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Suppose the allocation space is the unit square $A=[0,1] \times [0,1] \in \mathbb{R}^2$. The outcome is a single point $x\in A$. Assume all agents have single-peaked preferences. That is, each agent $i$ has a preference ordering $\succeq_i$ over the outcomes; there exists a point $p_i=(x_i, y_i) \in A$ such that for all $x\in A \setminus \{p_i\}$ and all $\lambda \in [0,1)$, we have $(\lambda x + (1 − \lambda)p_i) \succ_i x$.

The social choice $f$ takes in the agents' preferences $(\succeq_1,\dots,\succeq_n)$, and output $x\in A$.

Why is the function $f(\succeq_1,\dots,\succeq_n)=(x_i,y_i)$ where $x_i$ and $y_i$ are the medians of the $x$ and $y$ coordinates, respectively, not strategy-proof?

What I think: In one dimension, we know that the median scheme is strategy-proof. I don't see how taking the median over each dimension is not?

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Your condition is not strong enough to enforce structure on the preferences that result. For example, with three options (say (0,0), (1,0), and (0,1)) a voter could have any linear preference over these options (even if his "peak" was fixed at say (1,1)). Thus, impossibility results such as Arrow’s Theorem and Gibbard–Satterthwaite will apply.

One variant you could try to fix this situation is define your peaks "coordinatewise", i.e. options are preferred less if both their coordinates are further from the "peak". However, you can check that this still allows all preferences on (0,0), (1,0), and (0,1), all with peak (0.5, 0.5).

In general, the test "are all preferences on three options possible" is probably a good litmus test for whether "good" voting schemes will exist (at least for the more basic definitions of "good").

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In one dimension, under the median scheme, manipulating your preference peak doesn't get you anything: If you feign a peak on the side of the median where your peak actually lies, that doesn't change the median, and if you feign a peak on the other side, that only pulls the median even further away from your peak.

In more than one dimension, however, you might get a better result by shifting the result's $x$ coordinate further away from your peak's $x$ coordinate – while that causes the result to end up further away from your acutal peak, you're then on a different line $\lambda x + (1 − \lambda)p_i$, which might be more favourable to you, since nothing in the premise constrains the relative ordering of points along different lines through the peak, so a point on one line further from the peak might be preferable to a point on another line closer to the peak.

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