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  1. The decision maker has an ideal point in mind and chooses the alternative closest to it.

I am not sure if I am right, but in order to rationalize it, we first have to construct a choice function. So, in this question we can say that U(x)= min d(x, I) where I is the ideal point.

Now, we would have to see if it satisfies the conditions to be a preference relation (completeness, transitivity) and then we check if it can be rationalized (if we always choose the same alternative regardless of the size of the set).

Are the steps that we have to follow the ones mentioned above? How can we answer the question above?

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Ideal point and blisspoint are esentially the same. If you are always choosing the closest point, you could model it as if the blisspoint provided you with the highest utility and whatever is closer to it gives higher utility than whatever is farther.

There is no need to construct the minimum distance utility function to understand this problem since you can gain this result by classic functions as well.

Consider the quadratic utility function:

$$ U(\boldsymbol{x}) = (\alpha x_1 - \gamma x_1^2) + (\alpha x_2 - \gamma x_2^2) $$

You can visualize it in the following way:

Blisspoint

See that parameters $\alpha$ and $\gamma$ have to be the same in case of all the goods you consider. Then, this utility function is a good representation of what you asked. The consumer has some ideal point in mind which is the highest elevation of utility function and whatever is closer to it gives you higer utility meaning the consumer would choose the closest point he or she could choose.

Now, the main idea: If it can be modeled as utility function (!), then, we know the preferences are complete and transitive (just from definition).

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