# Infinite time and economics

I was just looking at an interesting post from Philosophy, et cetera. They are discussing value when we have ‘infinites’. Now if we are looking at points in time with infinite resources this would be pointless for economists, as economics is the study of scarcity. If there are infinite resources, consumption is infinite and value is infinite.

However, they also discuss infinite time. Now if we have a game which is played infinitely into the future how do we decide the optimal choice of an individual? In order to work out the optimal choice of an agent economists will often discount the agents future consumption decisions. This implies that, if the ordering of preferences is expected to stay constant over time, an agent will value a unit of consumption more now than they will at any given point in the future (Ultimately it means that the game provides a finite value even though we have infinite time, as a result these values can be ranked allowing us to choose an optimum). Now exactly how we discount is constantly discussed by economists, especially since the way we commonly discount doesn’t hold true in empirical tests.

Now, even though I have spent a lot of time discussing discounting, that is not what I want to talk about. I want to talk about why and infinite horizon game or choice problem is sensible. Now you might say that no game between agents will be played for an infinite amount of time because everything has an end. However, that is not the way I see it. Infinite time is the idea of unbounded time. If we do not have a definitive end-point then we can view our game going on into infinity.

Let me explain. Ultimately, agents in a game will associated some probability to the game ending during a certain period. In this case they will either believe that their is 100% probability that the game will have ended at some set point, which acts as a boundry and so makes the game finite, or they believe that there is a probability that the game will end at each point in time, but they never associate a 100% probability to the gaming ending at a set point. In the second case we need to look at a infinite horizon game.

The discussion of discount factors is important when we go to look at a person’s choice in this way. In a sense the discount factor represents the likelihood that the game or choice problem will still be going during future periods, as well as a general preference people may have for consumption now instead of consumption tomorrow. As a result, even if we think that saying the current value of a pie in a year is greater than the current value of a pie in 10 years is silly, the fact that I place a higher value on being dead in 10 years than in one means I will value the pie in a year more (as I may never consume the pie in 10 years). Now my consumption choice may still be bounded (as I associate 100% chance of being dead at 800 😉 ). However there are cases where there may be point where a 100% probability is implied, eg the survival of a firm such as the Warehouse.

All this teaches us, is that we have to be careful that the horizon we choose to look over things makes sense, and that infinite does not mean the game or choice problem will never end, just that we are completely uncertain when it will end.

Woohoo, I get to nit-pick back again 😉 I slightly disagree with you about an infinite horizon being the same as uncertainty about when games will end. Having an infinite horizon allows an infinite number of trigger strategy equilibria to exist. Even a small probability of the game ending will cause some of those equilibria to collapse since no punishment strategy will be able to support them. As the probability of the game ending rises, fewer of the equilibria will be sustainable. So the solution set of an infinite game is likely to be different from the solution set for a game where there is some probability of the game ending in each period. I think that conflating the two situations is misleading.

I do agree that infinite games are ones in which the time frame of the game is unbounded though; although, since that really just draws on the definition of infinity it’s a pretty uncontroversial statement.

Interesting. I’d like to ask a few question, as you know dynamics is definitely not my strong point 😉

Now the multiplicity of trigger strategies depends on the discount factor right? So the more you discount the future, then there will be a smaller number of credible trigger strategies in the game. I never defined what the probability of dieing looked like, however can’t we just say that the associated probability of dieing in different periods is like having a time dependent discount factor (note that for this to be an infinite horizon game there must be some non-negative probability of the individual never dieing, that is why I said it is more appropriate for a firm than an individual).

For example suppose the probability of dieing is history independent and constant. In fact say that in each period there is a 10% chance of dieing. Now if the payoff when you die is zero, then even if you do not discount the value of the future based on interest rates or a preference for current consumption, you will still value next periods game only 90% as much as you value this periods game. When you set up your bellman value function, isn’t this the same as having a discount factor of 0.9 out in front of the recursive term?

Now I’m not saying that this is what the probability of dieing does look like, it will differ depending on the object and depending on beliefs and historical knowledge. However, changing the functional form of the probability of dieing is the same as changing the functional form of the discount factor.

So as a result, the probability of dieing will only influence trigger strategies in the same way as a discount factor. In fact, the probability of dieing should be one of the factors that defines the value of the discount factor.

Finally, I know that the statement that an infinite horizon game is an unbounded one is completely not controversial. However, I still thought it was an important point because many people get confused. I’ve heard plenty of people say infinite horizon games are stupid because nothing goes forever, I just wanted to clear up the fact that the game doesn’t have to go forever for the ex ante choice to be made under the guise of an infinite horizon framework.

I asked Jacek and he said that you’re right. A game with a probability of dying in each period can be rewritten as an infinite horizon game with a discount rate incorporating the probability of death. Given that, it would be able to support trigger strategies although, as you say, there would be a reduced number of available punishment strategies with a higher discount rate.

The feel good comment of the year.