BLUFFCHESS

I invented a simple but dynamic game. I have not really decided on a name but for the moment I like to call it bluffchess. The has little in common with chess except that it is played by two player on a board with a pawn. The game is a lot about bluffing.

RULES

Here are the rules for bluffchess.

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:

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.

Remarks and variations

From the values of a and b it is only the ratio a/b that really matters. One could rescale the numbers a and b during the game (for example to prevent them from getting very small).

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.

Winning Positions

What are the winning positions in the game? I proved the following result:

Let k=Pi/(2n+1). Suppose that the pawn is on position t. Then player A can force a win if and only if

a/b>2cos(k)sin(k(n+t+1))/ sin(k(n+t))

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 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).

Open Problem: An Optimal Strategy?

Is there a (mixed) strategy for Player A which will force a win in at least 50% of the games, regardless what strategy Player B plays? Can one explicitly describe the strategy?

1/27/2002