# Can FOSD-transitivity replace the transitivity in utility representation theorem?

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.

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.

• 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. Dec 5, 2023 at 10:04
• May I ask what is the specific preference relation you have in mind for the set of all lotteries? Dec 5, 2023 at 10:06