Corona Games: Masks, Social Distancing and Mechanism Design
CCorona Games: Masks, Social Distancing andMechanism Design
Bal´azs Pej´o Gergely Bicz´okCrySyS Lab ∗ Budapest University of Technology and Economics { pejo,biczok } @crysys.hu Abstract
Pandemic response is a complex affair. Most governments employa set of quasi-standard measures to fight COVID-19 including wearingmasks, social distancing, virus testing and contact tracing. We arguethat some non-trivial factors behind the varying effectiveness of thesemeasures are selfish decision-making and the differing national imple-mentations of the response mechanism. In this paper, through simplegames, we show the effect of individual incentives on the decisionsmade with respect to wearing masks and social distancing, and howthese may result in a sub-optimal outcome. We also demonstrate theresponsibility of national authorities in designing these games prop-erly regarding the chosen policies and their influence on the preferredoutcome. We promote a mechanism design approach: it is in the bestinterest of every government to carefully balance social good and re-sponse costs when implementing their respective pandemic responsemechanism.
The current coronavirus pandemic is pushing individuals, businesses andgovernments to the limit. People suffer owing to restricted mobility, so-cial life and income, complete business sectors face an almost 100% dropin revenue, and governments are scrambling to find out when and how toimpose and remove restrictions. In fact, COVID-19 has turned the wholeplanet into a “living lab” for human and social behavior where feedback onresponse measures employed is only delayed by two weeks (the incubationperiod). From the 24/7 media coverage, all of us have seen a set of quasi-standard measures introduced by national and local authorities, includingwearing masks, social distancing, virus testing, contact tracing and so on. Itis also clear that different countries have had different levels of success em-ploying these measures as evidenced by the varying normalized death tollsand confirmed cases . ∗ Although in the same institute, authors have collaborated remotely. Johns Hopkins Coronavirus Resource Center. https://coronavirus.jhu.edu/map.html a r X i v : . [ ec on . T H ] J u l e believe that apart from the intuitive (e.g., genetic differences, medicalinfrastructure availability, hesitancy, etc.), there are two factors that havenot received enough attention. First, the individual incentives of citizens,e.g., “is it worth more for me to stay home than to meet my friend?”, havea significant say in every decision situation. While some of those incentivescan be inherent to personality type, clearly there is a non-negligible rationalaspect to it, where individuals are looking to maximize their own utility.Second, countries have differed in the specific implementation of responsemeasures, e.g., whether they have been distributing free masks (affecting theefficacy of mask wearing in case of equipment shortage) or providing extraunemployment benefits (affecting the likelihood of proper self-imposed socialdistancing). Framing pandemic response as a mechanism design problem,i.e., architecting a complex response mechanism with a preferred outcomein mind, can shed light on these factors; what’s more, it has the potentialto help authorities (mechanism designers) fight the pandemic efficiently.In this paper we model decision situations during a pandemic with gametheory where participants are rational, and the proper design of the gamescould be the difference between life and death. Our main contribution istwo-fold. First, regarding decisions on wearing a mask, we show that i)the equilibrium outcome is not socially optimal under full information, ii)when the status of the players are unknown the equilibrium is not to weara mask for a wide range of parameters, and iii) when facing an infectiousplayer it is almost always optimal to wear a mask even with low protec-tion efficiency. Furthermore, for social distancing, using current COVID-19statistics we showed that i) going out is only rational when it corresponds toeither a huge benefit or staying home results in a significant loss, and ii) wedetermined the optimal duration and meeting size of such an out-of-homeactivity. Second, we take a look at pandemic response from a mechanismdesign perspective, and demonstrate that i) different government policies in-fluence the outcome of these games profoundly, and ii) individual responsemeasures (sub-mechanisms) are interdependent. Specifically, we discuss howcontact tracing enables targeted testing which in turn reduces the uncer-tainty in individual decision making regarding both social distancing andwearing masks. We recommend governments treat pandemic response asa mechanism design problem when weighing response costs vs. the socialgood.The remaining of the paper is structured as follows. Section 2 brieflydescribes related work. Section 3 develops and analyzes the Mask Gameadding uncertainty, mask efficiency and multiple players to the basic model.Section 4 develops and analyzes the Distancing Game including the effectof meeting duration and size. Section 5 frames pandemic response as amechanism design problem using the design of the two games previously in-troduced as examples. Finally, Section 6 outlines future work and concludesthe paper. 2 Related Work
Here we briefly review some related research efforts. For a more completesurvey we refer the reader to [3].[13] modeled the behavioral changes of people to a pandemic using evo-lutionary game theory, and showed that slightly reducing the number ofpeople an individual is in contact with can make a difference to the spreadof disease. Moreover, both an earlier warning and when accurate informa-tion about the infection is disseminated quickly, people’s responses to theinformation can limit the spread of disease.[17] focused on the traveling habits of people between areas affected dif-ferently by the disease, and found conflict between the Nash equilibrium (in-dividually optimal strategy) and the Social Optimum (optimal group strat-egy) only under specific changes in economic and epidemiological conditions.In line with our mask game, [6] studied how individuals change theirbehavior during an epidemic in response to whether they and those theyinteract with are susceptible or infected. The authors show that there isa critical level of concern, i.e., empathy, by the infected individuals abovewhich the disease is eradicated rapidly. Furthermore, risk-averse behavior bysusceptible individuals cannot eradicate the disease without the preemptivemeasures of infected individuals.In line with our social distancing game, [14] studied how individualswould best use social distancing and related self-protective behaviors duringan epidemic. The study found that in the absence of vaccination or otherintervention measures, optimal social distancing can delay the epidemic untila vaccine becomes widely available.[2] studied the vaccination behavior of people. The designed model ex-hibits a “wait and see” Nash equilibrium strategy, with vaccine delayers re-lying on herd immunity, and vaccine safety information generated by earlyvaccinators. As a consequence, the epidemic peak’s timing is strongly con-served across a broad range of plausible transmission rates. Besides, anyeffect of risk communication at the start of a pandemic outbreak is ampli-fied.Finally, the game-theoretic model in [15] focuses on the various level ofdrug stockpiles in different countries, and find controversial results: some-times there is an optimal solution of a central planner (such as the WHO),which improves the decentralized equilibrium, but other times the centralplanner’s solution (minimizing the number of infected persons globally) re-quires some countries to sacrifice part of their population.
Game Notations
To improve readability, we summarize all the parameters and variables usedfor the Mask and the Distancing Game in Table 1.3ask Game Distancing GameVariable Meaning Variable Meaning C out Cost of playing out C Cost of staying home C in Cost of playing in B Benefit of going out C i Cost of being infected m Mortality rate C use Cost of playing use L Value of Life ρ Prob. of being infected ρ Probability of infection p Prob. of using a mask p Probability of meeting a Protection Efficiency t Time duration of meeting b Spreading Efficiency g Group size of meetingTable 1: The parameters of the games.
Probably the most visible consequence of COVID-19 are the masks: beforetheir usage was mostly limited to some Asian countries, hospitals, banks (incase of a robbery), and some other places. Due to the coronavirus pandemic,an unprecedented spreading of mask-wearing can be seen around the globe.Policies have been implemented to enforce their usage in some places, but ingeneral, it has been up to the individuals to decide whether to wear a maskor not based on their own risk assessment. In this section, we model thisvia game theory. We assume there are several mask types, giving differentlevel of protection: • No Mask: This corresponds to the behavior of using no masks duringthe COVID-19 (or any) pandemic. Its cost is consequently 0; however,it does not offer any protection against the virus. • Out
Mask: This is the most widely used mask (e.g., cloth mask orsurgical mask). They are meant to protect the environment of theindividual using it. They work by filtering out droplets when coughing,sneezing or simply talking, therefore hinders the spreading of the virus.They do not protect the wearer itself against an airborne virus. Thecost corresponding to this protection type is noted as C out > • In Mask: This is the most protective prevention gear designed formedical professionals (e.g., FFP2 or FFP3 mask with valves). Valvesmake it easier to wear the mask for a sustained period of time andprevent condensation inside the mask. They filter out airborne viruseswhile breathing in, however the valved design means it does not fil-ter air breathed out. Note that CDC guidelines recommend using acloth/surgical mask for the general public, while valved masks are only C in >> C out .Besides which mask they use (i.e., the available strategies), the playersare either susceptible or infected . The latter has some undesired conse-quence; hence, we model it by adding a cost C i to these players’ utility (whichis magnitudes higher than C in and C out ). Using these states and masks, wecan present the basic game’s payoffs where two players with known statusmeet and decide which mask to use. On the left of Table 2, the payoff ma-trix corresponds to the case when both players are susceptible. Note, thatin case both players are infected, the payoff matrix would be the same withan additional constant C i . On the right of Table 2 corresponds to the casewhen one player is infected while the other is susceptible. susceptible / susceptible susceptible / infectedno out in no out inno (0 ,
0) (0 , C out ) (0 , C in ) ( C i , C i ) (0 , C out + C i ) ( C i , C in + C i )out ( C out ,
0) ( C out , C out ) ( C out , C in ) ( C out + C i , C i ) ( C out , C out + C i ) ( C out + C i , C in + C i )in ( C in ,
0) ( C in , C out ) ( C in , C in ) ( C in , C i ) ( C in , C out + C i ) ( C in , C in + C i ) Table 2: Payoff matrices for the cases when only one (right) and both (left)player is susceptible.On the left side of Table 2 it is visible that both players’ cost is minimalwhen they do not use any masks, i.e., the Nash Equilibrium of the game whenboth players are susceptible is ( no , no ). This is also the social optimum,meaning that the players’ aggregated cost is minimal. The same holds incase both players are infected, as this only adds a constant C i to the payoffmatrix. This changes when only one of the players is susceptible (i.e., rightside of Table 2): the infected player using no mask is a dominant strategy forher , since it is her best response, independently of the susceptible player’saction. Consequently, the best option for the susceptible player is in , i.e.,the NE is ( in , no ). On the other hand, the social optimum is different:( no , out ) would incur the least burden on the society since C out << C in .In social optimum, susceptible players would benefit, through a positiveexternality, from an action that would impose a cost on infected players;therefore it is not a likely outcome. In fact, such a setting is common inman-made distributed systems, especially in the context of cybersecurity. Awell-fitting parallel is defense against DDoS attacks [9]: although it wouldbe much more efficient to filter malicious traffic at the source (i.e., out ),Internet Service Providers rather filter at the target (i.e., in ) owing to arational fear of free-riding by others. We simplify the well-known SIR model [8, 5] since in case of COVID-19 it is currentlyunknown how the human body behaves after recovery when exposed again to the virus. .1 Bayesian Game Since in the basic game no player plays out , we simplify the choice of theplayers to either use a mask or no (hence, we note the cost of a mask with C use ). To represent the situation better, we introduce ambiguity about thestatus of the players: we denote the probability of being infected as ρ . Weknow from the basic game that if both players are infected (with probability ρ ) or susceptible (with probability (1 − ρ ) ) they play ( no,no ), while if onlyone of them is infected (with probability 2 · ρ · (1 − ρ )) the infected playerplays no while the susceptible plays use .In this Bayesian game the players play no most of the cases (e.g., withprobability 1 − ( ρ · (1 − ρ ))). This chain of thought is too simplistic, as weassumed case-wise that the players know their status. Consequently, withuncertainty we must minimize the costs of the players: if both players areinfected with equal probability, the payoff for Player 2 is Equation (1) where p n is the probability that Player n plays use (otherwise she plays no ). Thepayoff for the other player is similar since the game is symmetric. In moredetail, the first two lines correspond to the case when Player 2 is not infected(hence the multiplication with 1 − ρ at the beginning), while the last linecaptures when she is infected. Either way, she plays use with probability p ,which costs C use . Otherwise she plays no , which has no cost except whenPlayer 1 is infected and she plays no as well (end of the second line). U = (1 − ρ ) · [ (1 − ρ ) · [ p · C use + (1 − p ) · ρ · [ p · C use + (1 − p ) · [ (1 − p ) · C i + p · ρ · [ p · ( C i + C use )+ (1 − p ) · C i ] (1)Since this formula is linear in p , its extreme point within [0,1] is at theedge. We take its derivative to uncover the function steepness: the conditionfor the function to be decreasing (i.e., higher probability for using a maskcorresponds to lower cost) is seen below. Consequently, the only scenariowhich might admit wearing a mask with non-zero probability correspondsto the availability of sufficiently cheap masks. ∂U ∂p < ⇔ C use C i < ρ · (1 − ρ ) · (1 − p ) ≤ In the basic game we assumed in provides perfect protection from infectedplayers, while out protects the other player fully. However, in real life thesestrategies only mitigate the virus by decreasing the infection probability(i.e., ρ ) to some extent. For this reason, we define a, b ∈ [0 ,
1] in a waythat the smaller value of the parameter corresponds to better protection; a measures the protection efficiency of the protection strategy, while b capturesthe efficiency of eliminating the further spread of the disease. Consequently,6 and b was set in the previous cases to a out = 0 , a in = 1 (both in preventsfurther spread fully, while out does not prevent it at all), b out = 1 ( out hasno effect on protecting the player) b in = 0 ( in fully protects the player).We simplify the action space of the players as we did in the Bayesiangame: in and out is merged into use , with efficiency parameters a and b .We set b = , as surgical masks on the infectious person reduce cold &flu viruses in aerosols by 70% according to [12], while a home-made maskachieves 2 / a is much harder to measure.It should be a ≤ b since any mask keeps the virus inside the players is moreefficiently than stopping the wearer from getting infected. For the sake ofthe analysis we set a = b = .We are interested in the mask-wearing probability of a susceptible playerwhen the other player is infected. The utility in such a situation is shownin Equation (3), where for simplification we defined p = p = p , i.e., bothplayers play a specific strategy with the same probability. With such a con-straint, we restrict ourselves from finding all the solutions; however, sincethe game is symmetric, an equilibrium of this reduced game is also an equi-librium when the players could use a different strategy distribution. U = p · ( C use + C i · a · b ) + p · (1 − p ) · ( C use + C i · a )+(1 − p ) · p · ( C i · b ) + (1 − p ) · ( C i ) U = p · ( C use + C i · . ˙2) + p · (1 − p ) · ( C use + C i ) + (1 − p ) · C i (3)From this we easily deduce that use corresponds to a smaller cost that no if C use C i < , which holds by default as C use (cid:28) C i (even for more inefficientmasks). Moreover, use (i.e., p = 1) is the best response most of the timebecause of the following.1. The utility is a second order polynomial, hence it has one extremepoint.2. This extreme point is a minimum due to U (cid:48)(cid:48) = · C i > p = · C i − C use C i due to U (cid:48) = C use − C i + · C i · p .5. The minimum point is expected to be above 1 due to C use (cid:28) C i .6. p ∈ [0 ,
1] is on the left of the minimum point, hence, a higher p corre-sponds to a smaller cost. The Bayesian game combined with efficiency is left for future work due to the lack ofspace. .3 Multi Player Game This game can be further extended by allowing more players to participate.In this extension, if there exists an infected player (who plays no as weshowed already), all the susceptible players should play in . This NE is theSO as well if the ratio of the infected (which is identical to the probabil-ity ρ of being infected) is sufficiently high: the accumulated cost when thesusceptible players play in (and the infected play no ) is less than the accu-mulated cost when the infected players play out (and the susceptible play no ) if C in C out < ρ − ρ . Although it is mathematically possible that the infectedplays no in the SO, but it is doubtful: both the cost of in is significantlyhigher than out , and the infection ratio ρ is low (at least at the beginningof the pandemic). Another thing most people has experienced during the current COVID-19pandemic is social distancing. Here we introduce a simple Distancing Game;it is to be played in sequence with the previously introduced Mask Game:once a player decided to meet up with friends via the Distancing Game, shecan decide whether to wear a mask for the meeting using the mask game.A summary of the used parameters are enlisted in Table 1: we representthe cost of getting infected with m · L , i.e., the mortality rate of the diseasemultiplied with the player’s evaluation about her own life. We set m = 0 . : 0 . ≈ { dead } { all cases } < m < { dead } { closed cases } ≈ . B . On the other hand, staying homeor missing a meeting could have some additional costs, denoted as C . Theprobability of getting infected is denoted as ρ . With these notations, theutility of the Distancing Game is captured on the left of Equation (4), where p is the probability of going out. Since this is linear in p , its maximum iseither at p = 0 (stay home) or p = 1 (go out). Precisely, the player prefersto stay home if the right side of Equation (4) holds. U = p · ( B − ρ · m · L ) − (1 − p ) · C B + Cρ · m < L (4) Example.
For instance, a rational American citizen should go out only if she valuesher life less than 2941(= . · . ) times the benefit (of going out) and theloss (of staying home) together if we approximate the infection ratio with This is an optimistic approximation, as besides dying the infection could bear othertolls on a player. { active cases } { population } = 0 . .
3% interest rate). This means a rationalaverage American should only meet someone if the benefit of the meetingand the cost of missing it would amount to more than 3842 USD (= . M ). One way to improve the above model is by introducing meeting durationand size. Leaving our disinfected home during a pandemic is risky, and thisrisk grows with the time we spend outside. Similarly, a meeting is riskierwhen there are multiple participants involved. In the original model, wecaptured the infection probability with ρ = 1 − (1 − ρ ). This ratio increasesto 1 − (1 − ρ ) g · t when there are g possible infectious sources for t time. Since g and t are interchangeable, we merge this two together under a commonnotation: z = g · t .This extended model can be used to determine the optimal durationand size of a meeting, once the player decided to go out according to thebasic distancing game. We define 0 < z < Let us assume, a bus’ tank is leaking; the owner is loosing 1 USD of petrolevery t time, i.e., C ( z ) = 1 · t (as the owner must visit a mechanic and meetone person). In the meantime, the benefit of leaving the house is constant,e.g., B ( z ) = 1. The middle of Figure 1 corresponds to this setting. It isvisible, that going out for a short or long period (e.g., z = 5 and z = 100)is only rational for someone who values her life below 3553 and 3499 USD,respectively. 9igure 1: A few examples for various benefit and cost functions of the lowerlimit on the life value which would ensure that a rational American wouldstay home (i.e., the formula inside max in Equation (5)). x axis represents z , while y axis represents the utility. Pandemic response is a complex affair. The two games described abovemodel only parts of the bigger picture. We refer to the collection (and interplay) of measures implemented by a spe-cific government fighting the epidemic in their respective country as mech-anism . Consequently, decisions made with regard to this mechanism con-stitutes mechanism design [10]. In its broader interpretation, mechanismdesign theory seeks to study mechanisms achieving a particular preferredoutcome. Desirable outcomes are usually optimal either from a social as-pect or maximizing a different objective function of the designer.In the context of the corona pandemic, the immediate response mech-anism is composed of e.g., wearing a mask, social distancing, testing andcontact tracing, among others. Note that this is not an exhaustive list: fi-nancial aid, creating extra jobs to accommodate people who have just losttheir jobs, declaring a national emergency and many other conceptual vesselscan be utilized as sub-mechanisms by the mechanism designer, i.e., usually,the government; we do not discuss all of these in detail due to the lackof space. Instead, we shed light on how government policy can affect thesub-mechanisms, how sub-mechanisms can affect each other and, finally, theoutcome of the mechanism itself. We illustrate the importance of mecha-nism design applying different policies to our two games, and adding testingand contact tracing to the mix. Here we analyze the impact of commonly seen policies: compulsory maskwearing, distributing free masks, limiting the amount of people gatheringand total lock-down. 10 ompulsory mask wearing and free masks. If the government de-clares that wearing a simple mask is mandatory in public spaces (such asshops, mass transit, etc.), it can enforce an outcome ( out , out ) that is in-deed socially better than the NE. The resulting strategy profile is still notSO, but it i) allocates costs equally among citizens; ii) works well under theuncertainty of one’s health status; and iii) may decrease the first-order needfor large-scale testing, which in turn reduces the response cost of the gov-ernment. By distributing free masks, the government can reduce the effectof selfishness and, potentially, help citizens who cannot buy or afford masksowing to supply shortage or unemployment. Limiting the amount of people gathering and total lock-down. Ifthe government imposes an upper limit l for the size of congregations, thiswill put a strict upper bound on the “optimal meeting size” g ∗ , and theresulting group size will be min( l, g ∗ ). Note that if l < g ∗ then it createsan “opportunity” for longer meetings (larger t ), as Equation (5) maximizesfor z = gt . On the other hand, if the chosen restrictive measure is a totallock-down, both the Distancing Game and the Mask Game are renderedmoot, as people are not allowed to leave their households. Testing and contact tracing. It is clear that the Distancing and theMask Games are not played in isolation: people deciding to meet up invokethe decision situation on mask wearing. On the other hand, so far we havelargely ignored two other widespread pandemic response measures: testingand contact tracing.With appropriately designed and administered coronavirus tests, medi-cal personnel can determine two distinct features of the tested individual: i)whether she is actively infected spreading the virus and ii) whether she hasalready had the virus, even if there were no or weak symptoms. (Note thatdetecting these two features require different types of tests, able to showthe presence of either the virus RNA or specific antibodies, respectively.) Ingeneral, testing enables both the tested person and the authorities to makemore informed decisions. Putting this into the context of our games, test-ing i) reduces the uncertainty in Bayesian decision making, and ii) enablesthe government to impose mandatory quarantine thereby removing infectedplayers. Even more impactful, mandatory testing as in Wuhan completelyeliminates the Bayesian aspect rendering the situation to a full informationgame: it serves as an exogenous “health oracle” imposing no monetary coston the players. To sum it up, the testing sub-mechanism outputs resultsthat serve as inputs to both the Distancing and the Mask Game.Naturally, a “health oracle” does not exist: someone has to bear the costsof testing. From the government’s perspective, mandatory mass testing is ex-tremely expensive. (Similarly, from the concerned individual’s perspective,a single test could be unaffordable.) Contact tracing, whether traditionalor mobile app-based, serves as an important input sub-mechanism to test-ing [7]. It identifies the individuals who are likely affected and inform both11 ocial OutcomeGovt. CostSocial Distancing Govt. Policies Contact Tracing Testing Mask Wearing Response Mechanism Figure 2: Pandemic response mechanism as influenced by government policy(dotted lines) and the interplay of sub-mechanisms (solid lines)them and the authorities about this fact. In game-theoretic terms, for suchplayers, the benefit of testing outweigh the cost (per capita) with high proba-bility. From the mechanism designer’s point of view, contact tracing reducesthe overall testing cost by enabling targeted testing , potentially by orders ofmagnitude, without sacrificing proper control of the pandemic. Note thatmobile OS manufacturers are working on integrating contact tracing intotheir platform to eliminate adoption costs for installing an app [1]. The big picture . As far as pandemic response goes, the mechanism de-signer has the power to design and parametrize the games that citizens areplaying, taking into account that sub-mechanisms affect each other. Aftergames have been played and outcomes have been determined, the cost forthe mechanism designer itself are realized (see Figure 2). This cost functionis very complex incorporating factors from ICU beds through civil unrest toa drop in GDP over multiple time scales [11]. Therefore, governments haveto carefully balance the – very directly interpreted – social optimum andtheir own costs; this requires a mechanism design mindset. In this paper we have made a case for treating pandemic response as amechanism design problem. Through simple games modeling interactingselfish individuals we have shown that it is necessary to take incentives intoaccount during a pandemic. We have also demonstrated that specific gov-ernment policies significantly influence the outcome of these games, and howdifferent response measures (sub-mechanisms) are interdependent. As an ex-ample we have discussed how contact tracing enables targeted testing whichin turn reduces the uncertainty from individual decision making regardingsocial distancing and wearing masks. Governments have significantly morepower than traditional mechanism designers in distributed systems; there-fore it is crucial for them to carefully study the tradeoff between socialoptimality and the cost of the designer when implementing their pandemicresponse mechanism. Limitations and future work . Clearly, we have just scratched the sur-12ace of pandemic mechanism design. The models presented are simple andare mostly used for demonstrative purposes. Also, the mechanism designconsiderations are only quasi-quantitative without proper formal mathe-matical treatment. In turn, this gives us plenty of opportunity for futurework. A potential avenue is extending our models to capture the tempo-ral aspect, combine them with epidemic models as games played on socialgraphs, and parametrize them with real data from the ongoing pandemic(policy changes, mobility data, price fluctuations, etc.). Relaxing the ra-tional decision-making aspect is another prominent direction: behavioralmodeling with respect to obedience, other-regarding preferences and risk-taking could be incorporated into the games. Finally, a formal treatment ofthe mechanism design problem constitutes important future work, incorpo-rating hierarchical designers (WHO, EU, nations, municipality, household),an elaborate cost model, and analyzing optimal policies for different timehorizons. 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