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Strategyproof Mechanisms for Connecting Impassable Regions

Hau Chan, Jianan Lin, Chenhao Wang

arXiv:2609.08488v1cs.GT

TL;DR

The paper asks how to connect two regions separated by an obstacle while eliciting agents’ private locations truthfully and approximately minimizing travel costs. It characterizes pathways and develops deterministic and randomized strategyproof mechanisms. It reports tight deterministic maximum-cost performance, a population-independent randomized social-cost guarantee, and strengthened bounds for a related real-line model.

  • Problem

    The paper studies how to construct a pathway between obstacle-separated regions when agents have private locations and the goal is to minimize maximum or total travel cost.

    Source: §1 · pp. 1–2

  • Method

    It characterizes optimal pathways and analyzes deterministic and randomized strategyproof mechanisms, including a power-proportional mechanism, using expected costs for randomized mechanisms.

    Source: Randomized Mechanisms · p. 6, §7 · p. 29

  • Results

    The paper reports an exact deterministic maximum-cost ratio, a randomized social-cost guarantee independent of population size, a tight factor of 3 for its randomized mechanism at k = 0, and improved bounds for the real-line model.

    Source: §5.3 · p. 23, §7 · p. 29

  • Takeaways & Limitations

    The reported randomized social-cost guarantee does not depend on the number of agents, while the deterministic social-cost bounds are linear in population size when k = 0.

    Source: §7 · p. 29

    Open questions include tightening randomized bounds in the obstacle model and resolving the n −1 deterministic SP social-cost lower bound for the real-line model when n ≥ 4.

    Source: §7 · p. 29

Abstract

from arXiv · show

We study strategyproof mechanisms for building a pathway between two regions of a line segment separated by an obstacle. Each of the $n$ agents has a private location within its region and may use either its original route to a facility or the new pathway, whose traversal cost is a fraction $k\in[0,1)$ of its length. We seek strategyproof (SP) and group-strategyproof (GSP) mechanisms that approximately minimize maximum cost or social cost. After characterizing optimal pathways for both objectives, we establish a tight deterministic maximum-cost approximation ratio of $\frac{2}{1+k}$ and a deterministic social-cost upper bound of $\frac{n}{1+k(n-1)}$, together with complementary lower bounds. Both upper bounds are achieved by GSP mechanisms. We then study randomized mechanisms under strategyproofness in expectation. A power-proportional mechanism achieves a social-cost approximation ratio at most $5$, independent of $n$ and $k$, with a tight guarantee of $3$ for this mechanism when $k=0$. We prove randomized lower bounds of $\frac{3+2k}{2+3k}$ for maximum cost and $\max\big\{1,\frac{285}{263+385k}\big\}$ for social cost, the latter for $n\ge7$. Finally, we improve several bounds for the real-line pathway model of [Chan and Wang, AAMAS 2023]. Our deterministic maximum-cost lower bound of $2$ matches the upper bound obtainable from [Qin, Fang, and Liu, COCOA 2024]. We strengthen the deterministic social-cost lower bound from $\frac32$ to $2$ under SP and to $\max\{2,n-1\}$ under GSP. For randomized social cost, we sharpen the guarantee of Chan and Wang's proportional mechanism from $6$ to $3$ and raise their lower bound from $1.02$ to $\frac{285}{263}\approx1.08365$ for $n\ge7$.

1 Introduction

The paper studies how to connect regions separated by an obstacle when agents’ locations are private and pathway choices affect their travel costs. It characterizes optimal pathways and develops strategyproof mechanisms with approximation guarantees, while distinguishing its model from earlier pathway and facility-location work.

1 Introduction

  • Disconnected regions can restrict access to facilities across an obstacle, motivating a pathway that connects the regions and expands agents’ accessible options.The paper frames this as a mechanism-design problem because agents’ starting locations are private and can be misreported.Source: §1 · pp. 1–2

1.1 Our Contributions

  • The model places agents on opposite sides of an obstacle on a line segment and charges pathway users a traversal cost scaled by k.Agents may instead use their original route to a facility; the objectives are maximum cost and total social cost.Source: §1.1 · pp. 2–3
  • For both objectives, an optimal pathway can be chosen with one endpoint at an original facility, but the paper’s optimal mechanisms are not strategyproof.The structural basis is that agents in at most one region can strictly benefit from any fixed pathway.Source: §1.1 · p. 3
  • CriticalExtreme is GSP and matches the deterministic maximum-cost lower bound, while GSP TwoExtreme achieves the stated deterministic social-cost upper bound.The paper also gives complementary deterministic SP lower bounds, including max{2, n −1} at k = 0 and Λ_m(k) for admissible active populations.Source: §1.1 · p. 3
  • PowerProportional samples agents according to a power of their maximum possible cost reduction and is strategyproof in expectation, with a social-cost ratio at most 5 for 0 < k < 1.At k = 0 it reduces to proportional sampling, whose guarantee of 3 is tight for that mechanism.Source: §1.1 · pp. 3–4
  • The paper improves bounds for the real-line pathway model, including the proportional mechanism’s social-cost guarantee and the randomized social-cost lower bound for n ≥ 7.It also strengthens deterministic social-cost lower bounds under SP and GSP.Source: §1.1 · p. 4

1.2 Related Work

  • The related algorithmic work connects entire regions, whereas this paper considers finitely many agents’ private locations and mechanism design.The closest mechanism-design work differs in the distance agents care about; the paper also relates its setting to a real-line pathway model with different feasible outputs and reporting domains.Source: §1.2 · pp. 4–5

2 Preliminaries

The model fixes public region membership and lets agents report locations only within their original region. It defines pathway costs and the maximum-cost and social-cost objectives, then formalizes strategyproofness and group-strategyproofness.

2 Preliminaries

  • Agents lie on [0,1] on one of two sides of obstacle o, and a feasible pathway has endpoints in opposite regions.Region membership remains fixed under reports.Source: §2 · p. 5
  • A pathway user pays the distance through the edge plus k times its length, while each agent chooses the cheaper of that route and the original route to a facility.The objective is either the sum of agents’ costs or the largest individual cost.Source: §2 · pp. 5–6
  • Strategyproofness rules out beneficial individual misreports, and group strategyproofness rules out joint misreports that strictly benefit every coalition member.Every SP mechanism is partial GSP, where the coalition is restricted to agents at the same location.Source: §2 · p. 5

Randomized Mechanisms.

The randomized analysis evaluates expected costs and uses structural properties of pathways to support its incentive arguments. A pathway can strictly benefit agents in only one region, which also constrains coalitions that could profit from misreporting.

Randomized Mechanisms.

  • A randomized mechanism is evaluated by expected social cost or expected maximum cost, and averaging feasible deterministic pathways cannot improve on the unrestricted optimum.The expectation version of partial group strategyproofness follows by moving colocated agents’ reports one at a time.Source: Randomized Mechanisms · p. 6
  • For any pathway, agents who strictly benefit from using it must all belong to the same region.Thus at least one region has no agent who strictly benefits from that pathway.Source: Randomized Mechanisms · p. 6
  • If a deterministic mechanism admits a coalition deviation that strictly benefits every member, all coalition members must lie on the same side of the obstacle.A strictly beneficial deviation must place each member’s cost below the cost of taking the direct route.Source: Randomized Mechanisms · p. 7

3 Optimal Pathways

Optimal pathways for both objectives can be chosen with one endpoint at an original facility. The paper characterizes optimal maximum-cost and social-cost pathways, while showing that the resulting optimal mechanisms need not be strategyproof.

3 Optimal Pathways

  • For both objectives, an optimal pathway exists with a = 0 or b = 1.At least one region has no agent who strictly benefits from a fixed pathway, so extending the pathway to an original facility need not increase costs.Source: §3 · p. 7

3.1 Maximum Cost

  • Mechanism 1 is optimal for maximum cost, choosing (a*, 1) or (0, b*) according to whether xr + yl ≥ 1.Its two methods use the extreme reported locations in the two regions.Source: §3.1 · pp. 7–8
  • Mechanism 1 is not strategyproof; a two-agent profile in the left region admits a profitable misreport for every 0 ≤ k < 1.The example gives a strict cost reduction of (1 + k)/20.Source: §3.1 · p. 8

3.2 Social Cost

  • Mechanism 2 is optimal for social cost by comparing the two endpoint families and minimizing over finite candidate endpoints.The candidates include agent locations and route-switching breakpoints.Source: §3.2 · pp. 8–9
  • Mechanism 2 is not strategyproof: near the obstacle, a right-region agent can misreport and change the chosen pathway, lowering its cost.The reported-profile objective favors the alternative by (1 + k)ϵ², so the output change does not depend on tie-breaking.Source: §3.2 · p. 9

4 Deterministic Strategyproof Mechanisms

This section studies deterministic strategyproof mechanisms for maximum-cost and social-cost objectives, giving GSP mechanisms with approximation guarantees and establishing lower bounds for strategyproof mechanisms. For social cost, it also presents stronger bounds under group strategyproofness.

4.1 Maximum Cost

  • For a location profile x, Mechanism 3 (TwoExtreme) returns (xr, yl) and Theorem 3 states it is group strategyproof; the excerpt’s achievement metric is incomplete.Source: §4.1 · p. 10
  • 2/(1+k) is the tight maximum-cost approximation ratio: CriticalExtreme is GSP, and no strategyproof mechanism can do better for n ≥ 2.The lower bound holds for every 0 ≤ k < 1.Source: §4.1 · pp. 11–12

4.2 Social Cost

  • n/[1+k(n-1)] is the social-cost approximation ratio achieved by GSP TwoExtreme, and its guarantee is asymptotically tight for this mechanism.For k = 0 the bound is linear in n; for k = Ω(1) it is constant.Source: §4.2 · pp. 13–14
  • For social cost, SP mechanisms have lower bounds including Λ_m(k) for admissible 3 ≤ m ≤ n and a bound of n − 1 when k = 0.The Λ_m(k) bound applies when 0 < k < (m−2)/m.Source: §4.2 · pp. 15–16
  • Λ_n(k) is a social-cost approximation lower bound for every strategyproof mechanism.The proof also extends the bound to Λ_m(k) for 3 ≤ m < n by fixing the other agents at the facility.Source: §4.2 · pp. 17–18
  • 2 under SP and max{2, n−1} under GSP are deterministic social-cost lower bounds.At k = 0, the bound max{2, n−1} matches the upper bound for n = 2 and is asymptotically tight as n grows.Source: §4.2 · p. 18
  • lim_{k→0+} Λ_m(k) = m−1, strengthening β(k) for sufficiently small positive k when m ≥ 4.For n = 6 and k = 0.01, Λ_6(k) ≈ 2.993, while β(k) ≈ 1.944.Source: §4.2 · p. 18

5 Randomized Strategyproof Mechanisms

This section studies randomized strategyproof mechanisms for approximating social cost and maximum cost. It gives a social-cost guarantee independent of the number of agents, including at k = 0, and establishes lower bounds for randomized social-cost and maximum-cost approximation, with a deterministic upper bound serving as a benchmark.

5.1 Maximum Cost

  • (3+2k)/(2+3k) is a lower bound on the maximum-cost approximation ratio, versus the deterministic upper bound 2/(1+k).At k = 0, the respective bounds are 3/2 and 2.Source: §5.1 · p. 19

5.2 Social Cost: A Power-Proportional Mechanism

  • 5 is an upper bound on Mechanism 5’s social-cost approximation ratio, independent of n and k, and the mechanism is strategyproof in expectation.At k = 0, it selects an agent with probability proportional to t_i; the exponent accounts for positive pathway cost.Source: §5.2 · pp. 20–21

5.3 A Sharper Guarantee for Zero Pathway Cost

  • 3 is a tight social-cost approximation ratio for Mechanism 5 when k = 0, with tightness as n grows.The mechanism is strategyproof in this setting by Theorem 9.Source: §5.2 · p. 20, §5.3 · p. 23

5.4 A Randomized Social-Cost Lower Bound

  • max{1, 285/(263+385k)} is a lower bound on the social-cost approximation ratio for randomized strategyproof mechanisms when n ≥ 7.Source: §5.4 · p. 24
  • The ratio is at least max{1, 285/(263+385k)}, and at k=0 the bound 285/263 holds for every fixed admissible λ.Source: §5.4 · p. 25

6 Improved Bounds for the Real-Line Pathway Model

The real-line model permits arbitrary reports and pathway endpoints, and results for a restricted fixed-facility model do not automatically transfer to it. The section establishes deterministic lower bounds and randomized social-cost guarantees directly for the real-line setting.

6 Improved Bounds for the Real-Line Pathway Model

  • The fixed-facility cost correspondence preserves costs only for restricted locations and pathways, so real-line results are justified directly.Replacing a mechanism’s outputs with pathways incident to 0 need not preserve strategyproofness; the lower-bound proofs allow arbitrary endpoints.Source: §6 · pp. 25–26

6.1 Deterministic Mechanisms

  • 2 is the tight deterministic maximum-cost approximation ratio in the real-line model.The lower bound matches the upper bound transferred from the fixed-facility model.Source: §6.1 · p. 26, §6 · p. 26
  • The deterministic social-cost lower bound improves to 2 under SP and max{2, n −1} under GSP.The GSP proof uses group strategyproofness to eliminate a positive pathway-fee term; its coalition includes agents at two different locations.Source: §6.1 · pp. 27–28
  • max{2, n −1} is a lower bound on deterministic GSP social-cost approximation, while deterministic SP mechanisms have a lower bound of 2.For arbitrary deterministic SP mechanisms, the n −1 lower bound remains unproved for n ≥4; it follows under the additional requirement that pathways include 0.Source: §6.1 · pp. 27–28

6.2 Randomized Mechanisms

  • For randomized social cost, the proportional mechanism’s guarantee improves from 6 to 3, while the optimal randomized SP ratio for n ≥7 remains between 285/263 and 3.The factor 3 is tight for this mechanism, not established as optimal across all randomized SP mechanisms.Source: §6.2 · pp. 28–29
  • The proportional mechanism is SP in expectation and achieves a tight social-cost approximation ratio of 3 as n grows.It selects agent j with probability proportional to |x_j| and outputs (0, x_j).Source: §6.2 · pp. 28–29
  • For n ≥7, every randomized SP mechanism has a real-line social-cost approximation ratio of at least 285/263.The proof handles arbitrary pathway endpoints using the nonnegative fee term h.Source: §6.2 · p. 29

7 Conclusion

The paper characterizes optimal pathways and develops deterministic and randomized strategyproof mechanisms for the obstacle model, while also improving several bounds for the real-line model. It leaves gaps in randomized bounds and in deterministic SP social-cost lower bounds for the real-line setting.

7 Conclusion

  • The deterministic maximum-cost approximation ratio is exactly 2/(1+k), and the social-cost bounds are linear in population size when k = 0.For the real-line model, the deterministic maximum-cost lower bound of 2 matches the upper bound, and the deterministic social-cost lower bounds are strengthened under SP and GSP.Source: §7 · p. 29
  • The power-proportional mechanism gives a social-cost guarantee independent of n and k, with a sharper factor of 3 when pathway traversal cost is zero.The paper also obtains randomized lower bounds for both objectives.Source: §7 · p. 29
  • Several gaps remain: randomized obstacle-model bounds can be improved, and the n −1 lower bound for arbitrary deterministic SP mechanisms in the real-line model is unresolved for n ≥4.For randomized social cost, sharper positive-k guarantees and stronger lower bounds are also needed, especially away from k = 0.Source: §7 · p. 29

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