75 Round 10 0.846 Round 15 0.879

These correlations are all median values of 11 runs of the simulation program.

It seems to me that these values for the correlations are satisfactory: the best teams are likely, but not certain, to come out on top.It so happens that Brian Senior selected what he considered to be the four best teams: they all finished in the top 6. On the other hand there is no enjoyment in a competition if the result is a foregone conclusion, and it is well that the weaker teams should be in with a chance, otherwise why should they bother to enter?

#### Shuffling

In the European championships and the Swiss Teams, all the boards were dealt at the table. Would the results have been different if random computer-dealt hands had been used?

### Should you bid thin games?

#### At IMPs scoring

If you are a cautious bidder,and stay out of a makeable game, you lose 250 points (6 IMPs) non-vulnerable, or 450 points (10 IMPs) vulnerable. On the other hand, if you bid a game which fails by one trick, you lose 170,180 or 190 points (5 IMPs) non-vulnerable, or 220,230 or 240 points (6 IMPs) vulnerable.

Since the penalty for underbidding is greater than the penalty for over bidding, it pays to strain for game. Indeed you should bid a non-vulnerable game if you estimate that your chance of making it is greater than 5/11(45.5%) and a vulnerable game if you estimate your chance of making it is greater than 6/16 (37.5%).

For example, you should always bid a game which depends on a simple finesse, but not one which depends on a finesse plus a 3-2 division of the outstanding trumps.

#### At pairs with match-point scoring

At pairs, Sheehan's rule applies: Estimate your own percentage chance of making the game, and also estimate what percent of the other declarers will make the game. If and only if the two percentages add up to more than 100 should you bid the game. Note that it is totally immaterial whether or not your competitors bid the game.