# A Bayesian In-Play Prediction Model for Association Football Outcomes
## 5 Application to In-Play Betting
### Betting Performance  
In order to further verify the out-of-sample performance of our model in real-time prediction capability (e.g., during a match), we made bets every five minutes with the intention of calculating the average return. Bets were accordingly made at the following time points: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85. The actual return of bets during the 2013/8-2015/5 and 2016/8-2017/5 seasons can be determined with reference to a range of \(\tau\)-values. Vigorish is standard in betting markets: if the bookmakers are accurate in their probability specifications, they will have an in-built 'take' that corresponds to their expected profit. To win money from bookmakers, in the sense of gaining a positive expected return, requires a determination of probabilities which are sufficiently more accurate than those obtained from the odds so as to overcome the bookmakers' take. In Figure 4 we present the returns of betting on over-under market for various values of \(\tau\). Both models' average returns are presented as the full curves and are compared with the negative average return of 3.04%, the bookmaker's take. The detailed mean return calculation procedure is as follows. Firstly, for a specific match, there are 17 time points that we need to consider whether to bet one unit for per time point according to Equation (25). Then we can obtain the total net return and total staked for a match, further, total net return and bet of all matches obtained. Finally, total net return divided by total bet of all matches is the mean return. It will be noted that our model under the common strategy does not only achieve a return that exceeds \(-3.04\%\) but also generates a positive return, upon the condition that threshold \(\tau\) exceeds zero. However, the returns of the birth process model are considerably less than \(-3.04\%\). As \(\tau\) increases, with the concomitant establishment of a stricter betting regime, the returns of our model also increase while the returns of the birth process model decrease. One reason for returns decreasing as \(\tau\) increases may be that more and more small probability events with very high odds are  
Figure 4: Average return from betting on over–under market for various values of the threshold \(\tau\). chosen to be staked, with the consequence that we calculated the average betting right rate. The results demonstrate that, as \(\tau\) increases, our model's average betting right rate only decreases from about 48% to 47%; however, this can be directly contrasted with the average betting right rate for the birth process model, which dropped from about 46% to 36%--this in turn verifies the anticipated results. Using the same simple betting strategy, our model gains positive returns while the birth process model does not produce positive returns, which suggests that the returns mainly emanate from the model. Figure 5 demonstrates the returns that derive from betting on the point-spread market for various values of \(\tau\). The average returns of both model are presented as full curves and they are compared with the negative average return of 2.47% of the bookmaker's take. It will be noted that both model cannot gain positive returns. With small threshold values, our model performs better than the birth process model, however, both model perform worse than the bookmaker. With large threshold values, our model performs worse than the birth process model, and both models perform better than the bookmaker. The above phenomena implies that our model fail to enhance point spread prediction accuracy. Figure 6 describes the returns that derive from betting on match outcomes for various values of \(\tau\). Average returns of both model are presented as the full curves and them are compared against the negative bookmaker's take 5.11%. It is immediately obvious that our model obtains returns that far exceed the average negative bookmaker's take \(-\)5.11%.