Pandemic Chronicles: Vaccination and the Tipping Point to Herd Immunity

Society is at a tipping point with two critical issues: climate change and the COVID-19 global pandemic. A tipping point is a phenomenon where enough people have collectively changed their social behavior to push society towards either a positive or negative direction. The climate change crisis is similar to the current public health pandemic in that it requires coordination among individual members of society in order to avert a disaster.

Ideally, countries would make good on their treaty commitments by investing in clean energies and take steps to meet greenhouse gas reduction targets. But since the Paris climate change treaty is voluntary and non-legally binding, what actually happens is some countries set meaningful national targets and takes steps to get there while other countries free ride by staying their current course.

In a Nature article “Tipping Climate Cooperation,” the authors use game theory to explain the global coordination dilemma between nations to address a problem that might become dangerous some day but not immediate.Their insight shows that When the uncertainty is small, however, societies are much more likely to coordinate efforts and avoid the tipping point.

In climate change forums, countries like America and China are strategic actors who weigh the choice of cutting or not cutting greenhouse gases depending on their payoffs. Each country’s pay-off (or expected benefits) in treaty adoption are dependent on the other country’s choice. As the payoff matrix above shows, there are two states of the world that represents Nash equilibria. A Nash equilibrium is a condition where neither country can improve its payoff or outcome by changing its strategy.

One Nash equilibrium captures the ideal scenario in which both countries don’t renege on their treat commitments and take actions at home to cut CO2 emissions. But the other state of the world (i.e. Nash equilibrium) is where neither country chooses to invest in clean energy or measures to cut carbon emissions. That’s a less desirable scenario because society is worse off. Yet, given then uncertainty around climate change risks in the short term, there is no immediate pending disaster.

Because of short-sightedness or lack of domestic support, free rider countries are less likely to jump on the bandwagon. To reach the ideal scenario, several things need to happen, including directly observing the real catastrophic impact of extreme weather patterns, or improving scientific research to reduce the uncertainty about the tipping point. These social, political, and economic considerations are similar to the current state of the COVID-19 pandemic.

Herd Immunity – Coordination Problem

At the beginning of May 2021 the U.S. government and the Centers for Disease Control and Prevention updated its guidance to allow Americans to go about their daily lives without a mask if they are fully vaccinated while calling out those unvaccinated to get vaccinated in order to get the same treatment in common public squares. After an intensive COVID-19 vaccination campaign to reach millions of Americans, experts believe America is at an inflection or tipping point to averting the raging pandemic. Previously, there were a lot of uncertainty whether we would reach the end of the tunnel on this dark chapter in the modern public health crisis. But since vaccinations and aggressive public health measures have taken place, more people will have an incentive to get vaccinated. Why? The tipping point.

Tipping point in collective behavior: over half of the U.S. are vaccinated. The unvaccinated are now the minority. Monthly cases and deaths are trending down. Society, governments, and organizations are rewarding the vaccinated to prompt social change in the unvaccinated population.

There are places that people would like to attend, like places of worship, grocery stores, and night clubs. These places have different levels of restrictions for those who are not vaccinated while allowing those who are vaccinated to enter without wearing a mask.

Chart: CDC's New Mask Guidance for Vaccinated, Unvaccinated People
Source: CDC Guidance

For churches, since it’s a place of faith, those centers recommend attendees who aren’t vaccinated to follow health guidelines by wearing masks and maintaining social distance. Worship centers tend to not impose hard requirement to show attendees are vaccinated, rather it’s an honor system.

Church Mask Policy Change

Nightclubs and Ride Shares

Nightclubs, on the other hand, are social spaces that attract people of all backgrounds. People interact closely and crowds tend to be more concentrated. Those places have an incentive to impose a requirement for people to show evidence of vaccination to enter. For example, Rumi is a nightclub in New York City populated with millenials and Genzers. To enter, customers go through two checkpoints: (1) age verification and temperature check; (2) vaccination card. These same people often commute using ride sharing services including Uber and Lyft, both of which have stated that they will maintain current mask restrictions for all customers (or no service).

Shopping Centers

Grocery stores approach the new government guidance in different ways, ranging from requiring all shoppers to still wear masks and maintain social distance to allowing the vaccinated to be maskless. Because of either hard requirements or public pressures from a large number of people who are vaccinated, people who are unvaccinated will have an incentive to get vaccinated. The combination of these policies carried out by private organizations, accompanied by local ordinances and requirements, as well as sustained commitment and guidance by the government to end the pandemic, will help push American society over the tipping point.

Consider the scenario below, where each Customer can choose to get vaccinated or not get vaccinated. Their individual choices and payoffs are dependent on what the other Customer do. If Customer 1 takes an effort to vaccinate, Customer 2 could be a free rider and not vaccinate. Why? Customer 2 may not want to take the effort, and may want to take advantage of others getting vaccinated and free ride off a perceived herd immunity. The expected payoff for C1 and C2 is (1,2), respectively.

There are two important outcomes from this decision matrix. The first is where both Customer 1 and Customer 2 vaccinate, which is the desired Nash Equilibrium, since neither Customer could increase its payoff by changing their strategy. Both Customers 1 and 2 choose to vaccinate after seeing a certain threshold of the population get vaccinated with minimal risk, while the vaccinated have the benefits of attending more more venues and activities. If unvaccinated customers wish to get the benefit accruing to vaccinated customers, then the unvaccinated have an incentive to get vaccinated. If enough unvaccinated people think that way, this establishes a tipping point in the health pandemic that inches us close to herd immunity. The other scenario, the undesirable Nash Equilibrium, is where Customer 1 and Customer 2 do not vaccinate. Here the expected payoffs are (2,2) for Customer 1 and 2, respectively. The social drivers for customers making this choice could be the following:

  1. if they’re skeptical of vaccines;
  2. if the unvaccinated don’t think there’s enough supply
  3. if they believe the pandemic will worsen (no light at the end of the tunnel)
  4. if the unvaccinated think their other people will also stay unvaccinated.

If the above drivers are strong enough, then it pushes society away from the tipping point and further away from herd immunity. The choices that we as individuals make based on our own expected payoff can put our friends, neighbors and overall society at risk.


Lenton, Timothy M. 2014. Game Theory: Tipping Climate Cooperation. Nature Climate Change 4 (1): 14–15

Why People Hoard and How Game Theory Can Help Stop a Crisis?

This is part of my pamphlet series Pandemic Chronicles that looks at the Coronavirus public health and economic crisis with the lens of game theory.

Hoarding behavior is not new. It happens when people feel the need to accumulate things regardless of its perceived value whether they’re sports cards or antique pens. During a normal environment, collecting goods for private consumption does not harm anyone; it reflects that person’s individual preference. But during a public health crisis, like the COVID-19 pandemic or the Spanish Influenza 100 years ago, hoarding behavior negatively affects other people and society. There are several reasons why:

  • Amassing goods takes away the supply of goods for people who need it more like the elderly, women and children.
  • Piling on large quantities causes supply disruptions and artificially increases prices and thus opening opportunities for black markets. Black markets draw in actors with perverse incentives to manipulate and take advantage of other consumers.

To understand what drives hoarding and potential ways to address it, we can draw lessons from game theory and the economics of stockpiling.

Shoppers vying for last unit of paper towels.

The Psychology of Hoarding

Individual hoarding can impact society when it becomes a larger panic buying scenario that causes rationing and irrationally high prices. In a crisis, where people, markets, and government may not have all the answers, it fuels a desire to hoard. When the COVID pandemic struck with the first wave in 2020, it wasn’t uncommon to see empty shelves at American stores, especially for consumer goods such as hand sanitizer and a range of toiletries. At the store, customers are behaving in strategic interactions of whether to stock up on certain goods or engage in normal buying based on their needs for the day or week. However, in a pandemic, thew news and social media can fuel anxiety and the perception of scarcity. If panic sets in, customers have reasons to stockpile.

Consider a scenario where two customers walk into a store:

In this situation the customers make independent decisions whether they should engage in normal buying behavior or hoarding. They both have all information available to make a decision, including the price and quantity on the shelf. It’s been a tense few weeks since COVID cases have spiked and the area is still in locked down except for essential businesses. Social media communities have shared posts of stores running low on supplies. On the way in there was a line of customers who had already filled up their shopping cart with supplies.

With that backdrop, Customers 1 and 2 arrived at their relevant aisle. For Customer 2, if she thinks that Customer 1 will try to hoard, then Customer 2 best hoard because if she acts business as usual, she would not able to get the item she needs. Vice versa, when Customer 1 thinks that Customer 2 will hoard, he has an incentive to panic buy. When both customers choose to hoard, the whole store is worse off. Ideally, both customers adopts normal buying behavior but in a non-cooperative situation where newspaper and social media propagate fear and anxiety, grocery stores had to take drastic actions by imposing quota per customer which is an enforcement mechanism against individual irrational actions that harm the public good.

Stores impose rationing policy with 1 unit/household cap

Hoarding more than you need is not necessarily irrational behavior. Like other strategic situations, game theory helps us understand the social psychology of a person’s thinking in response to how other’s act. If you see others panic buy, your instinct is to buy the goods before the other person buys it. Each person’s optimal strategy is to beat the other guy to the paper towels or pasta package. That is represented by a Nash Equilibrium where each player has not incentive to deviate from their strategy. It is a no regret strategy given what they know about the situation. But the dreaded Nash Equilibrium in this case is sub-optimal for society. If enough people stockpile, the result are supply shortages and hefty prices. In 2020, we saw a demand surge for essential goods, masks, hand sanitizers, certain medicines, and various groceries. This demand run up of consumer goods, coupled with the economic lockdown of non-essential businesses, disrupted supply chains, which further put pressure on store supplies.

How do we stop panic buying?

There needs to be ways to counter people’s greed and fear. There’s social psychology, a belief that others are greedy. If that’s true, that incites panic buying, in order to not be the sucker (i.e. sucker’s payoff). That social psychology becomes a negative self-fulfilling prophecy that begets further hoarding.If everyone buys just what they need, there would not be shortages. These solutions could involve encouraging kinship, socially responsible behavior, explaining to people that this is a long-term game, signaling to shoppers that supplies are coming.

EConomics of Hospital Stockpiling

Large quantity purchases also caused supply shortages for healthcare front line workers. Masks, gloves, and respirators, which are personal protective equipment (PPE) for medical staff and hospitals, became scarce during Q1 2020. In the medical mask market, hospitals and states resorted to alternative channels to only find themselves competing against hucksters, profiteers, governments, and individual hoarders. Facing a global demand surge and scarce market, buyers faced the choice of acting as a good neighbor or panic buy. Just like customers in the stores, medical supply purchases do not want to be the sucker and miss out on supplies. There are reports of people stockpiling medical masks and selling it on the secondary market. That drove up prices which attracted copycats who sold poor quality medical masks. When official government guidance came out recommending mask wearing in public spaces, the general population went to alternative markets online and other shady networks, which exacerbated the problem but presented more opportunities for profiteers.

Solutions to Medical Supply Shortages

The motivation for stockpiling medical equipment is the same as collecting large quantities of household goods. However, countermeasures to the medical supply problem must be broader and forceful given the impact on the hospitals, doctors and people who are the nation’s first line of defense.

In a broken market with competing demands, one option is to rely on the federal government to serve as an efficient coordinator to purchase and manage distribution of supplies. The executive branch can invoke the National Defense Production Act to ramp up the manufacturing of masks, face shields, and other PPE. Historically, presidential administrations have used this authority to purchase critical military equipment to support infrastructure repairs and impacted citizens following natural disasters. In the same way, the government can leverage that authority to centralize supply management on strategic goods and services to deal with a global health pandemic. The government can deploy this policy strategy to fight a raging crisis by compelling private entities to produce medical supplies. When society is on a path to recovery, the same authority allows the government to guarantee purchases of the vaccine supply and manage its distribution.

For a broader crisis fighting strategy, strong leadership and communication to inform society and individuals. Hoarding behavior is rooted in panic and uncertainty about the future. That can be part of public health campaign to inform and address people’s doubt, fear and uncertainty. Private organizations also found creative ways to institute quotas on goods and services. For example, purchases of high demand products such as masks are subject to a cap per household. Stores used its soft power by posting signs that encourage customers to respect one another by not panic buying while maintaining a social distance from other shoppers.


With any crisis, decision-makers need to understand social psychology with empathy and putting themselves in people’s shoes. Game theory could be useful to understand how people make decisions in reaction to other people’s actions and what they see on the news. Once we understand how social psychology can break markets and worsen the crisis, governments and practitioners can use a combination of hard power like presidential authority and public health information campaigns to address the pain points so that our critical infrastructure and front line staff can continue to perform their duties to get society back to normal.

Game Theory of Political Negotiation

This is a more analytical approach to my post on the Dinner Bargain of 1790 that depicted the drama between Secretary of Treasury Hamilton and his political rivals over proposals to resolve state war debts and set up a national bank. Check out that post for a refresher on the story!

This piece will look at the bargaining process between the Federalists, led by Alexander Hamilton, and anti-Federalist coalition, from a game theory perspective. There are several aspects to consider:

Game Design

Human conflict is very complex to predict but game theory can be a useful tool to glean insights into the thought process of the Founding Fathers as they went through the negotiation process. It’s a high impact game and the course of the young republic were tied to the outcome. Game theory applies math to think through how the situation evolves based on strategic interactions between the players.

On one side, there’s Hamilton who is the standard-bearer for Federalists, advocating for the federal government to exert a strong influence on economics and politics. On the other end, Madison and Jefferson represent the coalition of anti-federalists and the Southern bloc who believe power resides with states. Jefferson, a wealthy land-owner who believes in a pastoral America, wants to delay government intervention even facing a fiscal crisis.

Chicken Game

The Chicken dilemma game is one type of game that illustrates the showdown between Hamilton and his opposition. It’s like a modern-day version of political logrolling or horse trade. In Chicken, each player has two strategies: swerve or go straight. Each play does not have a dominating strategy, unlike a prisoner’s dilemma game. Each player adopts a counter-coordination strategy against the opponent. If the Player 2 swerves, then Player 1 would go straight. If Player 2 goes straight, then Player 1 would swerve. Each player is testing each other’s patience and stays resolve until the other player gives up and back out.

When each player enacts their optimal strategy given the strategy taken by the other player, this is referred to as a Nash equilibrium. In the chicken game, there are two pure strategy Nash equilibria:

  1. Player 1 goes straight and P2 swerves. The payoff matrix results in +1 for Player 1 since she stayed strong and -1 for Player 2 since he backs out. 
  2. Player 2 goes straight and Player 1 swerves. There’s an incentive for a player to continue straight while the other player swerves. In a mirror outcome, the payoff matrix results in +1 for Player 2 since he stayed strong and -1 for Player 1 since she could not stomach it.  

But suppose the two scenarios above don’t happen, then it’s possible both players would swerve. If both don’t swerve, then both continue straight and ultimately collide. The worst payoff results from the two players running straight at each other. 

A strategy where both players go straight, both players have an incentive to change strategy given it’s the worst outcome for both players. If collectively choose to keep going, one would switch strategy by swerving, for example player 2 swerve because 0 is better than negative utility if gone straight. Player 1 has an incentive to change strategy to swerve. Keep going and Keep going will not be played strategy in a pure strategy Nash equilibrium. If P1 goes straight and P2 swerves, P2 doesn’t have an incentive to change strategy by going straight because it would yield a negative utility of -5.

What exactly is a Nash Equilibrium then? For a player, this entails a set of strategies such that the player has no incentive to change strategy given what the other players are doing. Although we know someone will swerve, but we may not know which one will serve in a Chicken Game. However, in reality, we know Hamilton had more interests in cooperating or making bargains in order to get his economic proposals approved by Congress.

Just like in the standard Chicken design, there are two pure strategy Nash equilibria in this game:

(1) Federalist swerve and Anti-federalist goes straight; 

(2) Anti-federalist goes swerve while Federalists go straight. It’s an anti-coordination game.

Mutual defects by both parties would be disastrous for America already in shaky conditions. Since Founding Fathers were statesmen who believe in moving the country forward,  it’s better for a player to swerve (i.e. cooperate) rather than all players defect. Applying chicken to the debate among our Founding Fathers, it’s a classic Congressional gridlock.  It’s mutually beneficial for the Federalists and anti-Federalists to adopt different strategies. Federalists wanted a strong government and preemptive federal assumption of state government debt. Anti-federalists rather the federal government disappear from the picture even amidst the crisis. Madison led the Southern state coalition, wanted to pass the Permanent Residence Bill to establish permanent Federal Government and national capital outside of Virginia. Hamilton, “Federalist coalition”, wanted to pass the Funding bill allowing the federal government to assume state debt. 

This is a repeated game since policy debates and negotiations are ongoing even before the 1790 Bargain. Each side has reputational incentive to win/lose. The ultimate loser is the U.S. economy and possibly its ability to forge ahead as a new country under acute fragile economic conditions and fractured leadership. Breaking the gridlock was Secretary of State Jefferson who set up a dinner between Madison and Hamilton to knock out a bargain. 

The game of chicken has played out over the course of history between presidents and Congress. In recent history, to force the other side to accept a downsized government budget, President Reagan ordered the suspension of government services which impacted 800,000 federal employees. That same day the U.S. Congress resolved the situation by approving funding of the government.

Decades later in 1995, President Bill Clinton and his Treasury Secretary Robert Rubin had a record showdown against Congressional Republicans. Congressional leaders opposed the President’s offer to keep the government open during the negotiation. During President Barack Obama’s showdown with Republican leaders in Congress, the government also temporarily laid off 800,000 federal employees.

Prisoner’s Dilemma

Rather than the Chicken Game, we can dissect the political negotiation process using a slightly different game design known as the Prisoner’s Dilemma (PD). Suppose two criminals from Al Capone’s gang are caught. Each are seated in a separate room. This is a single game with no communication between the two players. The FBI offers each a choice: either rat out the gang or stay silent. By design, this is a non-cooperative game with dominant strategies where each player has a preferred strategy. The dominant strategy for each player is to defect since it gives the highest payoff regardless of what the other player does. 

Player 1’s preferred strategy is to defect, ratting out the gang first since he wants a reduced sentence, thinking that the other guy will likely rat out the gang anyways. Player 2’s preferred strategy is to defect. When both guys rat out the gang, they indict the entire gang. So both players’s default strategy is to defect given what the other player is doing in the holding room. The expected outcome of Prisoner’s Dilemma is defect-defect or otherwise known as the Nash Equilibrium. However, it’s a Pareto suboptimal outcome–not an ideal result. The optimal outcome would be both players cooperating by staying silent so the police can’t indict anyone. Collectively that would be an efficient result but individual preferred strategy leads them astray.

Whether it’s the Chicken Game or the Prisoner’s Dilemma, there is a broader takeaway for U.S. economic policy. We already saw the dynamics between Hamilton and Madison are similar to modern fiscal showdowns in Congress with the decision of whether to approve the budget to keep the government open or shutdown government services. It’s clear that by Constitutional design, separation of power between the Executive Branch and Congress, as well as checks and balances between political parties, can lead to brinkmanship. This phenomenon is secular to the party in power or political affiliation of the US President.