Strategies and Situations

Most of the strategies that can be played (as described below) originated from situations we encountered while playing the game ourselves, and from reasoning about possible tactics and other situations.

Suspicion Strategies

There are two strategies that can be used to determine which suspicion an agent could/should make when they get their turn. We named the first the 'default' strategy, and the second the 'complex' strategy.

Default Suspicion Strategy

A very simple strategy: the agent chooses a card per category of which they do not know whether it is in the envelope, and then make that set their suspicion.

Complex Suspicion Strategy

A more complex suspicion strategy is to take into account the information of the other players. To do this, an agent first makes a list per category of the cards it does not yet know. Instead of just choosing one from this list, they check to see how many agents know something about that card. If agent A knows that agent I knows that card is in the envelope, or that agent I knows that card is held by agent A, that is seen as 'knowing something about that card'. If the other agent already knows something about these cards, they can gain more information if there are cards shown by an agent who has to respond to the suspicion.

An example: Agent A suspects Mrs. White and the candlestick. However, I know that agent A has Mrs. White, because they showed me that card in a previous round. Agent B, who is asked to respond to the suspicion, shows agent A a card. Because I know that Agent B cannot have Mrs. White (because I know Agent A has it), the card shown must be the candlestick. In the game of Cluedo the goal is to gain as much information as possible, while giving the other agents the least possible information. So, if it can be avoided, any agent wants to play the strategy that will give them new information, without giving more information than necessary to the other agents.

Response Strategies

If another player makes a suspicion, an agent can be asked to respond to that query. This means that if the agent has one or more of the cards queried, they must show one of them. In all cases, if an agent only has one of the cards in the suspicion, they have to show that one. When they have more than one, there are different strategies that can be played.

Default Response Strategy

The simplest strategy that can be played is just to choose a card from the suspicion to return at random. If the agent only has one, this is easy, and otherwise, they choose one at random.

Simple Response Strategy

A slightly more complicated strategy is to check whether you have already shown one of the cards to the querying agent. If you have (if you know that they know that you have that card), then just show that card again, that gives then no new information at all. If the agent hasn't shown any of the cards to the querying agent, then just choose a random card.

Optimal Response Strategy

This can be added on to the simple strategy, so if the agent has more than one of the cards in the suspicion, and has shown neither of them to the other agent before. In this case, you still want the opponent to gain the least amount of information. A way to do this is to reason about how many information showing each of these cards would give the other agent. If you know that they know almost all weapon cards, you would be better off to show a person card. However, if there are six possible weapons and four possible suspects, and you know that they know half of each category, it is better to show a weapon, because that gives the least amount of information, relatively.

Accusation Strategies

There are several strategies that can be used to determine whether to make an accusation. The first is obvious one. The second is an alternative strategy that can be played by winning-driven agents. This is one we encountered while playing.

Default Accusation Strategy

If you know all the cards that are in the envelope (be eliminating all other possibilities), make an accusation containing those cards.

Risky Accusation Strategy

It can happen that you know all the cards in the envelope, except for one. So of one of the categories you still hold 2 different cards for possible. If you make an accusation then, you can lose if you choose the wrong card. However, if you are that far in the game, another agent might be too, and all other agents have a chance to make an accusation before your turn comes again. If they are correct, you also lose. So in this situation, it could be a good strategy to guess the card.

Other Situations

Using one's own cards in a Suspicion

Initially we wanted to implement a strategy in which an agent uses their own cards in a suspicion. For example, if agent A holds the candlestick, they could make a suspicion with Mrs. White and the candlestick. One would do this to try to gain information specifically on Mrs. White. While playing the game ourselves we all used this strategy. However, it is not rational. It is defendable that we use it, but, as also described by H. van Ditmarsch in his thesis 'Knowledge Games' (2000), we only use it because humans have a limited processing capability and limited short-term memory.

This means that we cannot hold the entire model of possible states and knowledge in our heads, and therefore try to limit the number of possible states. As the system we built is capable of computing the entire model (for a smaller number of categories), we did not implement this strategy. Perfect epistemic logicians would not play this, but would inquire after that which they know least about.

Knowledge through the lack of accusations

A rational agent would, whenever they knew which cards are in the envelope, make an accusation. This means that if a rational agent does not make an accusation, that they cannot yet know the exact dealing of the cards. Because if they would know the dealing of the cards, they would have made an accusation.

This means that if an agent were to make a suspicion to which no one responds, one would expect the agent to make an accusation, unless the agent has one of those cards themselves. From this information, the other players might be able to draw new conclusions about the cards the agent has. Therefore, a truly rational agent would have to take into account that they might give away more information than intended in this situation.

Currently, this situation is not build into our tool, since not all players have to be rational. For example, a human player is not necessarily rational. This means that we cannot assume that, when a player does not make an accusation, they cannot know what cards are in the envelope. However, if we could make this assumption, it would be possible to construct a strategy that would take this into account.

We could do this by creating a strategy that, beside looking at the knowledge that the other players have about the cards now, also looks at the knowledge the other players would gain after the announcement that the current player does not know the cards in the envelope. This strategy could then also be used to answer the question whether or not this situation does actually occur, which is a question that was posed by Hans van Ditmarsch and Barteld Kooi in The Secret of my Success (2006)

Non-rational strategies and knowledge

Although most players would want to win a game, there are situations imaginable where one would want to consciously lose a game. A situation like this could be where a grandparent is playing with their grandchildren, who are still young and might not like it if the grandparent would win all the time, so the grandparent could chose to use a different playing style, which would lead to a style of play that is different from a rational agent.

Using this style of play, most strategies would stay fairly similar, but instead of trying to give the least amount of information possible, the player would want to give the other players as much information as possible. This means that the new strategies would use the same kind of knowledge, but instead of selecting the card that gives the least amount of knowledge, selecting the card that gives the most amount of knowledge.

It would also be possible to favour one specific player over the other players. Here one must make a distinction between actively helping them out (selecting the card that would give that player the most information) and not caring whether or not that player gets more information (not taking them into account when calculating how much information the other players get). Whereas the first method is probably preferable for the grandparent in the previous example, the other strategy might be used when playing with people who play the game for the first time or not very well. It could also be used as a tiebreaker, for example if the optimal suspicion strategy would need to decide between two cards, they could use a weighted ranking, depending on how close they think the players are to winning the game.