# 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