Fig. 4: The stability of the optimal solution depends on the number of vaccine-rich countries.

The optimal solution does not depend on the number of vaccine-rich countries, N, as long as \({v}_{{{\max }}}\) is held constant; however, the optimal solution is stable for a wider range of parameters if N is smaller. Each colored rectangle on the top panel represents the solution for the corresponding values of λ and \({v}_{{{\max }}}\). In regions I, II, and III (yellow shades), \({s}_{i}=2\) by all vaccine-rich countries is optimal. In regions IV, V, and VI (blue shades), \({s}_{i}=1\) by all countries is optimal. In region VII (black), \({s}_{i}=0\) by all countries is optimal. The only difference in parameters between the panels is that \(N=2\) in panel (a) and \(N=10\) in panel (b). \(\alpha =0.4\) in both panels. The case of \(N=6\) is demonstrated in Fig. 2. Note that in panel (a), where \(N=2\), the optimal solution is stable (Nash equilibrium or self-enforcing international agreement (SEA)) for all values of \({v}_{{{\max }}}\) and λ; but in panel (b), where \(N=10\), there are parameter regions in which the optimal solution is unstable (regions II and V).