Question

Two gas station owners (Kid and Play) have stations that are just across the street from one another. Each morning they must decide on a price without knowing what the other will do, and they are stuck with that price for the whole day. If they both price high, they each make $6,000 that day. If one prices high and the other prices low, the one with the high price makes $1,000 and the one with the low price makes $10,000. If they both price low, they each make $3,000. (Drop the thousands to make the math simpler.) a. Show the normal-form of the game. b. If this is a one-shot game (say both stations are going out of business tomorrow), what is the Nash equilibrium of the game? c. Now assume that they will continue to compete forever. If the interest rate is 20%, can they collude and each charge a high price? What strategy would maintain this collusion?

Answer 1

GAME THEORY

A) Because this is a simultaneous game, we will use a matrix. In this problem we are going to designate Kid and Player 1 and Play as Player 2. The options are set the price HIGH, or LOW.

The decision and pay-off of Player 1 can be read vertically and player 2 decision are horizontal. Payoff are like this (0,1) the first number (0) denotes player 1’s payoffs and the second number (1) denotes player 2 payoffs.

With the information of the problem, your game theory matrix should look li the next one.

B) To obtain the Nash equilibrium you must analyze what is the best response for each strategy from the other player.

Let’s start with player 1:

If player 2 chooses HIGH, the possibilities for player 1 is to set the prices HIGH (6) or LOW (10). Player 1 will choose to set the prices LOW because is the highest payoff in this situation. Player 1 decision is circle in red.

If player 2 chooses LOW, player 1 will choose again LOW (3) which gives him a higher payoff that if he chooses to HIGH.

We can also note that player 1 dominant strategy is to remain prices LOW.

Player 2:

If player 1 chooses HIGH, the possibilities for player 2 is to set the prices HIGH (6) or LOW (10). Player 2 will choose to set the prices LOW because is the highest payoff in this situation. Player 2 decision is circle in blue.

If player 1 chooses LOW, player 2 will choose again LOW (3) which gives him a higher payoff that if he chooses to HIGH.

player 2 dominant strategy is to remain prices LOW is also LOW.

Nash Equilibrium is LOW, LOW

C) If there is an interest rate for both players and both players know in the future their cost will increase, they can collude or agree to set the higher price so both can handle the increased expenses. The strategy is to set in both gas stations the prices (HIGH, HIGH) getting 6 each as payoff.