Skip to main content

Optimality of the coordinate-wise median mechanism for strategyproof facility location in two dimensions


We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in \(\mathbb {R}^2\). We show that for the p-norm (\(p \ge 1\)) objective, the coordinate-wise median mechanism (CM) has the lowest worst-case approximation ratio in the class of deterministic, anonymous, and strategyproof mechanisms. For the minisum objective and an odd number of agents n, we show that CM has a worst-case approximation ratio (AR) of \(\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}\). For the p-norm social cost objective (\(p\ge 2\)), we find that the AR for CM is bounded above by \(2^{\frac{3}{2}-\frac{2}{p}}\). We conjecture that the AR of CM actually equals the lower bound \(2^{1-\frac{1}{p}}\) (as is the case for \(p=2\) and \(p=\infty\)) for any \(p\ge 2\).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. Since \(sc(\mathbf {p}, z)^p\) is convex as a function of z and the composition of a convex function with a nondecreasing function is quasi-convex, \(sc(\mathbf {p}, z) = (sc(\mathbf {p}, z)^p)^\frac{1}{p}\) is quasi-convex as a function of z.

  2. When \(n=2m\) is even, the version of the coordinate-wise median mechanism given by \(c(\mathbf {p}) = (\text {median}(-\infty ,\mathbf {x}),\text {median}( -\infty ,\mathbf {y}))\) has worst-case approximation ratio equal to \(\sqrt{2}\). This follows from the bound in Lemma  3 and the worst-case profile \(\mathbf {p}\) where \(p_1=p_2 \dots p_m=(1,0)\) and \(p_{m+1}=p_{m+2} \dots p_{2m}=(0,1)\).

  3. The set \([p_i', g(\mathbf {p}')] \cap \Gamma\) is non-empty because \(g(\mathbf {p}')\) cannot be in the same quadrant as \(p_i'\). Any point in the same quadrant as \(p_i'\) subtends an angle of less than \(90^{\circ }\) with the other two points and hence it cannot be the geometric median.

  4. The lower bound actually holds more generally in that if f is a deterministic, strategyproof mechanism defined for all n, then \(\sup _{n \in N} AR(f) \ge 2^{1-\frac{1}{p}}\). If f is anonymous as well, the bound is a corollary of Theorem 3 due to the optimality of Coordinate-wise median (Theorem 1) for any n. For any f, the argument used to prove Lemma 4 in Feigenbaum et al. (2017) extends to this setting as well and is in the appendix proof.

  5. The same argument applies if we just take \(n=2\) but we wanted to illustrate that the result holds even under the restriction to odd number of agents.


  • Alon N, Feldman M, Procaccia AD, Tennenholtz M (2010) Strategyproof approximation of the minimax on networks. Math Oper Res 35:513–526

    Article  Google Scholar 

  • Barberà S, Gul F, Stacchetti E (1993) Generalized median voter schemes and committees. J Econ Theory 61:262–289

    Article  Google Scholar 

  • Bespamyatnikh S, Bhattacharya B, Kirkpatrick D, Segal M (2000) Mobile facility location (extended abstract). In: Proceedings of the 4th international workshop on Discrete algorithms and methods for mobile computing and communications—DIALM ’00, vol. 4, ACM Press, Boston, Massachusetts, United States, pp 46–53

  • Black D (1948) On the rationale of group decision-making. J Polit Econ 56:23–34

    Article  Google Scholar 

  • Border KC, Jordan JS (1983) Straightforward elections, unanimity and phantom voters. Rev Econ Stud 50:153

    Article  Google Scholar 

  • Bordes G, Laffond G, Le Breton M (2011) Euclidean preferences, option sets and strategyproofness. SERIEs 2:469–483

    Article  Google Scholar 

  • Chan H, Filos-Ratsikas A, Li B, Li M, Wang C (2021) Mechanism design for facility location problems: a survey, arXiv:2106.03457 [cs], arXiv: 2106.03457

  • Cheng Y, Zhou S (2015) A survey on approximation mechanism design without money for facility games. In: Gao D, Ruan N, Xing W (eds) Advances in global optimization, vol 95. Springer Proceedings in Mathematics & Statistics. Springer International Publishing, Cham, pp 117–128

    Google Scholar 

  • Dokow E, Feldman M, Meir R, Nehama I (2012) Mechanism design on discrete lines and cycles. In: Proceedings of the 13th ACM Conference on Electronic Commerce—EC ’12, vol. 13, ACM Press, Valencia, Spain, p 423

  • Durocher S, Kirkpatrick D (2009) The projection median of a set of points. Comput Geom 42:364–375

    Article  Google Scholar 

  • El-Mhamdi E-M, Farhadkhani S, Guerraoui R, Hoang L-N (2021) On the strategyproofness of the geometric median. arXiv:2106.02394 [cs], arXiv: 2106.02394

  • Feigenbaum I, Sethuraman J, Ye C (2017) Approximately optimal mechanisms for strategyproof facility location: minimizing L $_{\rm p }$ norm of costs. Math Oper Res 42:434–447

    Article  Google Scholar 

  • Feldman M, Wilf Y (2013) Strategyproof facility location and the least squares objective. In: Proceedings of the Fourteenth ACM Conference on Electronic commerce, vol. 14, ACM, Philadelphia Pennsylvania USA, pp 873–890

  • Feldman M, Fiat A, Golomb I (2016) On voting and facility location. In: Proceedings of the 2016 ACM Conference on economics and computation, vol. 17, ACM, Maastricht The Netherlands, pp 269–286

  • Fotakis D, Tzamos C (2014) On the power of deterministic mechanisms for facility location games. ACM Trans Econ Comput 2:1–37

    Article  Google Scholar 

  • Fotakis D, Tzamos C (2016) Strategyproof facility location for concave cost functions. Algorithmica 76:143–167

    Article  Google Scholar 

  • Gershkov A, Moldovanu B, Shi X (2019) Voting on multiple issues: what to put on the ballot? Theor Econ 14:555–596

    Article  Google Scholar 

  • Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41:587

    Article  Google Scholar 

  • Kim K, Roush F (1984) Nonmanipulability in two dimensions. Math Soc Sci 8:29–43

    Article  Google Scholar 

  • Kyropoulou M, Ventre C, Zhang X (2016) Mechanism design for constrained heterogeneous facility location. In: Fotakis D, Markakis E (eds) Algorithmic game theory, vol 11801. Lecture Notes in Computer Science. Springer International Publishing, Cham, pp 63–76

    Chapter  Google Scholar 

  • Lee Brady R, Chambers CP (2016) A spatial analogue of May’s Theorem. Social Choice Welf 47:127–139

    Article  Google Scholar 

  • Meir R (2018) Strategic voting, synthesis lectures on artificial intelligence and machine. Learning 12:1–167

    Google Scholar 

  • Meir R (2019) Strategyproof facility location for three agents on a circle. arXiv:1902.08070 [cs], arXiv: 1902.08070

  • Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35:437–455

    Article  Google Scholar 

  • Peters H, van der Stel H (1990) A class of solutions for multiperson multicriteria decision making. Oper Res Spek 12:147–153

    Article  Google Scholar 

  • Peters H, van der Stel H, Storcken T (1992) Pareto optimality, anonymity, and strategy-proofness in location problems. Internat J Game Theory 21:221–235

    Article  Google Scholar 

  • Peters H, Stel H, Storcken T (1993) Range convexity, continuity, and strategy-proofness of voting schemes. ZOR Methods Models Oper Res 38:213–229

    Article  Google Scholar 

  • Procaccia AD, Tennenholtz M (2013) Approximate mechanism design without money. ACM Trans Econ Comput 1:1–26

    Article  Google Scholar 

  • Satterthwaite MA (1975) Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. J Econ Theory 10:187–217

    Article  Google Scholar 

  • Sui X, Boutilier C (2015) Approximately strategy-proof mechanisms for (Constrained) facility location. In: Proceedings of the 2015 International Conference on autonomous agents and multiagent systems, Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems, AAMAS ’15, vol. 14, pp 605–613

  • Sui X, Boutilier C, Sandholm T (2013) Analysis and optimization of multi-dimensional percentile mechanisms. In: Proceedings of the Twenty-Third international Joint Conference on artificial intelligence, vol. 23, AAAI Press, IJCAI ’13, Beijing, China, pp 367–374

  • Tang P, Yu D, Zhao S (2020) Characterization of group-strategyproof mechanisms for facility location in strictly convex space. In: Proceedings of the 21st ACM Conference on economics and computation, pp 133–157, arXiv: 1808.06320

  • Walsh T (2020) Strategy proof mechanisms for facility location in Euclidean and Manhattan space. arXiv:2009.07983 [cs], arXiv: 2009.07983

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Sumit Goel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are grateful to Arunava Sen, Federico Echenique, Tom Palfrey, Omer Tamuz, Debasis Mishra, the Editor Clemens Puppe, and referees for this journal, as well as referees and participants at the Winter School of the Econometric Society at the Delhi School of Economics (2019), the Meeting of the Society for Social Choice and Welfare (2022), and the Symposium on Algorithmic Game Theory (2022) for helpful comments and suggestions. An earlier version of this paper circulated under the title “Coordinate-wise median: Not bad, Not bad, Pretty good”.


A Proofs for Section 4 (The minisum objective)

Theorem 2

For n odd, \(X=\mathbb {R}^2\), and \(sc(z,\mathbf {p})=\sum _{i=1}^n \Vert z-p_i \Vert\),

$$\begin{aligned} AR(CM)=\sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}. \end{aligned}$$


Define Centered Perpendicular (CP) profiles as all profiles \(\mathbf {p}\in (\mathbb {R}^2)^n\) such that

  • \(c(\mathbf {p})=(0,0)\)

  • for all i, either \(x_i=0\) or \(y_i=0\) or \(p_i=g(\mathbf {p})\)

  • if \(p_i' \in (p_i, g(\mathbf {p}))\), then \(c(p_i', p_{-i}) \ne (0,0)\)

Lemma 10

(CP) For any profile \(\mathbf {p}\in (\mathbb {R}^2)^n\), there exists a profile \(\mathbf {\chi } \in CP\) such that \(AR(\mathbf {\chi }) \ge AR (\mathbf {p})\).


Let \(\mathbf {p}\in (\mathbb {R}^2)^n\) be a profile. Let \(\mathbf {p}'\) be the profile where \(p_i' = p_i - c(\mathbf {p})\). Then \(\mathbf {p}'\) has the same approximation ratio and \(c(\mathbf {p}') = (0,0)\). Denote \(A = \{i: x_i=0\}\) and \(B = \{i: y_i=0\}\). Note that since \(c(\mathbf {p}') = (0, 0)\), it follows from the definition of \(c(\mathbf {p}')\) that \(A \ne \emptyset\) and \(B \ne \emptyset\). Let \(\Gamma = \{(x, y) \, : \, x = 0 \text { or } y = 0\} \cup g(\mathbf {p}')\). Starting from \(i=1\) and going until n, define \(p_i''\) to be the point in \([p_i', g(\mathbf {p}')] \cap \Gamma\) that is closest to \(g(\mathbf {p}')\) under the constraint that \(c(p_1'', p_2'', \dots , p_i'', p_{i+1}, \dots , p_n)=(0,0)\). Then \(\mathbf {p}'' \in CP\). Further, by lemma 7\(AR(\mathbf {p}'') \ge AR(\mathbf {p}') = AR(\mathbf {p})\); hence, taking \(\mathbf {\chi } = \mathbf {\mathbf {p}''}\) completes the proof.

Define Isosceles-Centered Perpendicular (I-CP) profiles as all \(\mathbf {p}\in CP\) for which there exists \(t \ge 0\) such that

  • \(p_1= \dots =p_m=(t,0)\)

  • \(p_{m+1}=(-t, 0)\)

  • \(p_{m+2}= \dots = p_{2m+1}=(0,1)\)

  • \(g(\mathbf {p})=(0,1)\).

Next, we prove some lemmas that will be useful in reducing the search space for the worst-case profile from CP to \(I-CP\).

First, we show that we can reduce the number of half-axes that the points lie on from (at most) four to (at most) three.

Lemma 11

(Reduce axes) Suppose \(\mathbf {p}\) and \(\mathbf {p}'\) are profiles which differ only at i where for some \(a > 0\), \(p_i = (0, -a)\) and \(p_i' = (-a, 0)\), and for which \(c(\mathbf {p}) = c(\mathbf {p}') = (0, 0)\) and \(y_g(\mathbf {p}) \ge x_g(\mathbf {p}) \ge 0\). Then \(AR(\mathbf {p}') \ge AR(\mathbf {p})\).


Again \(c(\mathbf {p}')=c(\mathbf {p})\) and \(sc(c(\mathbf {p}'), \mathbf {p}')=sc(c(\mathbf {p}), \mathbf {p})\). Thus, it is sufficient to show that \(sc(g(\mathbf {p}'), \mathbf {p}') \le sc(g(\mathbf {p}), \mathbf {p})\). For this, we just need to show that \(d(p_i', g(\mathbf {p})) \le d(p_i, g(\mathbf {p}))\). This follows from the following simple calculation:

$$\begin{aligned} d\left( p_i', g(\mathbf {p})\right) ^2&= \left( x_g(\mathbf {p}) + a \right) ^2 + y_g(\mathbf {p})^2\\&= x_g(\mathbf {p})^2 + 2 x_g(\mathbf {p})a + a^2 + y_g(\mathbf {p})^2\\&\le x_g(\mathbf {p})^2 + y_g(\mathbf {p})^2 + 2a y_g(\mathbf {p}) + a^2\\&= x_g(\mathbf {p})^2 + \left( y_g(\mathbf {p}) + a \right) ^2\\&= d\left( p_i, g(\mathbf {p})\right) ^2. \end{aligned}$$

Next, we show that we can combine points on each of the three half-axes while weakly increasing the approximation ratio.

Lemma 12

(Convexity) Let \(\mathbf {p}\in CP\) and let \(S \subseteq N\) be such that for all \(i \in S\), \(x_i > 0\) and \(y_i = 0\). Let \(p_S\) be the mean of the \(p_i\) across \(i \in S\). Let \(\mathbf {p}'\) be the profile where

  1. 1.

    \(p_j' = p_j\) for \(j \notin S\) and

  2. 2.

    \(p_j' = p_S\) for \(j \in S\).

Then \(AR(\mathbf {p}') \ge AR(\mathbf {p})\).


It is immediate that \(c(\mathbf {p}') = c(\mathbf {p})\). Further, \(sc(c(\mathbf {p}'),\mathbf {p}')=sc(c(\mathbf {p}),\mathbf {p}')=sc(c(\mathbf {p}),\mathbf {p}))\) and \(sc(g(\mathbf {p}'),\mathbf {p}')<sc(g(\mathbf {p}),\mathbf {p}') <sc(g(\mathbf {p}),\mathbf {p})\) where the last inequality follows from convexity of the distance function.

The same argument applies for any of the other strict half axes.

Next, we show that we can move all the points that are on the geometric median to the axis in a way that weakly increases the approximation ratio.

Lemma 13

(Double Rotation) Let \(\mathbf {p}\) and \(\mathbf {p}'\) be profiles that differ only at \(i_1\) and \(i_2\), such that for some \(a \ge 0\)

  • \(c(\mathbf {p}) = (0, 0)\),

  • \(y_g(\mathbf {p}) \ge x_g(\mathbf {p}) > 0\),

  • \(p_{i_1} = (-a, 0)\),

  • \(p_{i_1}' = (a + 2 x_g(\mathbf {p}), 0)\),

  • \(p_{i_2} = g(\mathbf {p})\), and

  • \(p_{i_2}' = (0 , d(g(\mathbf {p}), (0, 0)))\).

Then \(c(\mathbf {p}') = (0, 0)\) and \(AR(\mathbf {p}') \ge AR(\mathbf {p})\).


The first claim is immediate.

For the second claim, let

$$\begin{aligned} A&= \sum _{i \ne i_1}{d\left( p_i, c(\mathbf {p})\right) }\\ B&= \sum _{i \ne i_2}{d\left( p_i, g(\mathbf {p})\right) }. \end{aligned}$$

By Lemma 3,

$$\begin{aligned} A + d\left( p_{i_1}, c(\mathbf {p})\right) \le \sqrt{2} B. \end{aligned}$$

Hence, it follows that

$$\begin{aligned} \left[ A + d\left( p_{i_1}, c(\mathbf {p})\right) \right] d\left( p_{i_2}', g(\mathbf {p})\right) \le \sqrt{2} B d\left( p_{i_2}', g(\mathbf {p})\right) . \end{aligned}$$

But since \(y_g(\mathbf {p}) \ge x_g(\mathbf {p})\), it follows that \(d(p_{i_2}', g(\mathbf {p})) \le \sqrt{2} x_g(\mathbf {p})\). Hence,

$$\begin{aligned} \left[ A + d\left( p_{i_1}, c(\mathbf {p})\right) \right] d\left( p_{i_2}', g(\mathbf {p})\right)&\le 2 B x_g(\mathbf {p})\\&= B \left( d\left( p_{i_1}', c(\mathbf {p})\right) - d\left( p_{i_2}', c(\mathbf {p})\right) \right) . \end{aligned}$$

From this it follows that

$$\begin{aligned} \left( A + d\left( p_{i_1}, c(\mathbf {p})\right) \right) \left( B + d\left( p_{i_2}', g(\mathbf {p})\right) \right)&= AB + B d\left( p_{i_1}, c(\mathbf {p})\right) + \left[ A + d\left( p_{i_1}, c(\mathbf {p})\right) \right] d\left( p_{i_2}', g(\mathbf {p})\right) \\&\le AB + B d\left( p_{i_1}', c(\mathbf {p})\right) \\&= \left( A + d\left( p_{i_1}', c(\mathbf {p})\right) \right) B \end{aligned}$$

and hence

$$\begin{aligned} AR(\mathbf {p})&= \frac{A + d\left( p_{i_1}, c(\mathbf {p})\right) }{B}\\&\le \frac{A + d\left( p_{i_1}', c(\mathbf {p})\right) }{B + d\left( p_{i_2}', g(\mathbf {p})\right) }\\&= \frac{A + d\left( p_{i_1}', c(\mathbf {p}')\right) }{B + d\left( p_{i_2}', g(\mathbf {p})\right) }\\&\le AR(\mathbf {p}'). \end{aligned}$$

Lemma 14

(Geometric to axis) Suppose that \(\mathbf {p}\) is a profile such that there are \(a \ge 0\) and \(b, c > 0\) and subsets \(L, R, U \subseteq N\) with \(L \cap R = L \cap U = R \cap U = \emptyset\), \(L \cup R \cup U = N\), \(|L| = 1\), \(|U| = |R| = m\), and

  • \(p_i = (-a, 0)\) for \(i \in L\)

  • \(p_i = (0, b)\) for \(i \in U\)

  • \(p_i = (c, 0)\) for \(i \in R\)

and so that \(x_g(\mathbf {p}) > 0\).

Then, there exists another profile \(\mathbf {z}\) such that \(AR(\mathbf {z}) > AR(\mathbf {p})\).


We’ll consider two separate cases.

First assume that \(a>0\). Let \(\mathbf {p}(t)\) be the profile which is the same as \(\mathbf {p}\) for \(i \notin U\) and which has \(\mathbf {p}_i(t) = g(\mathbf {p})+t\left( -x_g(\mathbf {p}),b-y_g(\mathbf {p})\right)\) for \(i \in U\). Then there exists \(\epsilon > 0\) such that for \(t \in [0, 1+\epsilon ]\),

$$\begin{aligned} AR(\mathbf {p}(t))&= \frac{(a + (1-t)x_g(\mathbf {p})) + m(c - (1-t)x_g(\mathbf {p})) + mb_g(\mathbf {p})+mt(b-y_g(\mathbf {p}))}{d((-a, 0), g(\mathbf {p})) + m d((c,0), g(\mathbf {p})) + mt d((0, b), g(\mathbf {p}))}\\&=\dfrac{t\left( (m-1) x_g(\mathbf {p})+m(b-y_g(\mathbf {p}))\right) +a+x_{g}(\mathbf {p})+m(c-x_{g}(\mathbf {p})+y_g(\mathbf {p}))}{tmd((0, b), g(\mathbf {p}))+d((-a, 0), g(\mathbf {p})) + m d((c,0), g(\mathbf {p}))} \end{aligned}$$

Note that \(\mathbf {p}(1)=\mathbf {p}\). Now since the denominator of \(AR(\mathbf {p}(t))\) is strictly positive for \(t \ge 0\) and since both the numerator and the denominator are linear in t, \(AR(\mathbf {p}(t))\) is monotonic on \([0, 1 + \epsilon ]\). There are three possibilities.

If \(AR(\mathbf {p}(t))\) is strictly increasing, then \(AR(\mathbf {p}(1+\epsilon )) > AR(\mathbf {p})\).

If \(AR(\mathbf {p}(t))\) is strictly decreasing, then \(AR(\mathbf {p}(0)) > AR(\mathbf {p})\).

If \(AR(\mathbf {p}(t))\) is constant, consider the profile \(\mathbf {z}'\) obtained by putting t such that \(-a=(1-t)x_g(\mathbf {p})\). Then, under \(\mathbf {z}'\), we have 1 agent at \((-a,0)\), m agents at \((-a,bt+(1-t)y_g(\mathbf {p}))\) and m agents at (c, 0). Also, \(AR(\mathbf {p})=AR(\mathbf {z}')\). We can translate this profile by a to the right and get a profile of points in the case where essentially we have \(a=0\). We’ll deal with this case now.

Now let’s consider the case where \(\mathbf {p}\) is such that \(a=0\). To begin, note that since \(g(\mathbf {p})\) must be in the convex hull of (0, 0), (0, b) and (c, 0), if \(x_g(\mathbf {p}) \ge \frac{c}{2}\) and \(y_g(\mathbf {p}) \ge \frac{b}{2}\), then \(g(\mathbf {p})=(\frac{c}{2}, \frac{b}{2})\). But then \(\sum _{i=1}^n \dfrac{p_i-g(\mathbf {p})}{\Vert p_i-g(\mathbf {p}) \Vert }=\dfrac{(c,b)}{\sqrt{c^2+b^2}} \ne 0\), contradicting the characterization of the geometric median in Lemma 2. Hence, it must be that either \(x_g(\mathbf {p})<\frac{c}{2}\) or \(y_g(\mathbf {p}) < \frac{b}{2}\).

Suppose that \(x_g(\mathbf {p})<\frac{c}{2}\). Let \(\mathbf {z}\) be the profile obtained from \(\mathbf {p}\) by moving the point at (0, 0) to \((x_g(\mathbf {p}) - \frac{c}{2}, 0)\) and moving one of the points at (c, 0) to \((\frac{c}{2} + x_g(\mathbf {p}), 0)\). This transformation leaves coordinate-wise median unchanged, as well as leaving the sum of distances to the coordinate-wise median unchanged. However, the sum of distances to \(g(\mathbf {p})\) strictly decreases, since for the unaltered points the distance to \(g(\mathbf {p})\) remains the same, and the sum of the distances from the altered points to \(g(\mathbf {p})\) is

$$\begin{aligned} d((0, 0), g(\mathbf {p})) + d((c, 0), g(\mathbf {p}))&= d\left( \left( 2x_g(\mathbf {p}), 0\right) , g(\mathbf {p})\right) + d((c, 0), g(\mathbf {p}))\\&> 2d\left( \left( x_g(\mathbf {p}) + \frac{c}{2}, 0 \right) , g(\mathbf {p})\right) \\&= d\left( \left( x_g(\mathbf {p}) - \frac{c}{2}, 0\right) , g(\mathbf {p})\right) + d\left( \left( x_g(\mathbf {p}) + \frac{c}{2}, 0\right) , g(\mathbf {p})\right) , \end{aligned}$$

where the inequality follows from convexity of \(d(\cdot , g(\mathbf {p}))\). Since \(AR(\mathbf {z})\) is bounded below by the ratio of the sum of distances to the coordinate-wise median to the sum of distances to \(g(\mathbf {p})\), it follows that \(AR(\mathbf {z}) > AR(\mathbf {p})\).

Next, suppose that \(y_g(\mathbf {p}) < \frac{b}{2}\). Let \(\mathbf {z}\) be the profile obtained from \(\mathbf {p}\) by moving the point at (0, 0) to \((0, y_g(\mathbf {p}) - \frac{b}{2})\) and moving one of the points at (0, b) to \((0, \frac{b}{2} + y_g(\mathbf {p}))\). By essentially the same argument just given, \(AR(\mathbf {z}) > AR(\mathbf {p})\).

Finally, the following lemma shows that we can use convexity to make the triangle formed by the three groups of points isosceles.

Lemma 15

(Isosceles) Let \(\mathbf {p}\) be a profile such for which are m points at (a, 0), 1 point at \((-b,0)\) and m points at (0, c), and for which \(g(\mathbf {p})=(0,c)\) and \(c(\mathbf {p})=(0,0)\). Let \(\mathbf {p}'\) be the profile where there are m points at \(\left( \dfrac{ma+b}{m+1},0\right)\), 1 point at \(\left( -\dfrac{ma+b}{m+1},0\right)\), and m points at (0, c). Then, \(AR(\mathbf {p}') \ge AR(\mathbf {p})\).


Note that \(c(\mathbf {p})=c(\mathbf {p}')=(0,0)\). Since \(ma+b=m\frac{(ma+b)}{m+1}+\frac{ma+b}{m+1}\), we get that the numerator in \(AR(\mathbf {p})\) and \(AR(\mathbf {p}')\) remains the same. Thus, we only need to argue that the denominator goes down as we go from \(AR(\mathbf {p})\) to \(AR(\mathbf {p}')\).

Even though \(g(\mathbf {p}')\) may not be equal to \(g(\mathbf {p})\) we have that \(sc(g(\mathbf {p}),\mathbf {p}') \le sc(g(\mathbf {p}),\mathbf {p})\) by the convexity of the distance function which would imply \(sc(g(\mathbf {p}'),\mathbf {p}') \le sc(g(\mathbf {p}),\mathbf {p})\) by definition of \(g(\mathbf {p})\). Thus, we have that \(AR(\mathbf {p}') \ge AR(\mathbf {p})\).

Now, we use above lemmas to reduce the search space to I-CP.

Lemma 16

(ICP) For every \(\mathbf {p}\in CP\), there exists \(\chi \in I-CP\) such that \(AR(\chi ) \ge AR(\mathbf {p})\).


Without loss of generality, consider any profile \(\mathbf {p}\in CP\) such that \(y_g(\mathbf {p}) \ge x_{g}(\mathbf {p}) \ge 0\). Applying Lemma 11 to all points on the \(-y\)-axis gives a profile \(\mathbf {p}'\) with a weakly higher approximation ratio. In \(\mathbf {p}'\), we have all points on \(+x\)-axis, \(-x\)-axis, \(+y\)-axis and the geometric median. Using lemma 12, we can combine the points on \(+x\)-axis, \(-x\)-axis, \(+y\)-axis to some \((a,0), (0,b), (-c, 0)\) while weakly increasing AR. Let this profile be \(\mathbf {p}''\). Now, we use lemma 13 to move points on the geometric median to \(+y\)-axis. Using 12 again, we get a profile \(\mathbf {p}'''\) with m points on some (a, 0), 1 point on \((-c, 0)\) and m points on (0, b).

So we know that there must be a worst-case profile that takes this form. From Lemma 14, we can say that if the geometric median of such a profile is not on the y-axis, it cannot be a worst-case profile. Thus, there must be a worst-case profile \(\mathbf {z}\) with m points on some (a, 0), 1 point on \((-c, 0)\) and m points on (0, b) and \(x_g(\mathbf {z})=0\). Further, such a profile must have \(y_g(\mathbf {z}) = b\), since otherwise, the profile with m points on (a, 0), 1 point on \((-c, 0)\), and m points on \((0, y_g(\mathbf {z}))\) would have a strictly higher approximation ratio than \(\mathbf {z}\). By Lemma 15, since \(\mathbf {z}\) is a worst-case profile, it must be that \(c = a\).

Now, since \(\mathbf {z}\) is a worst-case profile, the profile \(\chi\) with \(\chi _i = \frac{1}{b}z_i\) is also a worst-case profile, and since \(\chi \in I-CP\), the result follows.

Using Lemma 16, we can now restrict attention to profiles in \(I-CP\). Define

$$\begin{aligned} \mathbf {\eta }_t = \left( p_1^t, \dots , p_{2m+1}^t\right) , \end{aligned}$$


$$\begin{aligned} p_i^t = {\left\{ \begin{array}{ll} (t, 0) &{} i = 1, \dots , m \\ (-t, 0) &{} i = m+1 \\ (0, 1) &{} i = m+2, \dots , 2m+1 \end{array}\right. } \end{aligned}$$

Then, \(I-CP = \left\{ \mathbf {\eta }_t \, : \, t \ge \sqrt{\frac{2m + 1}{2m - 1}} \right\}\). Defining \(\alpha (t) = \frac{(m+1)t + m}{(m+1) \sqrt{t^2 + 1}}\), we get that for \(t \ge \sqrt{\frac{2m + 1}{2m - 1}}\), \(AR(\mathbf {\eta }_t) = \alpha (t)\), and that \(\alpha (t)\) is maximized at \(t^* = \frac{m+1}{m} > \sqrt{\frac{2m + 1}{2m - 1}}\), from which it follows that

$$\begin{aligned} \text {Approximation ratio of CM} = \alpha \left( \frac{m+1}{m} \right) = \sqrt{2}\dfrac{\sqrt{(2m+1)^2+1}}{(2m+1)+1} = \sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}. \end{aligned}$$

Thus, we get that the worst case approximation ratio is \(\sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}\) as required.

B Proofs for Section 5 (p-norm objective)

Theorem 3

For \(X=\mathbb {R}^2\) and the p-norm objective with \(p \ge 2\),

$$\begin{aligned} 2^{1-\frac{1}{p}} \le \sup _{n \in \mathbb {N}} AR(CM) \le 2^{\frac{3}{2}-\frac{2}{p}} \end{aligned}$$


We’ll prove that the lower bound actually holds for any deterministic, strategyproof mechanism (defined for all \(n \in \mathbb {N}\)) and hence, it holds for the coordinate-wise median mechanism. So suppose f is any deterministic, strategyproof mechanism. With \(n=2m+1\) agents,Footnote 5 for any profile \(\mathbf {p}\) such that m agents have ideal point \(\alpha\), \(m+1\) agents have ideal point \(\beta \ne \alpha\), and \(f(\mathbf {p}) = \beta\),

$$\begin{aligned} AR_f(\mathbf {p}')&=\dfrac{sc(f(\mathbf {p}),\mathbf {p})}{sc(OPT(\mathbf {p}),\mathbf {p})}\\&\ge \dfrac{\left( m*\Vert \alpha - \beta \Vert ^p\right) ^\frac{1}{p}}{\left( (2m+1) (\Vert \alpha - \beta \Vert /2)^p\right) ^\frac{1}{p}}\\&= 2^{1-\frac{1}{p}} \cdot \left( \dfrac{m}{m + 1/2}\right) ^\frac{1}{p} \cdot \end{aligned}$$

To see that such a profile always exists, consider the profile \(\mathbf {p}\) where agents 1 through m have ideal point \((-1, 0)\) and agents \(m+1\) through \(2m+1\) have ideal point (1, 0). If \(f(\mathbf {p}) = (1, 0)\), then \(\mathbf {p}\) is such a profile; if not, let \(\mathbf {p}'\) be the profile where agents 1 through \(m+1\) have ideal point \(f(\mathbf {p})\) and agents \(m+2\) through \(2m+1\) have ideal point at (1, 0). Since \(\mathbf {p}'\), every agent’s ideal point is either the same as under \(\mathbf {p}\) or equal to \(f(\mathbf {p})\), it follows from strategyproofness that \(f(\mathbf {p}') = f(\mathbf {p})\), and so \(\mathbf {p}'\) is such a profile.

Thus, for any \(n = 2m+1\),

$$\begin{aligned} AR(f) \ge 2^{1-\frac{1}{p}} \cdot \left( \dfrac{n-1}{n}\right) ^\frac{1}{p}, \end{aligned}$$


$$\begin{aligned} \sup _n{AR(f)} \ge 2^{1-\frac{1}{p}}. \end{aligned}$$

Now we show that the asymptotic AR of coordinate-wise median mechanism is bounded above by \(2^{\frac{3}{2}-\frac{2}{p}}\). Consider any profile \(\mathbf {p}\in (\mathbb {R}^2)^n\). Let \(g(\mathbf {p})=(x_g(\mathbf {p}), y_g(\mathbf {p}))\) and \(c(\mathbf {p})=(x_c(\mathbf {p}),y_c(\mathbf {p}))\). Then, we have that

$$\begin{aligned} sc(g(\mathbf {p}), \mathbf {p})^p&=\sum _{i=1}^n \Vert g(\mathbf {p})-p_i \Vert ^p\\&\ge \left( \sum _{i=1}^n \Vert x_g(\mathbf {p})-x_i \Vert ^p+\sum _{i=1}^n \Vert y_g(\mathbf {p})-y_i \Vert ^p\right) \\&\ge \left( \sum _{i=1}^n \Vert OPT(\mathbf {x})-x_i \Vert ^p+\sum _{i=1}^n \Vert OPT(\mathbf {y})-y_i \Vert ^p\right) \\&\ge \dfrac{1}{2^{p-1}}\left( \sum _{i=1}^n \Vert x_c(\mathbf {p})-x_i \Vert ^p+\sum _{i=1}^n \Vert y_c(\mathbf {p})-y_i \Vert ^p\right) \\&\ge \dfrac{2^{1-\frac{p}{2}}}{2^{p-1}}\sum _{i=1}^n \Vert c(\mathbf {p})-p_i \Vert ^p\\&=2^{2-\frac{3p}{2}}sc(c(\mathbf {p}),\mathbf {p})^p \end{aligned}$$

Thus, we get \(AR(CM) \le 2^{\frac{3}{2}-\frac{2}{p}}\) for \(p \ge 2\) as required.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Goel, S., Hann-Caruthers, W. Optimality of the coordinate-wise median mechanism for strategyproof facility location in two dimensions. Soc Choice Welf (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: