Markov Chain

The theoretical probability of winning can be computed using a Markov Chain. The states are unique combinations of attackers and defenders, denoted as $(a,d)$, with $a$ as number of attackers and $d$ as number of defenders. The transition matrix can then be created and the theoretical probability of winning is the stationary distribution from the respective start state. The theoretical probability of winning with $0 \ge a \le 10$ attackers and $0 \ge d \le 10$ defenders are illustrated with the heatmap to the left. This method is beneficial when calculating probabilty for many different states since transition probabilities only needs to be calculated once. However, for estimating the winning probability from a single state this method is not as fast since all transition probabilities for all possible states have to be calculated.

Monte Carlo simulations

Another method to estimate the probability of winning is through Monte Carlo Simulations. With this method a large number of attacks are simulated whereafter the probability distribution is estimated. As seen on the heatmap to the left, the estimated probabilties are close to theoratical probabilities using 20.000 simulations per uunique combination of no. attackers and no. defender $(a,d)$. This method is preferred when estimating the winning probability for a large number of troops when we are interested in only a single combination. However, when calculating probabilities for many different combinations (to e.g. create this heatmap) Monte Carlo simulations are slow since 20.000 simulations need to be performed for every single combination and we can't use previous estimated probabilities.