A Bradley-Terry Artificial Neural Network Model for Individual ...

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18 Joshua Menke, Tony Martinez Table 1 Bradly-Terry ANN Enemy Territory Prediction Errors Heuristic None Inflation Certainty Both Prediction Error 0.1034 0.0635 0.0600 0.0542 5 Results and Analysis The results are shown in table 1. Each column gives the prediction error resulting from a different combination of heuristics. The first column uses no heuristics, the second uses only the inflation prevention heuristic (see section 3.6), the third column uses only the rating certainty heuristic (see section 3.5), and the final column combines both heuristics. Each heuristic improves the results and combining both gives the lowest prediction error at 5.42%. This value means that a given estimate is, on average, 5.42% too high or 5.42% too low. Therefore, when the model predicts a team has a 75% chance of winning, the actual % chance of winning may be between 70- 80%. One question is on which side does the prediction error tend for a given probability. To examine this, the predictions for the case that combines both heuristics are plotted against their prediction errors in figure 2 taking a 5% interval histogram of the results. The figure shows which direction the error goes for a given prediction. Notice that with only two low-error exceptions on the extremes, the prediction error tends to be positive when the prediction probability is low, and negative when it is high. This means the prediction is slightly extreme on average. For example, if the model predicts a team is 75% likely to win, they will be on average only 70% likely to win. A
An ANN Model For Individual Ratings in Group Competitions 19 error −0.05 0.00 0.05 0.0 0.2 0.4 0.6 0.8 1.0 prediction Fig. 2 Prediction vs. Prediction Error team predicted to be 25% likely to win will be on average 30% likely to win. Since the estimate is extreme, applying a form of empirical Bayes may be appropriate since it may move the maximum likelihood estimate closer to the true estimate on average. This would require gathering data from several servers and finding the prior distribution that generates the mean ratings of each server. This will be attempted in future work. In addition to evaluating the model analytically, the ratings it assigns to the players can be asssessed from experience. Table 2 gives the ratings of the top ten rated players from the 3,379 games. Several of the players listed in the top ten are well-known on the server for their tenacity in winning maps, and therefore the model’s ranking appears to fit intuition gained from
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