Let n>1 be a fixed integer. The game is played on a board which consists of an odd number of consecutive fields, numbered -n,1-n,...,n-1,n (so there are 2n+1) fields.
There are 2 players, let us call them A and B. The game starts with a pawn placed on the field labeled 0, the middle of the board. At each stage of the game, each player is assigned a real number, his/her energy. Let a be the energy of A and b be the energy of B. At the start, each player gets 1 unit of energy, so a=b=1.
The game is played in rounds. In each round the following happens:
Player A wins if the pawn reaches field n. Player B wins if the pawn reaches field -n. If the game lasts infinitely many rounds, the outcome is considered a draw.
Below is the initial position on the board for n=3.
A discrete variant of the game would be the following: In the beginning each player gets a fixed number of points, say 100. All bets have to be integers.
An outdoors version would be the following: The two players hold the ends of the rope. The middle of the rope is marked. Lines are marked on the ground with numbers -n,1-n,...,n. At the start the middle of rope is above the line marked with 0. The rules of the game are as before. The game can now be considered a mental version of the usual tug of war. One could call it a mental tug of war. The bluffchess game somewhat simulates a real tug of war: If you pull harder than your opponent than the rope will move in your direction, but you also lose more energy than your opponent.
Let k=Pi/(2n+1). Suppose that the pawn is on position t. Then player A can force a win if and only if
Here are some examples of winning ratios for various n and various positions:
If the pawn is in position 0 and ais at least 2b then Player A can force a win (regardless the size of the board).
If the pawn is in any position (except -n) and a is at least 4b Then player A can force a win (again, regardless of what n is).