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?

I thought I would (as a public service) post a comment about the term "zero sum", which has been misused and misapplied in this thread.

a) Two player zero sum just means that whatever I win you lose, and vice versa. The two players, in total, never gain or lose by playing a game.

b) Therefore, it is false to say that "two rational players will choose only to gamble if the game is zero-sum". Consider a non-zero sum game where you and I play, and each of us wins a dollar every time. I'm sure both of us would be willing to play this game.

c) Also, it would be false to say that "two rational players are indifferent about playing a game if it is zero sum". Consider a zero sum game where you and I play, and I always win a dollar and you always lose a dollar. The game is zero sum, but I doubt I can con you into playing.

d) Finally, a non-symmetric zero sum game can have a non-zero payoff even if the total of all possible payoffs is zero. For instance, consider a game where you and I both call either heads or tails. You get $24 if we both say heads and $8 if we both say tails. I get $16 if we disagree. This game I probably could con you into playing, because on the surface it looks like a fair game. In fact, it is very much to my advantage. I will say Heads with probability 3/8 and Tails with probability 5/8, and you will (on average) loss one dollar each time we play.

*Edited on ***April 6, 2008, 11:25 am**