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I have noticed that a dominance relation acts like a strict poset with the confidence interval equal to 0 but very differently with non-zero confidence intervals. For example, 1D2 and 2D3 always implies 1D3 under the confidence interval of zero but not so with non-zero confidence intervals. My friend said that this is because the events act like point estimation when the confidence interval is zero while not so with non-zero confidence interval. I could not understand this because of the word "point-estimation".

Please, explain the system with non-zero confidence interval.


System with the confidence interval equal to 0: working like a strict poset

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System with the non-zero confidence interval: not working like a strict poset

The transitivity does not work here as expected: the blue cases do not imply the red case because of the non-zero confidence interval.

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What is the technical interpretation for this observation with different confidence intervals?

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