Games with infinite horizons have a calculation problem. For example, suppose one outcome of a game gave a payoff of 3. If a player received that payoff an infinite number of times, then his payoff for the game would be infinity. Now imagine that another outcome of a game gave a payoff of 2. If he received that payoff an infinite number of times, then his payoff for the game would also be infinity. This is strange, as clearly the player should prefer the first outcome to the second one. Working with infinite payoffs in a coherent way requires a solution.
- We solve the calculation problem by discounting each period’s payoff by δ, the Greek lower case letter delta. It is a value between 0 and 1.
- Discount factors incorporate a few different ideas: time value of money, a person’s underlying impatient preferences, and an exogenous probability that the game might end before the next period. Thus, they provide a helpful mathematical solution while also maintaining empirical accuracy.
- Suppose that a player receives a payoff of 3 for every period. Then, with discounting, we write that player’s payoff as 3 + 3δ + 3δ2 + 3δ3 + …
- This infinite stream of payoffs forms what is called a geometric series, which has some convenient properties that are the subject of the next lecture.