I'm exploring a scenario where a public facility needs to be located along a street segment represented by the interval $[0, 1]$. In this setting, there are $n$ agents, each having their preferred location $p_i \in (0, 1)$ for the facility. The utility of agent $i$ is determined by the distance between their ideal location and the chosen facility location, denoted as $-|p_i - k|$, where $k$ represents the facility's location.
To decide the facility's location, a voting game is employed, where each agent $i$ casts a vote by selecting a location $x_i \in [0, 1]$. The facility's final location is determined as the average of all the votes: $$\frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}$$
I'm trying to prove that no agent has a weakly dominating strategy here. I think I have understood the intuition but facing some difficulty in formally writing down the solution. In my understanding, there are three possible cases.
#CASE_1 when $x_i < p_i$. In this case player $i$ can do better by voting something between $x_i$ and $p_i$
#CASE_2 when $p_i < x_i$. In this case player $i$ can do better by voting something between $p_i$ and $x_i$
#CASE_3 when $x_i = p_i$. In this case player $i$ can do better by voting $p_i - \epsilon$ where $\epsilon$ is positive and arbitrarily close to zero.
We're demonstrating that each agent's utility has always been somewhat higher while playing pi than when playing any other $x_i ≠ p_i$ regardless of what the other players do. I saw another solution where they did this by unilateral deviation such that $U(x_1, x_2, . . ., x_n) = p_i < 0$. I failed to understand this bit.
Is the way I did it correct? If not, please help me understand where I went wrong.
You are generally going in the right direction. However, your proof is not complete.
More specifically, your argument for the individual cases is not correct: For example, if player $i$ votes for $x_i \lt p_i$ (case 1), it is not necessarily true that they can improve by voting for some location in $(x_i,p_i)$ (just consider a situation where the other players voted in such a way that the outcome happens to be $p_i$ (if player $i$ votes for $x_i$). Then, voting for anything else will actually decrease the utility for player $i$.
However, in order to show that a strategy is not weakly dominant, you don't need to show that the player can always improve by deviating, anyway. It suffices to show that in at least one situation (=strategy profile of the other players) your player can improve by deviating.