1 Team ranking using least squares There are many ways to rank sports teams and

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1 Team ranking using least squares There are many ways to rank sports teams and players (and other competitive entities, such as universities and researchers-8D), besides the simple win-lose count. These rankings are a source of intense interest and sometimes controversy. In this exercise, we will use the least-squares method to rank sports teams Suppose that there are four teams in a particular sports league, iden- tified as team 1, 2, 3 and 4, respectively. Consider the competition results shown in table 1. Let x, denote the 'ranking score' or 'rating' Home team Away team Home-team score Away-team score 67 52 82 98 68 49 94 79 41 81 4 2 Table 1: Competition results. of team i. If the home-team rating is ri and the away-team rating is xj, then, ideally, we would like zi-ay si-si, where si IS the home-team score and sj is the away-team score. For the competition results shown in table 1, this corresponds to 1 57 This is an overdetermined linear system that cannot be solved. Nev- ertheless, we can try to find a least-squares solution. But (1) does not have a unique least-squares solution, as it is possible to add a

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Solution : Function for quadratic equation -

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