by Louis Nel

Qualification for the Knock-Out (KO) stage of a tournament is often primarily based on the number of games won in the preceding Block Play stage. In case of a single block this works well when supplemented by a satisfactory tie-breaking method. If more than one block is present, then the fairness becomes dependent on how equally challenging the blocks are. The greater the number of blocks, the more difficult it becomes to ensure fairness in this respect.

In the 2007 World Championships, 8 blocks were used. In that situation equally strong blocks are virtually impossible to obtain. The element of unfairness that creeps into the proceedings this way (despite the sincerest efforts to minimise it) has been accepted as inevitable. It is indeed inevitable if no practical and credible alternative is available. The purpose of this article is to bring to light such an alternative. It also addresses a more hidden issue of inherent unfairness illustrated by a case in the most recent World Championship.

The Concept of Event Performance Grade (EPG)

How could the performance of a player in a qualifying event be measured so as to allow fair comparison of all the players? The EPG algorithm does this through successive approximations. Start by simply guessing a performance Grade for the player. Call it a Trial Grade. Using this Trial Grade and the actual Grade of each opponent, calculate an adjustment for every game, much as is done in the World Ranking system. The adjustment is positive (upwards) for a game won, negative for a game lost. After doing this for all block games of the player, we end up with a total Net Adjustment. One Trial Grade is deemed better than another if its resulting Net Adjustment is closer to zero. The EPG of a player is defined to be that Trial Grade that gives a Net Adjustment exactly equal to 0. But how on earth will we find that unique real number? Actually we don’t have find it precisely. We merely need to find a good approximation – for example, one within the Tolerance of 0.01 could be deemed good enough. That can readily be done by using the following three facts: (1) a Trial Grade that is too low gives a positive Net Adjustment; (2) one that is too high gives a negative Net Adjustment; (3) the midpoint between two such Trial Grades will always be better than the worst one of the two. By taking a sequence of successive midpoints as described, the computer can find a good enough approximation typically within about six trials and can do that in just a fraction of a second.

EPG is nothing but a reincarnation of the Period Performance Grade (PPG) introduced in section 2 of my recent article “Bayesian Ranking for Croquet”. So those who may want to know more computational details will find them in that article. One noteworthy difference between PPG and EPG is that it would be unpractical to demand five Moderate Disparity wins as well as five such losses for calculation of EPG. We have to make do with the games at our disposal. That calls for an adaptation in the case of players who either won no games or lost no games. Fortunately, such players are not really the focus of our present scrutiny – they either obviously qualify or obviously don’t qualify for the KO, so the adaptation (details below) is largely a formality.

Note that EPG does not compare a player only with those in the same block. It compares all players involved in the block play stage with each other. This feature puts it in a good position to address inherent unfairness arising from unequal blocks.

An Illustration of EPG at Work

Let us turn to the historic data of the most recent World Championship and look at the EPG of the players after the block play stage. Players without losses were simply assigned an EPG of 50 more than the highest calculated EPG in existence; those without wins were assigned 50 below the lowest calculated EPG (the number 50 is chosen as a matter of arbitrary convenience).

In the table to follow, GW = Games Won, GL = Games Lost and the entry “qualified” in the last column indicates a player who qualified for the KO under the official wins-in-block criterion (in some cases after a play-off game as tie-breaker). The top 32 listed would have qualified under EPG ranking as criterion. One player qualified officially despite an EPG ranking of 47.

Any two different qualification criteria can be expected to produce different outcomes, especially for players near the cut-off point. However, the players far from this point that ended on different sides of the cut-off under the two systems ask for closer inspection. So let us look at the detailed records of these players. The table to follow shows for each player his EPG followed by the Bayesian Grades of opponents encountered in his Wins followed by those encountered in his Losses.

Is there anybody out there who would argue that qualifiers Stark or McInerney performed better in block play than either of the non-qualifiers McBride or Roberts?

Let me emphasise that I am not complaining about what was done in the past. Indeed, were I to do that, I would in a sense be complaining against myself, as I chaired the International Seeding Method Committee whose recommendations were adopted by the WCF. There is always room for improvement and that is the spirit in which I am writing. Note however, that the mentioned committee did not invent the idea of block-by-block promotion to the KO. That practice was inherited from earlier years and left unchanged - at the time there was no feasible alternative in sight. The committee introduced a number of other changes, all aimed at assuring fairness to all players.

McBride and Roberts each were involved in 3-way tie for two KO places.   From the point of view of EPG both of these two players should easily have qualified.  The procedure prescribed by the present WCF regulations works well only when both of the following two circumstances are present:

1. All blocks  have equal strength
2. No crucial ties occur in a block

Unfortunately even the best efforts of organizers cannot guarantee (a) and (b).

This vulnerability  can be addressed by replacing block by block promotion by a procedure, such as EPG,  in which all players are compared together.

Other advantages of EPG

1. It would eliminate tie-breaking play-off games. This will facilitate the task of the organisers. Occasionally, play-off games can be difficult to accommodate. (In the unlikely event that the rounded EPG of two players result in a tie, one could arrange for every EPG to be printed to with decimal digits. So the possibility of an EPG tie can be virtually eliminated, except at the top and bottom where it does not matter).
    
2. It would give formatting flexibility. A particular venue may be able to accommodate 7 blocks of 10 players but not 8 blocks of 10, for lack of time and space. The elimination of play-off games will already ease up the time constraint a little. Since it is awkward to let 32 players qualify out of 7 blocks when the qualification is primarily done per block, the venue in question would likely offer to accommodate 8 blocks of 8 under customary qualification but 7 blocks of 10 under EPG (the number of blocks being immaterial under EPG). Given that blocks of 10 are strongly preferable to blocks of 8, this flexibility could occasionally be quite valuable.

The 2006 Australian Open provides an opportunity to see how EPG handles blocks of different sizes. There were 4 blocks of 8 and 1 block of 7. It can be seen that EPG did not favor every 5 – 2 record over every 4 – 2 record, nor vice versa. The qualification list of EPG differs from the official list in only one inconsequential case (McDonald instead of Bailey).

Other Potential Uses of EPG

In some tournament formats  – particularly a Flexible Swiss – it  becomes necessary to rank players on the basis of  a different number of games  played  against  opponents of different strengths. EPG is particularly suitable for use in such circumstances, as along as the players to be ranked have played an adequate number of games. So it is potentially of interest to all event organisers.

Software

Obviously, EPG needs to be implemented on a computer.  These days laptops  are frequently seen at tournaments, so that should not be a problem. If the need arises, I would gladly provide the WCF or its event organiser with the required software in the form of an executable file that could run on a Windows XP platform, along with instructions for its use.

Louis Nel
20 January 2007