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?

Hard to imagine that a comment made by me on the puzzle to which this is 'linked' inspired this twist.  [That credit is now probably lost from the server since review has been fulfilled, but thanks to Brian Smith for that acknowledgement].

What should be my strategy?
I am initially thinking about a game where one is told to converge on a number somewhere below N, clues being Higher or Lower.

The first question:
Knowing the number of players in the game, and knowing N, I could reasonably assume (but not be positive) that if I was the first player I could select the number that is one more than the number of players.  Thereafter, on my turn, I would increment my call by one more than the player before me. 
This would mean that I'd either lose 1 point for breaking (going bust) or land on the Reserved number and lose nothing. After that I am in the hands of the other players.
[seems to be too conservative to be decent.]

The second question:
Again if I play first and survive then I can implement a strategy.
I will score best if I am farthest from R when it is called or broken.

I somehow have to maximise the margin between myself and the
previous player.  I am thinking that if there are 'y' players my ante is y/2. This would hopefully, if I was in the 'limit' of R against which I am always gambling, place me on or either side of it without creating too much jeopardy for me.

Ok. Supposing I play these strategies, surely the other players are going to realise in subsequent rounds what I am doing?

Shoot!

Posted by brianjn on 2007-04-25 08:19:21