Before revision:  We find that among the class of time-varying, affine portfolio strategies, the intra-personal equilibrium does not exist when \alpha>1/2, leads to zero investment in the risky assets when \alpha<1/2, and is a portfolio insurance strategy when \alpha=1/2. We then compare the intra-personal equilibrium strategy in the case of \alpha=1/2, namely under median maximization , to some other strategies and apply it to explain why more wealthy people invest more precentage of wealth in risky assets .
After revision:  We find that among the class of time-varying, affine portfolio strategies, the intra-personal equilibrium does not exist when \alpha>1/2, leads to zero investment in the risky assets . The median maximization strategy gives a simple explanation to why wealthier people invest more precentage of wealth in risky assets .