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 opponentís 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..).
Whatís your view of these three arguments and how do things change if the players, while still aware that the deck is finite, donít know what are the lowest and highest numbers?
(In reply to re(3): Zero Sum Confusion
by Dej Mar)
This is my final posting on this thread. EXPECTED PAYOFF OF ZERO is not the same as being ZERO SUM.
A rational player looks at his or her EXPECTED PAYOFF for a game, and do not play if it is negative.
ZERO SUM, however, does not mean that all players have a zero expected payoff. Please look it up if you disagree.
Many zero sum games have negative expected payoffs for one of the players, and that player rationally would not play the game.
And many games which are not zero sum have positive expected payoffs for all players, and rational players will play.
It is true that any non-zero sum game can be turned into a zero sum game by including the universe as an extra player, but the universe doesn't get a choice about whether or not to play. If the rest of the players, each of whom have a postive expected payoff, choose to play, then the game is played.