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Let $\succsim$ be a complete, non transitivity preference relation. I wonder if FOSD-transitivity implies transitivity.

The primitive is the space of lotteries $p_1,p_2,p_3,...$. We say the preference satisfies FOSD-transitivity if $p_1\succsim p_2 \succsim...\succsim p_n$ implies $p_n$ does not FOSD $p_1$.

I think the answer is no, as we can easily find three lotteries $p,q,r$ such that they don't FOSD each other. Then we let $p\succsim q\succsim r\succsim p$.

However, when finding a specific numeric example, I failed.

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Take as example the set of lotteries over the outcomes $(1, 5, 10)$ consider the lotteries $L_1, L_2, L_3$ where, $$ \Pr(L_1 = 1) = 0.5\\ \Pr(L_1 = 5) = 0\\ \Pr(L_1 = 10) = 0.5 $$

$$ \Pr(L_2 = 1) = 0.1\\ \Pr(L_2 = 5) = 0.9\\ \Pr(L_2 = 10) = 0 $$

$$ \Pr(L_3 = 1) = 0.2\\ \Pr(L_3 = 5) = 0.5\\ \Pr(L_3 = 10) = 0.3 $$

Lottery $L_i$ FOSD $L_j$ if $\Pr(L_i \le x) \le \Pr(L_j \le x)$ for all $x$.

Note that

  • $\Pr(L_1 \le 5) = 0.5 < \Pr(L_2 \le 5) = 1$ and $\Pr(L_2 \le 1) = 0.2 < \Pr(L_1 \le 1) = 0.5$, so there is no FOSD between $L_1$ and $L_2$.

  • $\Pr(L_2 \le 1) = 0.1 < \Pr(L_3 \le 1) = 0.2$ and $\Pr(L_3 \le 5) = 0.7 < \Pr(L_2 \le 5) = 1$, so there is no FOSD between $L_2$ and $L_3$.

  • Finally, $\Pr(L_3 \le 1) = 0.2 < \Pr(L_1 \le 1) = 0.5$ and $\Pr(L_1 \le 5) = 0.5 < \Pr(L_3 \le 0.5) = 0.7$, so there is no FOSD between $L_1$ and $L_3$.

I think you need lotteries on at least 3 outcomes to get such examples. If there are only two outcomes, the FOSD relation is complete.

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  • $\begingroup$ Great answer I verify that it works (finding three lotteries not FOSD each other). Though just one humble cent: lottery usual means a probability distribution of a real interval, say $[0,1]$. What you have here is called simple lottery: the lottery with finite number of outcomes, which is the counterpart of simple random variable. $\endgroup$
    – High GPA
    Dec 5, 2023 at 10:04
  • $\begingroup$ May I ask what is the specific preference relation you have in mind for the set of all lotteries? $\endgroup$
    – High GPA
    Dec 5, 2023 at 10:06

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