Manipulation-resistant facility location mechanisms for ZV-line graphs
aa r X i v : . [ c s . G T ] D ec Manipulation-resistant facility location mechanisms for
Z V -line graphs
Ilan Nehama ∗ Taiki Todo Makoto YokooDecember 11, 2018
Abstract
In many real-life scenarios, a group of agents needs to agree on a common action, e.g., on thelocation for a public facility, while there is some consistency between their preferences, e.g., allpreferences are derived from a common metric space. The facility location problem models suchscenarios and it is a well-studied problem in social choice. We study mechanisms for facility locationon unweighted undirected graphs, which are resistant to manipulations ( strategy-proof , abstention-proof , and false-name-proof ) by both individuals and coalitions and are efficient ( Pareto optimal ).We define a family of graphs, ZV -line graphs , and show a general facility location mechanism forthese graphs which satisfies all these desired properties. Moreover, we show that this mechanismcan be computed in polynomial time, the mechanism is anonymous, and it can equivalently bedefined as the first Pareto optimal location according to some predefined order.Our main result, the ZV -line graphs family and the mechanism we present for it, unifies the fewcurrent works in the literature of false-name-proof facility location on discrete graphs, including allthe preliminary (unpublished) works we are aware of. Finally, we discuss some generalizations andlimitations of our result for problems of facility location on other structures. ∗ Corresponding authorWe would like to thank Kentaro Yahiro and Nathana¨el Barrot for their comments which helped us to improve thepresentation of this work. This work was partially supported by JSPS KAKENHI Grant Numbers JP17H00761 andJP17H04695, JST Strategic International Collaborative Research Program, SICORP, and Israel Science Foundation Grant1626/18. ontents Introduction
Reaching an agreement could be hard. The seminal works of Gibbard [10] and Satterthwaite [22] showthat one cannot devise a general procedure for aggregating the preferences of strategic agents to a singleoutcome, besides trivial procedures that a-priori ignore all agents except one (that is, the outcome isbased on the preference of a predefined agent) or a-priori rule out all outcomes except two (that is,regardless of the agents’ preferences, the outcome is one of two predefined outcomes). The problemis that agents might act strategically aiming to get an outcome which they prefer, so there might bescenarios in which for any profile of actions (a possible agreement) at least one of the agents will preferchanging his action. Note that while we refer to a procedure and later to a mechanism , this impossibilityis not technical but conceptual. We identify a procedure with the conceptual mapping induced by theprocedure from the opinions of the agents to an agreement, while the procedure itself could be complexand abstract, e.g., to have several rounds or include a deliberation process between the agents (cheap-talk). For simplicity of terms, we refer to the direct mechanism which implements this mapping. Thatis, we think of an exogenous entity, the designer , who receives as input the opinions of the agents andreturns as output the aggregated decision. But in many natural scenarios, it is exogenously given that the preferences satisfy some additionalrationality property, i.e., the mechanism should not be defined for any profile of preferences, givingrise to mechanisms that are not prone to the above drawbacks. Two prominent examples are
VCGmechanisms and generalized-median mechanisms . VCG mechanisms [26, 4, 21, 11] are the mechanismswhich are resistant to manipulations like the ones described above for scenarios in which the agents’preferences are quasi-linear with respect to money [15, Def. 3.b.7], and monetary transfers are allowed. The second example,
Generalized-median mechanisms , do not include monetary transfers and havemore of an ordinal flavor. Generalized-median mechanisms [16] are the mechanisms which are resistantto manipulations like above when it is known that the preferences are single-peaked w.r.t. the realline [3]. That is, the outcomes are locations on the real line, each agent has a unique optimal location, ℓ ⋆ , and her preference over the locations to the right of ℓ ⋆ is derived by the proximity to ℓ ⋆ , and similarlyfor the locations to the left of ℓ ⋆ . For example, in the Euclidean single-peaked case, the preferences forall agents are minimizing the distance to their respective optimal locations. The facility location problem
A natural generalization of the second scenario is the facility location problem. In this problem, we aregiven a metric space over the outcomes (that is, a distance function between outcomes) and it is assumedthat the preference of each of the agents is defined by the distance to her optimal outcome: An agentwith an optimal outcome ℓ ⋆ prefers outcome a over outcome b if and only if a is closer to ℓ ⋆ than b . Forease of presentation, throughout this paper we assume that there are finitely many agents and finitely For the properties we study in this work, this assumption does not hurt the generality, as according to the revelationprinciple [18], any general procedure is equivalent (w.r.t. the properties we study) to such a direct mechanism. the facility .In this work, we seek mechanisms which satisfy the following desired properties:
Anonymity:
The mechanism should not a-priori ignore agents and moreover it should treat themequally in the following strong sense. The mechanism should be a function of the agents’ votes (whichwe also refer to as ballots ) but not their identities. Formally, the outcome of the mechanism shouldbe invariant to voters exchanging votes, i.e., to a permutation of the ballots. In practice, most votingsystems satisfy this property by first accumulating the different (physical) ballots, thus losing the voters’identities, and next applying the mechanism on the identity-less ballots.
Onto: The mechanism should not a-priori rule-out a location, and each location should be an outcomeof some profile. Formally, the mapping to a facility location should be an onto function. Moreover, themechanism should respect the preferences of the agents and aim to optimize the aggregated welfare ofthe agents.
Pareto optimality:
The mechanism should not return a location ℓ if there exists another location ℓ ′ s.t. switching from ℓ to ℓ ′ will benefit one of the agents (move the facility closer to her) while nothurting any of the other agents. In particular, if there exists a unique location which is unanimouslymost-preferred by all agents, then it must be the outcome. Note that any reasonable (monotone) notionof aggregated welfare optimization entails Pareto optimality. (Note that it is unreasonable to requirethat all locations are treated equally due the inherent asymmetry induced by the graph.) Strategy-proofness:
An agent should not be able to change the outcome to a location she strictlyprefers by reporting a location different than her true location. In the social choice literature [1], this property is referred to as
Citizen sovereignty or Non-Imposition . bstention-proofness: An agent should not be able to change the outcome to a location she strictlyprefers by not casting a ballot.
False-name-proofness:
An agent should not be able to change the outcome to a location she strictlyprefers by casting more than one ballot.False-name-manipulations received less attention in the classic social choice literature, since in mostvoting scenarios there exists a central authority that can enforce a ‘one person, one vote’ principle(but cannot enforce participation or sincere voting). In contrast, many of the voting and aggregationscenarios nowadays are run in a distributed manner on some network and include virtual identities oravatars, which can be easily generated, so a manipulation of an agent pretending to represent manyvoters is eminent.
Resistance to group manipulations:
We also consider generalizations of the above three propertiesdealing with manipulations of coalitions of agents. We define the preference of a coalition as theunanimous preference of its members, that is, a coalition C weakly prefers an outcome a over anoutcome b if all the members of C weakly prefer a over b , and require that a coalition should not beable to change the outcome to a location it strictly prefers by its members casting insincere ballots,abstaining, or casting more than one ballot. We note that for onto mechanisms this property entailsPareto optimality. Nevertheless, we prefer to think of Pareto optimality apart from this property dueto the different motivations. Our contribution
Besides the work of Todo et al. [24], who characterized the false-name-proof mechanisms for facilitylocation on the continuous line and on continuous trees, we are not aware of other works dealing withcharacterizing false-name-proof mechanisms on a graph. Moreover, as far as we know, a false-name-proofmechanism is known to the community only for very few simple graphs, and the current knowledge isstill highly preliminary. (When starting to work on this problem, we initially devised mechanisms forfew of the examples we describe below - cycles, cliques, and the 2 × n grid. We are not aware of anyother previously-known positive results besides these graphs or small perturbations of them.)In this paper we present a family of unweighted undirected graphs, which we name ZV -line graphs,and show a general mechanism for facility location over these graphs which satisfies the desired properties.To the best of our knowledge, this is the first work to show a general false-name-proof mechanism fora general family of graphs. Our mechanism for the ZV -line graphs family unifies the few mechanisms In the voting literature (e.g., [5, 17, 9]) this property is also referred to as voluntary participation and the no-showparadox . This property is also equivalent to individual-rationality which takes a different point of view of mechanismdesign. Hence, C strictly prefers a over b if ( i ) all the members of C weakly prefer a over b ( C weakly prefers a over b ), and ( ii ) at least one member of C strictly prefers a over b ( C does not weakly prefer b over a ). ZV -line graph there are two types of locations Z and V (and we refer tothem as Z -vertices and V -vertices, respectively), and the facility is ‘commonly’ (except if all agentsunanimously agree differently) located on a Z -vertex. For instance, the Z -vertices could representcommercial locations for locating a public mall, or a set of status-quo outcomes.For example, consider the following family of graphs (which is a sub-family of ZV -line graphs andcaptures the gist of our mechanism). Let G = hV , E i be a bipartite unweighted undirected graph withvertex set V and edge set E . That is, there exists a partition of the vertices V = V ˙ ∪ Z s.t. there areno edges between V -vertices and no edges between Z -vertices. In addition, we require that ( a ) thereexists a predefined order over the Z -vertices, which we refer to as left-to-right order, and that ( b ) anyof the V -vertices is connected to an interval (according to the order) of Z -vertices. Similarly to thesingle-peaked consistency case [3], one can think of this constraint as a homogeneity constraint over theagents’ preferences. Our mechanism for such graphs: ◮ The mechanism returns the leftmost Pareto optimal Z -vertex, if one exists. ◮ If no location in Z is Pareto optimal, then necessarily all agents voted for the same location, andthe mechanism returns this location.For example, bi-cliques (full bipartite graphs) can be represented as a ZV -line graph in which each V -vertex is connected to all the Z -vertices as follows (and we use below (cid:13) for Z -vertices and (cid:7) for V -vertices): (cid:7) ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬ (cid:7) ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ (cid:7) ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ (cid:7) ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ (cid:7) ❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❝❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Our mechanism for this case: ◮ If all agents voted unanimously for the same location, the mechanism returns this location. ◮ If all agents voted for V -vertices, the mechanism returns the leftmost Z -vertex. ◮ Otherwise, the mechanism returns the leftmost Z -vertex that was voted for.Notice that in this case the order over the Z -vertices is arbitrary (as well as the choice of one of thesides to be the Z -vertices) in the sense that it is not derived from the graph but a parameter of themechanism. For instance, the order might represent the social norm of the society.A second example is the discrete line graph, which can be represented as a ZV -line graph in whichevery two consecutive Z -vertices are connected by a unique V -vertex, (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ ··· (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) ··· (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ (cid:7) ④④④ . Inparticular, we show strategy-proof, false-name-proof, Pareto optimal mechanisms which are far from generalized-median mechanisms (for instance, in the common case the output of the mechanism belongsto a subset consisting of only half of the locations), in contrary to the characterization of these mecha- That is, there exists no other location ℓ in the graph s.t. switching the outcome to ℓ benefits one of the agents whilenot hurting any of the other agents. ZV -line graph representation of) the discrete linegraph are (cid:7) ⑦⑦⑦ ❅❅❅ (cid:7) ⑦⑦⑦ ❅❅❅ (cid:7) ⑦⑦⑦ ❅❅❅ (cid:13) (cid:13) (cid:13) (cid:13) (cid:7) ❅❅❅ ⑦⑦⑦ (cid:7) ❅❅❅ ⑦⑦⑦ (cid:7) ❅❅❅ ⑦⑦⑦ , in which every two consecutive Z -vertices are connected by two V -vertices,and the 2 × n grid (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)(cid:7) (cid:7) (cid:7) (cid:7) (cid:7) which can be represented as a ZV -line graph in which every threeconsecutive Z -vertices are connected by a unique V -vertex, i.e., (cid:7) sssssss ❑❑❑❑❑❑❑ (cid:7) sssssss ❑❑❑❑❑❑❑ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:7) sssssss (cid:7) ❑❑❑❑❑❑❑ sssssss (cid:7) ❑❑❑❑❑❑❑ .A common property to all the above examples is their regularity: All the V -vertices have the samedegree and similarly all the Z -vertices have the same degree. An example we encountered of a non-regular graph for which a mechanism exists is (cid:7) (cid:7)(cid:7) (cid:7) (cid:7)(cid:7) (cid:7) (cid:7) , which can be represented as a non-regular ZV -linegraph as (cid:7) ⑦⑦⑦✐✐✐✐✐✐✐✐✐✐✐✐ ❅❅❅ ❯❯❯❯❯❯❯❯❯❯❯❯ (cid:13) (cid:13) (cid:13) (cid:13) (cid:7) ❅❅❅ ⑦⑦⑦ (cid:7) ❅❅❅ ⑦⑦⑦ (cid:7) ❅❅❅ ⑦⑦⑦ .In the definition of the ZV -line graphs family we extend the above family (and extend the mechanismaccordingly) in two different ways: allowing edges between the Z -vertices (under a similar intervalconstraint), and replacing vertices by a tree, a clique, or any other ZV -line graph. For example, (cid:7) ▲▲▲▲▲▲ ✿✿✿✿✿✿✿✿rrrrrr☎☎☎☎☎☎☎☎ (cid:7) ▲▲▲▲▲▲ ✿✿✿✿✿✿✿✿ ✶✶✶✶✶✶✶✶✶✶✶✶✶ (cid:7) rrrrrr☎☎☎☎☎☎☎☎✌✌✌✌✌✌✌✌✌✌✌✌✌ (cid:7) ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ (cid:7) ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤qqqqqqqqqqqqqqqq③③③③③③③③③③③③③③③③③③ (cid:7) ❯❯❯❯❯❯❯❯❯❯❯❯❯ ✿✿✿✿✿✿✿✿ (cid:7) ✐✐✐✐✐✐✐✐✐✐✐✐✐☎☎☎☎☎☎☎☎ (cid:7) ▲▲▲▲▲▲ ✿✿✿✿✿✿✿✿ (cid:7) rrrrrr☎☎☎☎☎☎☎☎ (cid:7) ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ (cid:7) ✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤qqqqqqqqqqqqqqqq (cid:7) ▲▲▲▲▲▲ (cid:7) rrrrrr (cid:7) ▲▲▲▲▲▲ (cid:7) rrrrrr (cid:7) ✺✺✺✺✺✺✺✺✺✺✺ ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ (cid:7) ✠✠✠✠✠✠✠✠✠✠✠❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (cid:7) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ PPPPPPPPPPPPPP (cid:7) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ PPPPPPPPPPPPPP (cid:7) ♦♦♦♦♦♦♦♦♦♦♦♦♦ PPPPPPPPPPPPPP (cid:7) (cid:7) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ PPPPPPPPPPPPPP (cid:7) ❈❈❈❈❈❈❈ (cid:7) ④④④④④④④ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:7) PPPPPPPPPPPPPP ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:7)
PPPPPPPPPPPPPP ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:7) ❖❖❖❖❖❖❖❖❖❖❖❖❖ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:7) ❈❈❈❈❈❈❈ (cid:7)
PPPPPPPPPPPPPP ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ (cid:7) (cid:7) rrrrrr (cid:7) (cid:7) ▲▲▲▲▲▲ (cid:7) ✠✠✠✠✠✠✠✠✠✠✠ (cid:7) qqqqqqq (cid:7) (cid:7) ▲▲▲▲▲▲ .In particular, the ZV -line graphs family includes all trees, cliques, block graphs [12], cycles of sizeup to 4 (note that there is no manipulation-resistant Pareto optimal anonymous mechanism for cyclesof size larger than 5), and all graphs for which (as far as we found) a false-name-proof mechanism isknown to the community. Related work
Problems of facility location on discrete graphs were also studied by Dokow et al. [6], who characterizedthe strategy-proof mechanisms for the discrete line and discrete cycle. Other variants of the facility loca-tion problem were also considered in the literature. For instance, Schummer and Vohra [23] consideredthe case of continuous graphs, Lu et al. [14, 13] studied variants in which several facilities need to belocated and scenarios in which an agent is located on several locations, and Feldman et al. [8] studiedthe impact of constraining the input language of the agents.False-name-proofness was first introduced by Yokoo et al. [27, (based on a series of previous con-7erence papers)] in the framework of combinatorial auctions. In this work, the authors showed thatthe VCG mechanism does not satisfy false-name-proofness in the general case, and they proposed aproperty of the preferences under which this mechanism becomes false-name-proof. A similar conceptwas also studied in the framework of peer-to-peer systems by Douceur [7] under the name sybil attacks.Later, Conitzer and Sobel [5] analyzed false-name-proof mechanisms in voting scenarios, Todo et al. [25]characterized other false-name-proof mechanisms for combinatorial auctions, and Todo et al. [24] char-acterized the false-name-proof mechanisms for facility location on the continuous line and on continuoustrees. In a recent work, Ono et al. [19] showed, in the framework of facility location on the discrete line,a relation between false-name-proofness and the property of population monotonicity .The characterization of manipulation-resistant mechanisms for facility location is highly relatedto problems in
Approximate mechanism design without money [20]. In these problems, agents arecharacterized using cardinal utilities and the designer seeks to find an outcome maximizing a desiredtarget function (e.g., sum of utilities, product of utilities, or minimal utility). These works bound thetrade-of between the target function and manipulation-resistance, that is, they bound the loss to thetarget function due to manipulation-resistance constraints. Similar bounds were derived for false-name-proof facility location mechanisms on the continuous line and tree by Todo et al. [24], strategy-prooffacility location on the continuous cycle by Alon et al. [2], and for strategy-proof facility location on thediscrete line and cycle by Dokow et al. [6].
Approximate mechanism design
In this work we do not analyze the approximation implications of the characterization and in particularwe do not assume a specific cardinal representation of the agents’ preferences. Yet, we claim that formost natural representations and target functions the approximation ratio is expected to be bad. Forexample, recall the above bi-clique example. In this mechanism, the facility might be located on an‘extremely’ left Z -vertex. Moreover, the facility might be very far from the vast majority of the agents,resulting in a very bad approximation ratio for most reasonable target functions. This phenomenon isnot specific for the bi-clique graphs. For most ZV -line graphs, (due to the false-name-proof requirement)the mechanism might be located on a location extremely far from almost all agents, resulting in a verybad approximation ratio (roughly, the number of agents times the girth of the graph) for most reasonabletarget functions. 8 Model
Consider a graph G = hV , E i with a set of vertices V and a set of, neither weighted nor directed, edges E ⊆ (cid:0) V (cid:1) , and we refer to the vertices v ∈ V also as locations and use the two terms interchangeably.The distance between two vertices v, u ∈ V , notated d ( v, u ), is the length of the shortest path connecting v and u , and the distance between a vertex v ∈ V and set of vertices S ⊆ V , d ( v, S ), is defined as theminimal distance between v and a vertex in S . We define B ( v, d ), the ball of radius d > v ∈ V , to be the set of vertices of distance at most d from v , B ( v, d ) = { u ∈ V | d ( v, u ) d } .We say that two vertices are neighbors if there is an edge connecting them and notate by N ( v ) the setof neighbors of a vertex v .An instance of the facility location problem over G is comprised of n agents who are located onvertices of V ; Formally, we represent it by a location profile x ∈ V n where x i is the location of Agent i .Given an instance x , we would like to locate a facility on a vertex of the graph while taking into accountthe preferences of the agents over the locations. In this work, we assume the preference of an agentis defined by her distance to the facility: An agent located on x ∈ V strictly prefers the facility beinglocated on v ∈ V over it being located on u ∈ V iff d ( x, v ) < d ( x, u ).A general facility location mechanism (or shortly a mechanism ) defines for any profile of agents’locations a location for the facility. We require the mechanism to assign a location for the facility for anyprofile and any number of agents. Hence, we represent the mechanism by a function F : S t > V t → V .We also think on F as a voting procedure: Each agent votes (and we also refer to his vote as a ballot )for a location, and based on the ballots F returns a location for the facility. We say that a mechanism is anonymous if the outcome F ( x ) does not depend on the identities of the agents, i.e., it can be definedas a function of the ballot tally, the number of votes for each of locations. Manipulation-resistance
A strategic agent might act untruthfully if she thinks it might cause the mechanism to return a loca-tion she prefers (that is, a location closer to her). In this work we consider the following manipula-tions:
Misreport : An agent might report to the mechanism a location different from her real location;
False-name-report : An agent might pretend to be several agents and submit several (not necessarilyidentical) ballots;
Abstention : An agent might choose not to participate in the mechanism at all. Amechanism in which no agent benefits from these manipulations, regardless to the ballots of the otheragents, is said to be strategy-proof , false-name-proof , and abstention-proof , respectively. Wealso consider a generalization of these manipulations to manipulations of a coalition, and say a mech-anism is group-manipulation-resistant (shortly manipulation-resistant) if no coalition can changethe outcome, by misreporting, false-name-reporting, or abstaining, to a different location which theyunanimously agree is no worse than the original outcome (i.e., if they vote sincerely) and at least one For simplicity, we assume the graph is connected.
9f the coalition’s members strictly prefers the new location.
Definition 1 (Group-manipulation-resistant) . A mechanism F is group-manipulation-resistant if thereexists no coalition of agents C ⊆ { , . . . , n } , a vector of locations x ∈ V n , and a set of ballots A ∈ S t > V t s.t. ( i ) all the members of C weakly prefer F ( A , x − C ), that is, the outcome when the agents outsideof C do not change their vote and the agents of C replace their ballots by A , over F ( x ) and ( ii ) atleast one of C ’s members strictly prefers F ( A , x − C ) over F ( x ).We note that for C = { i } being a singleton, this general manipulation coincides with misreport for | A | = 1, with false-name-report for | A | >
1, and with abstention for A = ∅ . The revelation principle
One could consider more general mechanisms in which the agents vote using more abstract ballots,and define similar manipulation-resistance terms for the general framework. Applying a simple directrevelation principle [18] shows that any such general manipulation-resistant mechanism is equivalentto a manipulation-resistant mechanism in our framework: The two mechanisms implement the samemapping of the agents private preferences to a location for the facility, and since the above propertiesare defined for the mapping they are invariant to this transformation. That is, given some generalmechanism M that maps abstract actions to a location for the facility and a behavior protocol D thatmaps types of the agents (i.e., locations) to actions of M , if D satisfies the generalized desiderata, thenthe direct mechanism M ◦ D satisfies our desiderata (w.r.t. truth-telling). Efficiency
So far, we defined the desired manipulation-resistance properties for a mechanism. On the other hand,we would also like the mechanism to respect the preferences of the agents. We would like to avoid ascenario in which, after the mechanism has been used, the agents can agree that a different location ispreferable. Given a location profile x ∈ V n , the set of Pareto optimal locations,
P O ( x ), is the set ofall locations which the agents cannot agree to rule out. Formally, given two locations v, u ∈ V , we saythat u Pareto dominates v (w.r.t. a location profile x ) if ( i ) all agents weakly prefer u over v and ( ii ) at least one agent strictly prefers u over v . We say that v is Pareto optimal ( v ∈ P O ( x )) if it is notPareto dominated by any other location. We say a mechanism is Pareto optimal if for any reportprofile x (and assuming truthful reporting) F ( x ) ∈ P O ( x ). In particular, Pareto optimality entails unanimity , if all the agents unanimously vote for the same location then the mechanism outputs thislocation, and citizen sovereignty , the mechanism is onto and does not a-priori rule out any location. For simplicity of notations, we give the formal definition for anonymous mechanisms. Main Result
In this work, we define a family of graphs, ZV -line graphs , and present a general mechanism for thisfamily. This family is defined by introducing a simple combinatorial structure - partition to two types ofvertices and connectivity constraint. One could think of the partition as representing a social agreementaccording to which the mechanism is defined, e.g., a subset of status-quo locations or an a-priori priorityhierarchy over the locations. The connectivity constraint (as the graph in general) represents thehomogeneity over the agents’ preferences which allows us to show a manipulation-resistant mechanism. Definition 2 ( ZV -ordered partition) . Given an unweighted undirected connected graph G = ( V , E )and a sequence of non-empty sets of vertices Z, V , . . . , V k ⊆ V , we say that the sequence Z, V , . . . , V k ( k >
0) is a ZV -ordered partition if the following holds.1. The sets V i are disjoint, V i ∩ V j = ∅ for i = j .2. The sequence is a cover of V , Z ∪ V ∪ · · · ∪ V k = V , and no sub-sequence of it is a cover of V .3. For i = 1 , . . . , k , there is a unique vertex in V i which is closest to Z . We refer to it as the root of V i and denote it by R ( V i ), R ( V i ) = argmin v ∈ V i d ( v, Z ) .4. All paths between vertices of V i and vertices outside of V i pass through the root R ( V i ) andthrough Z .5. Last, Z is equipped with an order (that is, an injective mapping from Z to ℜ ). For simplicityof description, we refer to this order as an order from left to right. We call a subset A of Z an interval if A is the preimage of an interval in ℜ , , i.e., if it is a sequence of vertices according tothe order.We use the notions V i -subgraphs, V -vertices, and Z -vertices for the respective sets of vertices. Notethat we do not require the sets of the Z -vertices and the V -vertices to be disjoint. For instance, inthe last example of the introduction the 9-clique includes the rightmost Z -vertex. Notice that from thethird condition it is clear that for all i the intersection V i ∩ Z is of size at most one.Given a graph G = ( V , E ) with a ZV -ordered partition, Z, V , . . . , V k ⊆ V , and a sequence ofmechanisms F i : S t > ( V i ) t → V i for i = 1 , . . . , k , we define the following mechanism F ⋆ : S t > V t → V : Definition 3 ( F ⋆ ) . Given a vector of reports x ∈ S t > V t ◮ If all the ballots belong to the same V i -subgraph, return F i ( x ). ◮ Otherwise, return the leftmost Pareto optimal location in Z .It is not hard to see the following: • F ⋆ is well defined (If it does not hold that all ballots belong to thesame V i -subgraph, then necessarily P O ( x ) ∩ Z = ∅ ), • F ⋆ runs in polynomial time, and • If F , . . . , F k can be defined as the first Pareto optimal location according to some order, then an equivalent way to11efine F ⋆ is as the first Pareto optimal location in the following order: First, go over the vertices of Z from left to right, and then on the vertices of the V i -subgraphs in some order s.t. for each subgraph theorder over its vertices matches the order of F i .Next, we define ZV -line graphs by introducing a connectivity constraint. Definition 4 ( ZV -line graph) . An unweighted undirected connected graph G = ( V , E ) is a ZV -linegraph w.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ), if ( a ) h Z, V , . . . , V k i is a ZV -ordered partition of G , ( b ) for anyvertex z ∈ Z , B ( z, ∩ Z is an interval in Z , and if k > i = 1 , . . . , k ( c ) the induced graph G i = h V i , E ∩ ( V i × V i ) i is a ZV -line graph, ( d ) R ( V i ) is a Z -vertex of G i (that is a Z -vertex in therepresentation of G i ), and it is the leftmost Z -vertex of f G i , and last ( e ) B ( R ( V i ) , ∩ Z is an intervalin Z .For instance, for any ℓ > ℓ vertices, K ℓ , is a ZV -line graph w.r.t. Z = V and anyorder over the vertices.Given a ZV -line graph G = hV , E i , applying Def. 3 recursively on G and its V i -subgraphs gives usa mechanism F ⋆ : S t > V t → V . Note that F ⋆ depends on the representation of G w.r.t. a specific ZV -ordered partition, and in case that G can be represented as a ZV -line graph w.r.t. several ZV -orderedpartitions, they might result in different mechanisms. Our main result shows that this mechanismsatisfies the desired properties. Theorem 5 (Main result) . Let G = ( V , E ) be a ZV -line graph w.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ) and let F ⋆ : S t > V t → V be the result of applying Definition 3. recursively on G . Then F ⋆ is an anonymousPareto optimal mechanism and F ⋆ satisfies:For any vector of locations x ∈ ( V i ) n , a coalition of agents C , and a set of ballots A ∈ S t > ( V i ) t , A is not a beneficial deviation for C (That is, C does not strictly prefer F ⋆ ( A , x − C ) over F ⋆ ( x ) . Note that the theorem does not hold for weighted graphs (that is, when edgeshave non-uniform length). Consider the following weighted graph and a profilein which Alice is located on z r and Bob on v . Then, the outcome is z r , but Bobcan move the facility to a preferred location z ℓ both ( i ) by misreporting z ℓ , hence F ⋆ is not strategy-proof, and ( ii ) by false-name-reporting z ℓ in addition to hissincere report, hence F ⋆ is not false-name-proof. ?>=<89:; v ✟✟✟✟ GFED@ABC z ℓ GFED@ABC z r ✻✻✻✻ V = { v } Z = { z ℓ , z r } We note that there are trivial mechanisms which satisfy subsets of these properties: • The fixedmechanism , which locates the facility on a pre-defined location ignoring the votes of the agents, istrivially manipulation-resistant and anonymous, but it is not onto and hence not Pareto optimal. • A dictatorship , e.g., the mechanism that always locates the facility on the location reported by the firstagent, is not anonymous but clearly it is manipulation-resistant. • The median mechanism , which Since F ⋆ is an anonymous mechanisms, we define A as a set of ballots ignoring identities. Since F ⋆ is onto, this property entails Pareto optimality. Yet, we prefer to state explicitly Pareto optimality as adesired efficiency property. While we did not formally define false-name-proofness for non-anonymous mechanisms, assuming a false-name votecannot be counted as the vote of the first agent, no agent can benefit from casting additional ballots. • The mean mechanism , which minimizes the sum of squares of the distances betweenthe facility and the ballots, is anonymous and Pareto optimal but might not be strategy-proof or false-name-proof even against one agent, e.g., for the discrete line graph (it is abstention-proof, though).
By applying the main result to a recursive graph family, we can generate a recursive (and hence com-monly simple) mechanism which satisfies our desiderata. For instance, a corollary of our result isa manipulation-resistant mechanism for the following family of rooted graphs (that is, hV , E, r i s.t. E ⊆ (cid:0) V (cid:1) and r ∈ V ). Definition 6 ( F ) . • h{ v } , ∅ , v i ∈ F . • For any k, ℓ >
1: If {hV i , E i , r i i} ki =1 are in F (and the V i are disjoint), then also the followinggraph is in F . { b r j } ℓj =1 ˙ ∪ k ˙ [ i =1 V i , {h b r j , r i i} i =1 ...k,j =1 ...ℓ ˙ ∪ k ˙ [ i =1 E i , b r I.e., adding a new layer of pre-roots, a bi-clique between them and the roots of the graphs of theprevious stage, and defining one of the pre-roots to be the new root.
Claim . The anonymous Pareto optimal mechanism F ( x ) = argmin v ∈ P O ( x ) d ( v, r ), which returnsthe Pareto optimal location closest to the root and breaks ties according to a predefined order, ismanipulation-resistant.Note that by setting ℓ = 1 in the second step of the definition we get a recursive definition of rootedtrees . Hence, we get that for any tree G the mechanism that returns the lowest common ancestor of theballots (with regard to some root) is a manipulation-resistant mechanism (These are also the mechanismswhich Todo et al. [24] characterized as the false-name-proof, anonymous, and Pareto optimal mechanismsfor the continuous tree.). Proof.
We prove the claim by induction over, h ( G ), the number of steps needed to generate G .If h ( G ) = 0, i.e., G = h{ v } , ∅ , v i consists of a single vertex and the trivial mechanism satisfies allthe desired properties.If h ( G ) >
1, then G is a ZV -line graph w.r.t. Z = { b r j } ℓj =1 and V i = V i . Note that for all V i -subgraphs h ( hV i , E i , r i i ) h ( G ) −
1. Hence, our recursive mechanism returns one of the pre-roots ofthe ‘lowest’ subgraph which includes x when ties are broken according to the (arbitrary) order over thepre-roots. 13 second example is connected block graphs [12] . A connected graph G = hV , E i is a block graph ifthe following equivalent conditions hold: • Every biconnected component of G is a clique. (Since for any graph the structure of its biconnectedcomponents is described by a block-cut tree, connected block graphs are also called clique trees .) • The intersection of any two connected subgraphs of G is either empty or connected. • For every four vertices u, v, w, x ∈ V , the larger two of the distance sums d ( u, v ) + d ( w, x ), d ( u, w ) + d ( v, x ), and d ( u, x ) + d ( v, w ) are equal.Our mechanism for a connected block graph G returns the closest Pareto optimal location to an arbi-trarily predefined location, breaking ties according to an arbitrarily predefined order over the locations. Proof sketch . G is connected block graph and hence all bi-connected components of G are cliques.The block-cut tree of G is a tree T ( G ) which is defined in the following way. In T ( G ) there is a vertex( component-vertex ) for each maximal biconnected component of G and a vertex ( intersection-vertex )for each vertex in G which belongs to more than one maximal biconnected component. There is an edgein T ( G ) between each component-vertex and the intersection-vertices belonging to this component.Following the inductive structure of T ( G ), and recalling that a clique is a ZV -line graph w.r.t. allvertices of the clique being Z -vertices and any order over them, we get that our mechanism is definedby an arbitrary predefined component-vertex of T ( G ), R , and a series of arbitrary predefined ordersover the locations of each of the components. The mechanism is: ◮ If all ballots belong to the same component, return the first location (according to the order) thatwas voted for. ◮ Otherwise, choose the component closest to R s.t. one of the locations of the component is Paretooptimal, and return the first location (according to the order) in this component.Last, we note that an equivalent definition of this mechanism is returning the closest Pareto optimallocation to some location v ∈ R , breaking ties according to a concatenation of the orders over thecomponents. 14 Proof of Main Result (Thm. 5)
We prove a stronger result which shows a general method for generating a mechanism F ⋆ (satisfyingthe desired properties) for a given graph from mechanisms for its subgraphs, F i . Theorem 5 is animmediate special case of this lemma. The same proof shows that also for weaker manipulation-resistanceproperties, e.g., against individual agents, against misreporting, or against abstentions, manipulation-resistance of the mechanisms for the subgraphs F i , result in the same manipulation-resistance notionfor the mechanism of the graph F ⋆ . Lemma 8.
Let G = ( V , E ) be a graph with a ZV -ordered partition V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ) and let F i : S t > ( V i ) t → V i be a sequence of mechanisms s.t. for i = 1 , . . . , k • F i is anonymous and Pareto optimal; • For an infinite number of τ ∈ N , there exists a profile x ∈ S t > ( V i ) t in which all locations in V i were voted for at least τ times and F i ( x ) = R ( V i ) ; and • For any vector of locations x ∈ ( V i ) n , a coalition of agents C , and a set of ballots A ∈ S t > ( V i ) t , A is not a beneficial deviation for C (That is, C does not strictly prefer F i ( A , x − C ) over F i ( x ) ). ( ⋆ ) Then, for F ⋆ : S t > V t → V as defined in Definition 3, F ⋆ is an anonymous and Pareto optimalmechanism and ( I ) If G is a ZV -line graph w.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ) , then F ⋆ satisfies ( ⋆ ) . ( II ) If R ( V i ) ∈ Z for all i = 1 , . . . , k , and the mechanism F Z : S t > Z t → Z which returns the leftmostPareto optimal location satisfies ( ⋆ ) , then also F ⋆ satisfies ( ⋆ ) .Proof. The anonymity of F ⋆ is an immediate corollary of the mechanisms F i and F Z being anonymousmechanisms.Notice that if all agents are in the same V i -subgraph, then all of them strictly prefer R ( V i ) over anylocation outside of V i , so P O ( x ) ⊆ V i . Moreover, any location v ∈ V i \ P O ( x ) is Pareto dominated bya location y ∈ P O ( x ) ⊆ V i . Hence, the Pareto optimal set when considering only the locations in V i equals to the Pareto optimal set when considering all locations. Since, the mechanisms F i are Paretooptimal mechanisms we get that also F ⋆ is Pareto optimal.In order to prove the main part of the theorem, we assume towards a contradiction that there existsa vector of locations x ∈ V n , a coalition of agents C , and a set of ballots A ∈ S t > V t , s.t. C can, byvoting A , get an outcome F ⋆ ( A , x − C ) which it strictly prefers, that is, all of its members weakly prefer F ⋆ ( A , x − C ) over F ⋆ ( x ) = F ⋆ ( x C , x − C ), and at least one of C ’s members, Agent i for i ∈ C , strictlyprefers F ⋆ ( A , x − C ) over F ⋆ ( x ). F ⋆ ( x ) ∈ P O ( x ) and in particular the coalition of all agents does notstrictly prefer F ⋆ ( A , x − C ) over F ⋆ ( x ). Hence, there exists an Agent j , for j / ∈ C , who strictly prefers F ⋆ ( x ) over F ⋆ ( A , x − C ). Since F i (and later F ⋆ ) are anonymous mechanisms, we define A as a set of ballots ignoring identities. F ⋆ ( x ) is not in Z : Then necessarily, all the locations in x and F ⋆ ( x ) belong to the same V i -subgraph, w.l.o.g. V , so F ⋆ ( x ) = F ( x ). Since F is resistant to false-name manipulations of Agent i and since Agent i can achieve R ( V ) by casting enough false ballots, we get that Agent i weakly prefers F ⋆ ( x ) over R ( V ) and hence Agent i strictly prefers F ⋆ ( A , x − C ) over R ( V ). Since for any u outsideof V it holds that d ( x i , R ( V )) < d ( x i , u ), we get that F ⋆ ( A , x − C ) ∈ V \ R ( V ) ⊆ V \ Z . Hence, A ⊆ V and F ⋆ ( A , x − C ) = F ( A , x − C ), and we get a contradiction to the false-name-proofness of F .Similarly, if F ⋆ ( A , x − C ) is not in Z : Then necessarily, F ⋆ ( A , x − C ) and all the locations in A and x − C belong to the same V i -subgraph, w.l.o.g. V , so F ⋆ ( A , x − C ) = F ( A , x − C ). Since F is resistant tofalse-name manipulations of Agent j and since Agent j can achieve R ( V ) by casting enough false ballots,we get that Agent j weakly prefers F ⋆ ( A , x − C ) over R ( V ) and strictly prefers F ⋆ ( x ) over R ( V ). Sincefor any u outside of V it holds that d ( x j , R ( V )) < d ( x j , u ), we get that F ⋆ ( x ) ∈ V \ R ( V ) ⊆ V \ Z .Hence, x ⊆ V and F ⋆ ( x ) = F ( x ), and we get a contradiction to the false-name-proofness of F .If both F ⋆ ( x ) and F ⋆ ( A , x − C ) are in Z : We deal with this case using two different argumentationsfor the two scenarios of the theorem.( I ) G is a ZV -line graph w.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ): We first prove the following two auxiliarylemmas. Lemma i.
For any v ∈ V and d > , B ( v, d ) ∩ Z is an interval in Z .Proof. We prove the lemma by induction over d .For d = 0, B ( v, ∩ Z equals to { v } if v ∈ Z and to the empty set if v / ∈ Z .For d = 1, B ( v, ∩ Z is either the empty set or an interval in Z .For d >
2: If d < d ( v, Z ), B ( v, d ) ∩ Z = ∅ . If d > d ( v, Z ) > v / ∈ Z and is not aroot ), then there exists a location u (the root of the V i -subgraph v belongs to) s.t. all paths from v to locations in Z pass through u , 1 d ( v, u ) d ( v, Z ) d and B ( v, d ) ∩ Z = B ( u, d − d ( v, u )) ∩ Z which is an interval by the induction hypothesis.Otherwise, d ( v, Z ) < d and in particular B ( v, d ) ∩ Z = ∅ , and hence B ( v, d ) ∩ Z = ( B ( v, ∩ Z ) ∪ [ u ∈ N ( v ) s.t. d ( u,Z ) B ( u, d − ∩ Z .For any u ∈ N ( v ) s.t. d ( u, Z ) B ( u, d − ∩ Z and B ( v, ∩ Z intersect. • If u ∈ Z : u ∈ ( B ( u, d − ∩ Z ) ∩ ( B ( v, ∩ Z ). • If u / ∈ Z : then v ∈ Z and v ∈ ( B ( u, d − ∩ Z ) ∩ ( B ( v, ∩ Z ). An easy corollary of the definition of ZV -ordered partition is that for all V i -subgraphs d ( R ( V i ) , Z ) u ∈ N ( v ) s.t. d ( u, Z ) B ( u, d − ∩ Z and B ( v, ∩ Z are intersecting intervalsin Z . So B ( v, d ) ∩ Z is an interval as the union of intersecting intervals. Lemma ii.
Let x be a vector of locations s.t. F ⋆ ( x ) ∈ Z and let v ∈ Z be a location s.t. Agent i strictly prefers v over F ⋆ ( x ) . Then F ⋆ ( x ) is to the left of v .Proof. If x i ∈ Z then x i ∈ P O ( x ) ∩ Z and by the definition of F ⋆ , F ⋆ ( x ) is to the left of x i . Since F ⋆ ( x ) / ∈ B ( x i , d ( x i , v )) ∩ Z and since this set is an interval which includes x i , we get that F ⋆ ( x )is to the left of the interval and in particular to the left of v .Otherwise, x i / ∈ Z and there exists an Agent k for which x k is not in the same V i -subgraph as x i . Hence, there exists a location u ∈ Z s.t. u is on a shortest-path from x i to x k , u ∈ Z , and u ∈ P O ( x ). Hence, d ( x i , u ) d ( x i , v ) and so u ∈ B ( x i , d ( x i , u )) ∩ Z ⊆ B ( x i , d ( x i , v )) ∩ Z .The two sets are intervals in Z , F ⋆ ( x ) is to the left of u (or equal to it), and F ⋆ ( x ) / ∈ B ( x i , d ( x i , v )) ∩ Z .Hence, F ⋆ ( x ) is to the left of v .By applying Lemma ii for the profile x and Agent i , we get that F ⋆ ( x ) is to the left of F ⋆ ( A , x − C ); and by applying Lemma ii for the profile ( A , x − C ) and Agent j , we get that F ⋆ ( A , x − C )is to the left of F ⋆ ( x ). Hence, we get a contradiction.( II ) R ( V i ) ∈ Z and F Z satisfies ( ⋆ ): We notice that since R ( V i ) ∈ Z for all V i -subgraphs thepreference of an agent which is located in a V i -subgraph over the locations in Z and an agentwhich is located on the root, R ( V i ), are identical. Hence, for any profile y if F ⋆ ( y ) ∈ Z then F ⋆ ( y ) = F Z ( b y ) for b y being the profile generated from y by replacing each ballot outside of Z withthe root of its V i -subgraph. Therefore, for the profile b x ∈ Z n the coalition C can, by voting b A , getan outcome F Z (cid:16) b A , b x − C (cid:17) which it strictly prefers over F Z ( b x ), in contradiction to F Z satisfying ( ⋆ ). In this work, we presented a new family of graphs, ZV -line graphs, and a generic anonymous Paretooptimal manipulation-resistant mechanism for the facility location problem on these graphs. To thebest of our knowledge, the (very few) false-name-proof mechanisms which are currently known are forspecific graphs and this work is the first to show a generic false-name-proof mechanism for a largefamily, utilizing a broad graph property and unifying all existence results which we are aware of. Theconstruction of the mechanism is inductive: We derive a mechanism for a given ZV -line graph from17echanisms for its subgraphs. Hence, it is straightforward to derive from our construction generalmechanisms for recursive graph families.Two technical assumptions we had are connectivity of the graph and finiteness of the number ofagents and locations. Our results can be extended to the case of an infinite number of agents andlocations under common natural constraints like finite diameter of the graph, measurability of N ( v )and of coalitions, and the order over Z -vertices being a well-order. It is also not hard to see that thefollowing extension for graphs in which the connected components are ZV -line graphs will satisfy thesame desiderata. ◮ At the first stage, choose the first connected component according to some predefined order s.t.at least one agent voted for a location in this component. ◮ At the second stage, run our mechanism taking into account only agents who voted for locationsin the chosen component.The mechanism we presented is not the only mechanism satisfying the desired properties. Taking anyother order over the Z -vertices s.t. the constraints of Def. 4 hold and defining F ⋆ accordingly will alsosatisfy them. In particular, a mechanism which takes at the second stage of Def. 3 the rightmost Paretooptimal Z -vertex will also satisfy the same desiderata. We did not find any mechanism satisfying thedesiderata which is not of this template, and we conjecture that these are the only anonymous Paretooptimal manipulation-resistant mechanisms for facility location on a graph. Conjecture 9.
Let G = ( V , E ) be a ZV -line graph w.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ) and let F : S t > V t →V be a mechanism s.t. • F is anonymous and Pareto optimal; and • For any vector of locations x ∈ V n , a coalition of agents C , and a set of ballots A ∈ S t > V t , A is not a beneficial deviation for C .Then, for i = 1 , . . . , k : Whenever x ∈ ( V i ) n , also F ( x ) ∈ V i . Moreover, F is the outcome of applyingDef. 3 for some order over Z which satisfies the constraints of Def. 4 and mechanisms F i which aredefined by x ∈ ( V i ) n F ( x ) . Furthermore, unifying non-existence results for specific graphs we’ve found so far, we think thatthe partition to Z -vertices and V -vertices is a fundamental property of a false-name-proof mechanism.Consequentially, showing that a given graph does not have such structure could be an easy and efficientway to prove non-existence of a desired mechanism. Conjecture 10.
For almost any graph G = hV , E i , if there exists an anonymous and Pareto optimalmechanism F : S t > V t → V s.t.For any vector of locations x ∈ V n , a coalition of agents C , and a set of ballots A ∈ S t > V t , A is not a beneficial deviation for C .then there exists a sequence of non-empty sets of vertices Z, V , . . . , V k ⊆ V s.t. G is a ZV -line graphw.r.t. V = Z ∪ ( V ˙ ∪ · · · ˙ ∪ V k ) . (cid:7) ❯❯❯❯❯❯❯❯✐✐✐✐✐✐✐✐ (cid:7) (cid:7)(cid:7) ▲▲▲ (cid:7) rrr (and graphsderived from it, e.g., (cid:7) PPPPPPPPPPPP♥♥♥♥♥♥♥♥♥♥♥♥ (cid:7) ❦❦❦❦❦❦❦ (cid:7) ●● (cid:7) ✇✇ (cid:7) ✇✇●● (cid:7) (cid:7) (cid:7) ❙❙❙❙❙❙❙ (cid:7) ✇✇ (cid:7) ❙❙❙❙❙❙❙ (cid:7) ❇❇❇❇❇❇ (cid:7) ⑤⑤⑤⑤⑤⑤ ). It is not hard to verify that • a mechanism which returns thefirst Pareto optimal location according to one of the following orders - ❱❱❱❱❱❱❱❱❤❤❤❤❤❤❤❤ ▲▲▲ rrr , ❱❱❱❱❱❱❱❱❤❤❤❤❤❤❤❤ ▲▲▲ rrr , ❱❱❱❱❱❱❱❱❤❤❤❤❤❤❤❤ ▲▲▲ rrr ,or their rotations and reflections - is a manipulation-resistant mechanism and that • while all thesemechanisms are of the template of Def. 3 (for all vertices being Z -vertices), these representations donot satisfy the connectivity constraints of Def. 4 and the cycle of size 5 is not a ZV -line graph. Weconjecture that this is a representative extreme exception and intend to characterize the exception andreplace ‘almost’ in Conjecture 10 with an exact statement.Last, an important continuation of this work is analyzing the implications for approximate mechanismdesign without money [20]. That is, assuming the agents are accurately represented by a cost function(e.g., the distance to the facility or a monotone function of the distance) and analyzing implicationsof manipulation-resistance on the approximability of the minimization problem of natural social costfunctions, e.g., the average cost (Harsanyi’s social welfare), the geometric mean of the costs (Nash’s socialwelfare), or the maximal cost (Rawls’ criterion). For instance, assuming the two conjectures above, onegets that whenever there is a large disagreement in the population (i.e., the agents are dispersed overmany V i -subgraphs) an extreme status-quo alternative must be chosen by the mechanism, which resultsin a bad price of false-name-proofness . Nowadays, many aggregation mechanisms are highly susceptibleto double voting and to false-name manipulations in general (e.g., mechanisms over huge anonymousnetworks like the internet, but also other scenarios in which vote frauds are known to be easy). Wethink that such results should open a discussion on the costs of these protocols (since the benefits areclear). 19 eferences [1] Fuad Aleskerov. Chapter 2 categories of arrovian voting schemes. In Amartya K. Sen KennethJ. Arrow and Kotaro Suzumura, editors, Handbook of Social Choice and Welfare , volume 1 of
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