Braess’ Paradox is a classic example (though not an actual paradox) showing how difficult it might be to design mechanisms within an environment selfish routing. In essence, we consider a source $s$ and a destination $t$ connected by two distinct paths. Here $c(x)$ is a cost function describing how long it will take to navigate through an edge; the $x$ parameter here denotes how many other commuters are on each edge.

In the initial network, the equilibrium lay by evenly distributing traffic half-half between the top route and the bottom route, with mean cost of 1.5. Suppose the cost represents the number of hours it takes, and the city planners held this big meeting decrying the state of the transportation system: “it is completely unacceptable that it would take an average commuter an hour and a half to go to work,” and so the urban planners decide to build an ultra-efficient bridge between node $v$ and $w$ that takes nearly negligible time to traverse, believing that the improved infrastructure will alleviate some of the congestion. To their surprise, a few days later it was reported that after the addition of the new road, the average commute time somehow increased to two hours [1]! What happened?

It turns out that after adding the negligible cost capacity, commuters on the top ledge soon realized that if they all took the turnpike to $w$, their second leg of the trip will only ever cost $x < 1$, and soon, everyone began imitating them, so eventually the only equilibrium remaining is the $s \to v \to w \to t$ path. This classic example is used to showcase the counterintuitive phenomenon where adding additional capacity to a network where agents are selfishly choosing their own route may reduce efficiency. We call this loss of efficiency  due to the relegation of central control the price of anarchy [2].

However, it should also be interesting to note that this same counterintuitive phenomenon was observed in the epidemic game of infection, where the underlying network is a social contact graph [3]. While this should make theoretical sense, as the agents of this game are individuals who are selfishly attempting to prevent themselves from becoming infected, it seems a little strange. I will give a brief overview of the method and model studied in Zhang, et al.’s report.

Each agent will choose between three different actions: vaccination, which will cost the individual an expensive price of $c$ but will give them 100% immunity against infection; laissez-faire (their words, not mine), in which the individual do not pay any price but will be the most prone to the disease (at an infection rate of $\lambda$, discussed later); and self-protection, which will cost the individual a more moderate price of $b < c < 1$ (representing the cost of lowered social contact, increased cautiousness, what have you) but will only give them a $\delta < 100\%$ chance of staving off the infection. Their experiments show that the initial distribution of of which agent chooses which method does not affect the observation of their analogue of Braess’ Paradox.

During each epidemic season, we will put off the vaccinated agents as well as $\delta$ of the self-protected individuals into the immune group $I$, and the laissez-faire and the rest of the self-protected individuals into the susceptible group $S$, and we will model the rest of this epidemic season using the SIR model [4] with the immune and susceptible groups given above, and compute the set of infected individuals. We will add a cost of 1 from each infected individual. Now, this is where the actual game comes in. After each epidemic season, each agent will have a probability of switching to one of their neighbor’s action/policy on infection prevention inversely proportional to the cost they paid in the previous epidemic season.

They found that, as expected, when $\delta$, the chance of self-protection succeeding, is low, almost everyone either chose vaccination or laissez-fair because there’s negligible benefit of choosing self-protection; similarly, when $\delta$ is high, almost everyone chose to either self-protect or laissez-fair, as there’s little benefit to pay for the expensive vaccine, and similarly the proportion of the population that gets infected, $R^\circ$, declines at this extreme. The interesting dynamic occurs with mild/moderately low levels of $\delta$: here, people are beginning to notice the benefits of self-protection, and because it costs lower, it begins to be more wildly adopted (selfishly); unfortunately self-protection is still not very effective, yet it mostly steals away from the vaccination group, so the number of infected people actually begins increasing with increased effectiveness of infection protection!

Increase in disease prevention “paradoxically” increases the percent of the population infected. The pink zone is when self-protection is ineffective so there is no adoption, the blue zone is when increases in medical advances decrease the chance of being infected, the green zone represents this paradox.

In a sense, in a utilitarian equilibrium perhaps governed by a centralized health bureau, an increase in immunization effectiveness should either stay or reduce the number of infected individuals (based on whether the cost of self-protection is feasible). But because the agents are selfish, they wish to reduce their own expected cost by switching off of the expensive vaccination even when they can afford it, and we see a spike in infection rates as a result. The price of this selfishness is given by the spike in infection rates, and it is the price of anarchy.