CDC (Cindy and David’s Conversations) game: Advising President to survive pandemic

Summary Ongoing debates on anti-COVID19 policies have been focused on coexistence-with versus zero-out (virus) strategies, which can be simplified as “always open (AO)” versus “always closed (AC).” We postulate that a middle ground, dubbed LOHC (low-risk-open and high-risk-closed), is likely favorable, precluding obviously irrational HOLC (high-risk-open and low-risk-closed). From a meta-strategy perspective, these four policies cover the full spectrum of anti-pandemic policies. By emulating the reality of anti-pandemic policies today, the study aims to identify possible cognitive gaps and traps by harnessing the power of evolutionary game-theoretic analysis and simulations, which suggest that (1) AO and AC seem to be “high-probability” events (0.412–0.533); (2) counter-intuitively, the middle ground—LOHC—seems to be small-probability event (0.053), possibly mirroring its wide adoptions but broad failures. Besides devising specific policies, an equally important challenge seems to deal with often hardly avoidable policy transitions along the process from emergence, epidemic, through pandemic, to endemic state.

(1-b, 1) (1, 1-d) ( handicap) that ensures the honesty. For example, the peacock tail in flight is considered as a classic example of a handicapped signal of male quality, which signals the superior fitness of males to females in sexual selection. In human society, luxurious goods must be costly to be valuable (signaling social status).
In the case of the CDC game, there is a strategic cost (c) associated with miscommunication between Cindy and David. The SPS game not only provides a framework to model the strategic cost (c) capturing the possible noise communication between Cindy and David, but also offers measures to capture the costs associated with Cindy and David. For Cindy, there are two cost parameters (a, b) that are associated with underestimating or overestimating pandemic risk (m) level. For David, there is a cost parameter (d)  The 4x4 combinatorial asymmetric strategies of the discrete CDC game with the letters (A-P) for possible strategy pairs as shown in Figure 1 Sender strategies (Cindy's Advices) Responder's strategies (David's decisions) Four Basic Anti-Pandemic Policies, which should be the output from the previous CDC game model (see Table 2 for the corresponding CDC equilibriums)

Policy # Policy Description
Possible CDC equilibriums likely to output the policies Example iScience Article that can capture the cost associated with David's possible misjudgment. In short, SPS game and more accurately extended SPS game model have four cost parameters (a, b, c, d) that can capture the costs associated with their potential faults and their miscommunications. For strategic modeling, this structure of cost parameters is sufficiently specific to capture major cost determinants but general enough for inferring equilibrium analytically.
Third, the extended SPS game has a relatedness parameter k, which is termed inclusive fitness (payoff) and measures the cooperation tendency between two game players. This fifth parameter (k) of CDC game is hence designed to measure cooperation (or competition) tendency in the policy-making system.
Fourth, as mentioned previously, there is a sixth parameter-the pandemic risk (m), which is the counterpart of injury probability of Sir Philip's comrade in the classic SPS game. With the above-explained six parameters, the proposed CDC game is able to capture the estimation of pandemic risk (m), the costs (a, b, c, d) associated with reliability of the estimation, possible faults by Cindy and David, benefits (payoff) of the game strategies (analytically derived from the CDC game model). Finally, we have the relatedness parameter (k) that captures the intricacies of the strategic interactions between CDC chief and president. Although the president may ignore the chief's advice for political consideration, they do have shared interests measured by k. With the six parameters, a system of 8 differential equations is formulated to describe the dynamics of the CDC game, and possible equilibriums of the dynamic systems for 16 strategy iScience Article Box 1. The three key elements (Who, What and How) to formulate the CDC game, the transformation from SPS through to the CDC game, and the interpretations of CDC Parameters Three key elements (Who, What and How) to formulate the CDC game and the exposition of the transformation from SPS through to the CDC game (i) Who are the game players Sender (Cindy or Adviser) vs. Responder (David or Decision-maker). Sender (signaler) was Sir Philip's comrade, and Responder was Sir Philip Sydney in Maynard- Smith (1991) original SPS game. In the CDC game, Cindy is the message sender or adviser, and David is the responder or decision-maker. The essentiality of the role assignment lies in that who signals and who responds.
(ii) What is the content of message?
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 ''state-of-the-risk'' (injured in SPS or high-risk in CDC) with probability of m.
Through shouting or whispering in classic SPS game. In the CDC game, it is through some communication channel such as phone, person conversation, or media. However, the 'How' point does not matter much in the sense that it does not influence the mathematical logic of the game.

SPS Game
There were only four (2x2) strategies in classic (Maynard-Smith 1991) SPS game since it was considered sufficient for its initial design, i.e., to demonstrate the handicap principle or the reliability (honesty) of animal communications (Zahavi 1975(Zahavi , 1997 iScience Article interactions can be analytically obtained and simulations can be performed. Using differential equations to formulate games as dynamic system rather than the traditional matrix model is a huge advantage of the evolutionary game, and allows for obtaining evolutionary stable strategies, which means that ''mutation'' of strategies with time are considered, and the strategies should be sustainable over time in general from a practical perspective. Fifth, the SPS has an advantage over the familiar PD game, namely it can simultaneously capture all three fundamental processes (forces) driving biological and/or social evolution, whereas the PD game can only capture two (competition and cooperation) excluding communication. For example, in the PD game, there are no direct communications between the two players (criminals) and they are questioned by police officer separately. In the SPS game, both players have explicit communications, and in fact, the game is a requestresponse game. Furthermore, the communicated information can be asymmetrical (deception). This difference makes SPS game a more appropriate choice than PD game for our research purpose.
Intuitively, and also in fact, there are arguable only four major kinds of anti-pandemic, public-health policies regarding lockdown or reopen, including (1)  Hybrid equilibrium A hybrid equilibrium is some combination of separating equilibrium and pooling equilibrium.

Polymorphism
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.
Definitions for the stability properties Lyapunov Stable Assume a dynamical system which is described with _ x = f ðxÞ; x˛R n ; there is an equilibrium solution x˛R n , such that f ðxÞ = 0. The equilibrium solution x is stable in the sense of Lyapunov stable, if for given ε > 0, there exists d = dðεÞ > 0 such that, for any other solution yðtÞ, kyðt 0 Þ À xk < d ðt 0˛R Þ , then kyðtÞ À xk < ε for t > t 0 .
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.

Asymptotically 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 d = 0 with the radius r, such that the response, once entered the sphere, converges to the origin.
Neutrally Stable An equilibrium that is Lyapunov stable but not asymptotically stable is sometimes termed as neutrally stable. iScience Article policies or practices a country (region) may adopt and, in this article, we distinguish policies (referring to the above four) from strategies. Note that although we use the term policy in this article because we believe the usage is more natural in the context of public-health policy-making, the four policies described above and referred to in later sections, strictly speaking, should be termed strategy or meta-strategy in the terminology of game theory.
With the CDC game, we postulate that the four policies should be mapped to the output of the CDC game system, specifically, the ultimate decisions made by David with advice from Cindy. In other words, policies are the product or output of the decision-making process (system) such as the strategic CDC game. By distinguishing between the policy and strategy, one important gain is to shed light on the mechanism or process of decision-making process-how a particular policy can be reached by the advisor-decisionmaker body such as CDC-President taskforce. Furthermore, one can also investigate the influences of various pandemic and/or socio-economic factors (as captured by the previous mentioned six parameters) on the policy-making process and outcome.
We admit that the four-policy classification is the result of somewhat simplified view, which is necessary as explained previously; nonetheless we argue that it is sufficiently general to cover the full spectrum of currently adopted anti-COVID policies in the world. At present, the LOHC policy is adopted by most countries (regions) in the world, whereas no country has adopted its opposite (i.e., HOLC), which is irrational if not anti-intelligent. A handful of countries have adopted laissez-faire strategy (passive herd immunity), which is equivalent to AO policy. The strict AC policy appears impractical and no country (region) has claimed to adopt such a policy. However, arguably, the dynamic zero-out (infections) adopted by a handful of countries such as China and New Zealand may be considered as either AC (in terms of effects) or LOHC (in terms of implementation).

RESULTS-THE CDC GAME MODEL
The conceptual formulation of CDC game model After near two years of struggling against the COVID-19 pandemic, the vaccinations are bringing up, but the delta and emerging lambda are damping down, the hopes for reopening. Cindy, the CDC (center for disease control) chief scientist, is preparing her weekly briefing for the president, David, a newly elected president who has campaigned to bring peace and super humanity to the country and world. Cindy's team has been evaluating the nation's status of pandemic and vaccinations, and she realized that her team can only obtain a simple probability metric (m) to deliver her evaluation of the pandemic status, with m being worsen (high risk) and (1-m) alleviated (low risk). David, on the other hand, may need to balance the health versus wealth, the public interests versus other potentially conflicting interests. He may or may not follow Cindy's advice with potential loss (d) depending on the reliability of Cindy's advice and other factors on his side such as potential influences on economy and political correctness of his decision.
Furthermore, the trust between Cindy and David has been raising red flag, and their cooperative relationship is far from perfect. For example, given that Cindy was appointed by a previous administration and she might also be concerned with the risk of becoming a scapegoat in the case of catastrophic failure in antipandemic policy. In addition, Cindy's aptitude may be imperfect and can be faulty. In extreme cases, Cindy may delay or even hide her evaluation of the true state-of-the-pandemic. Cindy's potential faults, errors (benign faults) or dishonesty (malicious faults) can carry potential loss (cost) in the form of parameter a (underestimating risk m) or b (overestimate risk m), which influences her fitness or payoff.
Despite the potential communication issues between Cindy and David, both of them are certainly patriots and share common interests, and they tend to collaborate closely to maximize public and national interests. However, Cindy's care of her reputation as a scientist and David's care of his public approval rates are not necessarily always consistent, and may create competition. The balance between cooperation and competition between their interests can be measured by a relatedness parameter k, which is proportional to their tendency of cooperation. Furthermore, there is the cost (c) associated with strategic mistakes, the medical and socioeconomic consequences of misjudgment or miscommunication of the true state of pandemic, including consequences of possible cascading faults from both Cindy and David, i.e., cost of systemic failure.  9 , and Whitmeyer (2020) 14 , as well as Ma & Zhang's (2022) 13 ABD game. In formulating the CDC game (the bottom section), we adopted slightly different symbols with the classic SPS game, with Cindy for advisor (signaler or message sender) and David for decisionmaker (responder or message receiver). For the CDC game, Table 1 Figure 1 further illustrates the CDC as a decision-tree and strategy matrix. The differential equation system for the CDC game is introduced in the next sub-section.
As exposed in Table 1, Figure 1, and Box 1, with the strategy C 1 , we map ''open'' (low risk) in the CDC to signaling (healthy) in the classic SPS, i.e., C 1 = ''Cindy advises open only if she judges low-risk.'' This is equivalent to map ''closed'' to ''high risk.'' Regarding David's response, we map ''closed'' (high risk) in the CDC to ''no signaling'' in the original SPS, i.e., D 1 = ''Closed only if Cindy advises closed.'' Other strategy interactions can be mapped similarly. As further exposed in Table 1 and Box 1, there are three elements in trans-formulating the classic SPS into CDC game, who (are the game players), what (messages are sent and responded), and how (the message is transmitted). From the classic SPS to the extended SPS games including our CDC game, the strategy set is expanded to 16 (4x4) strategy interactions from 4 (2x2) interactions. Besides the three elements, a few additional points are particularly worthy of reiterations (see Box 1 for the details). First, once the first cell (C 1 , D 1 ) in the strategy matrix (set) is specified, the remaining 15 cells are set to keep consistent mathematic logic. Actually, once the ''who'' and ''what'' are set, the third element (''how'') may only influence the order of strategies in the strategic matrix, but not their mathematical logic. Second, although the classic SPS game is 2x2 asymmetric and not reciprocal, the extended SPS game including the CDC game is, in effects, equivalent to a reciprocal game given that virtually all 4x4 (16) possible strategy options are included in the strategy set (completeness). Owing to this completeness or reciprocal equivalence, approximately 1/2 of the strategies may be unnatural or senseless (without discernible meaning or purpose), if the other 1/2 is considered as natural or rational. We take the advantages of those unnatural strategy interactions to capture the idiosyncrasies in decision-making for anti-pandemic, some of which could be not only counterintuitive, but also unscientific, mirroring the various faults/noises in CDC explained previously.
In Table 1 and Figure 1, a series of 16 labels (A-P) for the 4x4 combinatorial strategies (strategy pairs or strategic interactions) of Cindy and David are assigned to facilitate the discussion on the CDC game. Furthermore, the bottom section of Table 1 also list the four basic anti-pandemic policies introduced in the introduction section, and their corresponding equilibriums in the CDC game, but further explanation of the relationship between policies and equilibriums are delayed to the section of results. Note that we use the letters (A-P) in bold font to denote the strategy pairs, which are the combinations of Cindy and David's strategies that are denoted with italic letters (C 1 -C 4 , D 1 -D 4 ). In addition, we use the bold italic fonts to denote those strategy pairs (out of the 16 possible pairs of A-P) with equilibriums (e.g., A, F). Strictly speaking, A-P represent 16 strategy pairs, i.e., the combinatorial strategies of Cindy and David, each of whom has 4 strategies (C 1 -C 4 , D 1 -D 4 ). In the following, whenever no confusion arises, we may call the strategy pairs (A-P) strategy for short. These conventions of letter usages are followed throughout the article.
For the mathematical model (systems of differential equations) of the CDC game model as well as the detailed parameter interpretations (designations), refers to the methods detail in the STAR Methods section.

Equilibriums and stabilities of the CDC game
In this section, we take advantage of Huttegger & Zollman (2010) 9 extensions and analytical results on the extended SPS game, and summarize the equilibriums of the CDC game in Table 2, and the payoff of the CDC game strategies in Table 3, respectively. From Equations 1 and 2, six types of equilibriums are derived and summarized in Table 2, and a brief description is presented below. With the strategy (pair or interaction) A of (C 1 versus D 1 ), Cindy advises for open only if she perceives low risk of pandemic, and David decides closed only if Cindy advises for closed. In this strategic interaction, David follows Cindy's advice, whereas Cindy seems rational with her strategy of ''open only if low risk.'' The strategy pair is Nash equilibrium and is asymptotically stable (see Box 2 for the terminology interpretations). Overall, the strategy pair A seems to be one of the most rational CDC strategy interactions from scientific perspective. Furthermore, because A is Nash equilibrium, nobody would regret for their decisions once they are committed. Strategy A appears to be the most natural and rational strategy.
Strategy A is nearly universally accepted by most countries in the world, and it is mapped to the number #1 anti-pandemic public health, i.e., the low (risk) open and high (risk) closed (LOHC) ( Tables 1 and 2). The LOHC is arguably the most rational among the four fundamental policies. The LOHC policy can be obviously made from strategy A with Nash equilibrium. However, the polymorphic hybrid strategy (PHS) can also lead to the LOHC policy, which may explain the idiosyncrasies occurred in various parts of the world regarding the anti-pandemic policy. That is, although the LOHC is adopted ultimately, the decision-making process to reach the policy could involve some irrational or unnatural strategy interactions, e.g., could be a trial-and-error process.
Strategy F is an ESS (evolutionary stable strategy), asymptotically stable, and can be mapped to LOHC policy With the strategy pair F of (C 2 versus D 2 ), Cindy appears irrational and advises for open if the risk is high, and perhaps she believes that the cost for closure is too high even though the pandemic risk is high. David appears more rational and overrules (reverses) the advice from Cindy, possibly either he does not trust Cindy's advice or he believes that the benefits from closure outweigh the cost when the risk is high. Because F is an Anti-pandemic Policy (also see Table 1) Asymptotically stable LOHC (low-risk open and high-risk closed) and is Asymptotically stable.
LOHC a J and K are behaviorally equivalent.
Neutrally stable AC (always closed) iScience Article ESS, their decisions (behaviors) may be mutated when the pandemic expands and, ultimately, they may be converted from one strategy to another. Also because F is asymptotically stable, their consensus (equilibrium) has a region (range), which means both Cindy and David have certain level of flexibility in their behaviors. Overall, in this strategy interaction, the president (David) seems to have an intuitive and more rational attitude to the risk and decision than his advisor, Cindy.   1 À c + kð1 À dÞ + mð À a + c + kdÞ 1 + kð1 À cÞ À d + mð À ak + ck + dÞ C -(C 1 : Open only if low-risk; Open only if low-risk; Open only if high-risk; Open only if high-risk;   iScience Article Strategy L is a pooling equilibrium and can be mapped to AC (always closed) policy With the strategy pair L of (C 3 versus D 4 ), Cindy always advises closure regardless of the risk level, and David always decides closure regardless of Cindy's advice. The equilibrium is neutrally stable. From a pure theoretic perspective, the equilibrium type L is more likely to become established because it has a larger basin of attraction, which measures the likelihood of evolving under standard evolutionary dynamics (Equations 1 and 2). From a pure health perspective without considering wealth, the strategy can be optimum. Overall, the strategy pair appears to mirror a pair of advisor and decision-maker who take totally same attitudes to the disease and risk. Because the equilibrium of strategy is a pooling equilibrium, Cindy's has no influences on David although Cindy's advice is always consistent with David's decision.
Obviously, this strategy interaction can be mapped to AC (always closed) policy for anti-pandemic. This strategy may be an optimum if the pandemic is over within a reasonably short period or economic loss is not a concern. Furthermore, if pandemic can be suppressed in a short period, the economic loss would not be a concern either. Mixed strategies of J and K are pooling strategies and both are behaviorally equivalent, and can be mapped to AO (always open) policy When Cindy adopted strategy C 3 (always closed), and David adopted a mixed strategy of D 2 (Closed only if Cindy advises open) with probability (1Àl) and D 3 (always open) with probability l, there is a pooling equilibrium that is neutrally stable. The condition for the equilibrium to occur is d > k½ma + ð1 À mÞb. David's mixed strategy is ð1 À lÞD 2 + lD 3 , and l R ½1 À c =ða À kdÞ determines the strategy of David.
Theoretically, when David hesitates, he can resort to l to make an optimal decision, i.e., choose D 2 when l R ½1 À c =ða À kdÞ, or D 3 when l < ½1 À c =ða À kdÞ. Mixed strategies of I and L are pooling strategies and both are behaviorally equivalent, and can be mapped to AC (always closed) policy When Cindy adopted strategy C 3 (always closed) and David adopted a mixed strategy of D 1 (Closed only if Cindy advises so) with probability (1Àm) and D 4 (always closed) with probability m, where m = 1 À ½c =ðkd À bÞ, there is a pooling equilibrium that is neutrally stable. Similar to previous J & K strategies, there is a pooling equilibrium that is neutrally stable, but here m determines the strategy of David. Intuitively, because Cindy always advises closed, David may hesitate: but in the end, he still decides to close. The condition to determine David's behavior seems to be opposite with the previous mixed strategy interaction of J & K. Strategies I and L are behaviorally equivalent with each other.
In practice, similar to strategy L, this mixed strategy of I and L maps to AC (always closed). The policy should be optimum when pandemic ends in a short period (''wealth'' is not a concern) or economic loss from closure is not a concern.
Polymorphic hybrid strategies (PHS) iScience Article First, how would Cindy make her recommendation? In fact, it is the parameter m that determines her recommendation to choose either C 2 or C 4 . Theoretically, when she hesitates, she can resort to m to make an optimal decision, i.e., choose C 2 when m > c=ðb À kdÞ, C 4 when m < c=ðb À kdÞ. When m = c= ðb À kdÞ, she tends to adopt mixed strategy lC 2 + ð1 À lÞC 4 . when l > fK½ma + ð1 À mÞb À d g=½ð1 À mÞðkb À dÞ . When l = fK½ma + ð1 À mÞb À dg=½ð1 À mÞ ðkb À dÞ, he tends to adopt mixed strategy mD 2 + ð1 À mÞD 3 . This hybrid equilibrium is Lyapunov stable, which implies that the actual stability may vary depending on the values of parameters. In other words, the stability of equilibriums may be broken when conditions such as risk level and/or cost change dramatically. Figure 2 shows an example of the decision-making in the phase space of the polymorphic hybrid equilibrium, obtained from the simulation study explained below. The polymorphic hybrid strategies, arguably, represent the most sophisticated options, which reflect the dynamic (evolutionary) nature of the masking-or-not decision-making.
In  iScience Article In summary, as listed in Table 1 (the bottom section) and Table 2 (the last column), arguably the most rational LOHC policy can be the output of equilibriums A & F; the AO policy can be the output of equilibriums J & K; the AC policy can be the output of equilibriums I & L or L. The polymorphic hybrid strategy (PHS) can produce any of the three previously discussed policies, i.e., LOHC, AO, or AL. However, none of the six types of equilibriums can produce the HOLC policy, which should be expected that given the policy is unlikely to sustain in practice.

Strategy interactions without equilibriums
For strategy interactions that do not produce equilibriums, there are the following mappings: strategy interactions D, H, L, P, N, and I can produce AC policy, and strategy interactions C, G, K, O, M, and J can produce AO policy. Because these strategy interactions, except for I, J, and K do not have known equilibriums, which implies that they may not be sustainable in practice, we do not discuss them further in this study.
Finally, strategies B & E can be mapped to irrational HOLC policy, but both the strategies do not have any equilibrium (Table 2), which should be expected and is consistent with the reality that the HOLC policy is so irrational and is unlikely to sustain practically.

Simulation experiments Simulation procedures and phase portrait
Because the analytic solutions (Tables 2 and 3) are less intuitive in interpreting the decision-making process of the CDC game, particularly, in understanding the influences of various CDC parameters (cost/benefit factors, risk level, cooperation tendency), we performed simulation experiments based on the replicator dynamics (Equations 1 and 2). Tables S1, S2, S3 and S4 contain the 4 lists of equilibriums corresponding to the three public health policies (LOHC, AC, AO), and the list of PHS, which may be mapped to one of the previous three policies, depending on the specific parameter values of the CDC game. The simulations were performed based on the replicator dynamics (Equations 1 and 2) with a program developed by . 13 Specifically, the equilibriums were simulated and determined based on the necessary conditions (listed in Table 2) and corresponding payoffs were computed based on the equations listed in Table 3, with simulation step length = 0.1 for all of the model parameters (a, b, c, d, k, m), all of which range from 0 to 1. A total of 10 8 simulations were performed to obtain the results, and the results were organized in terms of the policy category (online Tables S1, S2, S3 and S4).
The strength of simulations helps us to illustrate the complex interactions between advisor and decisionmaker intuitively, but the simulations cannot be exhaustive because of the continuous nature of the CDC model. In the following, we use the simulations to primarily address two questions: (1) The influences of various CDC parameters on the outcome (policies) of CDC game; those parameters approximate the cost/benefits, risk level, faults, miscommunications (dishonesty), etc in the decision-making system.  , and then decline steadily; that of adopting AC exhibited an opposite trend; and that of adopting LOHC policy is rather low but stable. The probability from adopting a policy generated by PHE (polymorphic hybrid equilibrium) is negligible. These probability trends appear natural. For example, when the cost (a) is relatively small (a<0.3), AO policy is favored (AC is disfavored), but when the cost is large (a>0.6), the trend is reversed. The cost parameter seems to have little influence on the likelihood of LOHC policy except when the cost is extremely high (a>0.7 approximately). The explanation for the negligible PHE probability is likely because of unduly complexity in implementing the PHE strategy. Indeed, we postulate that the small-probability associated with LOHC and PHE is because of their unduly complexities. Figure 3B plotted the estimated probabilities of adopting three policies (LOHC, AO, AC) from 10 8 simulations of the CDC game under different cost parameter b (overestimating the risk) of Cindy. With the rise of Cindy's cost to overestimate the risk (b), the probability of adopting AC increases gradually and reaches an asymptote; that of adopting AO exhibited an opposite trend; and that of adopting LOHC policy is rather low but stable. The probability from adopting a policy generated by PHE is negligible. These probability trends appear natural. For example, when the cost (b) of overestimation increases, AC policy is increasingly favored, and AO policy is increasingly disfavored. Figure 3C plotted the influence of strategic cost (c) on the likelihoods of the three policies (LOHC, AO, AC). With the rise of strategic cost (c), the probability of adopting AC increases steadily and reaches an asymptote; that of adopting AO exhibited the opposite trend; and that of adopting LOHC policy is rather low but stable. When the cost (c) > 0.6, the likelihood of adopting LOHC declines slightly, which may have to do with the possibly rising difficulty in effectively implementing the LOHC policy. The probability from adopting a policy (which could be any of the three policies, AC, AO, LOHC) generated by PHE (polymorphic hybrid equilibrium) is negligible, and little influenced by the strategic cost (c). These probability patterns again seem natural. For example, with the rise of strategic cost (c), the policy becomes more conservative and increasingly in favor for AC and against AO. Figure 3D illustrated the influence of David's cost parameter (d) on the probabilities of three policies (LOHC, AO, AC) from 10 8 simulations of the CDC game. With the rise of David's cost (d), the probability of adopting AO increases slowly, but declines when the cost is exceptionally high (>0.9); that of adopting AC exhibited the opposite trend; and that of adopting LOHC policy is low and decline slightly. The probability from adopting a policy generated by PHE (polymorphic hybrid equilibrium) is negligible. When David's cost is exceptionally high (d > 0.9), the policy is likely to reverse, which seems rather natural.  iScience Article Cindy and David (their collaboration tendency), the probability of adopting AO declines steadily; that of adopting AC increases gradually; and that of adopting LOHC is low but largely stable. The probability from adopting a policy generated by PHE (polymorphic hybrid equilibrium) is negligible. The finding seems natural again. For example, when the collaboration tendency increases, the AC policy is favored increasingly.

Overall likelihoods (probabilities) of the three policies (LOHC, AO, AC)
In the previous section, we discussed the influences of single factor (CDC parameter) on the policy-making. In this section, our focus is on the overall probabilities of the three policies. In Figure 4, we plotted the overall probabilities of adopting three basic policies (LOHC, AO, AC), respectively, across all simulated parameter settings (i.e., averaged from the values across all parameters or Figures 3A-3F). The fourth item-PHE-can be classified into one of the three basic policies, depending on complex interactions between Cindy and David. However, to classify each PHE into one of the three policies, additional 'real-time' (i.e., the threshold value of parameter l or m determines the choice of mixed strategy) interaction information is necessary, which requires additional extensive simulations. Because the total (combined) probability (p = 0.002) of the three policies from PHE is negligible, the lack of further classifications should only have negligible biases on the probabilities of the three policies drawn in Figure 4.
From Figure 4, findings on the overall likelihoods (probabilities) of the three policies include: (1) The near universally adopted LOHC, i.e., a middle-ground of AC and AO policies, intuitively should be preferred, A B Figure 4. The probabilities of adopting the three policies estimated across all possible parameter settings The fourth item-hybrid (i.e., PHE)-can be classified into one of the three basic policies, depending on complex interactions between Cindy and David; nevertheless, its probability is negligible and so is its practical implications. Figure 4A (pie chart) shows the 'partition' of the probability space ('pie'). Figure 4B shows  (Table 1), but they do not have equilibriums, which implies evolutionarily unstable and is unlikely to sustain in practice.

OPEN ACCESS
but can be too complex to enforce practically, as suggested by the only slightly larger than the probability of small event (p = 0.053).  Figure 4, is obviously anti-science and unnatural. Although LOHC may be generated from strategies B & E (Table 1), both strategies B & E do not have equilibriums, which means that they are evolutionarily unstable and unlikely to sustain in practice.

Payoffs of the CDC game players
In previous sections, our discussions are focused on the simulation of the influences of various CDC game parameters (that represent for various costs/benefits, risk and cooperation/communication factors) on the strategies (policies). In this section, our focus is the payoffs of CDC game. Figure 5 plotted the payoffs of Cindy and David under different policies, respectively. Notably, the order of payoffs is different for Cindy ( Figure 5A) and David ( Figure 5B). With Cindy's payoff, the descending order (from left to right) is AC>LOHC>AO>Hybrid, and with David's payoff, the descending order is LOHC>AO>AC > Hybrid. The orders are made based on statistically significant differences based on Wilcoxon tests (p-value < 0.05). The opposite payoff-orders suggest that Cindy prefers tighter AC policy, whereas David prefers more relax AO policy. This difference may be due to the reality that David should be more concerned with the health versus wealth tradeoff or dilemma.

Conclusions
The main insights from the CDC game can be summarized as follows: (1) Among the 16 (4x4) equivalently reciprocal strategy interactions of the CDC game, approximately ½ are unnatural or irrational, which allows us to capture not only appropriate strategies but also those unnatural (or irrational) strategies that might be generated by faulty judgment(s) and/or miscommunications in policy-making system. An advantage of our CDC game approach is the completeness of the 16-elements strategy set in a two-player game.
(2) The CDC game has six types of equilibriums, which are evolutionarily stable and can be sustainable in practice. Some of the equilibriums are Nash equilibrium, which means that no game player would regret for his or her strategy. Some are pooling equilibrium, which can be roughly considered as ''dictatorship-style''; in the case of CDC game, the pooling equilibrium implies that David would B A Figure 5. Cindy's and David's payoffs under different policies are illustrated as standard box charts respectively Notably, the order of payoffs is different for Cindy (the left chart) and David (the right). With Cindy's payoff, the descending order (from left to right) is AC>LOHC>AO>Hybrid, and with David's payoff, the descending order is LOHC>AO>AC>Hybrid. The orders are statistically significant per Wilcoxon tests (p-value < 0.05). Note the scatter points are omitted in the box charts to avoid overwhelmingly large number of points, which could lead to overly big file size of the graph. iScience Article ignore Cindy's advice. Some are separating equilibrium, which can be roughly considered as the opposite to pooling equilibrium, i.e., Cindy's advice does make differences. The so-termed polymorphic hybrid equilibrium (PHE) means that sometimes the equilibrium is pooling and sometimes separating.
(3) There are four basic anti-pandemic policies: LOHC, HCLO, AO, and AC. They can be mapped to the output of CDC game system, i.e., the ultimate decision made by David in the CDC game. The LOHC is arguably the most rational policy and has been adopted by the majority of countries (regions) in the world. The diametrically opposite HOLC is obviously the least rational, and it is unlikely being adopted unless fatal mistake occurs. Although the HOLC can be mapped to strategy interactions B & E, both do not have any equilibriums, mirroring the reality that the policy is unlikely to sustain in practice. In contrast, the LOHC policy can be mapped to strategy interactions, A, F, and PHS (polymorphic hybrid strategy), with A being Nash equilibrium, F being ESS (evolutionarily stable strategy), and PHS being hybrid (sometimes pooling and sometimes separating).
The simulation (of 100-millions times) with the CDC game revealed that the probability of LOHC is small-probability event (P = 0.053), whereas the probability of HOLC is zero. The small-probability event characteristic of LOHC appears to be counter-intuitive, and we postulate that practical implementation of LOHC can be too complex to be feasible, especially for the COVD-19 pandemic that has been recurring one wave after another. The multi-wave and extreme contagiousness of COVID make failure in one round of reopening sufficiently ruin the whole efforts, i.e., cascading failures. The small probability of LOHC seems to mirror a reality that, although most countries have adopted it, arguably and at least on the surface, obviously most countries have not achieved their expectations.
(4) The AO policy can be mapped to the equilibriums of mixed strategy of J & K, or PHS. The mixed strategy J & K are pooling equilibrium, which means that the CDC chief scientist's advice makes no difference or is ignored by the ultimate decision-maker, David. The hybrid strategy PHS may also generate AO policy, and in this case, sometimes the chief's advice matters and sometimes does not. The simulation of 100-million times with the CDC game revealed that the probability of AO policy is P = 0.412, and that of PHS is negligible (P = 0.002). Therefore, virtually all AO policies are generated from the mixed-strategy of J & K, which is from David's ''dictatorship-style'' decision (a characteristic of pooling equilibrium).
(5) The AC policy can be mapped to the equilibriums of mixed strategy I & L, or PHS. Simulations revealed that the probability of the AO policy is P = 0.533, and that of PHS is negligible (P = 0.002). Similar to the previous AO policy, the equilibriums of I & L are pooling, and therefore, Cindy's advice usually does not matter in the making of AC policy.
(6) Simulations of 100 million times suggested that the order of probabilities of three policies is AC>AO > LOHC, while HOLC = 0. Of interest, in terms of the payoff, Cindy and David have different orders. For Cindy, her payoff order is AC>LOHC>AO; for David, his payoff order is LOHC>AO>AC. Note that these three orders are of different nature. The first order is the likelihood of policy order, which is the product of the CDC game, or the outcome of Cindy and David strategic interactions. The second order is Cindy's preference (payoff) order, which should largely depend on her professional expertise; her profession as CDC chief obviously should have pushed her to prefer AC, i.e., her payoff (such as reputation or satisfaction from saving lives) is maximized. The third order is David's preference (payoff) order, which should largely depend on his duty as the president who needs to balance health versus wealth. For president, AC is likely his least favorite, whereas LOHC is ideally his top choice although it is often too complex to implement. Therefore, although both Cindy and David seem to have different preferred strategies, their interactions produce a single set of ''metastrategies'', i.e., the policy set of LOHC, AO, and AC.
(7) More or less, most countries (regions) in the world seemed to have adopted the LOHC policy, whereas a handful of countries have adopted laissez-faire strategy such as passive herd immunity that may be considered as AO policy. The dynamic zero-out policy adopted by several countries (notably China and New Zealand) seemed to be equivalent to AC in terms of the effects, and to ''precision'' LOHC operationally in terms of the implementation.
In conclusion, through rigorous analytic and extensive simulation analyses of the CDC game, we obtained reasonable interpretations for the three fundamental public-health policies against the COVID-19 iScience Article pandemic, i.e., AO, AC and their middle ground LOHC, precluding the obviously irrational or anti-intelligent HOLC policy. The policy of AO is likely to be wealth-optimal, while the policy of AC is likely to be health-optimal. The LOHC may achieve some level of balance between health and wealth, given it is a middle ground of both AC and AO. The HOLC generally favors neither health nor wealth, what we called antiintelligent and is not sustainable (corresponding to no equilibriums in the CDC game). In real world, most countries (regions) have adopted the LOHC policy, which seems to balance the health versus wealth dilemma and appears to be an optimal policy choice. However, simulations suggest that the policy may be too complex to implement successfully given its probability is only 0.053, a typical small probability event. In contrast, the two ends of the policy spectrum, AO and AC exhibited large probabilities (close to 0.41-0.53). A take-home message seems to be KISS (keep it simple strategically and persistently), and the large probabilities of AC and AO are likely due to their simplicity.

Discussion
Humans are believed to possess exceptional evolutionary capacity for decision-making, which should have contributed to the dominance of the humans on planet earth (Samson 2020). 33 Although on macroscopic scale, humans have been enormously successful from selecting the right food and shelter, to devising complex economic strategies and effective public health policies (Samson 2020) 33 , we have occasionally made expensive and painful mistakes, locally, regionally and globally, including the strategies and policies for fighting the COVID-19 pandemic. The challenge seems particularly serious in the era of disinformation (e.g., Bergstrom & West 2020). 34 Given the nature of the complexity and challenges, we formulate the decision-making problem for anti-COVID-pandemic as the CDC game, which is an extension to the classic SPS game. The latter achieved wide success in studying the reliability (honesty) of animal communications in evolutionary biology. The SPS game, especially its extensions (Huttegger & Zollman 2010, Whitmeyer 2020, Ma & Zhang 2022) are particularly powerful in modeling the processes that are driven by 3C forces (cooperation, competition and communication). 9,13,14 Since we are only interested in strategic decisionmaking, it is natural for us to focus on the stable equilibriums. Tactical or operational level analyses are not considered in this article.
There are 16 (4x4) possible strategy combinations (interactions) with the CDC game given it is an asymmetric four-by-four strategic-form game. Also given the completeness of the strategy interactions (4x4), the strategy interactions are in effects reciprocal. That is, there is likely an irrational (or unnatural) counterpart for each rational (or natural) strategy interaction. Given the equivalently reciprocal nature of the CDC game, it is expected that approximately 1/2 of the strategy interactions can be unnatural. Furthermore, there are six types of equilibriums (Table 2) among the 16 possible strategic interactions (Tables 1 and  2). The six equilibriums can be classified as separating (signaling), pooling, and polymorphic hybrid equilibriums, and they may possess different behavioral and payoff properties (see Table 3 The LOHC policy appears to be the most rational and apparently most effective, and it has obviously been adopted by most countries (regions) of the world for fighting against the COVID19 pandemic. Herd immunity or co-existence policies can be considered as AO policies. The strict AC policy, although medically optimum but economically risky, seems to be the least popular. In summary, co-existence with virus versus zero-out virus appears to mirror the AO versus AC policies, while the LOHC policy seems to be the middle ground of both AO and AC. iScience Article anti-pandemic policy-making. The explanation for the approximately ½ natural versus unnatural strategies or with/without corresponding equilibriums should be to do with the reciprocal nature of the strategy set (options) as explained previously. Although it is difficult to interpret all of the unnaturalness or the idiosyncrasy in half of the strategy set, there are some answers from the recent studies in behavior economics (Samson 2014(Samson , 2020. 33,37 Studies have revealed that humans are not always self-interested, benefits maximizing, and costs minimizing with stable preferences. Human minds are not always rational and may only possess bounded rationality, which suggests that human rationality is limited by brain's information processing capability, insufficient knowledge feedback, and time constraint (Ma 2015a, 2015b, Samson 2014, 2020), not to mention complex politics in the case of the CDC game. 33,[37][38][39] The fact that approximately ½ of the possible strategy options seem to demonstrate the rather high likelihood to devise an unsustainable (faulty) policy. In other words, behaving rationally should not be considered as granted for the CDC-president interactions.
In contrast, approximately half of the strategy interactions with possible equilibriums can be sustainable, and mapped to one of the three basic policies (AO, AC, & LOHC). However, equilibriums, stability, and rationality do not necessarily correspond to scientifically optimal, and may not even necessarily correspond to scientific correctness, since the CDC output (policies) may also be influenced by politics or other socioeconomic factors. This is determined by the aim of this study-identifying the potential cognitive gaps and possibly traps by emulating the reality of anti-pandemic policies with the CDC game model.
The AO policy is essentially the laissez-faire strategy (e.g., passive herd immunity), which was initially adopted by a handful of countries including Sweden and partially by UK. The strictly AC seems unrealistic intuitively, but theoretically it does corresponds to two strategies with stable equilibriums. In practice, the dynamic zero-out (infections) policy adopted in China and New Zealand can be considered as either precision version of LOHC in terms of implementation, or an equivalent AC policy in terms of the effects.
The dynamic zero-out policy does have a precondition, namely, it must be feasible to control the infection level below the threshold of Allee effect, and therefore, the infections level should be relatively lower in general to further push it below the threshold (of ''local extinction'').
The Allee effect is named after animal ecologist W. C Allee (Allee & Bowen 1932) 17 , and it is a theoretic threshold of population growth (such as the infections of COVID- 19), above which population growth may accelerate and below which population growth may decelerate and may go extinct ultimately (Kramer et al. 2017). 40 Its implications to epidemiology have been discussed in recent literature; for example, Friedman et al. (2012) studied the relationship between the Allee effect of disease pathogens and the Allee effect of hosts (e.g., humans) to answer a question of fundamental importance in epidemiology: Can a small number of infected individual hosts with a fatal disease drive the host population to go extinct (assuming a healthy stable host population at the disease-free equilibrium is subject to the Allee effect)? 41 Ma (2020) estimated the population aggregation critical density, the threshold for aggregated (clustered) infections of COVID-19 to occur. 42 Besides Allee effects, two other theories, i.e., metapopulation dynamics and tipping-point theories, are particularly important for supporting the dynamic zero-out policy. The metapopulation theory maintains that extinctions of local population are common events, and tipping point theory suggests that, at some critical points, population growth may transit to either outbreak or die-off from the tipping point (threshold) (Citron et al. 2021). 16 There are techniques for detecting tipping points by identifying some early warning signals (EWS), which can help to implement dynamic zero-out policy. These theories suggest that there are thresholds (tipping points) at which local pathogen (infections) may go extinct, depending on the dynamics of host-pathogen system and disease control measures. Therefore, public health policies designed to drive the infections (such as SARS-COV-2) to go extinct locally (below the threshold of Allee effect of the pathogen), whereas keeping the host population absolutely safe from the risk associated with the Allee effect of the hosts (this is, of course, not an issue for humans at all), should be feasible. In other words, the dynamic zero-out policy, which is equivalent to driving local infections to extinctions, is theoretically feasible when implemented properly. The practices of dynamic zero-out policy during much of the COVID-19 pandemic period by a handful of countries, notably China and New Zealand, have demonstrated its feasibility and success.

Limitations of the study
There exist a few limitations with our study, as rightly pointed out by the anonymous reviewers of our article.  [52][53][54][55][56] Although we do believe that these additional factors are critical for the decision-making of anti-COVID pandemic, the game theoretical analysis should focus on strategy level, and those additional factors can be investigated at tactical and operational levels.
In this aspect, the so-termed three-layer survivability analysis (Ma 2008, Ma & Krings 2011, in which traditional engineering reliability theory and survivability theory for survivable network systems were unified in a hierarchical framework consisting of the strategic, tactical and operational levels, may be applied to develop a decision-support system with strong operability for the anti-pandemic decision-making. 12,57 The three-layer survivability analysis framework consists of survival analysis (reliability analysis) at tactical level, dynamic hybrid fault models with the so-termed ''Byzantine Generals playing evolutionary game'', and the optimization supported with hedging principle and evolutionary computing. 12,57 The embedding of the CDC game modeling into the three-layer survivability analysis framework should be able to provide a blueprint for building a decision-support system for managing pandemics, including possible future public health crises.
In addition, what epidemiology studies are typical complex dynamic systems, from emergence, epidemic, through pandemic to endemic stage. Prediction per se is a rather challenging task, especially for dynamic systems such as COVID-19 pandemic (Vytla et al. 2021, Ma 2020, 2022. 42,[58][59][60] According to Vytla et al. (2021) survey, the prediction of the COVID-19 pandemic can be characterized as ''the kryptonite of modern AI'' and ''many predictions by AI and machine learning were neither accurate nor reliable.'' 58 Properly predicting the stage transitions and associated EWS (early warning signals) for the tipping points of epidemics/pandemic is another critical mission of decision-making body such as CDC, which is necessary for timely and smooth adaptations/transitions of anti-pandemic strategies/policies. To further illustrate the point, we end the article with recording following fictitious transcript of a phone conversation between Cindy and David.
We emphasize that the fictional scenario below related to the COVID pandemic is in no reference to any actual conversation between any real persons. It should also be reiterated that the dialogue only represents an idealized scenario out of 16 possible interactions of the CDC game, actually a possibly dynamic mixture of the interactions based on specific stages of the pandemic. Readers are free to infer their own thoughts from the conversation. However, the authors take no standings on the conversations and their intentions and interests are limited to the pure mathematical and simulation analyses of the CDC game model per se. If there is any implicit expression of personal opinions from the authors, that is the call for mutual understanding and reconciliation between the AO (always open) and AC (always closed) supporters. In the end, given that the virus-host interaction is a dynamic system, which is deeply interwoven with socio-economic systems, our strategies to intervene the interaction systems in favor for humans, whether it is to extinguish, contain, or accommodate the virus, should be dynamically adaptive with the evolution of virus, the development of economy and impacts from social and political environments. In other words, the optimal policies should be stage (emergence, outbreak, pandemic, endemic) dependent.
Assume the following contrived scenario: It's the year-end of 2022. President is planning for his New Year's speech, and he picks up the phone and calls Cindy. Cindy: Yes, Mr. President. I think it will be over next year, and this should be the last New Year's briefing they are still interested in this topic.
David: Right, Cindy. It has been three whole years. I still recall our first meeting three years ago. You were almost shouting, at me, ''Mr. President, it's time to shut down . and enact the quarantine immediately.'' You were right and saved all of us from the first wave! Cindy: Thank you! David, I have always been appreciative of your trust! You know, it was not only feasible, but also the most cost-effective strategy during the initial outbreak.
David: That's right! But not long later we had to enter the cycles of lockdown-reopen . one round after another. Frankly, I had somewhat lost patience occasionally, certainly not with you, Cindy, but with the cascading ups and downs, locally, nationally, and continently. You were telling me your LOHC repeatedly . economy began to suffer . people losing patience . In the end, we saved millions of lives while avoiding stagnation of the economy . that's a fact! I thank you for your persistence and service to our country.
Cindy: David, it was your wise calls! I only did what I was supposed to do.
David: You are too generous! It was your thorough, timely, and convincing analysis that persuaded me! I just wonder if we could have lifted all the controls earlier.
Cindy: I wish we could! But David, you know, it's all about timing! Omicron is a frenemy variant, and our immune systems can usually beat it just like overcoming the flu.
David: What about long-COVID and the possibility of a new vicious variant? Which seem to be a current focus of public concerns.
Cindy: Mr. President, I understand. Nobody has a crystal ball, but I believe the chance of a vicious variant is rather small. In case I am wrong, we may have to restart, but I am confident that we will do it even better! As to long-COVID, it is beyond my expertise, as you know, Mr. President. My suggestion would be to increase funding for research.

ETHIC APPROVAL
N/A because the study does not involve any wet-lab experiments or survey on human or animal subjects, and all analyzed datasets are already available in public domain, as mentioned above.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following: