The sum of a stream of discounted payoffs creates what is known as a geometric series. Fortunately, geometric series have a number of convenient properties. This lecture covers the identities that matter the most for our purposes and provides basic proofs for some of them.
- Consider the first n terms of a payoff stream: x + δ + δ2 … + δn-2 + δn-1. This is equal to x(1-δn)/(1-δ).
- Taking the limit of that expression as n goes to infinity, the sum of an infinite stream of x values equals x/(1-δ). This is important because it means we can analytically describe an infinite number of values in a single, compact expression.
- The sum of an infinite stream of x values except the first one equals δx/(1-δ). This is important for calculating what happens if the players take an action in the first period, do not like what happened, and switch to doing something else for the rest of time.
- The sum of every odd term of an infinite stream of payoffs equals x/(1-δ2). This may seem unimportant now but will be helpful later.
- The sum of every even term of an infinite stream of payoffs equals δx/(1-δ2).