Strategyproof Facility Location Mechanisms on Discrete Trees
SS TRATEGYPROOF F ACILITY L OCATION M ECHANISMS ON D ISCRETE T REES
Alina Filimonov
Technion - Israel Institute of TechnologyHaifa, Israel
Reshef Meir
Technion - Israel Institute of TechnologyHaifa, IsraelFebruary 5, 2021 A BSTRACT
We address the problem of strategyproof (SP) facility location mechanisms on discrete trees. Ourmain result is a full characterization of onto and SP mechanisms. In particular, we prove that when asingle agent significantly affects the outcome, the trajectory of the facility is almost contained in thetrajectory of the agent, and both move in the same direction along the common edges. We show tightrelations of our characterization to previous results on discrete lines and on continuous trees . We thenderive further implications of the main result for infinite discrete lines.
In facility location problems, a central planner has to determine the location of a public facility that needs to serve a setof agents. Once the facility is located, each agent incurs some cost. Importantly, in non-cooperative settings, agentsmay have an incentive to misreport their locations to decrease their costs. One key objective of the planner that receivedmuch attention in the multiagent systems literature is to design a mechanism that incentivizes agents to report their truelocations, i.e., mechanisms that are strategyproof (SP).An n -agent facility location mechanism on a domain D receives a profile of the agents’ locations a = ( a , . . . , a n ) ∈ D n and outputs a location in D depending on the profile. We refer to the agent’s location as her peak. We say that amechanism is SP if it is a weakly dominant strategy of every agent to report truthfully.The fundamental characterization result for strategyproof facility location was given by Moulin [1], who characterizedthe class of deterministic SP mechanisms on the real line when the preferences of the agents are single-peaked as“generalized median voter schemes" (g.m.v.s.’s). An agent with single-peaked preferences on a line prefers a closerlocation to her peak over a distant location on the same side of her peak.Border and Jordan [2] proved that the characterization also applies for cases where the preferences are “quadratic" (i.e.,symmetric and single-peaked)—the more common model in facility location used in this work as well. An agent withquadratic preferences on a line prefers a closer location to her peak over a distant locationAs quadratic preferences are a special case of single-peaked preferences, the class of SP mechanisms for quadraticpreferences may be larger. This is indeed the case e.g. for mechanisms on the discrete lines [3], but not on continuouslines [2].Schummer and Vohra [4] generalized the result of Border and Jordan to prove that an SP mechanism on a continuoustree, under quadratic preferences, is a consistent collection of g.m.v.s.As hinted above, the trigger for the current work is the observation by Dokow et al. [3] that results on continuous graphsdo not carry over to discrete graphs. In particular, while g.m.v.s entails that the trajectory of the facility is containedin the trajectory of the moving agent on a line (we later observe this also applies for continuous trees), Dokow et al.show it is only “almost contained" when the line is made of discrete vertices. Similarly, while a strategyproof ontomechanism on a continuous circle must be dictatorial [4], it is only “almost dictatorial" when the circle is discrete [3]. a r X i v : . [ c s . G T ] F e b hese extensions may be subtle, but they help us understand what in the characterization is inherent to the topology ofthe graph.Given these previous results, a natural question is whether a similar extension can be applied to discrete trees.As we will later show, a naïve extension of the properties defined in [3] fails. We therefore formulate similar propertiesto characterize the valid moves of the facility under SP, onto mechanisms on discrete trees. In particular, we provide adefinition of “almost Pareto efficient" mechanisms that might be of independent interest.Recent research by Peters et al. [5] provides a different characterization of randomized strategyproof voting mechanismson trees and on other graphs (which, of course, include deterministic mechanisms), under general single-peakedpreferences. However, the class of strategyproof mechanisms on discrete trees under single-peaked preferences is notequivalent to the one under quadratic preferences and therefore the characterization in [5] does not apply to our study. After some preliminary notation in Section 3, we provide an alternative characterization of onto, SP mechanisms oncontinuous trees in Section 4, based on the work of Schummer and Vohra [4].We then present our main result in Section 5—a full characterization of onto, SP mechanisms on discrete trees. Incontrast to the work of Dokow et al. [3], our proof also works for infinite trees (with bounded degree).In Section 6, we derive a characterization of SP and shift-invariant mechanisms on infinite discrete lines.
Following the initial work of Black [6], several researchers have developed characterizations of deterministic andprobabilistic strategyproof facility location mechanisms in various scenarios.Schummer and Vohra [4], beyond their work on trees, showed that any onto SP mechanism on the continuous cyclemust be a dictatorship and that any SP mechanism on a graph has a dictator in a subdomain. Their work was extendedto discrete cycles in [3].Additional variations of the problem include the multiple facility problem [7, 8], the obnoxious facility locationproblem [9, 10], the heterogeneous facility location problem [11, 12], and the activity scheduling problem [13].Todo et al. [14] extended Moulin’s work for characterizing the class of false-name-proof mechanisms on the continuousline. Their work was extended to discrete structures in [15, 16]. The motivation for designing such mechanisms is toprevent agents from submitting multiple reports under different identities, e.g., in internet polls by creating differente-mail addresses. A later work of Wada et al. [17] on variable and dynamic populations characterizes mechanisms thatincentivize the agents to participate in the reporting process.Finally, concrete cost functions also allow us to measure the social cost (e.g., as the sum or max of agents’ costs). Theresearch line of approximate mechanism design without money builds on characterizations such as those mentionedabove, and seeks the mechanisms that minimize the social cost among all strategyproof mechanisms. Incidentally, theiconic domain for this line of work, as reflected in the fundamental paper of Procaccia and Tennenholtz [18], is thefacility location problem. Their work was extended to the domain of continuous graphs by Alon et al. [19]. In thecontext of onto and strategyproof mechanisms on trees (either continuous or discrete), the question of minimizing theutilitarian social cost, defined as the sum of agents’ costs, is moot since there is a simple mechanism for trees (themedian voter) that is both strategyproof and socially optimal.
Consider an unweighted, undirected, bounded degree discrete tree T = ( V, E ) with a set V of vertices and a set E ofedges. The sets V and E can be infinite. For any two vertices v , v ∈ V , d ( v , v ) is the length of the unique pathbetween v and v . The distance between two sets of vertices A ⊆ V and B ⊆ V is the length of the shortest pathbetween any pair of vertices a, b , where a ∈ A and b ∈ B . We sometimes refer to a discrete tree as the set of its vertices.Consequently, the distance between two subtrees of a tree is the distance between the corresponding sets of vertices.For u, w ∈ V with u (cid:54) = w , [ u, w ] is the sequence of vertices v , . . . , v k on the unique path of length k between u and v s.t. v = u, v k = w . We denote by ( u, w ] the sequence v , . . . v k , and by ( u, w ) the sequence v , . . . v k − , where { v i , v i +1 } ∈ E for all i = 0 , . . . k − . We say that e = { u, w } ∈ [ a, b ] if [ u, w ] ∈ [ a, b ] . A line-graph is a tree with amaximum degree of 2. 2igure 1: In this tree, { v i | ≤ i ≤ } is the set of vertices. { v , v , v , v } is the set of leafs. The agents are located at a and a (i.e., not on any vertex). The facility is located at point f , which is closer to a .Let N = { , . . . , n } be the set of agents, and a = ( a , . . . , a n ) ∈ V n be a location profile, where a i ∈ V denotes thelocation of agent i for every i ∈ N . The location profile of all agents excluding agent i is denoted by a − i ∈ V n − . Adeterministic facility location mechanism on a discrete tree is a function f : V n → V , that maps a given profile of theagents’ locations to a single location.The notation v j (cid:31) i v k indicates that agent i prefers vertex v j over vertex v k . The notation v j (cid:23) i v k indicates that agent i prefers vertex v j over vertex v k , or is indifferent between the two.In this research we assume the agents’ costs are inversely related to their distance from the chosen location. We referto such cost functions as “quadratic" costs. The class of preferences induced by quadratic costs is single-peaked andsymmetric. Formally, for every agent i ∈ N , ∀ v j , v k ∈ V : v j (cid:31) i v k ⇐⇒ d ( a i , v j ) < d ( a i , v k ) Next, we give the standard definitions of mechanism properties:
Definition 3.1 (Strategyproof) . A mechanism f is strategyproof (SP) if an agent does not benefit from reporting afalse location. Formally, f is strategyproof if for every agent i ∈ N , every profile a ∈ V n and every alternative location a (cid:48) i ∈ V , it holds that d ( a i , f ( a i , a − i )) ≤ d ( a i , f ( a (cid:48) i , a − i )) . Definition 3.2 (Onto) . A mechanism f is onto , if for every location x ∈ V there is a location profile a ∈ V n s.t. f ( a ) = x . Definition 3.3 (Unanimous) . A mechanism f is unanimous if for every location x ∈ V , f ( x, . . . , x ) = x Clearly, every unanimous mechanism is onto. The following lemma provides a necessary condition for an onto, SPmechanism on any domain.
Lemma 3.1 (Barbera and Peleg [20]) . Every mechanism that is both onto and SP, is unanimous.The definitions above apply also for continuous trees. A finite continuous tree G = ( V, E ) is a connected, acycliccollection of curves of finite length. E is the set of curves and V is the set of the extremities and intersections of thecurves [4]. Let L ⊆ V denote the set of extremities only. For all p , p ∈ G , d ( p , p ) is the length of the unique pathbetween p and p . We denote by ( p , p ) the open segment between p and p , and by [ p , p ] the closed segmentbetween the two points. For any point p and a set S ⊆ G (which may itself be a segment), the notation [ p, S ] stands forthe segment [ p, s ] where s = arg min s ∈ S d ( p, s ) . For a mechanism on a continuous tree, the agents and the facility canbe placed on arbitrary points on the edges. A mechanism on a continuous tree is therefore a function f : G n → G . Thedefinitions are illustrated in Fig 1. Schummer and Vohra [4] provided a characterization of onto, SP mechanisms on continuous trees. They showed thatwhen the agents’ preferences are quadratic, every SP, onto mechanism on the continuous tree is based on a set ofgeneralized median voter schemes, defined in [1], satisfying a consistency condition.In this section, we provide an alternative characterization of onto, SP mechanisms on continuous trees, that relies onprevious works [2, 4].
Previous results on a Continuous Line
The following definition given in [4] describes onto and SP mechanisms ona continuous line. It is similar to the one introduced by Moulin [1] for single-peaked preferences and confirmed forquadratic preferences by Border and Jordan [2]. 3 efinition 4.1 (Generalized Median Voter Scheme [4]) . A function g xy is called a generalized median voter scheme (g.m.v.s.) on [ x, y ] if there exist | N | points in [ x, y ] , { α xyS } S ⊆ N such that:1. S ⊂ R implies that d ( α xyS , x ) ≤ d ( α xyR , x ) .2. α xy ∅ = x and α xyN = y .3. For all a ∈ [ x, y ] n , g xy ( a ) is the unique point satisfying d ( g xy ( a ) , x ) = max S ⊂ N min { ( d ( a i , x )) i ∈ S , d ( α xyS , x ) } The following is a key property of the class of g.m.v.s.’s defined in [2]. It implies that when an agent moves withoutcrossing the mechanism outcome, the facility does not move.
Definition 4.2 (Uncompromising [2]) . A mechanism f : R n → R is called uncompromising if for every a ∈ R n , i ∈ N , a (cid:48) i ∈ R , it holds that:1. a i > f ( a ) implies that f ( a − i , a (cid:48) i ) = f ( a ) for all a (cid:48) i ≥ f ( a ) a i < f ( a ) implies that f ( a − i , a (cid:48) i ) = f ( a ) for all a (cid:48) i ≤ f ( a ) Lemma 4.1 (Border and Jordan [2]) . Suppose that f : R n → R is SP and unanimous. Then f is uncompromising.As shown in [4], this also applies for every mechanism f : [ x, y ] n → [ x, y ] where [ x, y ] is a finite interval in R . Notethat every SP and onto mechanism is unanimous by Lemma 3.1. Therefore we can rely on uncompromisingness for ourcharacterization of onto and SP mechanisms. Previous results on a Continuous TreeDefinition 4.3 (Graph Restriction [4]) . For any subgraph G (cid:48) ⊂ G , the graph restriction of f : G n → G to G’ is thefunction f | G (cid:48) : G (cid:48) n → G s.t for all profiles a ∈ G (cid:48) n , f | G (cid:48) ( a ) = f ( a ) By [4], if mechanism f is SP and onto, then for every a ∈ [ x, y ] n , f | xy ( a ) ∈ [ x, y ] .The following property characterizes onto and SP mechanisms on a continuous trees. For all x, y ∈ L and a i ∈ G , letthe unique point in [ x, y ] closest to a i be denoted a i | xy = arg min z ∈ [ x,y ] d ( z, a i ) . Definition 4.4 (Extended Generalized Median Voter Scheme [4]) . A mechanism f is an extended generalized medianvoter scheme (e.m.v.s.) if1. For all w, x, y, z ∈ G , f | xy and f | wz are consistent g.m.v.s.’s2. For all a ∈ G n , f ( a ) is the unique point p such that for all x, y ∈ L , p ∈ [ x, y ] implies f | xy ( a | xy ) = p Theorem 4.2 (Schummer and Vohra [4]) . For any continuous tree G , a rule f is SP and onto if and only if it is ane.m.v.s.We omit the definition of consistency since it is not relevant for our purpose. We rely on the characterization in [4] to formulate our characterization for SP and onto mechanisms on continuoustrees, which is a conceptual step on the way to our main result on discrete trees. The following properties limit theeffect of an agent’s move on the outcome of a mechanism on a continuous tree.
Definition 4.5 (Tree Monotone) . A mechanism f on the discrete tree is tree monotone (TMON) if for every profile a ∈ G n , every agent i ∈ N , every location a (cid:48) i ∈ G and every segment [ x, y ] s.t. [ x, y ] ⊆ [ a i , a (cid:48) i ] ∩ [ f ( a ) , f ( a − i , a (cid:48) i )] , itholds that d ( a i , x ) < d ( a i , y ) ⇔ d ( f ( a ) , x ) < d ( f ( a ) , y ) Intuitively, TMON means that the facility moves in the same direction as the moving agent (if it crosses the agent’s pathat all).
Definition 4.6 (Trajectory contained) . A mechanism on a tree is
Trajectory Contained (TC) if for all a, a (cid:48) = ( a − i , a (cid:48) i ) it either holds that [ f ( a ) , f ( a (cid:48) )] ⊆ [ a i , a (cid:48) i ] , or f ( a ) = f ( a (cid:48) ) .4n words, either the trajectory of the outcome is contained in the trajectory of the agent, or the facility does notmove at all. In Fig. 1, when agent 1 moves from a to a (cid:48) , the facility moves from f to f (cid:48) . This violates TC since [ v , f (cid:48) ] (cid:54)⊆ [ a i , a (cid:48) i ] . This also violates TMON since the facility and the agent move in opposite directions in the segment [ f, v ] . Lemma 4.3.
Every onto and SP mechanism f on the continuous tree is TC. Proof.
Consider an SP, onto mechanism f : G n → G on a continuous tree. Assume by contradiction that there existsan agent i and two profiles a, a (cid:48) = ( a (cid:48) i , a − i ) s.t. f ( a ) (cid:54) = f ( a (cid:48) ) and w.l.o.g., that f ( a ) / ∈ [ a i , a (cid:48) i ] . By Theorem 4.2, f is an e.m.v.s. and therefore it is a collection of g.m.v.s’s. Let g xy denote the g.m.v.s. on [ x, y ] , where x, y ∈ L and [ f ( a ) , f ( a (cid:48) )] ⊆ [ x, y ] . Note that g xy = f | xy . By Lemma 4.1, g xy is uncompromising. From the second property of thee.m.v.s., it holds that g xy ( a | xy ) = f ( a ) and g xy ( a (cid:48) xy ) = f ( a (cid:48) ) We divide into two cases, according to the locations a i | xy , a (cid:48) i | xy . Note that for any other agent j (cid:54) = i , a (cid:48) j = a j and thus a (cid:48) j | xy = a j | xy .1. | [ a i , a i | xy ] ∩ [ a (cid:48) i , a (cid:48) i | xy ] | > : In this case, a i | xy = a (cid:48) i | xy and therefore g xy ( a | xy ) = g xy ( a (cid:48) | xy ) contradictingthe assumption that f ( a ) (cid:54) = f ( a (cid:48) ) .2. | [ a i , a i | xy ] ∩ [ a (cid:48) i , a (cid:48) i | xy ] | = 0 : In this case, the path between a i and a (cid:48) i must intersect the segment [ x, y ] andtherefore, [ x, y ] ∩ [ a i , a (cid:48) i ] = [ a i | xy , a (cid:48) i | xy ] . Recall that by our initial assumption f ( a ) = g xy ( a | xy ) / ∈ [ a i , a (cid:48) i ] , thus f ( a ) / ∈ [ a i | xy , a (cid:48) i | xy ] , contradicting the uncompromisingness of g xy . Lemma 4.4.
Every SP mechanism on the continuous tree is TMON.The proof follows from the definitions of TMON and SP.
Lemma 4.5.
Every TC and TMON mechanism on the continuous tree is SP.
Proof.
Assume by contradiction that there exists a TC, TMON mechanism f that violates SP. Consider a pair of profiles a, a (cid:48) = ( a − i , a (cid:48) i ) s.t. f ( a ) (cid:54) = f ( a (cid:48) ) . Since f is TC, it holds that [ a i , a (cid:48) i ] ∩ [ f ( a ) , f ( a (cid:48) )] = [ f ( a ) , f ( a (cid:48) )] and therefore by TMON, d ( a i , f ( a )) ≤ d ( a i , f ( a (cid:48) ) . Therefore no agent can benefit from reporting a false location.
Theorem 4.6.
An onto mechanism on the continuous tree is SP if and only if it is TMON and TC.
Proof.
The proof follows from Lemmas 4.3, 4.4 and 4.5.
In this section we provide a complete characterization of onto, SP mechanisms on discrete trees, generalizing the resultof Dokow et al. for discrete lines [3].Before presenting the main result, we show that a naïve extension of the properties defined for mechanisms on discretelines in [3] fails for trees. Their result implies that an agent can affect the outcome of the mechanism only in a way inwhich its trajectory intersects the trajectory of the facility in at least two consecutive points.The mechanism described in Fig. 2 is an example of an SP, onto mechanism that violates a naïve extension of thisproperty. Agent 1 is located at vertex y = 0 . Agent 2 is initially at and moves to . As a result, the facility movesfrom vertex to without intersecting the segment [3 , . 5or every ≤ x, y ≤ : if y=0 then f ( x, y ) = x mod elseif y=2 then f ( x, y ) = min { x, } else f ( x, y ) = min { x, y } endend a
01 2 a a (cid:48) Figure 2: An example of a mechanism that is SP and onto on a discrete tree, that violates the properties defined byDokow et al. [3].
Quadratic vs. single-peaked preferencesDefinition 5.1 (Single-Peaked [5]) . A preference of an agent i is single-peaked on a graph G if there is a spanningtree T = ( V, E ) of G such that for all distinct x, y ∈ V with a i (cid:54) = y , x ∈ [ a i , y ) ⇒ x (cid:31) i y The following example shows that under single-peaked preferences, the mechanism in Fig. 2 is not SP. Assume thepreferences of the agents are as follows:1. Agent 1: (cid:31) (cid:31) (cid:23) (cid:31)
2. Agent 2: (cid:31) (cid:23) (cid:31) (cid:31) Both agents have single-peaked preferences according to the definition in [5]. In particular, the preference of the firstagent is quadratic. The preference of the second agent is not, since she strictly prefers vertex 0 over vertex 1. If bothagents report truthfully, the facility will be located at vertex 1. However, if the second agent reports vertex 4 as herpeak, the facility will be located at vertex 0 and the agent will benefit.We conclude that similarly to the case of the line-tree, quadratic preferences allow more SP mechanisms than single-peaked preferences, and therefore the characterization of probabilistic SP mechanisms under single-peaked preferencesin [5] does not apply for quadratic preferences.
Here we define several new terms which are specific for mechanisms on discrete trees.
Definition 5.2 (Tree) . tree ( a → b, v ) is the subtree which includes only v and vertices which are accessible from v ,via the edges that are not in [ a, b ] . Definition 5.3 (Depth) . depth ( a → b, v ) is the distance of vertex v from [ a, b ] .We demonstrate the above definitions in Fig. 3: tree ( a → a (cid:48) , v ) contains v (at depth 0) and another node at depth 1. depth ( a → a (cid:48) , v ) = 2 in the subtree rooted by a .Our next definitions are intended to generalize the properties defined in [3]. Definition 5.4 ( m -tree step independent) . A mechanism f is m -tree step independent ( m -TSI) if for every a ∈ V n , i ∈ N , a (cid:48) i ∈ V s.t. d ([ a i , a (cid:48) i ] , f ( a )) > m , it holds that tree ( a i → a (cid:48) i , f ( a )) = tree ( a i → a (cid:48) i , f ( a − i , a (cid:48) i )) For m = 1 , the definition states that for every a ∈ V n , i ∈ N , a (cid:48) i ∈ V s.t. | [ f ( a ) , f ( a − i , a (cid:48) i )] ∩ [ a i , a (cid:48) i ] | ≥ it holds that d ( f ( a ) , [ a i , a (cid:48) i ]) ≤ Fig. 3 illustrates a violation of the property. Mechanism g violates -TSI since tree ( a → a (cid:48) , g ( a )) (cid:54) = tree ( a → a (cid:48) , g ( a (cid:48) )) and d ([ a , a (cid:48) ] , g ( a )) = 3 v a (cid:48) zv f ( a ) f ( a (cid:48) ) a a g ( a (cid:48) ) v a (cid:48) zv g ( a ) a Figure 3: Mechanism f violates ADR and TPAR w.r.t. profile a (cid:48) = ( a (cid:48) , a ) , but satisfies TPAR w.r.t. profile a .Mechanism g violates 1-TSI and DB. Definition 5.5 (Depth Balanced) . A mechanism f is depth balanced (DB) if for every a ∈ V n , i ∈ N , a (cid:48) i ∈ V , itholds that d ( tree ( a i → a (cid:48) i , f ( a )) , tree ( a i → a (cid:48) i , f ( a − i , a (cid:48) i ))) ≥| depth ( a i → a (cid:48) i , f ( a )) − depth ( a i → a (cid:48) i , f ( a − i , a (cid:48) i )) | Informally, DB means that when the facility moves as a result of a single agent’s deviation, the distance between thetree of the original outcome and the tree of the new outcome is bigger than the difference between the depths of theoutcomes. Fig. 3 illustrates a violation of the property by mechanism g . g violates DB since d ( tree ( a → a (cid:48) , g ( a )) , tree ( a → a (cid:48) , g ( a (cid:48) , a ))) = 1 < | depth ( a → a (cid:48) , g ( a )) − depth ( a → a (cid:48) , g ( a (cid:48) , a ) | = 3 Definition 5.6 (Tree Pareto Location) . Let
Int ( a ) be the set of interior vertices of the subtree defined by profile a : Int ( a ) = { v ∈ V |∃ a i , a j ∈ a s.t. v ∈ ( a i , a j ) } . A location x ∈ V is tree Pareto w.r.t. a if d ( x, Int ( a )) ≤ or x = a i for some i ∈ N .This definition generalizes the definition in [3] of a Pareto location on the discrete line. Note that it is weaker than thestandard definition of Pareto. Definition 5.7 (Tree Pareto Mechanism) . A mechanism f is tree Pareto (TPAR) if for every profile a ∈ V n , f ( a ) is atree Pareto location w.r.t. a .Mechanism f in Fig. 3 violates TPAR w.r.t. profile a (cid:48) since d ( f ( a (cid:48) ) , Int ( a (cid:48) )) = d ( f ( a (cid:48) ) , [ v , v ]) = 2 Definition 5.8 (Almost Depth Restricted) . A mechanism f is almost depth restricted (ADR) if for every a ∈ V n , i ∈ N , a (cid:48) i ∈ V s.t. f ( a ) (cid:54) = f ( a (cid:48) i , a − i ) and tree ( a i → a (cid:48) i , f ( a )) = tree ( a i → a (cid:48) i , f ( a (cid:48) i , a − i )) , the following holds: Let z be the unique point s.t. z = [ a i , f ( a )] ∩ [ a i , f ( a (cid:48) i , a − i )] ∩ [ f ( a ) , f ( a (cid:48) i , a − i )] Then 1. d ( f ( a ) , z ) = d ( f ( a (cid:48) i , a − i ) , z ) d ( f ( a ) , z ) = 1 Informally, ADR means that when the facility moves as a result of a single agent’s deviation, without intersecting thetrajectory of the agent in at least two points, the new outcome has the same parent as the original outcome in the treeinduced by the deviation. That is, either the facility does not move, or it moves to a sibling node. We can think of thisproperty as “approximate uncompromising" (replacing ‘1’ with ‘0’ in the definition would yield exact uncompromising).In Fig. 3, ADR is violated by mechanism f for the pair of profiles ( a, a (cid:48) ) , which differ by the location of agent , since d ( f ( a ) , z ) = 2 . Definition 5.9 (Almost Trajectory Contained) . A mechanism f is almost trajectory contained (ATC) if it is ADRand 1-TSI. 7inally, we use the term TMON defined in Section 4 to describe mechanisms on discrete trees as well.We now go on to characterize SP and onto mechanisms on discrete trees. Theorem 5.1.
An onto mechanism f on the discrete tree is SP if and only if it is TMON and ATC.First we show a weak characterization. We then use it to prove the tree Pareto property and the main property of “almosttrajectory containment", which consists of two properties, ADR and 1-TSI. To prove ADR, we first show that when anagent moves towards the facility, either the facility remains in place (as in the continuous case), or it moves exactly onestep towards the agent and one step away. To prove 1-TSI, we first show that when an agent moves to a neighboringvertex on edge e , the facility can intersect e only if it is at most one step away from e . DB and TMONLemma 5.2.
A pair of profiles violates SP if and only if it violates DB or TMON.
Proof. “ ⇒ " Every pair of profiles a, a = ( a − i , a (cid:48) i ) which violates SP, violates DB or TMON.Consider a mechanism f and a pair of profiles a, a = ( a − i , a (cid:48) i ) s.t. x := f ( a ); x (cid:48) := f ( a − i , a (cid:48) i ); and d ( a i , x ) > d ( a i , x (cid:48) ) For any mechanism f and profile a it holds that d ( a i , x ) = d ( a i , tree ( a i → a (cid:48) i , x )) + depth ( a i → a (cid:48) i , x ) (1)Assume that f is TMON. Then it follows that: d ( a i , x (cid:48) ) = d ( a i , tree ( a i → a (cid:48) i , x )) (2) + d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) ))+ depth ( a i → a (cid:48) i , x (cid:48) ) And thus from Eqs. (1), (2) above, d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) + depth ( a i → a (cid:48) i , x (cid:48) ) < depth ( a i → a (cid:48) i , x ) And therefore, d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) < depth ( a i → a (cid:48) i , x ) − depth ( a i → a (cid:48) i , x (cid:48) ) , contradicting DB.“ ⇐ " Every pair of profiles a, a (cid:48) = ( a − i , a (cid:48) i ) which violates TMON or DB, violates SP.TMON. Consider a mechanism f and a pair of profiles a, a = ( a − i , a (cid:48) i ) that violates TMON, i.e., ∃ i ∈ N, a, a (cid:48) i s.t. x := f ( a ) ∈ tree k ( a i → a (cid:48) i ); x (cid:48) := f ( a − i , a (cid:48) i ) ∈ tree j ( a i → a (cid:48) i ); and j < k It follows that d ( a i , x (cid:48) ) = d ( a i , tree ( a i → a (cid:48) i , x (cid:48) )) + depth ( a i → a (cid:48) i , x (cid:48) ) (3) d ( a i , x ) = d ( a i , tree ( a i → a (cid:48) i , x (cid:48) )) (4) + d ( tree ( a i → a (cid:48) i , x (cid:48) ) , tree ( a i → a (cid:48) i , x ))+ depth ( a i → a (cid:48) i , x ) . Assume by contradiction that f is SP. Then it follows that ∀ i, a, a (cid:48) i , d ( a i , x ) ≤ d ( a i , x (cid:48) ) , and thus fromEqs. (3),(4) above, d ( tree ( a i → a (cid:48) i , x (cid:48) ) , tree ( a i → a (cid:48) i , x )) + depth ( a i → a (cid:48) i , x ) ≤ depth ( a i → a (cid:48) i , x (cid:48) ) . From the contradiction assumption x and x (cid:48) are in different trees. Therefore, depth ( a i → a (cid:48) i , x ) < depth ( a i → a (cid:48) i , x (cid:48) ) . (5)Symmetrically, ∀ i, a, a (cid:48) i : d ( a (cid:48) i , x ) ≥ d ( a (cid:48) i , x (cid:48) ) from which we derive depth ( a i → a (cid:48) i , x ) > depth ( a i → a (cid:48) i , x (cid:48) ) , contradicting Eq. (5) above. Therefore the pair a, a (cid:48) violates SP.8B. Consider an SP mechanism f and a pair of profiles a, a = ( a − i , a (cid:48) i ) . For any mechanism f and profile a it holdsthat d ( a i , x ) = d ( a i , tree ( a i → a (cid:48) i , x )) + depth ( a i → a (cid:48) i , x ) . (6)From TMON: d ( a i , x (cid:48) ) = d ( a i , tree ( a i → a (cid:48) i , x )) (7) + d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) ))+ depth ( a i → a (cid:48) i , x (cid:48) ) . From strategyproofness, ∀ i, a, a (cid:48) i , d ( a i , x ) ≤ d ( a i , x (cid:48) ) , and thus from Eqs. (6),(7) above, depth ( a i → a (cid:48) i , x ) ≤ d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) + depth ( a i → a (cid:48) i , x (cid:48) ) Therefore, d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) ≥ depth ( a i → a (cid:48) i , x ) − depth ( a i → a (cid:48) i , x (cid:48) ) (8)Symmetrically, from strategyproofness, ∀ i, a, a (cid:48) i : d ( a (cid:48) i , x ) ≥ d ( a (cid:48) i , x (cid:48) ) from which we derive d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) ≥ depth ( a i → a (cid:48) i , x (cid:48) ) − depth ( a i → a (cid:48) i , x ) (9)And thus, from Eqs. (8),(9) above, d ( tree ( a i → a (cid:48) i , x ) , tree ( a i → a (cid:48) i , x (cid:48) )) ≥ | depth ( a i → a (cid:48) i , x (cid:48) ) − depth ( a i → a (cid:48) i , x ) | Therefore, any pair of profiles which violates DB and satisfies TMON, violates SP.Lemma 5.2 characterizes all pairs of profiles that violate SP. We later show a stronger characterization that describes thepairs that indicate that an onto mechanism is not SP. These pairs differ only in a single agent’s location, who does notnecessarily benefit from misreporting, contrary to the characterization in Lemma 5.2.
TPARLemma 5.3.
Every SP and onto mechanism f on the discrete tree is TPAR. Proof.
Assume by contradiction that an onto, SP mechanism f is not TPAR. Then there exists a profile a s.t. d ( f ( a ) , Int ( a )) > . Let v denote the first vertex on the path from f ( a ) to all other locations of a . Let z i := ( a . . . , a i , v, . . . , v ) . Note that z n is the profile a and z is the profile ( v, . . . v ) . By definition of v , wehave that f ( a ) = f ( z n ) ∈ tree ( a i → v, v ) . Since TPAR is violated, v has exactly one neighbor u on the path to anyother agent location a i (cid:54) = v . When moving from profile z i +1 to z i , it follows from TMON that ∀ ≤ i ≤ n − , f ( z i ) ∈ tree ( a i → v, f ( z i +1 )) = tree ( a i → v, v ) (10)From DB and (10) we have that depth ( a i → v, f ( z i )) = 1 (cid:54) = 0 = depth ( a i → v, v ) (11)It follows from (11) that f ( z ) = f ( v, .., v ) (cid:54) = v in contradiction to unanimity. ADR
Here we prove that ADR is a necessary condition for an onto, SP mechanism. We first show that if a pair ofprofiles violates the property, there exists a pair of profiles that violates the property in which the agent moves to avertex on the path between the two outcomes.
Lemma 5.4.
If an onto, SP mechanism f violates ADR, there exists a pair of violating profiles a, a (cid:48) = ( a − i , a (cid:48) i ) s.t. z = [ a i , f ( a )] ∩ [ a i , f ( a (cid:48) )] ∩ [ f ( a ) , f ( a (cid:48) )] = a (cid:48) i and d ( f ( a ) , z ) = d ( f ( a (cid:48) ) , z ) i a (cid:48) i c i ; z ; yy (cid:48) f ( a ) f ( a (cid:48) ) f ( c ) Figure 4: An illustration of the proof of Lemma 5.4. Here the violating pair of profiles is ( a, c ) . Note that this pair ofprofiles does not violate SP. Proof.
Assume by contradiction that there is a violating pair ( a, a (cid:48) ) for which f ( a ) and f ( a (cid:48) ) are different vertices inthe same tree w.r.t. the move a i → a (cid:48) i and a (cid:48) i (cid:54) = z . Consider the profile c where c − i = a − i and c i = z . If f ( c ) = f ( a ) or f ( c ) = f ( a (cid:48) ) the proof follows.Assume that f ( c ) (cid:54) = f ( a ) and f ( c ) (cid:54) = f ( a (cid:48) ) . Then it follows from TMON that f ( c ) ∈ tree ( a i → a (cid:48) i , f ( a )); and tree ( a i → a (cid:48) i , f ( a )) = tree ( a i → a (cid:48) i , f ( a (cid:48) )) Otherwise, if the tree that contains f ( c ) is closer to a i than the tree that contains f ( a ) and f ( a (cid:48) ) , mechanism f is notTMON w.r.t. the pair of profiles ( a, c ) . If the tree that contains f ( c ) is closer to a (cid:48) i , f is not TMON w.r.t. the pair ofprofiles ( a (cid:48) , c ) . For the same reason, f ( c ) belongs to the subtree of z , i.e. d ( f ( c ) , [ a i , a (cid:48) i ]) > d ( z, [ a i , a (cid:48) i ]) .Let y denote the unique point s.t. y = [ a i , f ( a )] ∩ [ a i , f ( c )] ∩ [ f ( a ) , f ( c )] and y (cid:48) the unique point s.t. y (cid:48) =[ a i , f ( a (cid:48) )] ∩ [ a i , f ( c )] ∩ [ f ( a (cid:48) ) , f ( c )] (see example in Fig. 4). From DB, we have that d ( f ( a ) , y ) = d ( f ( c ) , y ) and d ( f ( a (cid:48) ) , y (cid:48) ) = d ( f ( c ) , y (cid:48) ) .If d ( f ( c ) , y ) = d ( f ( a ) , y ) = 1 and d ( f ( c ) , y (cid:48) ) = d ( f ( a (cid:48) ) , y (cid:48) ) = 1 , it follows that y and y (cid:48) are the same vertex, incontradiction to the assumption that the pair ( a, a (cid:48) ) violates ADR. If d ( f ( a ) , y ) > , the proof follows for the pair ( a, c ) . Otherwise, d ( f ( a (cid:48) ) , y (cid:48) ) > and the proof follows for the pair ( a (cid:48) , c ) .The following lemma proves the necessity of ADR by showing that there is no violating pair of profiles in which theagent moves to a location on the path between the outcomes. Lemma 5.5.
Every onto SP mechanism on the discrete tree is ADR.
Proof.
Assume that there exists an agent i and two profiles a, a (cid:48) = ( a (cid:48) i , a − i ) s.t. x := f ( a ); x (cid:48) := f ( a (cid:48) ); and tree ( a i → a (cid:48) i , x ) = tree ( a i → a (cid:48) i , x (cid:48) ) From DB, we have that d ( x, z ) = d ( x (cid:48) , z ) , where z is the unique point s.t. z = [ a i , x ] ∩ [ a i , x (cid:48) ] ∩ [ x, x (cid:48) ] . We show thatADR holds when z = a (cid:48) i . Assume by contradiction that z = a (cid:48) i and d ( x, z ) = d ( x (cid:48) , z ) > (12)Among the violating pairs, let ( a, a (cid:48) = ( a − i , a (cid:48) i )) be one of the pairs that minimize Σ k (cid:54) = i d ( a k , [ a i , a (cid:48) i ]) . We let v denote the first vertex on the path from x to z and v denote the second vertex on the path from x to z (see Figs. 5.1, 5.2).Note that v and z is the same vertex if d ( x, z ) = d ( x (cid:48) , z ) = 2 .Since f is TPAR, there must be some other agent j s.t. x is sufficiently close to the path from agent j to agent i .Formally, ∃ j : a j ∈ tree ( a i → v , v ) \ v (see locations a j , a j , a j in Fig. 5.1). We define two profiles b = ( a − j , b j = v ) and b (cid:48) = ( a (cid:48)− j , b (cid:48) j = v ) , whichdiffer from a and a (cid:48) only by the location of agent j . Let y denote f ( b ) and y (cid:48) denote f ( b (cid:48) ) .For the pair of profiles ( a, b ) : If x is on the path from a j to v (location a j in Fig. 5.1), it follows from DB and TMONthat the facility will stay in the same tree w.r.t. the move a j → v at depth 0 (location y in Fig. 5.3) or move to tree ( a j → v , v ) and be located at depth 0 (location y in Fig. 5.3) or 1 (locations y , y in Fig. 5.3). Otherwise, if x is not on the path from a j to v (locations a j , a j in Fig. 5.1), x will stay in the same tree at the same depth w.r.t. themove a j → v (locations y , y in Fig. 5.3). Overall, the possible locations of y are x, v , v and the direct children of v . 10ocation y (cid:48) satisfies: d ( y (cid:48) , v ) ≤ d ( x (cid:48) , v ) . Otherwise, SP is violated for the pair of profiles ( a (cid:48) , b (cid:48) ) . The location y (cid:48) also satisfies y (cid:48) ∈ tree ( a i → a (cid:48) i , a (cid:48) i ); and depth ( a i → a (cid:48) i , y (cid:48) ) = depth ( a i → a (cid:48) i , y ) (13)Otherwise, SP is violated for the pair of profiles ( b, b (cid:48) ) (therefore y (cid:48) in Fig. 5.4 is not a valid location). We divide intotwo cases by the possible locations of y (cid:48) :1. y (cid:48) is a child of v in the tree ( a i → a (cid:48) i , v ) : In this case the pair ( a (cid:48) , b (cid:48) ) is a violation of SP, since d ( a j , y (cid:48) ) ≤ d ( a j , v ) + d ( v , y (cid:48) ) (14) = d ( a j , v ) + 1 (15) < d ( a j , v ) + 3 ≤ d ( a j , x (cid:48) ) (16)Eq. (14) follows from the triangle inequality, Eq. (15) follows from the case condition and Eq. (16) followsfrom Eq. (12).2. y (cid:48) is not a child of v in the tree ( a i → a (cid:48) i , v ) : It follows from Eq. (13) and the case condition that y (cid:48) / ∈ tree ( a i → a (cid:48) i , v ) \ { v } (17)Therefore, v ∈ [ a j , y (cid:48) ] . From Eq. (17), d ( v , y (cid:48) ) = d ( v , x (cid:48) ) (18)Otherwise, the pair ( a (cid:48) , b (cid:48) ) violates SP, since the nearest location to v between y (cid:48) and x (cid:48) is strictly closer toagent j whether agent j is located at a j or at v . From Eq. (13), depth ( a i → a (cid:48) i , y (cid:48) ) ∈ { depth ( a i → a (cid:48) i , x ) , depth ( a i → a (cid:48) i , v ) , depth ( a i → a (cid:48) i , v ) } (19)We divide into two cases:(a) depth ( a i → a (cid:48) i , y (cid:48) ) ∈ { depth ( a i → a (cid:48) i , v ) , depth ( a i → a (cid:48) i , v ) } (locations y (cid:48) , y (cid:48) in Fig. 5.4): d ( v , y (cid:48) ) ≤ d ( v , z ) + d ( z, y (cid:48) ) (20) < d ( v , z ) + d ( z, x (cid:48) ) (21) = d ( v , x (cid:48) ) contradicting Eq. (18). Eq. (20) follows from the triangle inequality and Eq. (21) follows from the casecondition.(b) depth ( a i → a (cid:48) i , y (cid:48) ) = depth ( a i → a (cid:48) i , x ) (locations y (cid:48) , y (cid:48) in Fig. 5.4):Since y (cid:48) is not a direct child of v , the pair ( b, b (cid:48) ) violates ADR in contradiction to the minimality of Σ k (cid:54) = i d ( a k , [ a i , a (cid:48) i ]) , since agent j is closer to the trajectory of agent i in profiles b, b (cid:48) than in profiles a, a (cid:48) .We have shown that there is no valid location for y (cid:48) , and therefore ADR is not violated for the case in which a (cid:48) i = z .From Lemma 5.4, every SP onto mechanism is ADR. Here we prove that 1-TSI is a necessary condition for an onto, SP mechanism. We first show that if there is aviolation of the 1-TSI property, then w.l.o.g. it occurs when an agent moves a single step.
Lemma 5.6.
If an onto, SP mechanism f violates 1-TSI, there exists a pair of violating profiles a, a (cid:48) = ( a − i , a (cid:48) i ) s.t. d ( a i , a (cid:48) i ) = 1 . Proof.
Assume that d ( a i , a (cid:48) i ) > and assume w.l.o.g. that d ( f ( a ) , [ a i , a (cid:48) i ]) > We denote by x, y the two adjacent vertices on the path from a i to a (cid:48) i s.t. x (cid:48) = ( a − i , x ); y (cid:48) = ( a − i , y ); and tree ( a i → a (cid:48) i , f ( x (cid:48) )) (cid:54) = tree ( a i → a (cid:48) i , f ( y (cid:48) )) (22)11 i zv v a j ; x a j a j a a (cid:48) i ; zv v a (cid:48) j a (cid:48) j x (cid:48) a (cid:48) j a (cid:48) b i zv ; y v ; y ; b j x ; y y x (cid:48) b y (cid:48) b (cid:48) i ; zv v ; b (cid:48) j y (cid:48) y (cid:48) b (cid:48) Figure 5: An illustration of the possible locations of agent j and the facility locations for the profiles a, a (cid:48) , b, b (cid:48) in theproof of Lemma 5.5. a i x y a (cid:48) i f ( a ) f ( y (cid:48) ) f ( x (cid:48) ) Figure 6: An illustration of the proof of Lemma 5.6. The pair ( x (cid:48) = ( a − i , x ) , y (cid:48) = ( a − i , y )) violates ADR.If there is more than one such pair then we select the pair x, y closest to a i (see Fig. 6). From Lemma 5.2, every SP,onto mechanism satisfies DB. Therefore, it holds that for every move a i → z where z ∈ [ a i , x ] , the facility stays in thesame depth w.r.t. its initial tree, i.e., depth ( a i → z, f ( a − i , z )) = depth ( a i → z, f ( a )) (23)Assume by contradiction that tree ( x → y, f ( x (cid:48) )) = tree ( x → y, f ( y (cid:48) )) (24)From Eq. (22), when agent i moves from x to y , the trajectory of the facility intersects the segment [ a i , a (cid:48) i ] in two points,and since d ( f ( a ) , [ a i , a (cid:48) i ]) ≥ , it holds that d ( f ( a ) , f ( y (cid:48) )) ≥ (25)On the other hand, from Eq. (24) and ADR we have that d ( f ( a ) , f ( y (cid:48) )) ∈ { , } (26)contradicting Eq. (25) (see locations f ( x (cid:48) ) , f ( y (cid:48) ) in Fig. 6). Thus, it follows that tree ( x → y, f ( x (cid:48) )) (cid:54) = tree ( x → y, f ( y (cid:48) )) . Therefore, the pair ( x (cid:48) , y (cid:48) ) violates 1-TSI by a one-step deviation, since d ( f ( x (cid:48) ) , [ a i , a (cid:48) i ]) > , and inparticular, d ( f ( a ) , [ x, y ]) > as required.The following lemma proves the necessity of 1-TSI by showing that there is no violating pair of profiles in which theagent moves to a neighboring vertex. Lemma 5.7.
Every onto, SP mechanism on the discrete tree is 1-TSI.
Proof.
Assume by contradiction that there exists a pair of profiles a, a (cid:48) = ( a − i , a (cid:48) i ) s.t. tree ( a i → a (cid:48) , f ( a )) (cid:54) = tree ( a i → a (cid:48) , f ( a − i , a (cid:48) i )) (27)12 ; a i a j ; f ( a ) a j a j a a (cid:48) i a (cid:48) j a (cid:48) j f ( a (cid:48) ) a (cid:48) j a (cid:48) y ; b i ; b j y y y b b (cid:48) j b (cid:48) i y (cid:48) y (cid:48) b (cid:48) Figure 7: An illustration of the possible locations of agent j and the facility locations y, y (cid:48) for the profiles a, a (cid:48) , b, b (cid:48) .Assume w.l.o.g. that d ( f ( a (cid:48) ) , [ a i .a (cid:48) i ]) > . We can assume that d ( a i , a (cid:48) i ) = 1 from lemma 5.6. Among these pairs, let ( a, a (cid:48) ) be the pair that minimizes Σ k (cid:54) = i d ( a k , [ a i , a (cid:48) i ]) . From TMON, we have that f ( a ) ∈ tree ( a i → a (cid:48) i , a i ) and f ( a (cid:48) ) ∈ tree ( a i → a (cid:48) i , a (cid:48) i ) Since d ( a i , a (cid:48) i ) = 1 , d ( tree ( a i → a (cid:48) i , f ( a )) , tree ( a i → a (cid:48) i , f ( a (cid:48) ))) = 1 and in order to satisfy the DB condition, the difference between the depths of f ( a ) and f ( a (cid:48) ) has to be at most 1.Therefore, depth ( a i → a (cid:48) i , f ( a )) ≥ Let p denote the first vertex on the path from f ( a ) to a i . From TPAR, there exists an agent j s.t. a j ∈ tree ( a i → a (cid:48) i , p ) \ p (see locations a j , a j , a j in Fig. 7.1).We define two profiles b = ( a − j , p ) and b (cid:48) = ( a −{ i,j } , b (cid:48) i = a (cid:48) i , b (cid:48) j = p ) .For the pair ( a, b ) , when agent j moves to p , it follows from DB that the facility can only move to location p , or to alocation z s.t. d ( p, z ) = 1 (28)For the pair ( a, a (cid:48) ) , when agent j moves to p , it follows from ADR that the facility can only move from f ( a (cid:48) ) to adifferent child of p in the tree induced by the move a i → a (cid:48) i (see Fig. 7.2, Fig. 7.4). Therefore it holds that y := f ( b ); y (cid:48) := f ( b (cid:48) ); y (cid:48) ∈ tree ( a → a (cid:48) i , a (cid:48) i ); depth ( a → a (cid:48) , y (cid:48) ) = depth ( a → a (cid:48) , f ( a (cid:48) )) ≥ (29)(See Fig. 7). We show that every possible location of the facility for the profile ( a − j , p ) violates SP or the minimalitycondition:1. y ∈ tree ( a i → a (cid:48) i , a (cid:48) i ) (location y in Fig. 7.3): DB is violated for the pair ( b, b (cid:48) ) since y, y (cid:48) are in the sametree w.r.t. the move a i → a (cid:48) i , but from Eq. (28), depth ( a i → a (cid:48) i , y ) = 0 while depth ( a i → a (cid:48) i , y (cid:48) ) = 2 fromEq. (29).2. y ∈ tree ( a i → a (cid:48) i , a i ) (locations y , y , y in Fig. 7.3): The pair ( b, b (cid:48) ) violates 1-TSI. This contradicts theminimality of Σ k (cid:54) = i d ( a k , [ a i , a (cid:48) i ]) , since agent j is closer to the trajectory of agent i in profiles b, b (cid:48) than inprofiles a, a (cid:48) . 13 .2 Our Characterization We now complete our characterization of onto, SP mechanisms.
Lemma 5.8.
Every TMON and ATC mechanism f on the discrete tree is SP. Proof.
Suppose f is TMON and ATC. By definition it is 1-TSI and ADR. Consider an arbitrary pair of profiles ( a, a (cid:48) ) s.t. a (cid:48) = ( a − i , a (cid:48) i ) for some i ∈ N, a (cid:48) i ∈ V . If tree ( a i → a (cid:48) i , f ( a )) = tree ( a i → a (cid:48) i , f ( a (cid:48) )) , it follows from ADR that depth ( a i → a (cid:48) i , f ( a )) − depth ( a i → a (cid:48) i , f ( a (cid:48) )) = 0= d ( tree ( a i → a (cid:48) i , f ( a )) , tree ( a i → a (cid:48) i , f ( a (cid:48) ))) (30)If tree ( a i → a (cid:48) i , f ( a )) (cid:54) = tree ( a i → a (cid:48) i , f ( a (cid:48) )) , it follows from 1-TSI that d ( f ( a ) , [ a i , a (cid:48) i ]) ≤ and d ( f ( a (cid:48) ) , [ a i , a (cid:48) i ]) ≤ (31)Eq. (31) implies that | depth ( a i → a (cid:48) i , f ( a )) − depth ( a i → a (cid:48) i , f ( a − i , a (cid:48) i )) | ≤ ≤ d ( tree ( a i → a (cid:48) i , f ( a )) , tree ( a i → a (cid:48) i , f ( a (cid:48) ))) From Eqs. (30), (31), f satisfies DB, and therefore f is SP from Lemma 5.2.We conclude that every onto mechanism f on the discrete tree is strategyproof if and only if it is TMON and ATC. Thisfollows from Lemmas 5.2, 5.5, 5.7 and 5.8. In this section, we show that on infinite discrete lines, the only anonymous, shift-invariant SP mechanisms are orderstatistics mechanisms—as on continuous lines [1]. Therefore, all additional mechanisms that are SP in the discretedomain must single out specific locations to satisfy Neutrality.We assume that the vertices of the line are indexed in increasing order and that the agents are ordered by their locationin profile a , i.e., if a i < a j , i < j . Definition 6.1 (Shift-Invariant) . A mechanism f on an infinite discrete line is shift-invariant if for every locationprofile a = ( a , . . . , a n ) and d ∈ N : f ( a + d, . . . , a n + d ) = f ( a ) + d . Definition 6.2 (Anonymous) . A mechanism f is anonymous if for every location profile a and every permutation ofagents π : N → N , it holds that f ( a , . . . , a n ) = f ( a π , . . . , a π n ) Definition 6.3 ( k th order statistic mechanism) . A mechanism f is the k th order statistic mechanism for some k ∈ N ,if for every profile a = ( a , ..., a n ) , it holds that f ( a ) = a k . Lemma 6.1.
If a mechanism f is onto, SP and shift-invariant, then for every profile a = ( a , . . . , a n ) , it holds that f ( a ) = a i for some i ∈ [1 , n ] . Proof.
Consider the profile a (cid:48) = ( a + 1 , . . . , a n + 1) . From shift-invariance, f ( a (cid:48) ) = f ( a ) + 1 . The profile a (cid:48) can beachieved by n moves of one step, one per an agent. For every such move of agent j ∈ N , resulting in profile a j , thefollowing hold:1. f ( a j ) ≥ f ( a ) from TMON.2. If f ( a j ) (cid:54) = f ( a ) , from shift-invariance and the previous statement, it holds that f ( a j ) = f ( a ) + 1 .The only way to satisfy these conditions without violating DB, is by demanding that f ( a ) = a i for some i ∈ N toassure that tree ( a j → a j + 1 , f ( a )) (cid:54) = tree ( a j → a j + 1 , f ( a j )) Lemma 6.2.
An onto, SP, anonymous, shift-invariant mechanism f for the line is a k th order statistic mechanism. This property is sometimes called “tops-only" or “peaks-only". roof. Assume by contradiction that f is not a k th order statistic mechanism, i.e. there exists a pair of profiles ( a, b ) s.t. f ( a ) is the j th agent location and f ( b ) is the k th agent location. Assume that j < k . Let d denote b n − a + 1 .Consider a profile c where c i = b i − d if d > and c i = b i otherwise. from shift-invariance, it follows that f ( c ) = c k .We now iteratively move each agent in profile c to a location in profile a . In the l th iteration we move the agent withindex n + 1 − l to the location a n +1 − l , getting a sequence of profiles ( c l ) nl =1 . In every iteration, the output is thelocation of an agent with an index higher or equal to k from TMON. After the n th iteration we reach a profile c n that isidentical to a , up to permutation of agents. Thus by anonymity f ( a ) = f ( c n ) .On the other hand, f ( c n ) ≥ ( c n ) k = a k > a j = f ( a ) , i.e. a contradiction. The proof is similar for the case j > k . Theorem 6.3.
An onto mechanism f is anonymous, shift-invariant and SP if and only if it is the k th order statistic forsome k ∈ N . Proof.
The first direction follows from Lemmas 6.1 and 6.2.We prove the second direction. Clearly, a k th order statistic mechanism is anonymous and shift-invariant. The only wayfor an agent i to change the chosen location is by reporting a location a i > a k if i < k or a location a i < a k if i > k .In both cases, the distance of the agent from the facility will increase. In this research, we provide a complete characterization of onto and strategyproof facility location mechanisms ondiscrete trees—under quadratic preferences. Interestingly, while a characterization for continuous trees exists due to [4],these are not easily compared, as the latter uses a collection of median-like rules rather than axiomatic properties. Thekey property that allows comparison of continuous and discrete mechanisms is trajectory containment (TC): whilethis property characterizes exactly the strategyproof onto mechanisms on continuous trees, it needs to be relaxed in aparticular way to apply for discrete trees.A different characterization for discrete trees and general single-peaked preferences [5] uses a third type of propertiesthat map agents to leafs of the tree. Thus a better understanding of how properties from the different models map ontoone another is important.One possible direction for further research is a characterization of strategyproof mechanisms on discrete weightedgraphs. Contrary to continuous graphs, there is a finite set of possible locations for each agent and outcome. Unlike thediscrete case, the distances between such two neighboring locations vary along the tree.Additionally, characterizations of SP mechanisms can promote the study of optimal SP approximation mechanisms, forexample for minimizing the Egalitarian cost.
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