A competent logician is playing cards with four of his friends, so that there are a total of 5 people sat around the table. Each player has the same 5 cards, showing A, 2, 3, 4, and 5 (Ace being low). Each player then has to choose a single card from their hand.
The objective of the game is to choose the lowest unique card that is displayed in the round. If two or more people show the same numbered card, they will not win. If no one chooses a card unique from anyone else, then the round is repeated until someone has a uniquely numbered card.
Each of them has 55 seconds to decide which card they want to draw, at which point they must each in turn show what they have picked. No one else knows which card anyone else has picked until they are displayed at the end of the time limit, but they are allowed to communicate with each other during that period.
The competent logician has to come up with a logical solution for which card to choose to maximize his chance of winning. What card should he choose? If the round needs to be repeated, should he choose the same card again?
Note: Whilst I do have a specific "correct" answer in mind, it is not necessarily the one that I will choose, as any card could reasonably be chosen and win, or lose. I'll accept the answer that is accompanied by the best logical argument for the choice of card.
Edit: To clarify, he has never played this game with any of his friends before, so he has no idea how they will initially think to play the game. He is unable to make any assumptions about their possible strategies, he is on the same playing field as they are, he simply needs his own strategy that he believes will give him an edge.
Edit 2: I should also clarify that this is only for the first game that they play. Only one game (with potentially multiple rounds) needs to be played until there is a winner. Once the game begins to be repeated the strategies will evolve based on what each person sees that the others are choosing. This question is for the first game only.
Answer
Assuming each player wants to maximize their personal win probability for the single game...
I immediately act, flipping over my 2,3,4, and 5; and putting the Ace face down in the center. "I've played an Ace."
Each player now knows that they cannot win by playing an Ace, individually, their only chance of winning is to play a non-ace, and hope someone else plays an Ace. It's becomes something that loosely resembles the Prisoner's Dilemma among my opponents.
The reason I chose this approach is
We have a 55 second window in which to discuss, which none of the other suggestions use. I believe the time window is critical, although it admittedly does give others opportunities to use external incentives to encourage diving on the Ace-Grenade.
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