Miscoverage-Regret Pareto Frontier

• The miscoverage-regret Pareto frontier is defined as the set of optimal mechanisms where no alternative can simultaneously lower manipulability (regret) and social deficit (miscoverage).
• Its structure is piecewise linear, convex, and continuous, allowing hybrid mechanisms to be constructed via convex combinations of adjacent optimal points.
• Algorithmic methods like LP FindOpt and interpolation techniques precisely locate supporting manipulability bounds to guide effective random mechanism design.

The miscoverage-regret Pareto frontier describes the explicit trade-off between incentive compatibility (“regret” from manipulation, quantified as approximate strategyproofness) and performance on another desideratum (“miscoverage,” measured as the deficit in achieving a social goal) in the design of random ordinal mechanisms. The frontier comprises the set of mechanisms that are Pareto-optimal: no other mechanism can simultaneously reduce manipulability (regret) and deficit (miscoverage). This foundational framework, as characterized in "The Pareto Frontier for Random Mechanisms" (Mennle et al., 2015), rigorously delineates the optimal boundary of these trade-offs and provides both mathematical formulations and computational algorithms for its construction.

1. Quantification of Manipulability and Deficit

Each mechanism $\varphi$ is uniquely identified (its “signature”) by two quantities:

• Manipulability $\varepsilon(\varphi)$: The minimal $\varepsilon$ for which the mechanism is $\varepsilon$-approximately strategyproof. Formally, for any agent $i$, preference profiles $(P_i, P_{-i})$, any misreport $P_i'$, and any utility function $u_i$,

$$\sum_{j \in M} u_i(j) \left[\varphi_j(P_i',P_{-i}) - \varphi_j(P_i,P_{-i})\right] \le \varepsilon.$$

Here, manipulating one's report can yield at most $\varepsilon$ improvement in expected utility.

• Deficit $(\varphi)$: The maximum shortfall, across all profiles, between the highest achievable social value (under a desideratum $d$) and the value achieved by $\varphi$:

$$(\varphi) = \max_{\bm{P}} \left\{\max_{j\in M} d(j,\bm{P}) - d(\varphi(\bm{P}),\bm{P})\right\},$$

where $d$ evaluates, e.g., efficiency (Plurality) or fairness (Veto). This “deficit” directly captures miscoverage: how far the mechanism falls short of the social objective.

This dual measure enables explicit analysis of the inherent tension between incentive properties and social optima.

2. Structural Characterization of the Frontier

The Pareto frontier admits a piecewise linear, convex, monotonic, and continuous structure:

• Supporting manipulability bounds:

$$0 = \varepsilon_0 < \varepsilon_1 < \cdots < \varepsilon_K = \bar{\varepsilon}$$

These partition the range of allowable manipulability. For $\varepsilon$ in $[\varepsilon_{k-1}, \varepsilon_k]$, every optimal mechanism is a convex mixture of two adjacent extremal mechanisms.

• Linear interpolation: When $\varepsilon$ is written as a convex combination,

$$\varepsilon = (1-\beta)\varepsilon_{k-1} + \beta\varepsilon_k,$$

the minimal deficit is

$$(\varepsilon) = (1-\beta) (\varepsilon_{k-1}) + \beta (\varepsilon_k).$$

The mapping $\varepsilon \mapsto (\varepsilon)$ is thus convex and piecewise linear; the Pareto frontier consists of the “corner points” and the mixtures interpolating between them.

3. Hybrid Mechanisms and Algorithmic Computation

Constructing mechanisms on the Pareto frontier exploits the concept of hybridization:

• Hybrids: If $\varphi$, $\psi$ are optimal for $\varepsilon_{k-1}$, $\varepsilon_k$, then

$$h_\beta(\bm{P}) = (1-\beta)\varphi(\bm{P}) + \beta \psi(\bm{P})$$

yields a mechanism with

$$\varepsilon(h_\beta) \le (1-\beta)\varepsilon(\varphi) + \beta\varepsilon(\psi), \quad (h_\beta) \le (1-\beta)(\varphi) + \beta(\psi).$$

• Algorithmic procedures:
  • LP FindOpt: A specialized linear program (see full formulation in the source) computes the optimal mechanism for a fixed $\varepsilon$.
  • FindLower: Identifies the lower-bound $\underline{\varepsilon}$ for manipulability.
  • FindBounds: Iteratively interpolates between known signatures to exactly locate the supporting manipulability bounds and “kinks” in the frontier.

This approach guarantees that the full frontier can be computed with finitely many LP calls.

4. Instantiating with Plurality and Veto Scoring

• Plurality: The desiratum is the proportion of first-choice votes, $d^{Plu}(j,\bm{P}) = n_j^1/n$. In the $n=3$, $m=3$ case, the frontier has only two supporting points.
  • $\varepsilon_0 = 0$: Random Dictatorship (strategyproof), deficit $1/9$.
  • $\varepsilon_1 = 1/3$: Uniform Plurality, deficit $0$.
  • Intermediate mechanisms are hybrids.
• Veto: The target is minimized last-place votes, $d^{Veto}(j,\bm{P}) = (n-n_j^m)/n$. The $n=3$ case yields four corner points.
  • $\varepsilon_0 = 0$: Random Duple.
  • $\varepsilon_3 = 1/2$: Uniform Veto.
  • Intermediate bounds $\varepsilon_1, \varepsilon_2$ require new mechanisms not decomposable into simple hybrids.

These examples demonstrate both trivial and nontrivial curvature in the frontier, highlighting the structural richness of the miscoverage-regret relationship.

5. Mechanism Design Implications

The Pareto frontier’s structural theorems establish that:

• The trade-off between manipulability and deficit is not complicated; it is piecewise linear.
• The first minor increase in acceptable manipulative regret typically yields the greatest reduction in deficit (i.e., improved coverage of the social desideratum).
• Small sacrifices in strategyproofness may generate substantial gains in efficiency or fairness, guiding mechanism designers where to “spend” incentive compatibility.
• For settings where the desideratum strictly competes with strategyproofness, the frontier precisely dictates how much efficiency or fairness can be “purchased” at each allowable increment of manipulation.

6. Formal Computation and Representation

Summary of formal apparatus:

Symbol	Meaning	Quantifies
$\varepsilon$	Max utility gain from manipulation	Regret / Manipulability
$(\varphi)$	Max deficit in desired social objective	Miscoverage / Deficit
LP FindOpt	Linear program optimizing deficit at fixed $\varepsilon$	Construction of Pareto-optimal mechanisms
$h_\beta$	Hybrid of two mechanisms	Convex combination for interpolation

All central constraints and mixture structures are stated explicitly in the source, allowing direct mechanistic and computational instantiation.

7. Mathematical and Empirical Insights

• The Pareto mapping $\varepsilon \mapsto (\varepsilon)$ is monotonic, decreasing, convex, and continuous.
• Every optimal mechanism can be constructed, for any $\varepsilon$, as a solution to LP FindOpt; between supporting bounds, as a hybrid.
• The piecewise linear nature of the frontier implies that after finding its supporting points (via algorithms), its entirety is known.
• Empirical illustrations (with computed plots) verify these properties in specific scoring systems.

This rigorous characterization allows both theoretical analysis and practical computation of the miscoverage-regret Pareto frontier in random mechanism design.