Syzygy

Sunday, February 17, 2008

Strategies for CLUE

So I was in the shower, thinking about game theory and wondering what some strategies might be for the board game known as CLUE.

For those who are unfamiliar or may need a refresher, the game involves guessing a specific combination of person, room, and weapon to solve a murder. Players are distributed equal proportions of the unused cards, so they can immediately cross off certain persons, rooms, and/or weapons off the suspected list. This information, is obviously private and distinct for each individual. Play begins with players navigating the board and making guesses after entering room (the guess must use the room that the player is in, so there is some navigational trickery involved). After making a guess for a combination, the person to the left is given a chance to disprove it by revealing a card to the player who made the guess. If the person has one of the cards that is part of the guess, they must show a card, but they may choose which card if multiple cards can be shown. The card is shown only to the player who made the guess. If the person to the left cannot disprove the guess, then the next person to the left is given the opportunity to disprove the guess. If no one disproves the guess, the player who made the guess is then given the option of making a formal accusation, at which point he looks at the hidden cards for the actual person, room, and weapon and reveals if he was correct or not. In the case of an incorrect accusation, he is prevented from further play but must still participate in revealing cards.

Now, the information that is available to each player can be divided into several levels:

1. direct information
- Each player can cross off the cards they hold in their hand as they cannot be part of the murder trifecta.
- Each player can cross off the cards that are revealed to them in the course of making guesses and having them disproven.

2. indirect information [this is probably key, since it can expand the amount of information known since you can learn something new with every person's guess, and not just your own.]

- When making a guess combination (such as Mr. Green, Kitchen, Candlestick), any person that doesn't reveal a card to disprove the guess (when given the opportunity) cannot have any of those cards in hand.
- The same is true when other people make guesses, you are given information about what cards are not in a person's hand if someone does not reveal a card to the player who made the guess when given the opportunity. Also, any reveal tells you that the person who revealed has one of the cards in the guess, keeping track of this can reveal what card is revealed if one knows where the other cards in the guess are located.

I also considered one strategy of using misleading guesses to throw off the other players, but doing this successfully requires not just keeping track of what one knows, by modeling the available information everyone else knows, so that they won't KNOW that you are misleading them. I'm also not entirely sure what you gain by misleading guesses, since if you make a guess that is composed solely of cards in your hand, it will be apparent if other players have seen specific cards in your hand. Given the amount of effort required to model everyone's information, I don't think misdirection is a really viable strategy.

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