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 2**n**+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:

- Each player secretly writes down a betting amount, which
is a nonnegative real number less than or equal than the player's
current energy.
So
**A**bets**x**say, and**B**bets**y**, where 0<=**x**<=**a**and 0<=**y**<=**b**. - The player who bets most may move the pawn one step in his/her direction.
So if
**A**bets more then**B**(**x**>**y**) then the pawn is moved up one place. If**y**>**x**then the pawn moves down one step. If**x**=**y**then the pawn stays at its current position. - For each player, the bet is subtracted from the current energy. So
**a**:=**a**-**x**and**b**:=**b**-**y**.

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**/(2**n**+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:

-2 | -1 | 0 | 1 | 2 |

inf | 2.618 | 1.618 | 1 | 0 |

-3 | -2 | -1 | 0 | 1 | 2 | 3 |

inf | 3.247 | 2.247 | 1.802 | 1.445 | 1 | 0 |

-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |

inf | 3.532 | 2.532 | 2.137 | 1.879 | 1.653 | 1.395 | 1 | 0 |

If the pawn is in position 0 and **a**is at least 2**b** 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 4**b**
Then player **A** can force a win (again, regardless of what
**n** is).

1/27/2002