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Hoarding and the Prisoner’s Dilemma

Santa Clara County Coronavirus update

With a few more data points, instead of using a 5-day moving average, I will be using a 2-week moving average to project future cases.

With that said, here is the latest graph of the daily rates of change in the number of cases each day:

One important and dangerous sign is that the red line has become more horizontal since my last post. If the red line is horizontal, the daily rate of change is constant meaning an exponential growth in the number of cases.

The green line shows the trend in the 2-week moving average. The hope is that this is closer to the future daily rate of change. The effect of the red line being nearly horizontal vs. the slightly faster rate of decrease of the green line, scan be seen in the projections shown in the graph below.

Hoarding

With many stores running out of supplies such as toilet paper, hoarding has become a problem. Assuming that there is a reduced but sufficient supply, then hoarding may cause an actual shortage and result in an insufficient supply. Fear that others may be hoarding toilet paper may cause you to be a hoarder too. The choice of whether to hoard toilet paper or to buy your normal quantity is akin to the problem known as the prisoner’s dilemma.

In the classic prisoner’s dilemma two persons working in partnership are caught committing a crime and arrested for it. The police do not have enough evidence to convict both for the maximum penalty of five years in prison, and without a confession from one of the prisoners, the police can only convict the prisoners for a lesser offense with a prison term on two years.

So the police offer each prisoner a plea deal where if he rats on his partner and his partner stays quiet, i.e. does not rat on him, the prisoner will be set free while his partner will be sentenced to the maximum five year sentence. If both rat on each other, then they will both be sentenced to three years. This offer is made to each in secret and the prisoners are unable to communicate.

From the point of view of the prisoner, if he stays quiet and does not rat on his partner, he will be sentenced to two years if the other prisoner also remains quiet, or five years otherwise. If he rats on his partner, he will be set free if his partner stays quiet, but will be sentenced to four years otherwise.

This can be shown in a payoff matrix:

Partner
QuietRat
PrisonerQuiet2 years5 years
Rat0 years3 years

The prisoner can reason that if he rats on his partner, he will be better off no matter what his partner does. He therefore rats on his partner.

However, his partner reasons the same way and they both end up ratting on each other. This results in each receiving three year sentences. But if both had remained quiet, both would have spent only two years in prison Thus, even though each prisoner chose the best option for himself, both did worse than they could have had they chosen their other option!

How is this related to hoarding toilet paper? Assume that when you shop for toilet paper you buy a two-week supply. You might imagine if the rest of the world or at least a large number of people are hoarding toilet paper, you may not be able to get any. So when you find toilet paper, you may either buy your usual two-week supply, or you may become one of the hoarders. Being a hoarder means that you go out earliier than you need to and buy more than you need, say a four-week supply. If the rest of the world is buying their normal supply, then there will be enough for you to buy four-week supply. If the rest of the world is hoarding the paper, you may have to search many stores and at best, a one-week supply.

The payoff matrix may look like this:

Rest of the World
Normal BuyHoard
YouNormal Buy20
Hoard4≤1

Just as in the prisoner’s dilemma, you can reason that if you hoard toilet paper, you are always better off than if you buy normally. And like you, the rest of the world can reason the same way resulting in everybody hoarding toilet paper. The end result is that while some people may succeed in building up a four-week supply (the early hoarders), you and many others end up with a one-week or smaller supply. And if everybody stuck with their normal buying pattern, everybody would have a two-week supply.

This is obviously an oversimplification of the problem, but I think it lends some insight into why instances of toilet paper hoarding has been popping up during the pandemic.

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