1 Introduction
2 Setup of the game
2.1 Threshold levels and optimal strategies

Case 1a: \(W<0\) and \(K>0\). The tradeoff between obey and disobey under a hard policy is positive, while it is negative under a soft policy. When politicians choose hard, citizens’ dominant strategy is obey, while if politicians choose soft citizens’ dominant strategy is disobey. This is a case in which the reduced probability of being fined under a soft policy makes it easier for people to break the rules in order to avoid the policy cost.

Case 1b: \(W<0\), \(W>K\), \(K<0\) and \(K<\gamma (1y_O)\). The tradeoff between obey and disobey is positive both in the presence of hard and soft policies, but the excess utility of obey with respect to disobey in soft is smaller than the peer effect. Therefore, in the case of a soft policy, if the number of obeying individuals is “large enough,” the peer effect may prevail over K and disobey is the citizens’ dominant strategy.

Case 1c: \(W>0\), \(K>0\) \(K>W\). The citizens’ dominant strategy is disobey both under hard and soft policy regimes. This result holds irrespectively of the peer effect value.

Case 1d: \(W=\gamma \) and \(K=0\). The tradeoff between disobey and obey is zero at point (0,1), i.e., soft and obey. At point (1,0), i.e., hard lockdown and disobey, the excess utility of disobey with respect to obey (net of the peer effect) is exactly equal to the peer effect. In other words, this is case in which citizens are indifferent toward the two strategies.

Case 2a: \(K<0\), \(K>\gamma (1y_O)\) and \(W>0\). Disobey is the dominant strategy in the presence of a hard policy, while obey is the dominant strategy under a soft policy; this is true since K prevails, in absolute value, over the peer effect.

Case 2b: \(K<0\), \(K>\gamma (1y_O)\), \(W<0\) and \(W<K\). Obey is the dominant strategy for citizens under soft and hard policies, if the peer effect is zero. However, if \(\gamma >0\), the peer effect could prevail over W, since \(W<K\). So, if the number of disobeying citizens is “large enough,” disobey is the dominant strategy for individuals under a hard policy.
3 The evolutionary game
Parameter  Case 1a  Case 1b  Case 1c  Case 1d—Bifurc  Case 2a  Case 2b 

B  10  10  10  10  10  10 
D  9  9  1  9  3  3 
\(C_h\)  7.5  7.5  7.5  7.25  5.6  5.4 
\(C_s\)  7.3  6.6  6.6  7.2  2.75  2.75 
\(\mu \)  0.7  0.7  0.7  0.7  0.7  0.5 
L  2  2  2  2  2  2 
\(\beta \)  0.7  0.7  0.7  0.7  0.5  0.5 
\(\theta \)  0.9  0.9  0.9  0.9  0.5  0.5 
\(\gamma \)  6.5  6.5  2  6.5  0.5  0.5 
F  7  7  7  7  2  2 
\(\sigma _h\)  0.7  0.7  0.2  0.7  0.7  0.7 
\(\sigma _s\)  0.1  0.1  0.1  0.1  0.2  0.2 
\(\omega _s\)  2  2  2  2  0.7  0.7 
\(\omega _h\)  0.4  0.4  0.7  0.4  0.3  0.3 
W  – 6  – 6  4.7  \(\) 6.5  0.3  \(\) 0.1 
K  0.2  \(\) 1.2  6  0  \(\) 2.4  \(\) 0.4 
WK  \(\) 6.2  \(\) 4.8  \(\) 1.3  \(\) 6.5  2.7  0.3 

Case 1a: As we mentioned in Sect. 2.1, this case is characterized by the fact that citizens’ dominant strategy under a soft policy is disobey, as suggested by \(K>0\), while the dominant strategy under a hard policy (net of the peer effect) is obey. The peer effect, however, is fairly large (\(\gamma =6.5\)) and may offset W, when the number of disobeying citizens is large enough.^{14} It follows that when the number of nonobeying citizens increases, the dominant strategy under a hard policy may become disobey. This leads to the dynamic shown in Fig. 2: The dynamics are represented by a growing spiral starting from the internal equilibrium, which converges to the only stable fixed point represented by (1, 0). Indeed, if we assume that obey and hard are frequent strategies (so that we are at a point northeast of the interior equilibrium), politicians will try to maximize their utility by choosing a soft policy, because this allows them to gain utility from increased popularity. Diversely, if politicians choose a soft policy, citizens’ dominant strategy is disobey, so \(y_O\) decreases. At this point, since the number of obeying citizens decreases, fines become more important for politicians; thus, they will try to maximize their utility by choosing a hard policy. However, at this point the dominant strategy for citizens becomes obey, and as long as the share of disobeying citizens is not very large, this remains the dominant strategy. As a consequence, the system continues rotating around the internal equilibrium, until it reaches the stable state (1, 0). Indeed, given this set of parameters, the eigenvalues associated with the equilibrium (0, 1) are one positive and one negative. However, those associated with the (1, 0) equilibrium are all negative, so that the system converges toward this equilibrium point. (More details can be found in “Appendix.”)

Case 1b: W and K are both negative. Therefore, obey, net of the peer effect, is a dominant strategy under both hard and soft lockdowns. However, the peer effect is quantitatively large and may offset the excess utility of obey compared with disobey, even when the share of disobedient citizens is low. The size of the peer effect is particularly visible when the policy implemented is a soft lockdown. When the lockdown is hard, the share of disobeying citizens must be quantitatively significant so that the peer effect offsets W. This case is shown in Fig. 3. Assuming that obey and hard policies are frequent strategies, such as at a point northeast of the interior equilibrium, politicians will try to maximize their popularity, given that the share of obeying individuals is larger than \(y_O\). However, since \(K<0\), citizens’ choice of obey becomes easier to make, and as long as the number of obeying citizens is large enough, i.e., \(K>\gamma (1y_O)\), the dominant strategy for citizens remains obey and the system converges toward (0, 1). In other words, all the politicians choose a soft policy and all the citizens obey. Diversely, when hard and obey are rare strategies, such as at a point southwest of the interior equilibrium, politicians will try to maximize their utility by imposing fines. Therefore, they will choose a hard policy since the number of obeying individuals is small. But for citizens, the dominant strategy under hard is to disobey, since the number of disobeying citizens is large and the (absolute value of the) peer effect \(\gamma (1y_O)\) offsets W. This leads the system to converge toward (1,0) where all politicians choose a hard policy and all citizens disobey. The set of parameters W and K both negative leads to the local stability of both equilibria (1,0), that is, hard and disobey, and (0,1), that is, soft and obey. Further, the eigenvalues of the first (1,0) equilibrium are both negative (\(\lambda _1 = DF\) and \(\lambda _2 = W\gamma \)), and so are the eigenvalues of the second (0,1) equilibrium (\(\lambda _1 = DL\) and \(\lambda _2 = K\)) (see “Appendix”).

Case 1c: W and K are both positive. Therefore, the dominant strategy under both soft and hard lockdowns is disobey, irrespective of the peer effect value. Furthermore, W is greater than K, i.e., the excess utility of disobey with respect to obey is greater under hard than soft. This case basically represents a situation in which it is never easier for citizens to respect policy rules; this is because the economic costs avoided and the psychological benefits of disobeying are greater than the potential benefits of not getting infected by complying with the confinement rules. Costs would include the risk of health loss due to the chance of contracting the virus, and the potential costs of fines. This scenario leads to a dynamic shown in Fig. 4. In this case, the equilibrium represented by soft and obey is locally unstable. Its associated eigenvalues are \(\lambda _1= DL\) and \(\lambda _2 = K\), which are negative and positive, respectively (see “Appendix”). Therefore, equilibrium (0, 1) is a saddle point. The equilibrium represented by hard and disobey, instead, has eigenvalues equal to \(\lambda _1=DF\) and \(\lambda _2=W\gamma \), which are both negative. Furthermore, whatever the initial conditions are, the system will converge, sooner or later, to the equilibrium (1, 0), i.e., to hard and disobey.

A special case 1d: edgetocorner bifurcation. It is worth noting that the equilibria (0, 1) and (1, 0) change their stability properties as a consequence of a bifurcation. When \(E_{b_1}\) and \(E_{b_2}\) are included in the player action space, then the equilibria (0, 1) and (1, 0) are stable. For specific sets of parameters when \(K=0\) and \(W=\gamma \), it may occur that \((0,1)=E_{b_1}\) and \((1,0)=E_{b_2}\) (the boundary equilibrium disappears through the transcritical bifurcation, despite the fact that these conditions are not necessarily simultaneously met), and equilibrium (0, 1) inherits the stability properties of \(E_{b_1}\) and equilibrium (1, 0) those of \(E_{b_2}\). When both bifurcations simultaneously occur, i.e., when \((0,1)=E_{b_1}\) and \((1,0)=E_{b_2}\), none of the equilibria are locally stable. (All the equilibria have at least one nonnegative eigenvalue at their associated Jacobian matrix.) Therefore, this special case is characterized by a perpetual dynamic characterized by a diverging spiral starting from the internal equilibrium. From an economic point of view, as we suggested in Sect. 2.1, in this case the excess utility of obey with respect to disobey (net of the peer effect) is zero, and a situation in which the excess utility of obey (net of the peer effect) under hard is equal to the peer effect. Herein, citizens diverge from the equilibrium (0, 1), i.e., soft and obey, because they are indifferent toward the two strategies. Since disobeying citizens do not exist, \(K=0\) means that no strategy is dominant at the (0, 1) point. This set of parameters also makes the other equilibrium (1, 0), i.e., hard and disobey, unstable (Fig. 5). This is true given that—at this point, since all citizens are disobeying the rules—no strategy is dominant. Indeed, the excess utility of obey with regard to disobey (\(W\)) is exactly equal to the peer effect. (The equality \(W=\gamma \) is verified.) Thus, once again, citizens are indifferent toward the two strategies.

Case 2a: In this case, as mentioned in Sect. 2.1, \(WK\) is positive and this leads to two real eigenvalues associated with the Jacobian computed at the interior equilibrium point. W is positive, and therefore citizens’ dominant strategy under hard is disobey, irrespective of the peer effect, while the dominant strategy under a soft policy, net of the peer effect, is obey. (Indeed K is negative.) The peer effect is small compared to the value of K, therefore it never offsets the excess utility of obey with respect to disobey. Further, it implies that under a soft policy the dominant strategy for citizens is obey. This situation gives rise to the dynamics shown in Fig. 6. Whatever the initial conditions are, the system will rapidly converge to either (0, 1) or (1, 0). The basin of attraction of the equilibrium (0, 1) is quite large, consisting in more than half of the surface of the action space representing the set of feasible actions.

Case 2b: In this case, as pointed out in Sect. 2.1, citizens’ dominant strategy (net of the peer effect) under both hard and soft lockdowns is obey; however, the peer effect may offset both K and W if the number of disobeying citizens becomes large enough. Therefore, assuming that hard and obey are frequent strategies, politicians will try to maximize their utility by seeking greater popularity from the obeying individuals; thus they will choose soft. Two cases exist. First of all, if the number of obeying individuals is large enough and the peer effect does not offset the excess utility of obey with respect to disobey under soft, the system will converge toward (0, 1). Secondly, instead—if the number of obeying individuals is small—then the peer effect will offset K so that citizens will start disobeying. As long as the number of disobeying citizens increases, politicians will try to increase their payoff by imposing fines and will therefore choose a hard policy. But when the number of disobeying citizens is large, the peer effect offsets W in absolute value, and citizens’ dominant strategy will become disobey. The system will end up converging toward (1, 0), i.e., to a situation in which all the politicians choose a hard policy and all the citizens disobey. This dynamic is represented in Fig. 7.