Reopening a Society During the Pandemic and the Revisited Prisoner’s Dilemma

At the last press conference, the Danish Prime Minister announced that, if everybody continues to behave according to the lockdown rules and the COVID-19 statistics in Denmark, then the society will start to reopen gradually in two weeks. This announcement is a clear signal that the long-term goals (healthy economy) should start to come into the picture along with the immediate goals of controlling the pandemic. This is a challenging problem since even the non-experts, such as myself, know that economy relies on the dynamics of people and goods, while the same dynamics may propel further infections and even a second wave of infections. Then, which strategy and policies should be adopted towards gradual reopening of the society?

The basic tradeoff in this process can be conveniently casted in the framework of Prisoner’s dilemma. Let us at first make a short digression to describe this classic example of a game in game theory; the word “prisoner” in this case is just an unfortunate association with the fact that during a pandemic lockdown we need to stay at home.

The Prisoner’s dilemma goes as follows. Two partners in crime, Alice and Bob, are imprisoned. Each of them is in her/his own cell and has no way to communicate with the other. The prosecutors talk separately to each of them. Each prisoner has two possible opportunities: (a) stay silent (a) defect and say that the other one has committed the crime. There are four possible outcomes:

(1) Alice silent, Bob silent: Each of them gets 2 years of prison.

(2) Alice silent, Bob defects: Alice gets 5 years of prison, Bob walks free.

(3) Alice defects, Bob silent: Alice walks free, Bob gets 5 years of prison.

(4) Alice defects, Bob defects: Each of them gets 4 years of prison.

Considering this, what should Alice do? The key element in the game is that Bob and Alice cannot communicate and coordinate; otherwise they would agree to stay silent and get a minimal prison.

This game is represented by the table on Figure 1. The pair (A, B) denotes that, for those chosen actions, Alice gets A years of prison and Bob gets B years of prison. The numbers - 2, -4, and -5 are utilities and in this example they are negative, since a lower prison sentence has a higher utility for the player (prisoner). The actual numbers chosen in the example are arbitrary, but the relationship between them plays a significant role. In this example Alice reasons as follows. At first she assumes that Bob stays silent: if she stays silent, she gets 2 years of prison and if she defects she gets 0; hence, she should defect. Next, she assumes that Bob defects: If she stays silent, then she gets 5 years of prison, while if she defects, she gets 4 years of prison; so she should always defect. Thus, both Alice and Bob conclude that they should always defect and this game ends with each of them getting 4 years of prison.

The intention here is not to analyze this game in detail, which has been exhaustively done in the literature, but just to point some of its features. For example, if the utility that one gets when both defect is 10 years of prison and we have (-10,-10) in the lower right corner, then the reasoning with the "defect" strategy, described above, does not work. Furthermore, if Alice and Bob somehow manage to communicate and cooperate, they can both agree to be silent and stay with 2 years of prison only.

Figure 1: Prisoner’s dilemma for Alice and Bob.

Now we change the roles of Alice and Bob, they are not prisoners, but just ordinary citizens under a pandemic lockdown. To simplify the model, let us assume that Alice and Bob play a game with only two available actions: StayHome of GoShopping. Let us call this the shopping game. Another key assumption in the game is that none of them has symptoms of the disease and thus does not know if she/he is infected or not. Finally, Alice and Bob do not know each other, so they cannot communicate and coordinate before going shopping.

At first, we define the utilities of Alice and Bob only by taking into account the chances to get infected; this is shown in Figure 2. Here the one that stays at home gets always the highest utility (2). The one that goes to the shop gets less utility (1) as she or he may be infected by, for example, virus that still resides on the shelves of the shop. But if both go to the shop then both get utility zero, translated as high chances for getting infected. Based on the utilities in Figure 2, it is clear that StayHome is the preferred action for both of them.

Figure 2: The shopping game with utilities defined based only on chances for infection.

The things get more complicated by considering that both Alice and Bob get an additional utility from going to the shop and satisfying some consumer need. Let us change the utilities to include consumer-related utility; an example is shown on Figure 3. With the chosen utility values, StayHome is not anymore the strategy that brings the highest utility. Note that if Alice knows that Bob is at home, she can get a higher utility by going to the shop.

Figure 3: The shopping game with utilities defined based on chances for infection and consumer need.

We need one more element in the utilities used in the shopping game: the overall economy. Figure 4 shows how the utilities would look like when there is no pandemic and no risk of infection. The rationale for choosing the utilities is as follows. If both Alice and Bob stay at home, the economy is stalled and none of them gets any utility on the short or long run. If only one of them goes shopping, she or he gets utility 2, but the one that stays at home also gets a positive utility. The latter is due to the fact that the economy is kept afloat and even the one that stays at home at this specific moment, gets some utility. When both of them are unconstrained to go shopping, then the economy goes well, the competition is high and the products are offered at good prices. Here the clear winner is GoShopping, that is, keep the economy going.

Figure 4: The shopping game in time when there is no pandemic and the utilities are defined based on the consumer need and the overall economy.

Thus, the key in the Shopping game is how the utility is determined, taking into account chances for infection, consumer satisfaction ,and the state of economy. Due to the large uncertainty during a pandemic, these utilities are changing from day to day. At the beginning of the pandemic, the utilities are close to ones from Figure 2, that is, the weight is put on staying healthy. As the pandemic starts to slow down, the utilities start to look as in Figure 3. But after a longer time, the utilities start to converge towards Figure 4, that is, there is an increased utility in both Alice and Bob choosing GoShopping.

Thus, Figure 3 represents the utilities when the pandemic slows down and the society needs to be reopened. Clearly, the desirable behavior here is to have Alice choose GoShopping when Bob chooses StayHome, and vice versa. Unlike the prisoner's dilemma, the shopping game can be optimized by the actions of the government and by the use of policies and techniques to coordinate the behavior of the population.

In the previous blog I wrote about how to use the principles of wireless communication protocols to create policies and induce behaviors that minimize the encounters of customers in a shop during a pandemic. In that spirit, the central question when reopening a society after a pandemic is: how to introduce policies that ensure minimization of chances for infection, while still keeping the economy running? These policies can be implemented by using, for example, digital technologies and moblie apps for coordinating the access of users to public spaces. However, in that case the utility should take into account the problems created for user privacy and restriction of citizens' freedom. Thus, the implementation of policies that will coordinate the population according to the utilities on Figure 3 is a challenging (but exciting) problem with multiple dimensions, where engineering methods can play a significant role.