Solution to two player fair game

The game

• A&B bid in a “winner takes all” game. Higher bid wins. Each has to give 25% of his bid amount to the other on both win & loss as fees

• What is the winning strategy when played by A & B repeatedly through a single sealed bid?

Pay Offs Graph for the Game

A’s Bid Amount

B’s Bid Amount

B = AB = 5 A

A = 5 B

Region where A will have positive payoffs

Region where B will have positive payoffs

Why is this A = 5B???

Dominant Strategy is to go on a strategy that identifies the opponent’s strategy and defeats it.

Same as

Early Identification of Opponent Pattern Wins

+ve / -ve

A / B

-ve / +ve

A / B

Expected Gain = 0 for

both

Player B

Player A

Random

RandomStrategy

Strategy

Winning Strategy

You can always find a strategy

that defeats the random strategy

e.g. stratified randomization

You can always find a strategy

that defeats the random strategy

e.g. stratified randomization

What about maximin & minimax?

• Maximizing minimum gain

– No change in the dominant strategy

• Minimizing maximum loss

– No change in the dominant strategy

Note the game as designed currently is a zero sum game and thus is strictly competitive. What if it is not a zero sum game?

For e.g. what if the original game is changed a little by making the 25% fees, payable to the club that is conducting the game instead of paying to the other player?