All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Paradoxes
Forced Win, Forced Loss, or Neutral? (Posted on 2008-02-08) Difficulty: 4 of 5
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?

  Submitted by FrankM    
Rating: 1.5000 (2 votes)
Solution: (Hide)
Simplicimus is right about the game being zero sum, but too lax in applying the information available to the players in a given round. For instance, it is obviously wrong to assume that a player will be indifferent to the bidding decision when he sees his opponent holding a 9. If, on the other hand, players are in the dark about the the highest and lowest cards in the deck, Simplicimusí argument becomes more persuasive, although still not conclusive. Consider, for instance, that with ever greater play experience players will find themselves able to make increasingly reliable hypotheses about the range of the deck.

Optimisticusí analysis is a bit deeper, but the notion that both players can expect positive average payout so long as they avoid betting against a 9 goes against the symmetric, zero sum nature of professor P's game.

Sceptisimus has the deepest insight. Yet his reasoning is also contradictory: At one point Sceptisimus claims there is a risk that his opponent, when seeing him holding (say) a 7 might chose to participate (to Sceptisimus' disadvantage). Later Sceptisimus claims that a rational opponent will refuse the chance to participate when he sees his opponent holding a 7. In addition, we should be suspicious of Sceptisimus' argument, as it seems to say that a player has no win chances for a very broad range of cases (all but seeing a 1). This would be as much a violation of the zero sum nature of the game as Optimisticus' argument above.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(5): Zero Sum ConfusionDej Mar2008-02-19 12:11:01
re(4): Zero Sum ConfusionCharlie2008-02-19 10:18:46
re(4): Zero Sum ConfusionSteve Herman2008-02-19 09:40:54
re(3): Zero Sum ConfusionDej Mar2008-02-19 04:54:11
re(3): Zero Sum ConfusionDej Mar2008-02-19 04:11:46
re(2): Zero Sum ConfusionSteve Herman2008-02-19 02:02:51
re(2): Zero Sum ConfusionCharlie2008-02-18 23:33:36
re: Zero Sum ConfusionDej Mar2008-02-18 22:49:12
Zero Sum ConfusionSteve Herman2008-02-18 11:14:26
SolutionIngeniousus Remarks..FrankM2008-02-13 20:16:45
SolutionBut game theory says (part II) ...Steve Herman2008-02-10 23:14:01
Some Thoughtsre: Some corrections in interpretation (Dej Mar)Dej Mar2008-02-10 14:58:02
Some ThoughtsQuestioning the assumptionFrankM2008-02-10 13:05:55
Some ThoughtsSome corrections in interpretation (Dej Mar)FrankM2008-02-10 12:52:29
Some ThoughtsBut game theory says ...Steve Herman2008-02-10 10:58:11
SolutionDej Mar2008-02-10 01:26:01
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information