I was reading the article "On option pricing bounds" by Ritchken(1985).

It uses linear programming to determine options upper and lower bounds.

Given a single period model, the stock price will have n states. ie.($s_1,s_2...s_n$)

According to the law of one price, the stock price can be written as $$S_0=\sum_{j=1}^n s_je_j$$ where $e_j$ is the current price of a pure state j security that pays out 1 at time T if state j occurs. (ie. $e_j$ is state-contingent claim )

We now assume that investors are risk-averse, and there exists a set of ex ante consensus probabilities ($\pi_i$)for the states. Specifically, we have $$e_j=\pi_jd_j$$ And since investors are risk-averse, so $d_1\geq d_2 \geq d_3 ... \geq d_n \geq 0$.

And my question is why $d_1\geq d_2 \geq d_3 ... \geq d_n \geq 0$ must hold when investors are risk-averse?enter image description here


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