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<math>\{c_J^*(x),x\}</math> is an SPNE iff:
:a) <math>c_J^*(x) is feasible\,</math>is feasible
:b) <math>x^*\,</math>maximizes<math>\quad u_e(x) + c_g^*(x) + c_h^*(x)</math>
:c) <math>\forall j (i \ne j) x^*\,</math>maximizes <math> \quad \{u_e(x) + c_j^*(x) + c_i^*(x)\} + \{u_j(x) - c_j^*(x)\} = \{u_e(x) + u_j(x) + c_i^*(x)\} </math>
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