HAT BRIDGE

(IS IT REALLY THEIR TURN TO WIN)

 

Every board in a Match Point Duplicate session is a simultaneous but separate competition for the NS and EW pairs. If pairs maintain their direction throughout the session, as with a Mitchell movement, then there must be two winning pairs.  To produce one winner the last  round  is arrow switched to reduce the advantage of possibly being on the same side as any weak players or reduce the liabilty of being on the same side as any good players. So the EW/NS scores for each board include the result of an alien pair  of reversed polarity as justification for merging the NS and EW pair results to produce an overall winner.

The board match points are calculated by comparing the NS scores on the traveller: 2 mp for every pair beaten and 1mp for every pair tied. It is the nature of match point scoring that the EW mp score is complementary to that of their NS table opponents, both scores add to a board TOP. The reconciliation of all match points produces a total for each pair that is converted to a percentage of the theoretic maximum that pair could have obtained.

 

Most players have an idea of the percentage required to win a match.    If they or anyone else wins with less, there is suspicion that the win was a chance result  because the score  is consistent with the win score of a random match and as such the result is not considered significant. Crudely this means that pairs did no better than if they had pulled their board scores out of a hat and that someone had to win. There are many different types of hat capable of producing random board scores (and match points) and the real interest in Hat Bridge lies with determining the win scores frequencies. This distribution allows us to calculate the score threshold above which it becomes unlikely that a win score is due to chance alone.  The particular hat mechanism that a player has in mind, when thinking about a random match, is probably vague so I will suggest two of them.

 

One method is to assume pairs always reach rational contracts and make rational plays but that the success of their contracts is entirely dependent on the placement of critical cards in the deal. So consider all boards come with a set of reasonable alternate results labelled A, B, C …M, N,…  etc. together with an individual appropriately marked spinner.  If a board is to produce a spread of results then many alternates appear on the spinner, or if the outcome is to be close then only a few alternates are marked. All NS pairs spin to randomly select a letter. The absolute (plus/minus) score value associated with any letter does not actually matter because in match point only the rank (and ties) of the scores is important. Therefore, the alternate scores could be 20 for letter A, 30 for B, 40 for C and so on and the same series can be used for all boards. So we do not need to bother with the letters, just the spinners, and they can be marked 20, 30, 40 etc. To jumble up stuff more we can put the spinners in a hat and on each board select a spinner at random. There is a subtlety at this point, as to whether we put the last used spinner back in the hat before picking the next. The selected board scores are transcribed to a traveller, mp scored, and then the spinning starts again with the next board. The match points are copied to a reconciliation sheet to obtain the match result. Therefore, despite all the rational behaviour mentioned at the beginning we are reduced to picking differently sized spinners at random from a hat.

 

The second suggestion is the round robin.  The match points are calculated as though each board had been a separate round robin competition anyway:  2mps won or shared at each challenge between the NS pairs on every board (2 mps for each pair beaten and 1 mp for each pair tied with).  The mp awards can be generated randomly by tossing a coin, albeit with a wren design. Three variations leap to mind.   Two-sided coin, the winner gets the 2mps. Three sided coin (make it thick). If the coin lands on its side, 1mp is given to each side. Double toss a two-sided coin with 1mp at stake on each toss. Here  the outcomes are random and all pairs have an equal chance of gaining match points. However, the coin round robin does allow  for the possibility that pair 1 can win their board challenge with 2, 2 can win their challenge with 3 and 3 can win their challenge with 1. The statistical profile of coin robin genrated results is therefore slightly different to the profile of  the random ranked board score comparisons from the spinners, but the mp score outcomes from the coin robin can be rationalised over each board set as a  traveller set constructed by spinners. This process is more abstracted from the real game by virtue of bypassing board results altogether and going straight to the mps. (But then ‘method one’ was pretty much abstracted by the end.) 

 

There are many other mechanisms for producing random results for a hatbridge competition. The preferred method is almost a matter of taste guided by basic probability principles and ease of coding a computer simulation. I selected the coin robin method because it is relatively straightforward to implement despite 2000 coin flips per match (a board played 9x requires 72 coin tosses if the 1 mp per flip method is used). MS EXCEL can toss coins very quickly and award the match points to the pair combinations on a set of pseudo travellers very easily. The reconciliation produces the winning score and the whole thing can be repeated 100s of times without adjustments. 

 

So explore, by downloading the spreadsheet. Click HERE  to see additional  inifo or just look at the pictures.  Sheet1(click to view then use BACK to return) is the workhorse -travellers and score reconciliation.  Sheet2(click) - scatter diagram and histogram) shows the results so, over 1000 of them. One set uses the 1mp double flip and the other   the thick coin with equal probability of 1/3 win, lose or draw. The movement is a complete nine table Mitchell 27 boards 3 per round with the last  round arrow switched.

 

It becomes obvious that the most unlikely event of all is that everyone gets 50%. The likelihood of a win pair succeeding with score of 51%, 52% or 53% is still very unlikely. The random outcomes are most likely to produce a win pair greater than 54% by chance alone.     A win pair by chance alone can achieve session scores of 60% or more. It does not happen often and the greater the win score the more unlikely it is.  

The occurrence of percentages greater than 70% are very rare both simulated and with real play which is of course why clubs make a bit of a fuss about it. It is almost impossible to win real life with a score of 50-51% and most likely, that the winning score with 9 tables or is going to 60%+.  

 

Once the distribution of the ‘chance alone’ win scores is known, it is possible to place ‘a statement about the significance of a real winning score’ in a statistical context.

 

Consider the suggestion (hypothesis is the proper language) that ‘a win pair that reaches or exceeds 56% achieves their win because, as better players, they  applied greater expertise’ : The better pairs can be thought of as owning a biased ‘smart’ coin* that favours them during the board round robins mini contests.  The probability of any win pair achieving 56% or more by chance alone (with honest 50-50 coins) is estimated from the (double toss) sheet2 histogram data to be 8% (yellow curve image- left  is based on histogram data and the   area of the shaded region on the right is proportional to the win by chance probability of a 56%+ win ) .

Then, whenever the win score is 56% or greater, we announce the winners as best players  but with a doubt or risk that 8% of time we could be in error and really it was just their turn† to win. Therefore significance is relative to the level of possible error.  Select an error of 3% then  a score greater than  57% is required to diminish thoughts of a random win.  Commercial practice is to select an error level of 10%, 5%, or 1% assuming reduced profit is the worst consequence of error. (If the distribution was the height needed to safely undertake a parachute jump, I would insist on a minimum error level of 0.000001%, but by then I probably would be unable to breathe the rarefied atmosphere). 

 There is a standard statistical procedure to calculate the threshold for a stated error, using the mean and standard deviation of the high score win data. A match winning result equal to or greater than the threshold is considered significant at the specified level of error.  The calculation (2nd top box in each sheet2 picture ) produces a threshold value of  56.2% is at a 5% level of error with the double flip data. (57.3% at 5% with the thick coin data .)

† ‘their turn to win’ is a loaded statement implying natural justice in the domain of probability (there is none!) but it does give incentive to inexperienced players ‘waiting their turn’. 

* the simulation principles remain intact.

 

The project was intended to illustrate the meaning of ‘by chance alone’ and the significance of scores. The project is not an authoritive derivation of the better players’ threshold score, however the sums are rigorous.  My judgement is the 1mp per flip is too ‘soft’ as a randomiser by producing a draw 50% of the time in the mini round robin contests associated with each board.  The three-sided coin with equal probabilities (1/3) of a win draw or lose seems to me a better   represention of the feast or famine quality of board score comparisions. That data supports a score threshold of 57 %. Above this the winners should be praised, below this  you should check out how many players wear hats.

 

 

 

CAVEAT

I cannot accept any responsibility for players obtaining poor results at the table or having disagreements with colleagues because of the material presented here. I also cannot accept any responsibility should any of the material offered cause problems to your machine or system. That said I don’t anticipate problems.  Download details are found   HERE  Please read carefully.