Fig. 3

The cross sections of the example \(a(m_1,m_2)\) payoff function when \(v=1\), \(c_0=4\), and \(\alpha =16\) holding \(m_1=m^*=\frac{5}{8}\) (blue line) and \(m_2=m^*=\frac{5}{8}\) (red line) and their first derivatives (purple and orange lines, respectively). The \(\mathbf {\times }\)’s demonstrate that the ESP conditions of Eq. (5) are satisfied at \(m^*=\frac{5}{8}\)