How to solve a pricing competition for complementary goods

Question

I am preparing for an exam and I have come across this question in my textbook but I am not quite sure how to solve it so would really appreciate some help!

Suppose that company L produces left shoes and company R produces right shoes. If L charges $p_L$ for a left shoe, and R charges $p_R$ for the right shoe, then the price for a pair of shoes is $p=p_L + p_R$. The quantity of pairs of shoes purchased is determined by the demand function $q = 100 - p$. The cost of production is $c>0$ per shoe. Both firms choose prices simultaneously and independently of each other. Formulate this situation as a game (specify the players, strategies and payoff functions).

So obviously there are two players in the game, company L and company R. In terms of strategies I am slightly confused but I assume they are the following three:
- $p_R < p_L$
- $p_R = p_L$
- $p_R > p_L$

Please can you tell me is this is correct? Secondly, I am extremely confused by the payoff. I understand it would be optimal for both firms to price their shoes at the same level. And I know that to calculate the payoff we need to calculate profit.

Profit = $(100-p)\cdot q - c\cdot q$ Where I am stuck is whether it is necessary to sub in the expression that we have for q?

Finally, I believe that the payoff when prices are equal is just half the total profit for each firm. However, I have absolutely no idea what the payoff would be when the prices don't align. Any hint would be really useful! Thank you!

Answer 1

Here are some hints in the spirit of helping you prepare for the exam by thinking about the solution yourself:

You are right about the two players.

---

Your formulation of strategies is incorrect. Think of a strategy as a complete set of instructions that a player writes down before the game begins. These instructions are handed to a lawyer and must contain the information that the lawyer needs to play the game on the player's behalf and in accordance with his wishes. In particular, the instructions can't ask the lawyer to use information he doesn't have access to and must tell the lawyer how to deal with any circumstances that might conceivably arise. Put yourself in the shoes of the lawyer: you are supposed to choose a price on $R$'s behalf. What is the set of instructions that you would understand and have enough information to implement? Note that "$p_R<p_L$" doesn't pass this test because

• it would require that the lawyer knows the value of $p_L$ before the game starts.
• it doesn't pin down which $p_R$ the lawyer should choose. Even if he knew that $p_L=10$, there are many choices of $p_L$ that satisfy $p_L<p_R$. The poor lawyer would be very confused!

---

The payoffs from a game are a function that takes as an argument (i.e. as an input) a strategy for each player and spits out as the result the corresponding payoff. So $L$'s payoff function will look something like this $$\underbrace{(\text{L's strategy},\text{R's strategy})}_{\text{input}}\mapsto \underbrace{\text{payoff for $L$}}_{\text{output}}$$.

Now, imagine that I have asked $L$ and $R$ what their strategies are and have them written down here in front of me: $p_L=\ldots,\ p_R=\ldots$. Your job is to give me a rule that I can use to take this (and only this) information and calculate what each player's payoff will be. That rule is the payoff function.

Once you found the right payoff function, the question of what happens when the two prices are not equal will answer itself because the function will output the payoffs for any valid input strategies.