Social Distancing: A Game of Two Locales

The coronavirus pandemic which began late last year in the Chinese city of Wuhan has now spread globally. It continues to take a huge toll on major Western economies including the hardest hit to date: Italy, Spain, the United Kingdom, and the United States.

The slowing down of the virus, which is now conservatively estimated as going to have fatalities in the US upwards of 200,000 when it is all over, is premised on continued social distancing and other safety measures being judiciously carried out, as advised by medical and healthcare professionals.

Globally, the word is also out - countries have to follow the template of social distancing to slow the spread of the virus. This is all the more pressing for developing countries with merely a fraction of the number and quality of healthcare provisions the West has, and yet is struggling to cope.

The media is filled daily with reports of hospitals and treatment facilities having to come up with protocols on who to treat and who not to treat, should it get to that point, due to not having enough treatment apparatus including mechanical ventilations, ICU beds, oxygen therapies etc. This has heightened the sense of desperation in many countries with far fewer capabilities, and very large and densely populated cities, which have so far been spared the onslaught of the virus, but have started showing to be in their early phases of being infected by the virus.

Governments in these nations have therefore taken the template of social distancing and sought to apply the same within their countries, but here is a difficulty that they face. Many of their citizens, in particular the poor, while having been explained to the need for the policy, are all the more perturbed about the fact that for them it is more or less an equally undesirable death sentence to comply. I illustrate this with a very basic game theoretic model below:

The Social Distancing Game

Consider a relatively low income developing nation with a large population, such as Nigeria or any other sub-Saharan African or South-East asian country with similar statistics. I will divide this country into two types of families - the rich and the poor.

My delineation is reflective of the very stark income inequality that persists where the rich are simply those who can, on short notice, stock up on food supplies and other basic necessities able to last them for a few to 14 days. The poor cannot do this, and have to live on a day to day basis from a social hustle that has come to define their existence.

The societal objective is to mitigate the overall impact of the pandemic by slowing the spread through social distancing that is effected through self isolation at home.

Both the Rich and the Poor are faced with two strategy choices: to self isolate at home (SI) or not to self isolate (N) over the next couple weeks. The payoffs from these choices are given in the table below:

Let me explain the structure of the payoffs and the nature of the game. The game is a static one, where each player (the rich or the poor) chooses the strategy that gives her the highest payoff, given what she knows the other player will choose to play. So there is perfect information in the model and payoffs are known.

The payoffs are numerical magnitudes to capture some normalized degree of pleasure or satisfaction associated with the strategy choice employed by a player, given the strategy employed by the other player, during and after the end of the two-week policy period. The row player's payoff is written first, followed by that of the column player's.

So if the Rich decides to self isolate and the Poor also does the same, the payoff to the Rich is 10, and to the Poor it is 1. If the Rich self isolates, and the Poor does not, the payoff to the Rich is 4, while the payoff to the Poor is 2. If the Rich decides not to self isolate, but the Poor self isolates, the payoff to the Rich is 5, while it is 0 to the poor. If they both do not self isolate, then the Rich gets a payoff of 3 and the Poor a payoff of 2.

One can think of these payoffs as reflecting certain realities about what self isolating means and the importance of it in the current pandemic. The (10,1) payoff reflects the fact that if everyone self isolates, the Rich are the main beneficiaries, as they survive both on their own sacrifice and that of the Poor. The Poor are worse off because they would rather be out of their homes on a social hustle, but they still benefit a bit because the social objective is achieved. Perhaps, social inequality gets also more entrenched.

The (3, 2) payoff reflects the fact that if they both do not self isolate, then they both lose, but the poor, at least get to enjoy some of their daily social hustle, which to them might be better off than having to die of starvation or from lacking life preserving necessities. The payoff of (5, 0) reflects a bleak outcome, and a low payoff for the Rich when they do not self isolate but the Poor do, because the former still suffer but the latter experience the worst outcome because they deprive themselves of their social hustle and have absolutely no gain as the societal objective is not met.

Finally, the (4, 2) payoff from the Rich self isolating and the Poor choosing not to, again reflects the lower payoff from the Rich having to self isolate, but the Poor don't. The outcome is bleak for both, but the poor get to enjoy their social hustle even though the societal objective is not met.

From the structure of payoffs both the Rich and the Poor have what we call dominant strategies, but they are opposite. A dominant strategy is one that is played regardless of what the other player plays. The Poor will always play N (not self isolate) and the Rich will always play SI (self isolate). This means Society will end up with an outcome (4, 2), that is suboptimal from an efficiency perspective but perhaps more equitable at the end of the day.

Introducing Transfers...

However this outcome need not occur if we allow for well designed transfers.

Consider a modified game with transfers, where we have food and basic necessities being provided for the poor by the rich - maybe a tax-transfer scheme being put in place for the 14 day period. This allows the Poor to have what it takes to self-isolate in lieu of attempting to make ends meet by social hustling, so the payoffs now look as follows:

Notice that the payoffs have changed significantly. (7, 4) for example, now reflects the explicit transfer needed to increase the wellbeing of the Poor who enjoy 4 now from self-isolating compared to 1 in the previous game. The Rich also lose in all their payoffs equivalent to the transfers. In this case they enjoy 7 when both they and the Poor self-isolate.

Further, the payoffs are reduced to 0 for the Poor when they choose to not self isolate even though they were provided the transfers, since that is being irrational in wasting these supplies and embracing higher risk from dying as they are more likely to be infected and to get passed over in the allocation of scarce treatments.

Now we see that the transfers makes self isolation a dominant strategy for both the Rich and the Poor. By analogous reasoning, the outcome (7, 4) prevails in this game and is also efficient and more egalitarian though the status quo class structure is still maintained. So the transfers are desirable, especially when compared with the payoffs that result from doing otherwise (4, 2) in the previous game.

Lessons for Market Economies

The foregoing has focused on developing countries with particular characteristics, but one can readily see applications around the world, both in the current coronavirus climate and generally even for capitalist economies. We are often wary of tax-transfer schemes because of the presumed efficiency loss from the taxation, and the potential for distorting incentives away from productive work on the part of the beneficiaries.

However an underlying argument for such tax-transfer schemes is for the provision of public goods that perhaps the Rich have a relatively much stronger preference for in seeing them provided, compared to the Poor. Have we considered that "social stability" is a public good. It is non-rival and non-excludable - the characteristics that define such goods.

Public goods by design will be inefficiently under-provided without any explicit mechanism in our institutional design to provide for them.

The rich need this "good" because on its provision does the capitalist system function, as that allows for property rights ownership to be well defined and enforced, and also market transactions to be carried out. While the poor also value this "good", perhaps not as much as the rich.

A price structure that needs to be put in place to ensure that this public good is provided for would call for a version of a tax-transfer scheme - The rich are taxed and a social welfare net provided for the poor in such a way as to ensure that social stability is provided by preventing the proverbial natives from growing restless as the capitalist system creates inevitable income inequality.

Perhaps that is the major justification for a progressive income tax structure more generally, and specifically, for why any stimulus being provided in advanced economies should be focused primarily on the relatively poorer populations and not as much on the rich, except on the corporations that need to be actively involved in provision of the basic necessities needed and in procuring the treatments needed to combat the virus.

This is not socialist, it is every bit capitalist, and it is efficient.