The Greater Than Game is a playable adaptation of Always Greater. The rules of the Greater Than Game are as follows:

Players
- At least two players are needed to play, plus one more person acting as an impartial moderator.

Gameplay
- A game consists of a series of rounds described as follows:
- The moderator announces an integer N and then secretly chooses a random integer R in the range 1 to N. The value of N should be much greater than the number of players.
- The first player then calls out an integer in the range 1 to N. In subsequent turns, the players alternate calling integers greater than the previously called integer but not more than N.
- The round ends when one player calls an integer which is greater than or equal to R, at which point the moderator will announce the player has 'busted', meaning he has matched or exceeded R and lost the round.

Scoring
- When a player busts, he loses points equal to the amount he went over R, if he said exactly R then there is no penalty.
- Each other player scores the difference between their last call and the call before that. If a player has not made any calls, then he scores 0.

Example game 1
Three players A, B, C. N=20, R=18
A:8, B:12, C:15, A:17, B:19=busted
A scores 17-15=2, B penalized 19-18=1, C scores 15-12=3

Example game 2
Three players A, B, C. N=20, R=7
A:4, B:7=busted
A scores 4-0=4, B penalized 7-7=0, C scores 0

What is a player's best strategy if he wants to simply avoid busting?
What is a player's best strategy if he wants to maximize his expected points for a round?

What is a player's best strategy if he wants to simply avoid busting?
If the winning strategy were to completely avoid busting, I will quote Joshua from the movie WarGames, “A strange game. The only winning move is not to play. How about a nice game of chess?” But as the question asks only for the best strategy to simply avoid busting, I would have to say, if no number from the range 1 to p can be selected, it is to simply pick the number just one unit higher than the person just prior. If available, the initial pick should be p. As each number greater than p has an equal chance of having been selected as the secretly chosen number, selecting a number just one unit higher each time, the ratio of the number of numbers effectively selected with respect to the other players can be minimized.   By selecting  p, the player increases his odds as he eliminates selecting a number from the possible range of bustable numbers from  p+1 to 2p-1.

What is a player's best strategy if he wants to maximize his expected points for a round?
To maximize his expected points, the player would follow the near same strategy. The only difference would be the player would initially select the lowest number he could. If he is first that number is 1 (one).  If the bustable number is p+1, then no player would win or lose any points, yet it increases the chances of selecting a second number, and hence, increases the chance to score greater than zero points.

Posted by Dej Mar on 2007-04-28 17:26:49