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:<math>x_j \in \arg \max u_e(x) + u_j(x)\quad</math>
Comparing this to the equilibrium where both players contribute and noting that for the agent <math>x^* \succsim x_g\,</math> and <math>x^* \succsim x_h\;</math> , and so it must be the case that <math>x_g , x_h\,</math> are off the equilibrium path.
Therefore the agent will choose <math>x^*\,</math> iff:
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