January 18, 2004

Cold Game Theory

I had the chance to become better acquainted with rhinoviruses and streptococci this week, i.e. I got a cold. Very annoying as it kept me from important work by making me more distractible than usual. Which includes analysing the game theory of the common cold, work and epidemics. In fact, I ended up doing a bit of evolutionary game theory while my immune system was doing its duty in its evolutionary game. Are discounts and fees the cure for the common cold?

In an ideal world everybody who had a cold would stay at home, keeping away from everybody else. This would reduce the spread of infection and make people in general healthier. In reality many people go to work anyway, work at somewhat reduced capacity (but earn money, keep up contacts etc) and increase the risk for everybody else.

This can be analysed as a mathematical game (for those of the readers who can't stand math, jump directly to policy implications).

Two Person Game

The simplest game might be between me and 'everyone else'. We both can choose between the staying at home when sick or going to work. Let's say the basic probability of getting a cold is P (about 3/52 per week for adults), and this is increased by alpha (a measure of infectiousness) if another player goes to work sick. Each week of healthy work gives utility 1 (wages, companionship etc), a sick week produces half of that (due to unavoidable absence when too sick etc) and a negative utility epsilon due to the sheer nastiness of being sick. If I stay at home completely I get utility -epsilon.

The utilities for me looks like this:

Others stay Others go
I stay(1-P) - epsilon*P(1-P-alpha)-epsilon*(P+alpha)
I go(1-P/2)-epsilon*P(1-(P+alpha)/2)-epsilon*(P+alpha)

For the other side it is identical, with the roles reversed. So if we say that epsilon is 1 (having a cold is bad), alpha is 10% of P (=3/52) we get the following numbers:

Others stay Others go
I stay0.88460.8731
I go0.91350.9048

So the smartest move I can do is to go to work when I have a cold, since even the worst outcome is better than the loss of income from staying at home. That others also think like this will lead to the bottom right corner, where we all get sicker but still are better off than in the altruist upper left corner! In this case, for these parameters, it is acceptable to give each other the cold.

What if we increase epsilon? For larger values of epsilon it becomes worse to be sick, and the extra utility for working does not compensate it. But it is also a Prisoners' Dilemma situation: if I know the others stay home, I'm better off working. Which they will also think, producing a situation where we all make each other worse off.

Others stay Others go
I stay0.36540.3019
I go0.39420.3337

Evolutionary Game Theory

This is OK for a simple analysis of what I guess most people's intuitions tell them. But what about having an entire population? And in reality sometimes we go, sometimes we don't: we use mixed strategies. I did some simulations of that for different values of alpha and epsilon (details and code at the end). Imagine a number of agents (who could be populations of people thinking identically), where each agent has an individual probability of staying home when sick. Each week agents become sick with probability P + alpha* number of sick agents last week. Each agent gets a score for that week equal to 1 if healthy and working, -epsilon if sick and home and 0.5-epsilon if sick and working. After 52 weeks the 20% agents with the lowest scores rethink their strategy and change it randomly, and the whole process is repeated.

Cold cases over one year, alpha=1/100, P=3/52, epsilon=1

So, what happens? First we see individual cold cases that tend to occur together in waves - epidemies. Agents that stay at home get less score than the workers, so they change their views and there is a trend towards higher likeliehood of working. The epidemics become more and more common, making it even worse to be a stay at home person. In the end nearly everybody works despite getting colds all the time, and the total utility has decreased to a low valyu. A real tragedy of the commons.

Utility and likeliehood to go to work for alpha=1.5/100

This is the case if alpha is too large. For lower alphas the tendency is markedly weaker. Going to work doesn't change so much, and the epidemics that really hit the stay-at-homes hard doesn't occur.

People who work when sick as a function of alpha and epsilon

One can also vary the value of the work done while sick at work. Assuming the value is very low of course makes it less enticing to go there, producing stay-at homes. If it is closer to normal utility then the opposite is true, and we get many who work and cause epidemics.

People working for 25% sick work utility (alpha vertical, epsilon horizontal)People working for 50% sick work utility (alpha vertical, epsilon horizontal)People working for 75% sick work utility (alpha vertical, epsilon horizontal)

Policy Implications

So, what are the policy implications? Sometimes it is rational to risk increased infection rates for oneself and others, especially when the disease is not very distressing, the increase is small or the loss of pay from being sick is large. In other situations people would be better off if they stayed at home but the competition from others cause them to go. How can this be avoided? One can try to lower alpha (like the Japanese with their masks), but it is likely that this is hard. Changing epsilon is likely impossible (better anti-cold medications?). But the loss of pay or utility from staying at home (or the pay while working sick) is amenable to change. If the loss of pay is low the incentive to work sick vanishes and the equilibrium is healthy stay at homes. Similarly, if the reward for coming sick to work is not near the ordinary reward people also stay home.

If you get paid sick leave, this model suggests a higher likeliehood of staying at home. But adding days without sick pay (5 "karensdagar" in Sweden) will promote going to work for some - and increase the risk of epidemics etc. Maybe a good solution for companies and organisations would be to either fine people who arrive sick - something that sounds quite cruel and would likely be impopular, but it might actually make everybody better off. Maybe these fines could be redistributed as a bonus for people who stayed at home when sick, reducing the bad effect of unpaid days. Still, it is a rather intrusive means and I wonder if it can ever be implemented (without some draconian Singapore style health dictatorship).

Being a libertarian I believe that one can have contracts about nearly everything, and I can certainly see and accept employment contracts that contain things like this. People who infect me should really pay for the losses they cause me. Since it is usually impossible to trace them, it actually makes a lot of sense of making a redistribution as above.

Maybe the same system can be used at the really infectious places in society, the place where the epidemics really start: at kindergartens and schools. Parents get discounts depending on whether they keep their kids at home when sick. Given how kids act as social network bridges that transmit disease across society, this might be a higher priority than fixing companies and other adult organisations.

Details and Code

I used Matlab to run the simulations - my typical quick and dirty simulations.

N=100; % Number of agents
xgo=rand(1,N); % Likeliehood of going
basesick=1/52; % Base prob of being sick
psick=ones(1,N)*basesick;  % Likeliehood of being sick next week
sick=zeros(1,N); % Who are sick
sickwork=0.5; % Loss of utility from work
epsilon=1; % How bad is it to be sick?
alpha=1.5*1/100; % How much infection does one person cause?

for tt=1:10*4
ff=[]; % Map of cases
for t=1:52
sick=(rand(1,N) ff=[ff; sick];
sickgoers=(sick>0).*(rand gowork=(sick==0)+sickwork*sickgoers; % They get paid
utility=utility-epsilon*sick+1*(gowork); % And we subtract the illness
psick=(gowork(1,N)>0)*(basesick+alpha*sum(sickgoers)); % Infection


ind=sortrows([utility' xgo']); % Sort the agents after score
xgo(rep)=rand(1,size(rep,2)); % Replace the worst 20%

meango=[meango mean(xgo(:))];
meanutility=[meanutility mean(utility)];


Posted by Anders at January 18, 2004 01:23 AM

Just an abstract that gives some useful information for the model:

J Occup Environ Med. 2002 Sep; 44(9): 822-9. Related Articles, Links

Productivity losses related to the common cold.

Bramley TJ, Lerner D, Sames M.

Health-related productivity assessments typically focus on chronic conditions; however, acute conditions, particularly colds, have the potential to cause substantial health-related productivity losses because of their high prevalence in working-age groups. This article presents the findings of a study conducted to estimate productivity loss due to cold by using a telephone-administered survey that measured three sources of loss: absenteeism, on-the-job productivity, and caregiver absenteeism. Each cold experienced by a working adult caused an average of 8.7 lost work hours (2.8 absenteeism hours; 5.9 hours of on-the-job loss), and 1.2 work hours were lost because of attending to children under the age of 13 who were suffering from colds. We conclude that the economic cost of lost productivity due to the common cold approaches $25 billion, of which $16.6 billion is attributed to on-the-job productivity loss, $8 billion is attributed to absenteeism, and $230 million is attributed to caregiver absenteeism.

The costs are impressive, although the numbers seem to be low (hours per cold? or hours per week?). In any case, useful for calibrating the model and extending it to handling children + parents.

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