Question about Locally non-satiated preferences

Question

If a consumer has locally non-satiated preferences, which of these 2 bundles is preferred and why?

Bundle A: (1,3)

Bundle B: (4,2)

This is what I've reasoned from my very limited understanding of LNS preferences.

The distance between the two bundles has to be less than the radius of a ball around one of the bundles for the other bundle to be preferred. A ball around bundle A can't have a radius larger than 1, thus bundle B isn't preferred. However, a ball around bundle B can have a radius large enough so that Bundle A is within the ball. So bundle A is preferred.

I'm very unsure of this reasoning and I'd appreciate it if anyone could correct me where I'm wrong or help improve my answer.

Answer 1

Consider a locally non-satiated preference represented by the utility function \begin{equation} u(x,y)=-(x-4)^2(y-2)^2 \end{equation} It is easy to verify that $u(4,2)=0>-9=u(1,3)$. Therefore $B$ is strictly preferred to $A$.

Likewise, you can construct a locally non-satiated preference, one represented by $$u(x,y)=-(x-1)^2(y-3)^2,$$ under which $A$ is strictly preferred to $B$.

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What local non-satiation does is that it rules out indifference "zones", so that we're left with indifference curves. Hence, with local non-satiation alone, it's impossible to tell whether one alternative is preferred to another.

The following figures from Jehle and Reny (2011) illustrate preferences with local non-satiation (Fig 1.3) and without it (Fig 1.2).

_{Axioms 1 through 4' are completeness, transitivity, continuity, and local non-satiation, respectively.}