A mathematical analysis of public avoidance behavior during epidemics using game theory

Abstract

The decision of individuals to engage in public avoidance during epidemics is modeled and studied using game theory. The analysis reveals that the set of Nash equilibria of the model, as well as how the equilibria compare to the social optimum, depend on the contact function that governs the rate at which encounters occur in public. If the contact ratio – defined to be the ratio of the contact rate to the number of people out in public – is increasing with the number of people out in public, then there exists a unique Nash equilibrium. Moreover, in equilibrium, the amount of public avoidance is too low with respect to social welfare. On the other hand, if the contact ratio is decreasing in the number of people out in public, then there can be multiple Nash equilibria, none of which is in general socially optimal. Furthermore, the amount of public avoidance in equilibrium with a decreasing contact ratio is too high in that social welfare can be increased if more susceptible individuals choose to go out in public. In the special case where the contact ratio does not vary with the number of people out in public, there is a unique Nash equilibrium, and it is also the socially optimal outcome.

Introduction

Recent modeling studies have shown that social distancing, which refers to measures taken by individuals or communities to reduce contacts between people, can be highly effective at curbing the spread of infectious diseases such as pandemic influenza (Caley et al., 2008, Davey et al., 2008, Glass et al., 2006, Halloran et al., 2008, Kelso et al., 2009, Milne et al., 2008). Naturally, the extent to which social distancing can contain an epidemic depends on the specific policies that communities choose to adopt and the incentives that individuals have for staying home and avoiding public places.

The decision of individuals to engage in public avoidance behavior during an epidemic need not be a straightforward one. On the one hand, avoiding public gatherings reduces the likelihood of contacts with infected individuals—and hence the chances of acquiring an infection. On the other hand, staying home means forgoing certain activities – such as going to work or school, shopping, vacationing, or socializing – that contribute to the well-being of an individual. Therefore, choosing whether to engage in public avoidance when there is an outbreak entails evaluating the relative cost and benefit of staying home and reducing contact with other people.

As pointed out in a recent paper on this topic (Chen et al., 2011), the result of this cost-benefit analysis for an individual regarding the merit of public avoidance depends critically on the actions of other people—specifically, the decisions of other people to engage in public avoidance themselves. There are two reasons for this: (i) other people's decision to stay home or go out in public affects the rate with which an individual out in public will come into contact with others; and (ii) other people's decision to stay home or got out can – if there is a significant difference in how much public interactions infected individuals and uninfected individuals engage in – affect the proportion of infected people who are out in public. Both factors influence a susceptible individual's probability of getting infected if the individual chooses to go out in public. Therefore, other people's actions affect an individual's cost of going out in public, and hence an individual's decision to avoid public places.

In the parlance of game theory, there is strategic interaction between the actions of individuals with regards to public avoidance behavior, and one major objective of this paper is to apply the concepts and tools from game theory to study the incentives that susceptible individuals have for engaging in public avoidance behavior during an epidemic. It is important to point out that this is a key distinction between the earlier paper on this topic (Chen et al., 2011) and the current paper. The earlier paper assumes that the players in the public avoidance game (‘game’ in the sense of game theory) have adaptive expectations regarding the behavior of other players: players use the past behavior of others to forecast their future behavior; more specifically, the behavior of players at any time period is a best response to the actions of other players in the previous time period. By contrast, in the current paper, the solution concept of a Nash equilibrium from game theory is employed to study the behavior of players in the public avoidance game: in any time period, the behavior of every player is a best response to the actions of other players in the same time period. As we will see, the results of the analysis depend critically on the assumptions that we make regarding players' expectations of how other players will behave in the public avoidance game.

A second objective of the paper here is to compare the equilibrium of the public avoidance game – the outcome that obtains when all players act out of self-interest – with the social optimum—the outcome that is best for the community as a whole. It is well known from the voluntary vaccinations literature that, in general, the equilibrium amount of vaccinations is too low relative to the socially optimal amount (Fine and Clarkson, 1986, Brito et al., 1991). This obtains since there is a positive externality associated with vaccinations: one's act of getting immunized confers a benefit on other people, but these external benefits do not factor into a self-interested individual's vaccination decision; hence there is a divergence between individual interest and collective interest. In particular, since the benefit to an individual of getting vaccinated is lower than the collective benefit of having that individual vaccinated, there is too little – relative to the socially optimal outcome – vaccination in equilibrium. We will see, in the analysis that follows, whether or not these insights and conclusions gleaned from the study of voluntary vaccinations carry over to the case of public avoidance behavior. Specifically, the paper considers how the equilibrium amount of public avoidance compares to that in the socially optimal outcome.

Although public avoidance behavior during epidemics is also analyzed in a recent paper by Reluga (2010), it should be noted that in his model, an individual's likelihood of getting infected depends only on the individual's own action and is completely unaffected by the actions of other people. This means, in particular, that in his analysis an individual's number of contacts, as well as the fraction of those contacts that are infected, are assumed not to depend on the behavior of other people. Therefore, the strategic interactions between the actions of individuals with regards to public avoidance behavior – one major focus of the paper here – are entirely absent in Reluga's model. In fact, as we will see later, the Reluga model can be considered a special case of the general model presented in this paper.

The analysis here reveals that the set of Nash equilibria of the public avoidance game, as well as how the equilibria compare to the social optimum, depend on the contact function that governs the rate at which encounters occur in public, specifically on how the rate of contact varies with the number of people who choose to go out in public. If the contact ratio – defined here to be the ratio of the contact rate to the number of people out in public – is increasing with the number of people out in public, then there exists a unique Nash equilibrium. Moreover, in equilibrium, the amount of public avoidance is too low with respect to social welfare. On the other hand, if the contact ratio is decreasing in the number of people out in public, then there can be multiple Nash equilibria, none of which is in general socially optimal. Furthermore, the amount of public avoidance in equilibrium with a decreasing contact ratio is too high in that social welfare can be increased if more susceptible individuals choose to go out in public. In the special case where the contact ratio does not vary with the number of people out in public, there is a unique Nash equilibrium, and it is also the socially optimal outcome.

The paper is organized as follows. Section 2 introduces the model. The set of Nash equilibria of the public avoidance game is characterized in Section 3. In Section 4, the socially optimal outcome of the public avoidance game is considered, and comparisons between the set of equilibria and the social optimum are made. A summary of the results, as well as concluding remarks, are given in Section 5.