The Great Toilet Paper Crisis coordination game

This week in my ECONS101 class, we've been covering game theory. Which means I no longer have to hold back on blogging about this article in The Conversation from a couple of weeks ago, by Alfredo Paloyo (University of Wollongong):

Shoppers in Australia, Japan, Hong Kong and the United States have caught toilet paper fever on the back of the COVID-19 coronavirus. Shop shelves are being emptied as quickly as they can be stocked.

This panic buying is the result of the fear of missing out. It’s a phenomenon of consumer behaviour similar to what happens when there is a run on banks.

A bank run occurs when depositors of a bank withdraw cash because they believe it might collapse. What we’re seeing now is a toilet-paper run...

Both banking and the toilet-paper market can be thought of as a “coordination game”. There are two players – you and everyone else. There are two strategies – panic buy or act normally. Each strategy has an associated pay-off.

If everyone acts normally, we have an equilibrium: there will be toilet paper on the shop shelves, and people can relax and buy it as they need it.

But if others panic buy, the optimal strategy for you is to do the same, otherwise you’ll be left without toilet paper. Everyone is facing the same strategies and pay-offs, so others will panic buy if you do.

The result is another equilibrium – this one being where everyone panic buys.

Let's work through Paloyo's example systematically. In this game, there are two players: you, and everyone else. However, instead I'm going to use two named players (Sam and Chris) - the result would be the same if we used Paloyo's players, but I just find it easier to refer to named players. There are two strategies: panic buy, and act normally. Assuming that this is a simultaneous game (both players' decisions about strategy are revealed at the same time), then we can lay out the game as a payoff table, like this:

The payoffs are measured in utility (satisfaction, or happiness), for Sam and Chris. If both players act normally, then no one misses out on toilet paper, and both players receive 'normal' utility (utility = 0). If one player panic buys and the other doesn't, then the panic buyer is worse off by a little (they have to pay the costs of storing their panic purchases; utility = -2), but the player acting normally is much worse off (there is a good chance that the store runs out of stock and they have to search around for toilet paper; utility = -10). If both players panic buy, then both are worse off (costs of storing, and a good chance that the store runs out of stock, but at least they have some toilet paper once they find some that is available to buy; utility = -5).

Paloyo identifies two equilibriums in this game (everyone acts normally, and everyone panic buys). They are what we call Nash equilibriums, and to confirm that they are Nash equilibriums in our game, we can use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the textbook definition of Nash equilibrium). In this game, the best responses are:

1. If Chris chooses to panic buy, Sam's best response is to panic buy (since -5 is a better payoff than -10) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If Chris chooses to act normally, Sam's best response is to act normally (since 0 is a better payoff than -2);
3. If Sam chooses to panic buy, Chris's best response is to panic buy (since -5 is a better payoff than -10); and
4. If Sam chooses to act normally, Chris's best response is to act normally (since 0 is a better payoff than -2).

Notice that, as Paloyo notes in his article, there are two Nash equilibriums in this game: (1) where both players choose to panic buy; and (2) where both players choose to act normally.

Which of these two equilibriums will obtain depends on what each player thinks the other player will do. So, if there is good reason to believe that the other player is acting normally, your best response is acting normally (and vice versa for the other player). But, if there is good reason to believe that the other player is panic buying, your best response is to panic buy too (and vice versa for the other player).

Notice that the acting normally equilibrium is better for both players than the panic buying equilibrium - everyone would be better off if everyone just acted normally. We refer to that as a Schelling Point (named after the Nobel Prize winner Thomas Schelling). In a coordination game (a game with more than one Nash equilibrium), a Schelling Point is the equilibrium that would be more likely to obtain - that's because everyone can see that the Schelling Point equilibrium is better for at least one player, and makes no player worse off (compared with any other Nash equilibrium). However, even though acting normally is a Schelling Point, and should be more likely (after all, it is the usual state of affairs), once people start panic buying, it suddenly becomes a better option for everyone to panic buy.

Which happens to be what we are seeing, with people desperately stockpiling toilet paper. Few people are stocking up on toilet paper because they think they need that much toilet paper. But no one wants to miss out on toilet paper because everyone else bought it up first. The outcome is everyone panic buying. Hopefully, some sanity will be restored by supermarkets, which are starting to impose limits on buyers. If acting normally is imposed on some buyers, and people can believe that others are being forced to act normally, then the Schelling Point equilibrium will be restored. And then the Great Toilet Paper Crisis will be over.