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

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22 Joshua Menke, Tony Martinez mix for each group. The higher-rated groups would be appropriate for real encounters, whereas balanced groups may be preferred for practicing or training purposes. 7 Conclusions and Future Work This paper has presented a method to learn Bradly-Terry models using an ANN. An ANN model is appealing because extensions can be added as new inputs into the ANN. The basic model is extended to rate individuals where groups are being compared, to allow for weighting individuals, to model “home field advantages”, to deal with rating uncertainty, and to prevent rating inflation. The results when applied to a large, real-world problem including thousands of comparisons involving thousands of players yields probability prediction estimates within 5% of the actual values. Besides the aforementioned application of empirical Bayes, two additional areas for future work will include 1) extending the model to incorporate temporal information for given objectives and 2) attempting to model higher-level interactions. In many applications, including sports, military encounters, and online games like Enemy Territory, there are known objectives that need to be accomplished in order for a given group to defeat another group. In sports, this may include scoring points, in military encounters, completing subobjectives for a given mission, and in Enemy Territory, each map has a known list of objectives that both teams need to accomplish to defeat the
An ANN Model For Individual Ratings in Group Competitions 23 other team. For example, on one map, the Allies must dynamite three known objectives. A useful piece of information is how much time passes before one of these objectives is accomplished. If they are accomplished in a faster than average time, the team may be more likely to win than expected. If a team takes a longer than average amount of time, their probability of winning may be lower than expected. In the simplest case, the model can be extended to take into account not only which team won, but how long it took to win. One of the next pieces of future research will extend the Bradly-Territory ANN to incorporate this time-based information. One of the nice properties of using an ANN as a Bradly-Terry model is that it can be easily extended from a single-layer model to a multi-layer ANN. The convergence guarantees change—namely the model is guaranteed to converge only to a local minimum, but the representational power increases. No currently known Bradly-Terry model incorporates interaction effects between teammates and between opponents. Current models do not consider that player i and player j may have a higher effective rating when playing together than apart. A multilayer ANN is able to model these interactions, and therefore a future Bradly-Terry model will be a multi-layer Bradly-Terry ANN. One of the downsides of using a multi-layer ANN is that it will be harder to interpret the individual player ratings. Therefore, statistical approaches including hierarchical Bayes will also be applied to see if a more identifiable model can be contructed. One way to develop an identifiable interaction model can extend from the fact that it is commmon
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