As described in the Results, we found that the difference between second-order and first-order agents was not as big as we expected and that the second-order agents did not seem to have a substantial competitive advantage over the first-order agents. This can be explained by the limitations we put on our implementation in the beginning since a lot of the second-order epistemic logic for this game includes strategic lying based on the knowledge of another players knowledge, which we excluded for this implementation since it would require both a weight function of weighing the potential additional knowledge gain of including another unknown card in a suggestion against the diminished knowledge gain for other players by including an already known card or card in a players own hand. This could only, for example, be implemented by either calculating the knowledge gain by each player with each possible combination of known and unknown cards at each turn or by employing some form of heuristic to predict future suggestions based on the current suggestion. Both approaches are computationally expensive to implement in combination with the rather ineffective way of storing the game knowledge in a logically sound Kripke model. It would be an interesting direction for future research to see if this could be achieved in a computationally effective way that does not limit the possibilities of playing the game too much. Another limitation that might lead to these results is that a guess can only be made when an agent is in the corresponding room, which might render higher-order deductions obsolete in some turns because the agent cannot reach the room he wants to include in his suggestion. In subsequent works, it might therefore be a worthwhile addition to include a rule that allows higher-order agents to relax this precondition if they are not able to reach their preferred room in a given turn, allowing them to enter another room close to it and making a suggestion based only on the other two properties since those can be made from any room.

A surprising result could be seen in the graphs of the available goal worlds in each turn, where the agents of all three different orders of knowledge seem to have very similar amounts of worlds left that they consider in each turn. This can, for one, be explained by the fact that these results are averaged over all games and agents; therefore, a single agent’s strategic advantages are not as pronounced. Secondly, this result also hints at the fact that Cluedo might be a game where a high amount of deduction is not strictly necessary since the changes in the game that reduce the number of worlds by the highest amount are usually presented as facts from one player to another so that, if it is assumed that all of these facts are remembered, it is not necessary or even possible to apply much further deduction to these results. However, the fact that the zero-order agents kept losing in almost all games against higher-order agents contradicts these results and implies that the few additional worlds ruled out by the additional deduction of the higher-order agents are critical to ultimately winning the game. This can also be seen in the end phase of the three graphs, where the higher-order agents consistently have a small number of worlds less left than the zero-order agents.

A point future work should consider is randomizing the playing order. The current simulations have not looked at possible advantages in the playing order, possibly clouding the results that have been obtained. Another point we did not implement was the fact that when a character is included in a suggestion, its playing piece is moved to the corresponding room, allowing players to possibly prevent other players from entering a room if they know that another player wants to enter it, by repeatedly suggesting him into another room. This could lead to more interesting dynamics within the second-order agents that can reason about the knowledge of other players but would also again require some form of weight function comparing the knowledge gain of the room a player wants to suggest against the knowledge loss for another player by preventing him from entering a room, making it challenging to implement.

In addition to the results from our simulations, we also implemented several extensions and optimizations for the mlsolver package, which we plan to bring back into the original package, so that they can also be of use for other subsequent projects provided the package owner will accept our changes.