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. They both pay 50 credits and receive 50 credits, for a net payout of 0.
- Both players pick 100. They both receive 100 credits and pay 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. They both pay 99 credits to the house and receive 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?
I would either pick 100 or 99. 100 if I felt like being nice. 99 if I felt like being selfish for that extra credit. Of course though, that may be motivated as well by the factor of anonymity here.
ReplyDeleteFor a given choice X of player A, the incentive for Player B is to always go one lower. This works out in the end to a Nash-equilibirum of (1,1).
ReplyDeleteHere the House gains 2*99 Credits per round, so a negative expectation-value for the players.
In the scenario you presented, I can't expect to be paired with a high-trust opponent. I would not play the game.
I expect if we were to play this game i a real-life setting, the house would loose money in the long run. In this case I pick x = 100.
Is this just the prisoner's dilemma with more choices?
There is a Nash equilibrium of (1, 1), in a sense.
DeleteHowever, anyone who chooses to play the game will not expect the result to be (1, 1). So the Nash equilibrium is not predictive. Presumably, anyone playing the game is playing to win some money from the house. And, they will expect the other player they are paired with to have the same motivation. So it is rational to expect the other player not to select 1.