A Strange Game

Suppose that you go into an alien casino on the planet Zreebnorf, and you are offered to play a game. It works like this. The casino will match you with another player at random. The players don’t know who they are playing with, so there is no way for them to coordinate their actions or reciprocate after the game. Both players secretly pick a number between 1 and 100. The outcome of the game is then calculated as follows.

If both players picked the same number, let X be the number they picked. Both players will receive X credits from the house and have to pay (100 − X) credits to the house.

If the players picked different numbers, let X be the smaller number. The player who chose X will receive (X + 2) credits, and the other player will pay (100 − X) credits to the house.

(Credits are standard Galactic currency worth approximately $1 USD each.)

Some examples:

• Both players pick 50. Each pays 50 credits and receives 50 credits, for a net payout of 0.
• Both players pick 100. Each receives 100 credits and pays 0, for a net payout of 100 credits each.
• Player A picks 100 and player B picks 99. Player A pays 1 credit to the house, and player B receives 101 credits from the house.
• Player A picks 1 and player B picks 2. Player A receives 3 credits and player B pays 99 credits.
• Both players pick 1. Each pays 99 credits to the house and receives 1 credit, for a net payout of −98 credits each.

Would you play this game?

If so, what number would you choose?

Is there a Nash equilibrium?

Does the house make money from this game?