Game Theory at Work: Italy’s “Gentle” Lockdowns

Why Rome might Fall…this Fall

Photo by Gabriella Clare Marino on Unsplash: Roman Forum

Introduction: Italy’s Gentler Approach to Lockdowns

As the COVID vaccine is only a few months away and the antibody cure and other treatment options are getting cleared for use, humanity is coming to a close on the war against the virus. Therefore, governments are determined to minimize the casualties on the last battleground and bring as many of us as possible the other side of COVID.

Although countries have prioritized public health over economic concerns, they want to do so while not letting too many businesses go down and too many workers suffer from job losses. In an attempt to strike the tricky balance, the Italian government set out a national lockdown policy which adjusts restrictions based on the rate of spread of the infection as captured by Rt values. The policy color-codes Italy’s 21 regions as yellow, orange, and red depending on the local Rt value, thus allegedly protecting the economy to a greater extent compared to the one-size-fits-all lockdowns which tamed the first wave of infections back in March 2020.

The Italian government has not paid heed to the plea made by the Italian medical board for a March-style lockdown to be rolled out nationwide. The economic damage from locking down the country must be also taken into account — has been the contention of the government sidelining the concerns of the doctors. On their part, doctors have argued that a policy of locking down and releasing — a road that the Israeli government went down with apparent success — might do an even better job at shielding businesses and jobs. Even minor restrictions such as curfews and limitations to regional mobility might be just too much, doctors have pointed out, for the food & travel industries to turn over profits. Firms operating in high-risk industries run on very tight margins and need scale to stay in business.
The cost-benefit assessment of different virus containment policies is not the main topic I will be touching on in this article though — if interested, check the rundown of economists Favero, Ichino, and Rustichini. Rather, I want to establish whether the Italian government’s stated objective of keeping people home by using a softer touch is within the reach of the policy itself. As a spoiler or as a disclaimer, the answer I give is that it probably won’t. However, my main goal is that of showing how far basic game theory can go to garner insights on what the potential fallouts of governmental policies are.

The main provisions envisaged for red zones are curfews starting from 10 PM, no movements across regions, public transportation running at 50% capacity, and no moving for reasons other than going to the doctor, jogging, shopping, and very few other activities. Restrictions are even milder for yellow and orange zones, but I will use red zone areas for my toy example.
Italians must have self-declarations handy when exiting their homes as proofs that they are not joining other households, hanging out for unnecessary reasons, or, worse, crossing regional borders. Perpetrators who fail to show a self-declaration or show one which does not list valid reasons for being out are charged a €400 fee which reduces to a €280 if cleared within three days of the issue.
What matters more, however, is that the legislator seems to have taken a much more lenient approach compared to March’s record 110,000 fines by refusing to dispatch the army and relying much more on citizen cooperation. The question for us is whether a combination of soft lockdowns and mid-sized, loosely issued fines is likely to achieve the bottom line of the government. I will crunch some basic game theory to show that it likely won’t, which stems from the very nature of COVID as a very infectious, not-so-fatal illness — I beg you to listen to this Peter Attia podcast if you haven’t done so.

The Game Theory: A Very Rusty Approach

Please consider the following setup. Carolina and Marco are representative of the features of the Italian population and sit at the median of the most important demographics (age, income, education, etc.). Carolina is female and 45 years-old, earns €20,000/year after taxes owning a high school diploma and is an overall healthy person. Likewise, Marco – who is a “he” – is 45, healthy, and lives an Italy’s middle-class lifestyle. They live next to each other and stop by when they meet.
On a Saturday afternoon, they are presented with the choice of watching shows on Netflix or going out for a walk. They really shouldn’t be heading out but they know that finding a reason to smash on that self-certification is not a hard find. Carolina forgot about grabbing that toothpaste the last time she went shopping, and Marco really craves those luscious veggies at the grocery’s, plus he has not been getting that much exercise recently, so why would police even interfere with his quest for good health? Isn’t that exactly what the government wants to nurture?
As these thoughts keep rummaging around in their minds, the Italian Fall season has the best of them and lures them into taking a chance at the legislator. These are Carolina’s and Marco’s choice parameters:

• U is the utility of taking a walk and is normalized to 1 (U=1);
• if they meet each other, that utility reduces by one-half;
• the negative utility of getting COVID is 10U;
• the utility of a COVID-free country is set to infinity (∞);
• the negative utility of getting fined is 3U (approx. 3 days worth of paid work);
• c̅ is the probability of getting COVID upon taking a walk, which is strictly less than 1 and very close to 0 (0<c̅≪1);
• c is the probability of getting COVID upon staying home, which is marginally less than c̅ (0<c<c̅≪1);
• f is the probability of getting fined when stopped by the police and found not having a valid self-declaration, which is less than c̅ and c (0<f<c<c̅≪1).

Figure 1 sketches out payoffs for each combination of responses. Marco and Carolina choose either one of the two options of staying home (stay) and going for a walk (go). Also, they cannot communicate and must act independently from each other.
Carolina chooses on the expectation that Marco chooses to go for a walk, which reduces her choice set to the conditional set shown in Figure 2; by the same token, Marco’s choice set reduces to that showing in Figure 3. The cooperative state of the world (stay, stay) requires that Marco and Carolina coordinate their actions; however, they harbor mutual expectations that the other party would defect; therefore, the state of the world which maximizes social utility is forfeited when parties act independently as utility-maximizing agents.

Figure 1: Payoff Matrix

Figure 2: Carolina’s Payoff Matrix

Figure 3: Marco’s Payoff Matrix

Let’s now look more closely into the individual choices. Carolina must choose one between stay and go conditional on Marco going. As mentioned before, each party expects the other one to defect, hence only the conditional sets are feasible.
Upon choosing to stay, Carolina would “earn” the negative utility of getting COVID with probability c (i.e., –10c). When choosing to go, she would take half of the benefit from going for a walk (i.e., 0.5), the negative utility of getting COVID with probability c̅ (i.e., –10c̅), and the negative utility of receiving a fine with probability f (i.e., 3f). Noticeably, the probability of getting COVID when staying home — through families, at the local grocery’s, etc. — is about equal to that of getting infected when going for a walk (i.e., c≈c̅). The condition which must be satisfied for Carolina to choose the stay option is:

U(stay) > U(go)

or –10c >0.5–10c̅ – 3f. Because c≈c̅, the 10c and 10c̅ terms cancel out; solving for f yields the condition f>1/6.
What that means is, the probability of getting fined must be at least as high as 17% for Carolina to choose the stay option. That figure is an unrealistically high probability which does not even come close to the f which is currently traded in Italy’s red zone regions. Note that the probability required gets smaller, the larger the fine gets (i.e., as the negative utility associated with a fine grows from 3U to 4U, and so on).

Therefore, rising fines might be the only viable alternative to keep Italians home under the current policy regime, but that runs contrary to the case supporting softer lockdowns. As a result, the Carolinas of Italy end up crowding up the streets of Italy and not flattening the curve.
Much in the same way, Marco expects Carolina to defect, which constrains his choice set to (go, stay) and (go, go). His decision-making mirrors Carolina’s and he also chooses to go. Overall, (go, go) is a Nash equilibrium, which implies that Italy’s gentle lockdowns fail to gently nudge the contagion curve down.

The combination of lenient policing and the nature of COVID, a highly-transmissible, low-fatality virus mostly taken from asymptotic infection, seems to be setting Italy on course for an inevitable hard lockdown. That will probably be the only way to bridge the gap between the now and the vaccine rollout avoiding generalized healthcare failure.

The Game Theory: A Rusty Approach

For the sake of clarity, I have “cheated” a little bit with the game theory in the example above. After all, the article’s purpose is that of providing some game-theoretic intuition to highlight some potential shortcomings of soft lockdown policies. However, we can get a little bit more polished than that by parametrizing the utility of a COVID-free country using behavioral social sciences.
In general, we discount future utility i) because it is inherently less valuable than present utility; and ii) because we fail to appreciate the true value of future utility. Although we really know that saving that extra dollar today and investing it in the stock market will grant us some significant lifetime returns, our brain is tricked by some form of presentism into hyperbolically discounting those future yields.
I did not throw in the word “hyperbolically” to show off but rather to introduce hyperbolic discounting. To model an investment going to maturity in thirty days, we need to factor in the time value of money — i.e., the value of one dollar tomorrow is δ less of what it is today— and our present bias — i.e., we are only willing to value a future amount β times the amount we would value it today. That returns the hyperbolically discounted utility function:

U(30)=β(1-δ)³⁰U

Tying this back with our main discussion, we might model the response of Carolina and Marco to Italy’s lockdown policies as affected by both time discounting and present bias. That allows us to parametrize the utility they pledge upon a COVID-free country thirty days from now. I will use some numbers that make sense, however the end goal is more so understanding than stretching our math muscles.
Because Carolina and Marco care a lot about their country, 100U is how much they value a COVID-free country one month from now. They discount future utility at 10% (δ=0.10) and are affected by present bias (β=0.2). Hence, the utility associated with that choice is given by:

U(30) = 0.2(0.9)³⁰(100)≈0.85

Please note that a value of 0.2 for the present bias parameter is far from irrealistic. Imagine of being presented with a choice between a COVID-free country now and the same scenario in thirty-day time. Not even those who deny the reality of COVID would probably turn that deal down. How many (median) Italians would buy into the second offering though? Whether it is because they lack information on the effectiveness of lockdown policies, because they think that the situation would turn for the better much sooner than that, or whatever the flavor of their present bias is, it is probably going to be a generous estimate that 20% of them would take the deal.

Having said that, let’s move on to the game theory again and look at the new payoff matrix in Figure 4. We don’t need to “cheat” this time to show that (go, go) is a Nash equilibrium.
Let’s do the mental math for Carolina. When Marco chooses to stay, she will have a choice between gaining 1–10c̅–3f≈1 and going or gaining 0.85–10c̅≈0.85 and staying. Because going grants a larger utility, Carolina is willing to tweak around her self-certification and go for that walk. Similarly, when Marco chooses to go, she will have a choice between earning the negativity utility from staying in, or going and earning 0.5–10c̅–3f≈0.5; therefore, she would choose to go. Marco follows suit and he would choose to go whether or not Carolina chooses to go, thus making of (go, go) the one and only Nash equilibrium of the game.

Figure 4: Payoff Matrix (Hyperbolic Discounting)

The most important remark follows. The social utility associated with the (stay, stay) equilibrium is much larger than that associated with the (go, go) equilibrium of the private utility maximization. Unfortunately, this state cannot be achieved when uncoordinated parties transact “utilities” in the market.

U(stay, stay) = 2(0.85–10c̅) ≈1.7
U(go, go) = 2(0.5–10c̅ –3f)≈1

Overall, behavioral social sciences give us some more perspective on what previously taken away. This is by no means a deep plunge in game theory but one that shows the wide range of its applications. In our case, the interplay of behavioral incentives and lacking information might fatally hamper a policy predicated upon a compromise between hard lockdowns and a free-for-all.

Conclusions

In this article, I have applied basic game theory to shed light on the potential fallouts of Italy’s Fall 2020 lockdowns which heavily — “overly” is probably the better word here — relies on citizenship cooperation. Because of the nature of COVID, a much different infection to Ebola and even SARS-CoV with a fatality rate low enough not to bunker people down in their homes, soft lockdowns seem rather ineffective assuming the objectives of flattening the curve and softening the blow on the healthcare system.
Most likely, shorter, harder lockdowns as suggested by the Italian medical board are in a better position to achieve those objectives without having to sacrifice a much larger deal of the economy. This is because the economy cannot operate even close to efficiently with any sort of lockdown. Therefore, a couple more haircuts and espressos this fall will probably bound Rome to fall.