Professor Paradoxicus has invented a new game. A single card is dealt to each of two players. The cards are consecutive and come from a 9 card deck, each card bearing a distinct integer between 1 and 9. Each player then holds his card (number facing outward) against his forehead; ie, each player can see his opponent’s card, but not his own. The players are then asked whether they want to bet, and if both agree, they examine their own card, with the player who drew the lower card paying his opponent the amount on the higher card.

Professor Paradoxicus has invited three students to analyse the game.

Simplicimus notes that the game is symmetric and zero sum. He asserts that players will be indifferent as to whether or not to bet.

Optimisticus points out that whenever a player sees his opponent holding a 9 he can count on losing. On the other hand, Optimisticus reasons, if the opponent is holding any card N<9, then that player has a 50% chance of winning N+1 and a 50% chance of losing N. Optimisticus asserts that a player will quit when he sees his opponent holding a 9, and choose to play otherwise.

Finally, Sceptisimus expresses the view that a player will refuse to bet unless he sees his opponent holding a 1. He justifies his peculiar opinion as follows: Suppose a player sees his opponent holding a 9. He will certainly refuse to bet. Now suppose a player sees his opponent holding an 8. Clearly, he himself will be holding either a 9 or a 7. In the former case, he can be sure that his opponent will refuse to bet. He concludes that he has nothing to gain by agreeing to bet, but could have something to lose. He therefore should refuse to bet.

Sceptisimus next reapplies this argument repeatedly, with 7 (,6,5..) taking the role of 8 (,7,6..).

What’s your view of these three arguments and how do things change if the players, while still aware that the deck is finite, don’t know what are the lowest and highest numbers?

The fault in Simplicimus' observation is, unlike in poker, because both players must agree to the bet for the winner to collect the bet, neither intelligent player would bet if they saw the other player holding a 9, thus the game is not truly a symmetrical, zero-sum game.

Optimisticus' reasoning and assertion is not valid either. Though he correctly recognizes that a player may gamble if the opponent is not holding a 9, the probability is not a 50% chance of winning or losing, but 45% chance of winning and 55% chance of losing.

Sceptisimus' analysis can be summarized as "no player will bet unless they have 100% probability of winning". This argument is faulty in that intelligent people are known to gamble, even if the odds are against them.   In addition, his view that if a player sees his opponent holding a 8 that he will have a 9 or 7 is invalid. He could be as likely holding a 6, 5, 4, 3, 2, or 1.


If the players did not know the lowest and highest numbers then...

Simplicimus statement that the game is symmetrical, zero-sum is valid.  The players may be indifferent as to whether to bet or not.

Optimisticus reasoning remains invalid, as no longer is it certain that the 9 is the highest number, and only for the first game does his argument of the odds bode correctly. (After the first and subsequent games, each player will have more knowledge as to what the lowest and highest cards are, and can make logical guesses to the probabilities of winning or not)

Sceptisimus' analysis also remains invalid, as, again, even intelligent people are known to gamble. 

Posted by Dej Mar on 2008-02-10 01:26:01