How to solve a pricing competition for complementary goods
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 "$p_R&lt;p_L$" 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 $p_L&lt;p_R$. 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.