# PLAY;Ranking in Round-robin tournament

Having examined the method and theory of scheduling single round-robin tournament, the example of ranking systems of it will be covered in the following sections.

Most common way to rank players or teams in round-robin tournament Round-robin tournament is widely applied in competitions which are able to be held for a long period and need a comprehensive ranking of the players or teams. In order to acquire the ordering of teams and player, the most convenient and common way is to rank the participants according to their total victories. Nevertheless, tie may appear in an individual competition of the whole tournament. Considering three possible outcomes of a match, different scores are rewarded for having a winning, tie and losing game. The method used in the well-known English professional football league, Premier League, can be taken as an example here. Premier League is constructed by the competitions between the 20 top football clubs in England, in which double round-robin tournament is used, implying each team competes with all other teams twice. Three, one and zero point will be rewarded for a win, tie and lose in each competition. In the end of the league, a ranking list will be mainly arranged by the total scores of each team.

Problems arisen by ranking with total scores
Yet, this method may not be a perfect one when taking account of the outcome and fairness. Firstly, the total scores of the top two teams may be the same, showing that comparing only score is possibly inadequate to decide the champion. Secondly, the ranked ordering does not accurately reflect each team’s ability as a victory against a stronger team is more worthy than a win against a weaker team. (M. Stob,1985)

Solution to the problem of tie in ranking
To tackle the first problem, other criteria can be added into the arrangement of ranking by considering the nature of that sport itself. Referring to figure 1, the top two clubs have the same points but different ranking in the Premier League 2007/2008, because if there are same points between teams, the goal difference (Number of goal scored – Number of goal conceded) will be the second comparing factor, and the total goal scored will be the third factor. Lastly, if there is really such a coincidence that two important teams are still with same point, an extra match between them will be launched to solve the ranking problem. (Deloitte LLP, 2012) Likewise, other types of competition can consider extra factors other than just counting total points to avoid same ranking of teams.

Solution to the problem of fairness
For the second problem, it is much more complicated to deal with it. Goddard (1983), who had a deep study in ranking players in round-robin tournament, has proposed the concept of upsets to reflect the fairness of a ranking system. Upset in an ordering refers to a team with lower ranking defeats another team with higher ranking according to that ordering. (M. Stob,1985)

From the table, every row refers to the competition performance of each team. A team gets 1 point if it wins and 0 point if it loses. There are a total of two upsets in this ranking system, in which team 7 (weaker team) defeats team 1 (stronger team) and team 6 (weaker team) defeats team 2 (stronger team) are exactly the two upsets.

Having figured out the measure of fairness, Goddard suggested a new ranking approach called p-connectivity matrix which takes the different importance of victory into account. It actually works like how Google calculating the importance of web pages. (Springer US. 2005) Firstly, it takes the ranking of comparing total points as the basic ranking. Then, every victory in the tournament should be weighted in accordance with the primary ranking of the losing team, in other words, winnings against teams with higher primary ranking have a higher weight. The second ranking can thus be constructed with weighted winning. Similarly, the third ranking is built by using weighed victory which is adjusted according the second ranking. This process then repeats many times, in which generates many different ranking and the one with minimum upsets should be chosen as the final ranking. (M. Stob,
1985)

Nevertheless, neither the calculation process nor result of this ranking method is convinced to the teams and players. Even though the importance of different winning games is considered, players and teams may not be convinced to the fact that they get more victories but lower ranking than another team with less wins. Therefore, most of the round-robin tournament still adopt ranking without weighted victory. All in all, it is not easy to find the fairest ranking system. Even until now, there is not a so-called fairest ordering method.