key: cord-0261661-0u0rb2mx authors: Ma, Z.; Zhang, Y.-P. title: To mask, or not to mask, Alice and Bob's dating dilemma date: 2022-04-16 journal: nan DOI: 10.1101/2022.04.14.22273886 sha: 342c2e3b0f5b5638c62b82f1b5ea137f649747fc doc_id: 261661 cord_uid: 0u0rb2mx Face masking in current COVID-19 pandemic seems to be a deceivingly simple decision-making problem due to its multifaceted nature. Questions arising from masking span biomedicine, epidemiology, physics, and human behaviors. While science has shown masks work generally, human behaviors (particularly under influences of politics) complicate the problem significantly given science generally assumes rationality and our minds are not always rational and/or honest. Minding minds, a legitimate concern, can also make masking legitimately confusing. To disentangle the potential confusions, particularly, the ramifications of irrationality and dishonesty, here we resort to evolutionary game theory. Specifically, we formulate and analyze the masking problem with a fictitious pair of young lovers, Alice and Bob, as a Sir Philip Sydney (SPS) evolutionary game, inspired by the handicap principle in evolutionary biology and cryptography figures in computer science. With the proposed ABD (Alice and Bob's dating dilemma) as an asymmetric four-by-four strategic-form game, 16 strategic interactions were identified, and six of which may reach equilibriums with different characteristics such as separating, pooling, and polymorphic hybrid, being Nash, evolutionarily stable or neutrally stable. The six equilibrium types seem to mirror the diverse behaviors of mask believers, skeptics, converted, universal masking, voluntarily masking, coexisted and/or divided world of believers and skeptics. We suggest that the apparently simple ABD game is sufficiently general not only for studying masking policies for populations (via replicator dynamics), but also for investigating other complex decision-making problems with COVID-19 pandemic including lockdown vs. reopening, herd immunity vs. quarantines, and aggressive tracing vs. privacy protection. vs. reopening, herd immunity vs. quarantines, and aggressive contact-tracing vs. privacy protection in facing the unprecedented challenges from the COVID-19 pandemic. Here we formulate and analyze the dilemma with a fictitious pair of young lovers, Alice and Bob, who were 'professionally hackers' before the COVID-19 pandemic (http://cryptocouple.com/). After a few weeks of lockdown during the COVID-19 pandemic, Alice sent Bob a "miss you" message and they decided to meet somewhere. Alice could have two states: either healthy or infected by COVID-19. However, Bob had no clue on Alice's state, and indeed, he did not even know his own health status. Before departing for the dating, Bob fell in a dilemma-should he wear a mask? He might be concerned with traditional social stigma such as being conceived as timid, unfavorable perception from Alice, and/or the infection risk. Alice might also fell in a dilemma, in particular, should she be honest to Bob about her health (COVID-19) status? Time back to five centuries ago, British poet-soldier Sir Philip Sidney (1554-1586), when fatally injured in a battle, with the immortal words "thy necessity is greater than mine", passed his water bottle to one fellow soldier who was also a casualty. Whether or not this well-known English story is true is less relevant for the topic of this article since the other scenarios are well covered by famous evolutionary biologist John Maynard- Smith (1920 Smith ( -2004 , who transformed the story into a rather successful evolutionary game for investigating the honesty in animal communications. With his words, "the story deserves to be true, although it was based on the claims of a close friend of Sir Philip Sidney". Maynard-Smith (1991 , 2003 formulated the Sir Philip Sidney (SPS) game with an objective to resolve a then hotly debated hypothesis-the handicap principle first proposed by Zahavi (1975 Zahavi ( , 1997 , who tackled a fundamental problem in evolutionary biology with deep humanity implications. The handicap principle maintains that animal signaling (communication) must be costly to be reliable in the existence of conflicts of interests. For example, the tail of a male peacock is costly because it attracts the attention of its predator, but it also acts as a reliable (honest) signal to female to demonstrate the male's fitness (genetic quality). Similarly, in human society, the luxury goods must be costly to demonstrate its value. The handicap principle also shed light on a question of humanity: are we human beings the only creatures that can lie? If not, how the honest signaling was evolved in animal world? The Sir Philip Sidney (SPS) game is an action-response game and proceeds in two stages with two participants (Huttegger & Zollman 2010 , Whitmeyer 2020 : signaler (the message sender or Sir Philip Sidney's comrade in this case) may send a request to donor (responder or message receiver: Sir Philip) for water bottle and the responder may or may not respond by donating the bottle. Signaler could be in one of two states: healthy with probability of (1-m), wounded (needy) (m). If he receives the water (resource) from the responder, he will survive. Otherwise, his survival probability is (1-b) if healthy, and (1-a) if wounded (a>b). In addition, the signaler may need to pay for the cost (c) of sending the request. On the responder side, he can either donate the water or do nothing. In the former case, his survival probability is (1-d), and in the latter case, the survival probability is unchanged. Note that the 'cost' refers to 'strategic cost' in general, which may include the consequence from dishonest signaling, such as the increased risk of infection or decreased survival probability in the case of ABD problem. There is a relatedness parameter (k), which is defined as the proportion of shared 'genes' between a signaler and a potential responder. Both the sender and responder behave to maximize their own 'inclusive fitness' (from kinship theory), which are the k times the payoff of the other player plus his or her own payoff. This parameter is critical for the SPS game because it defines the level of common interests (the opposite of conflicts of interests) and therefore potentially has a critical impact on the reliability of signaling (Maynard Smith & Harper 2003 , Cooper et al 2018 . When the k-value is larger, the players are more closely related, the signaler is more likely to signal honestly and the donor is more likely to donate. In our ABD model, the relatedness parameter (k) represents the closeness of the love relationship between Alice and Bob, and measures their shared interests in masking-or-not decision-making. The SPS game was initially formulated to demonstrate the equilibriums underlying the handicap principle, i.e., signaling must be costly to be honest in the existence of conflict of interests. The equilibriums (see Box 3) could be evolutionarily stable, neural or unstable. For example, if the signaler only signals when in need and if the responder only donates the water in response to a signal, there is a signaling Nash equilibrium. It could be easily derived that c=(b-rd) is the minimum cost to maintain this equilibrium, i.e., the honest signaling. When b0.9541) is slightly lower than selecting A4 (Never masking with payoff>0.9542), but significantly higher than selecting the other two strategies (A1 or A3). This means that when the cost of infection is low (c<0.187), Alice would tend to select behavior of "not masking" even if the infection risk is high (m=0.7 in All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org/10.1101/2022.04.14.22273886 doi: medRxiv preprint this case). Second, the payoff that Bob should select B2 (Not masking only if Alice does not mask with payoff>1.092) is the highest, and his next favorable strategy is B3 (Always masking with payoff>1.0837), but their payoff difference is insignificant. This means that when the strategic cost (c) of infection is low, Bob would still select masking. In real world, the above interpretations of Alice and Bob's behaviors appear counter-intuitive. However, one possible explanation for their apparently contradictory behaviors could be related to the politics of COVID-pandemic, for example, the 'left' (always masking, like Bob in this case) vs. 'right' (never masking, like Alice). Nevertheless, beyond the pandemic politics, as humans, we have certain level of cooperative tendency. For Alice and Bob, their romantic relationship (which is represented by relatedness parameter k) should also play a role in their decision-making. It has been argued that one of the most important reasons why we humans have come to dominate the earth is attributed to our exceptional evolutionary capacity for decision-making (Samson 2020) . Although on macroscopic scale, we have been enormously successful from selecting the right food and shelter, to devising complex economic strategies and effective public health policies, it is fair to say, we occasionally make expensive and painful mistakes on both an individual and a group level (Samson 2020) . The COVID-19 pandemic has been one of the biggest challenges to the health and well-beings of humans in recent decades. It is obviously in the best interests of whole society and each individual to make right decisions to minimize the pandemic impacts. The challenging decision-making in facing the COVID-19 pandemic crisis involves both groups and individuals, and history tells us groups are not necessarily more likely to make right decisions, especially when the communication and interactions between individuals could not be controlled (Sunstein & Hastie 2015 , Samson 2020 . The bounded rationality and lack of full honesty may be inherent with Homo sapiens, which makes some apparently simple decision-making such as masking disproportionally complex. The challenge seems particularly real in the era of disinformation (e.g., Bergstrom & West 2020) . It was these properties that prompted us to apply the SPS game and underlying handicap principle, which has achieved enormous success in analyzing the evolution of animal communications, to study the ABD problem. Since we are only interested in strategic decision-making on masking, it is natural All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 for us to focus on the stable equilibriums and corresponding payoffs under variously possible strategies. Tactical or operational level analyses are obviously beyond the scope of this article. In previous sections, we formulated the ABD as an evolutionary game, specifically as a SPS game. There are 16 possible options or strategy combinations given that SPS game is an asymmetric four-by-four strategic-form game. We summarized a total of six types of equilibriums (Table 2 ) from the 16 possible strategic combinations and their payoffs (inclusive fitness) (Table 3) . Those equilibriums can be distinguished as separating (signaling), pooling, and polymorphic hybrid equilibriums, and they may have different behavioral and payoff properties (see Box 3, Box 4 and Table 3 ). Furthermore, their stabilities can be dramatically different (Table 2) , which may have different theoretical and practical ramifications. Theoretically, those ramifications have been extensively studied and well documented in the existing literatures (e.g., Maynard-Smith & Harper 2003 , Huttegger & Zollman 2001 , Bergstrom & Lachmann 1997 , 1998 , Biernaskie et al 2018 , Madgwick & Wolf 2020 , Whitmeyer 2020 ). However, their practical ramifications are much more complex, which we briefly discuss below. In our opinion, the ABD provides a powerful abstraction for analyzing the masking behavior in the COVID-19 pandemic. However, the ABD can only reveal possible behavior types (game strategies). To fully understand various masking (or not masking) behavior decisions, we must also consider the cultural and humanity dimensions of the problem. While medical science studies have generally support the masking or even universal masking strategies (e.g., Klompas et al. 2020 , Gandhi et al. 2020 , Peeples 2020 , in reality, from the very beginning of COVID-19 pandemic until today, protests have never stopped in some regions of the world. Indeed, masking still seems to be one of few most controversial issues in public health policy surrounding COVID-19 (Peeples 2020). In the remainder of this article, we try to shed light on the ABD of masking from the perspective of behavior economics. According to behavior economics (Samson 2014 (Samson , 2020 , it is not always true than humans are self-interested, benefits maximizing, and costs minimizing with stable preferences. Our minds may only possess bounded rationality, which suggests that human rationality is limited by brain's information processing capability, insufficient knowledge feedback, and time constraint. For example, our thinking and decisions may be strongly influenced by readily available information in memory, automatically produced affection, and salient information in the environment. Humans live in the moment, often resist to changing and may be bad in predicting All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 future behavior. Last but not the least important, humans are social animals with social preferences, including those embedded in trust, reciprocity and fairness. We are also emotional animals and are subject to social norms, caring reputation and self-consistency (Ma 2015a , 2015b , Samson 2014 ). An obviously different attitude has emerged between the Eastern and Western towards the using of facial mask at the very beginning of the COVID outbreaks. We postulate that the difference might be related to holistic verse analytic thinking styles. It has been argued that the differences in thinking styles may have a profound influence on the tensions between the psychology of Homo economicus and Homo sapiens, being relatively weaker in Eastern-Asian cultures (Nisbett et al. 2001) . The holistic reasoning is more likely context-dependent, in which people tend to use their intuition more if it is in conflict with formal rationality and tend to accept variations across scenarios (Nisbett et al. 2001 , Samson 2014 . This may explains relatively less resistance to the advocacy of using masks in public places in the Eastern Asian. It might be disappointing that we can produce few concrete recommendations to the general public from the present study regarding the masking dilemma, other than highlighting its complexity and its multidimensional nature (including science, culture, information availability, and etc), for which existing studies have presented confirmative recommendations in general (e.g., Peeples 2020). Instead, our contribution, hopefully being intellectual, lies in the formal abstraction of the problem and presented formal strategic options as well as their stabilities and payoffs in 16 possible strategic options (scenarios) and 6 types of equilibriums. Whereas the number of scenarios (16), equilibrium types (6) with various stability features may vary with other alternative abstract models, we believe our ABD game model is sufficiently general for further scientific investigation of the dilemma from a multi-dimensional perspective. As to the reason why we cannot generate tangible recommendations at this stage, it becomes clear if one notices how the society has been dealing with the impact of cigarette smoking, even after the hazards of second-hand smoking have been revealed by many scientific studies (Samson 2014 (Samson , 2020 ). If we choose to make one recommendation, our suggestion would be that nudging could be a more effective strategy in advocating masking. In recent years, nudging or nudge philosophy has been used in many public-policy devising such as consumer welfare (Thaler & Sunstein 2008) . The nudge philosophy treat people as Homo behaviouralis, shifted from Homo economicus, in which policy or advocacy is designed to modify the context in which decisions are made without changing the constraints faced while the freedom of choice is preserved All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101/2022.04.14.22273886 doi: medRxiv preprint (Samson 2020) . For example, rather than posting "no mask, no service", perhaps an alternative post with "please keep 2 meters of distance if you forget to bring mask" could be equally effective. In conclusion, despite the demonstrated complexities from our ABD game model, it seems that the six types of equilibriums theoretically derived from the game model adequately explain the diverse "norms of masking" evolved in much of the world with the progression of COVID-19 pandemic. Those norms represent diverse behaviors of mask believers, skeptics, converted, universal masking, voluntarily masking, coexisted and/or divided world of believers and skeptics, and their evolutions (formations) have obviously influenced by various scientific, social and other complex factors, and most importantly the severity of the pandemic. They are of different level of stabilities (resilience) at present and they are expected to continue evolve with the progression of the pandemic. As a science paper, we stick to the validity of mathematic logic underlying the ABD game model. Nevertheless, we do not pronounce a position for the right or wrong of those norms (including those that may be counterintuitive, unnatural or irrational ones) since we believe mutual respects and accommodations are part of the humanity. We are hoping that the breakthroughs in medical sciences such as vaccines or new treatments will ultimately make the issue of masking moot. The only regret (in our opinion) might be that Alice and Bob are likely to stick to working at home even after their dilemmas disappear! Of course, we wish them a lifetime of love and happiness! It is the summer of 2021 already when we are revising this manuscript, while the COVID-19 pandemic has been going on wave after wave for near two years, despite the expanding vaccination efforts worldwide. The emergence and spreading of significantly more vicious δvariant and somewhat disappointing protective efficacy of vaccines have prompted some countries (regions) to reenact some level of mask mandates, together with vaccination promotions and even mandates, while uncertainties surrounding the pandemic refuse to fade away. Public attitudes to masks have gone through about-face changes in many parts of the world, and mostly turned to accepting it as an effective personal protection equipment. From the perspective of ABD game modeling, the rising infection risk (m) should have shaped the evolution of masking behaviors since the start of the pandemic; yet we believe other social, cultural, economic and political factors have certainly played significant roles too. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 Through this article, we have minimized the discussion on the politics surrounding masking behaviors/policies and COVID-19 at large. Nevertheless, in reality, politics are hardly avoidable in making public health policies. We realize that some of the peculiar, idiosyncratic behaviors simulated by the ABD game might be rightly explained by science denialism, politics or political economy of pandemics. In fact, during the submission processes of this manuscript, we were advised by an anonymous expert reviewer in another journal to formulate the ABD game as "one in which the signaler's state is instead her political leaning (left or right, say), …" We do believe that was an excellent idea and may pursue it in future. While avoiding or minimizing politics in the science of anti-pandemic is crucial for making sound public health policies such as masking, vaccination, quarantines, and/or lockdown, it is equally important to avoid the "counter-revolution of science" (Hayek 1955) . The "counterrevolution" of science refers to the attempt to remove the human factor in order to obtain objective, strictly controlled results by using the so-termed hard science approaches, or the attempt to measure human action itself with the soft science approaches. The limitations of such attempts can be particularly serious when the scientific reasoning is based on incomplete information or knowledge. It is for this reason, we caution that the results of ABD modeling such as the equilibriums should only be used as explanatory/exploratory purposes at strategic level, rather than used for advising public at tactic levels. We argue that the methodology we used to model masking behaviors with the ABD game should also be inspirational for exploring other public health policies (measures) such as social distance, lockdown/reopen and quarantines (abbreviated as DLQ hereafter). These measures are designed to physically contain the spreading of pathogens by isolation (distancing or containment) and can be characterized as either "open vs. closed", i.e., without vs. with enforcing isolation (containment) measures such as DLQ. Obviously, masking also belongs to isolation. According to the metapopulation (i.e., population of local populations) theory (Citron et al. 2021 , Ma 2020 , infectious diseases such as COVID-19 can be modeled as a metapopulation of infectious pathogens, i.e., consisting of many local (regional) populations of pathogens (carried by human hosts) such as the local or regional outbreaks of COVID-19 (e.g., outbreaks in different countries). Also according to classic ecological theories (Hilker et al. 2009 , Friedman et al. 2012 , Ma 2020 , the extinctions of local populations can be common events, although the global metapopulation is usually stable and resilient against global (total) extinction. Hilker et al. (2009) demonstrated theoretically that the disease dynamics could be rather sensitive to All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 perturbations such as disease control methods even when the basic reproductive number (R 0 ) exceeds 1 significantly (R 0 >>1). R 0 is often used to evaluate the potential for disease invasion and persistence, to forecast the extent of an epidemic, and to infer the impact of interventions and of relaxing control measures (Shaw & Kennedy 2021) . The Allee effect is named after animal ecologist W. C Allee, and it refers to a theoretic threshold of population size, above which population growth may accelerate and below which population may go extinct (Kramer et al. 2017) . Friedman et al. (2012) They may or not cooperate with each other by adopting the same or different strategies. Similar to previously discussed ABD game for masking behaviors, their tendency to cooperate can be measured by parameter k, indicating their shared interests. The strategic cost (c) in the ABD game can be used to characterize the consequence (such as the fatality and/or economic loss) of failure in controlling local outbreaks. The parameter (m) can be treated as a probability function of R 0 (basic reproductive number) of the pathogen. Parameters a, b, and d could be treated as All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 probability functions that may lower the resilience (survivability) of society. With such an evolutionary game model inspired by the ABD game (let us call it "open or close dilemma": OCD), we can expect that the complex behaviors of two countries similar to Alice and Bob be generated from the OCD model. Both countries can, in fact, represent two "metapopulations" of countries (local populations), and within each metapopulation, different countries (local populations) may actually adopt different strategies (similar to Alice and Bob each has four different strategies). With the above conceptualization, the suggested OCD model can be used to analyze the interactions of two countries (metapopulations); and the effects of the key epidemiological and ecological parameters such as the thresholds of Allee effects and R 0 can be investigated at regional/global scales. Within the context of the suggested OCD game scheme, many countries may adopt coexistence (between humans and the pathogen) strategy, some may pursue zero-infection goal dynamically, and still others may adopt laissez-faire strategy (e.g., passive herd immunity). Intuitively, zeroinfection strategy is apparently unrealistic, but it should be possible locally at the minimum, if Allee effects of the pathogen can be exploited effectively. In fact, the eradication of smallpox is a successful example of zero infection strategy. Each of the strategies may possess some unique merits, and some of which could be exclusive. There may not be a simple criterion (or even a set of criteria) to evaluate various strategies objectively given the enormous complexities of the scientific and technological, geographic, cultural, economical and political factors underlying the host-pathogen dynamic system. For example, Sy et al (2021) estimated the R 0 of 1,151 US counties with the medium of R 0 =1.66, ranged from 0.38 to 12.44. One particularly important point we would like to emphasize from their findings is that the range of [0.38, 12.44] was largely dependent on local (county) population densities Their R 0 numbers were estimated before more contagious SARS-CoV-2 variants emerged. With the recent δ-variant, the upper limit of the range might be one order of magnitude higher, i.e., possibly >100 in some of the counties with highest population densities in the US, according to some recent reports on the R 0 of δvariant. Of countries, there are many countries in the world that have higher population densities than the US. For example, while the inter-residence distances in rural USA are usually hundred of meters away, and the distance can be as few as a few meters in the apartment settings in super big cities or in countries with higher population densities. The distance differences could easily exceed 2-3 orders of magnitude. If this assumption of the difference in residence distances holds, and if the "amplification" effects of population density (especially the residence settings) discovered by Se et al. (2021) is also applicable outside the US, then the R 0 of SARS-CoV-2 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 in some densely populated countries may be raised by another 1-2 orders of magnitudes. The higher R 0 also implies that the occurrences of super-spreading can be more frequent and severer. Given potentially 2-3 orders of differences in R 0, the same or similar co-existence (tolerance) policy in different countries can produce dramatically different consequences. While coexistence policy in a sparsely populated country may be a rational and even advantageous policy, the same policy may lead to disastrous outcomes in a densely populated country. Obviously, population density is certainly not the only major factor that influences R 0 , and of course R 0 is not the only critical parameter of pandemics. For example, the threshold of Allee effects is another critical parameter for suppressing pandemic or eradicating a pathogen. Therefore, exploring the policies and strategies for fighting pandemics should sufficiently consider complex scientific, technological, and socioeconomic factors that shape the pathogen-host dynamic systems. Game-theoretic models such as ABD game can offer important cognitive tools for the decision-making in fighting pandemics. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org/10.1101 https://doi.org/10. /2022 Box 1. Three Key Elements (Who, What and How) to Formulate SPS/ABD Games and the Exposition of the Transformation from SPS to ABD game Who are the game players? message sender (=signaler, or bottle requester) vs. responder (=message receiver, or donor of water bottle). Sender was Sir Philip's comrade, and responder was Sir Philip Sydney in Maynard-Smith (1991) original SPS game. In the ABD game, Alice is the sender, and Bob is the responder. The essentiality of the role assignment lies in that who initiates the action and who responds to the message. What is the "content of message (request)"? It is apparently the "request for water bottle" for the message sender, and the responder could either honor or ignore the request (water bottle) in the classic SPS game. The essentiality here is the message of "unhealthy state" (injured in SPS or infected in ABD) with probability of m (see Box 2 for explanation). How is the message sent (signaled)? Through shouting or whispering in the original SPS game. In the ABD game, it is through masking or not during their dating. However, the 'How' point does not matter much in the sense that it does not influence the mathematical logic of the game, although it is important for the narrative of the 'story' (Sir Philip Sydney's selfless heroism or Alice and Bob's romance). However, the heroism or romance is not necessarily reliable (honest), and dishonest carries a huge strategic cost (c: the handicap) in the classic SPS game, but this is not necessarily true in the ABD game. There were only four (2x2) strategies in the classic (Maynard-Smith 1991) SPS game since it was considered sufficient for its initial motivation for devising the SPS game, i.e., to demonstrate the handicap principle or the reliability (honesty) of animal communications (but, we are not sure if Maynard-Smith was aware of the other possible strategy options). The initial hypothesis of the handicap principle was that to communicate honestly, there must be a handicap in the communication system that enforces the honesty. With the handicap principle, there were only four strategies that make sense. We use the term "make sense" to imply that the strategies are broader than being "rational" since evolutionary game theory does not enforce "rationality." Instead, as long as there are ESS (evolutionary stable strategy) equilibriums, the strategies make sense. Actually, the strategies in the original SPS game not only make sense, but also are rational, including sending dishonest signal (selfish behavior) since payoff was high for doing so. It was Huttegger & Zollman (2010), Whitmeyer (2020) (and several other groups of scholars), approximately a decade later after Maynard-Smith (1991) seminal work, which significantly expanded the classic SPS game, most notably, expanding the strategies from 4 to 16. With the expansion (extensions), those strategies that do not make sense (senseless) in the classic SPS game were brought back. To formulate the ABD game, we adopt those extensions. The extended SPS allow us to capture various kinds of idiosyncratic masking behaviors: some are rational; some are not only irrational, but also senseless (without discernible meaning or purpose). Table 1 and Fig 1 listed the 16 possible strategies of the ABD game. The strategy matrix (Table 1) starts with (A1, B1) strategy: Alice signals that she is healthy by "not masking only if healthy;" Bob responds with "Not masking only if Alice masks". Once the first strategy cell (A1, B1) is specified, the remaining 15 strategies should keep consistent with the first element in the matrix. Our consideration for selecting (A1, B1) as the first element (cell) was to choose a strategy pair that seems most rational. That is, Alice does not mask, which sends signal of her healthy status; Bob does not mask when Alice masks since Bob feels unnecessary for him to wear one when Alice is protected (wearing mask). Before finalizing the ABD strategy matrix as exhibited by Table 1 & Fig 1, we tried several alternatives. We realized that, after elements (i: Who) & (ii: What) are specified, the setting of (iii: How) does not matter in the sense that the setting only affects the order of the 16 combinatorial (interaction) strategies. It influences neither the validity of various strategy equilibriums (Table 3 ) nor the completeness (Table 1 or Fig 1) of the strategy set (options). That is, whatever order is specified, all possible strategy options (behaviors) are All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 covered by the game (i.e., completeness). Therefore, the setting of element (iii), which in ABD game is "mask" or "not mask", does not influence the validity of mathematical logic underlying the ABD game (also see the following discussion on asymmetric vs. symmetric. (1) In the classic SPS game of Maynard-Smith (1991) , the terminology used was "honest communication," which is often used interchangeably with "cooperation" in the later literatures. However, communication is not necessarily honest (cooperative), particularly in the extended SPS game. In the ABD game, we use "communication" as alternative to "cooperation". In the meantime, we use "interaction" and "dating" interchangeably with "communication" in various contexts. (2) Although the classic SPS (with 4 strategy pairs) is asymmetrical, in the extended SPS game, when the 12 ignored strategy pairs (due to the non-reciprocal or asymmetrical nature of SPS) were brought back (e.g., Huttegger & Zollman 2010 , Whitmeyer 2020 , the asymmetrical nature changed. In effects, the extended SPS and ABD games cover all possible strategy options (completeness, as explained previously), becoming somewhat equivalent to a symmetrical game given the completeness of the strategy set. (3) The primary reason why the first strategy pair (A1, B1) was specified as it is was its rationality or naturalness. However, given the equivalently symmetrical nature of the ABD game, approximately ½ of the 16 strategies could be or close to irrational or senseless, mirroring the various idiosyncratic behaviors of masking in the current pandemic. For example, the case of lowered payoff due to masking may occur, if his or her lover's masking behavior is senseless, or their "romantic" relationship (captured by the relatedness parameter k) may lead to shared loss or gain in their payoffs. Once the first strategy pair (A1, B1) is specified, the remaining 15 pairs are set to keep logic consistence. The logic consistency refers to the validity of mathematic logic (truth table of a predicate statement), which determines the rigorous inferences of the various equilibriums listed in Table 3 . The equilibriums indicate that the corresponding strategies (masking behaviors) may become established in populations. Nevertheless, the validity of the mathematic logic of those strategies does not guarantee the rationality or naturalness of strategies. That is, valid strategies can be senseless, mirroring various idiosyncratic masking attitudes or behaviors during the pandemic. Expositions to Clarify the Logic Underlying the ABD Strategies (4) Although the ABD game appears to be a two-player game, it is in fact a game of populations due to the nature of evolutionary game model, which uses replicator dynamics (described in differential equations) rather than algebraic payoff matrix. In fact, the ABD not only can represent for two populations of replicators (Alice and Bob) and her or his own "fans or followers" (offspring of the replicators), but also represent for the 16 subpopulations corresponding to 16 different strategy pairs. Therefore, the ABD model is sufficient general to study the complex, full spectrum of the masking behavior at population or societal level. Actually, the method is also applicable to other decisionmaking problems beyond masking, such as lockdowns vs. open. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 Box 2. Interpretations of the parameters (a, b, c, d, m & k) in the SPS/ABD Games a, b, c Parameters a, b, and c are "conditional cost" parameters directly associated with Alice (sender of signal in the original SPS game) since they may affect the payoff (fitness) of Alice. The reason we term them "conditional cost" parameters is because of the following mechanism: whether or not their values affect ("kick in") the payoff of Alice depends on two factors, i.e., her infection status and the response behavior of Bob (receiver of signal in the original SPS game). If Bob choose to cooperate with Alice, the parameter a and b would be equal to zero (accruing no cost to Alice). However, if Bob does not cooperate, then the values of a and b would be non-zero (accruing cost to Alice). Furthermore, if Alice is infected, her survival probability would be (1−a), otherwise (1−b), with a condition a>b. Therefore, parameter a and b can be considered as conditional probabilities (costs) that may lower the survivability of Alice, but whether or not they "kick in" depend on Bob's behavior and Alice's infection status. Parameter c is the strategic cost that Alice may need to pay if she lied (conceal her infection status), and it can also be considered as another conditional probability (cost) that may lower her survival probability. d Parameter d is a "conditional cost" parameter associated with Bob. It reduces Bob's survival probability to (1−d) if Bob chooses to cooperate (honor the request of Alice), and d=0 if he refuses to cooperate. Parameter m and k are different from the previous previously defined parameters a, b, c, and d in the sense that they may influence 'cost', but they are not 'cost' per se. The 'cost' (a, b, c, d) can be considered as the reduction of survival probability of Alice or Bob. Parameter m is the infection probability of Alice, and parameter k is the relatedness or closeness of Alice and Bob's relationship. Parameter k represents their shared interests (shared genes in evolutionary biology) and it has a direct impact on their inclusive fitness (=k times the fitness of the other player plus his or her own fitness). It should be pointed out that the above interpretations are primarily based on the design of the original SPS game by Maynard-Smith (1991) . In the ABD, which is based on the latest extensions of SPS game by Huttegger & Zollman (2010) , Whitmeyer (2020) and others, all 16 possible combinatorial strategies are considered. The 16 possible strategies cover some of the strategies that were initially considered as 'inapplicable' or 'useless' in the original SPS game. Consequently, although the same parameters (a, b, c, d, m, k) are used in the original SPS, extended SPS as well as ABD games, their interpretations could sound 'awkward' for some of the strategies initially ignored. The reason that some of the strategies were considered as 'inapplicable' and apparently ignored in the original SPD game at Maynard-Smith (1991) era might be to do with his motivation to conceive the SPS game-a simple game-theoretic model to illustrate the feasibility of the handicap principle in evolutionary biology. In the ABD game, we actually take advantages of the rich strategy interactions extended from the classic SPS game by Huttegger & Zollman (2010) , Whitmeyer (2020) and others. As a side note, in the evolutionary game setting, Alice (similarly Bob) as a 'player' actually represents a population of 'players' with possibly different strategies, for example, Alice=(Alice-1, Alice-2, Alice-3, Alice-4) with corresponding strategy vector (A1, A2, A3, A4). This further complicates the interpretations of the ABD parameters when extending the interpretations from the classic SPS game. In summary, the previous basic interpretations of the ABD game parameters based on the classic SPS game could become counter-intuitive for some of the strategy interactions in the extended SPS games. Nevertheless, the controversies, peculiarities and idiosyncrasies surrounding masking behaviors in the COVID pandemic era, some of which may be explained only by science-denialism, do deserve a game-theoretic model that can describe such rich behaviors beyond rationality and/or scientific understanding. Finally, all of the ABD parameters described above fall within the interval of [0, 1]. A Nash equilibrium (i.e., "no-regret" strategies) is a solution to a noncooperative game involving two or more players, in which each player is supposed to be aware of the equilibrium strategies of the other players, and no player is better off by unilaterally changing own strategy. In other words, when each player has selected a strategy and no player may benefit from changing strategies whereas the other payers keep theirs fixed, then the current set of strategies and the consequent payoffs (fitness) constitute a Nash equilibrium. An evolutionarily stable strategy (ESS) is a strategy that, if adopted by a population of players in a given environment, cannot be invaded by any mutational (alternative) strategy, which is usually scarce initially. It is a Nash equilibrium that is "evolutionarily stable or resilient." In other words, an ESS is a strong version of the Nash equilibrium, and the Nash equilibrium is a weak version of ESS. A pooling equilibrium is an equilibrium in which, regardless of the types of sender with differing characteristic signals, the receiver shall choose the same response. In the case of the ABD game, Bob does not care whether or not Alice wears a mask when he makes his own decision. A separating equilibrium is opposite to pooling equilibrium: a receiver's response can be different depending on the type of sender's signal. In the case of ABD game, Alice's signal influences Bob's decision. A hybrid equilibrium is some combination of separating equilibrium and pooling equilibrium. A polymorphism is a mixed Nash equilibrium where sender mixes between being honest and dishonest signaling, and responder may also adopt a mixed strategy accordingly. Polymorphisms, which were ignored in biological signally until Huttegger & Zollman (2010) extensions of the original SPS game, correspond to hybrid equilibriums in this study. Polymorphic equilibrium allows for partially meaningful (honest) communication, and therefore grants the SPS much realism. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted April 16, 2022. ; https://doi.org /10.1101 /10. /2022 Box 4. Definitions for the Stability Properties in the SPS/ABD Games Lyapunov Stable Assume a dynamical system which is described with there is an equilibrium solution € x ∈ R n , such that Lyapunov stability is a rather weak requirement on equilibrium points. Especially, it does not require that trajectories starting close to the origin approach to the origin asymptotically. In a simplified interpretation, if the solutions that start out near an equilibrium point x forever, then x is Lyapunov stable. A time-invariant system is asymptotically stable if all the eigen-values of its system matrix (A) possess negative real parts. In the case of asymptotic stability, there is a sphere S, centered around = 0 with the radius r, such that the response, once entered the sphere, converges to the origin. An equilibrium that is Lyapunov stable but not asymptotically stable is sometimes termed as neutrally stable. Social efficiency deficit deciphers social dilemmas The evolution of cooperation Communications in plants. Neuronal aspects of plant life Signalling among relatives I. Is costly signalling too costly? 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The copyright holder for this preprint this version posted Honest signaling: the Philip Sidney game Animal Signals A (2001) Culture and systems of thought: Holistic versus analytic cognition What the data say about wearing face masks The Behavioral Economics Guide The Behavioral Economics Guide 2020 (with an Introduction by Colin Camerer) What the reproductive number R 0 can and cannot tell us about COVID-19 dynamics. 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