Tiling Type Condorcet Domains

• The paper demonstrates that tiling type Condorcet domains are constructed through geometric rhombus tilings of zonogons, providing a visual interpretation of linear orders.
• It reveals algebraic composition methods, such as concatenation and never-last composition, that combine smaller domains and refute earlier maximality conjectures.
• The study highlights that these domains support acyclic majority aggregation and strategy-proof voting rules, ensuring robust and consistent preference aggregation.

Condorcet domains of tiling type are highly structured sets of linear orders that admit a geometric realization via rhombus tilings of zonogons, yielding rich algebraic and combinatorial characterizations with direct implications for aggregation theory, preference diversity, and the design of robust voting rules.

1. Geometric Construction of Tiling Type Condorcet Domains

A tiling type Condorcet domain is constructed from a zonogon $Z_n$ formed by the Minkowski sum of %%%%1%%%% vectors $\xi_1, \ldots, \xi_n$ arranged in the plane: $$Z_n = \left\{ \sum_{i=1}^{n} a_i \xi_i : 0 \leq a_i \leq 1 \right\}$$ The zonogon is then partitioned by a rhombus tiling $T$, where each tile corresponds to the Minkowski sum of two basic segments $[0,\xi_i]$ and $[0,\xi_j]$ (an $ij$-tile).

A snake in $T$ is a directed, upward path from the bottom vertex $b$ to the top vertex $t$ crossing each track (associated to $\xi_i$) exactly once; each such snake thereby induces a permutation $\sigma \in \mathcal{L}([n])$ by recording the sequence of tracks traversed. The collection $\Sigma(T)$ of all such linear orders constitutes the tiling type Condorcet domain.

These domains have a natural poset structure linked to the geometric configuration:

• The set of tiles to the left of a snake corresponds bijectively to the inversion set of the associated permutation.
• The weak Bruhat order on permutations manifests as geometric containment: $L(\sigma) \subseteq L(\tau)$ if and only if $$\operatorname{Inv}(\sigma) \subseteq \operatorname{Inv}(\tau)$$.
• For any $T$, $\Sigma(T)$ is a maximal (complete) Condorcet domain; this acyclicity is verified locally by restricting to every triple of alternatives and confirming the presence of either the hump or hole configuration (i.e., “never best” or “never worst” conditions).

2. Algebraic Compositions and Cardinality Bounds

The structure of tiling type Condorcet domains permits algebraic operations for constructing larger domains while preserving Condorcet properties.

• Concatenation and Shuffle: For domains $\mathcal{D}_1$, $\mathcal{D}_2$ on disjoint sets, with distinguished orders $u \in \mathcal{D}_1$, $v \in \mathcal{D}_2$, the tensor product is

$$(\mathcal{D}_1 \otimes \mathcal{D}_2)(u,v) = (\mathcal{D}_1 \odot \mathcal{D}_2) \cup (u \oplus v)$$

The formula for the size of the composition is:

$$|\mathcal{D}_1 \otimes \mathcal{D}_2| = |\mathcal{D}_1||\mathcal{D}_2| + \binom{m+n}{m} - 1$$

where $m = |A|$, $n = |B|$ are the sizes of the respective alternative sets. This construction was used to produce counterexamples to longstanding conjectures regarding maximal domain size (Danilov et al., 2010, Slinko, 2020).

• Never-last (nl) Composition: Given two smaller Condorcet domains $\mathcal{D}_1$ on $\{1,\ldots,n-1\}$ and $\mathcal{D}_2$ on $\{2,\ldots,n\}$, their nl-composition is

$$\mathcal{D}_1 \diamond \mathcal{D}_2 = \{ u\,n : u \in \mathcal{D}_1 \} \cup \{ v\,1 : v \in \mathcal{D}_2 \}$$

Sufficient conditions on obstructions and underlying copiousness guarantee that the composition yields a maximal Condorcet domain (Keehan et al., 13 Dec 2024).

These algebraic frameworks reveal that the product of “large” domains can exceed the size of Fishburn’s alternating domain. Explicit counterexamples demonstrate that the Fishburn–Galambos–Reiner–Monjardet conjectures on maximal cardinality are false for tiling type domains (Danilov et al., 2010, Slinko, 2020).

3. Unification of Structural Approaches

Tiling type domains subsume and unify earlier constructions:

• Abello’s maximal chains in the Bruhat lattice,
• Galambos–Reiner’s pseudo-line arrangements (second Bruhat order),
• Chameni-Nembua’s distributive sublattices.

The equivalence is characterized via three perspectives:

1. Domains of tiling type (maximal chains in the Bruhat lattice),
2. Semi-connectedness (distinguished orders $\alpha$ and $\omega$ connected within the domain’s Bruhat graph),
3. The hump–hole condition on restrictions to triples (never best/never worst).

For $n=3$, all maximal Condorcet domains arise as tiling domains corresponding to the hump and hole configurations. More generally, all hump–hole domains are precisely those of tiling type (Danilov et al., 2010).

4. Majority Rule and Aggregation Properties

Condorcet domains of tiling type possess enhanced majority aggregation characteristics:

• For profiles drawn from $\Sigma(T)$, majority aggregation yields acyclic relations (prevents Condorcet cycles) (Danilov et al., 2010, Danilov et al., 2020).
• Each domain is closed under the majority operation; resulting majority orders (for odd-sized profiles) remain within the domain (Puppe et al., 2015).
• The median voter rule is well-behaved: the majority/median operation on inversion sets (or tiles) preserves tiling structure if aggregation is performed within Condorcet super-domains (Danilov et al., 2020).

Condorcet super-domains (CSDs)—collections of tilings—ensure that for profiles of ballots (as tilings), the majority rule yields another tiling, maintaining acyclicity at a higher organizational level (Danilov et al., 2020).

Additionally, the structure of the domain (viewed as a median graph) allows for the implementation of monotone Arrovian aggregators, yielding strategy-proof social choice functions and ensuring immunity to strategic manipulation under broad conditions (Puppe et al., 2015).

Recent computational advances enable explicit computation of majority outcomes for uniform vote tallies on tiling domains using poset-theoretic and word-reduction techniques (Reiner et al., 23 Sep 2025).

5. Diversity, Ampleness, and Copiousness in Tiling Domains

Preference diversity in tiling type domains is analyzed via abundance parameters:

• A domain is $(k,s)$-abundant if, for every $k$-subset $A$, its restriction $D(A)$ contains at least $s$ distinct linear orders.
• Ampleness is $(2,2)$-abundance; copiousness is $(3,4)$-abundance.
• For large $n$, the local diversity (number of distinct suborders on subsets of fixed size) is universally bounded above by $2^{k-1}$ for $k$-subsets (Karpov et al., 22 Jan 2024).
• Black's single-peaked domain attains optimal local diversity, matching this upper bound. Tiling type domains, irrespective of global size, cannot surpass this ceiling in their restrictions.

This implies that structural regularity from tilings may yield domains with large overall cardinality, but the local diversity for fixed-size subsets does not exceed the classical single-peaked and group-separable benchmarks.

6. Implications, Applications, and Open Directions

Condorcet domains of tiling type serve as foundational designs for voting systems that robustly avoid cyclical majorities:

• Their geometric and combinatorial structures support tractable aggregation algorithms for profile evaluation, median and majority operations, strategy-proof rule design, and monotonic aggregation (Danilov et al., 2010, Danilov et al., 2020, Puppe et al., 2015).
• Algebraic composition techniques suggest modular construction of large domains from smaller ones, with controllable copiousness and ampleness properties (Slinko, 2020, Keehan et al., 13 Dec 2024).
• Empirical and algorithmic methods are now available for explicit computation of majority outcomes in large domains (Reiner et al., 23 Sep 2025).

A plausible implication is that future research may focus on optimizing the trade-off between global cardinality and local diversity, extending the geometric tiling paradigm to higher dimensions (cubillages) and investigating which tiling configurations yield Condorcet domains with application-specific aggregation or stability properties (Danilov et al., 2020).

7. Historical Context and Resolution of Conjectures

The geometric realization of Condorcet domains via tilings not only unifies prior combinatorial constructions but also resolves key conjectures:

• The maximal size conjecture for connected Condorcet domains, previously believed to be attainable by alternating (Fishburn) domains, is refuted (Danilov et al., 2010, Slinko, 2020).
• The composition and stacking arguments show that stacking tilings can yield domains whose cardinality exceeds all known alternating domain cardinalities for sufficiently large $n$.

In summary, Condorcet domains of tiling type exhibit a deep interplay of geometry, combinatorics, and aggregation theory. Their paper facilitates both theoretical insights and practical methodologies for ensuring consistent preference aggregation in social choice settings.