Minimal-Revenue Equilibrium States

• Minimal-Revenue Equilibrium States are defined as the worst-case revenue outcomes achievable under equilibrium constraints imposed by market design.
• They arise in competitive bundling and oligopoly settings, where mechanisms use structured bundling and pricing to secure revenue floors benchmarked against optimal welfare.
• Their analysis informs robust auction and market design strategies by delineating conditions that result in tight lower bounds on principal revenue.

Minimal-revenue equilibrium states formalize the worst-case revenue outcomes achievable across equilibrium concepts where the primary mechanism designer—such as a seller or system operator—faces incentive, competition, or combinatorial constraints. The term is most precisely instantiated in two domains: competitive bundling equilibrium (CBE) in combinatorial markets, and in competitive oligopoly pricing against single-item rivals. Both lines of research characterize the minimum revenue guaranteed in equilibrium under structural supply and demand constraints, and identify the conditions and constructions that realize the lower bound, often in terms of the optimal welfare (OPT) as a benchmark. The minimal-revenue equilibrium is always defined by the worst-case among all equilibrium outcomes permitted by the mechanism’s rules or the competitive structure.

1. Definitions and Formal Model Structure

A minimal-revenue equilibrium state is the equilibrium of a market game or mechanism design problem that achieves the lowest possible revenue for the principal (e.g., market operator, seller) among all possible equilibria under the design constraints.

Competitive Bundling Equilibrium (CBE)

Let $M$ denote a set of $m$ indivisible goods and $N = \{1, \ldots, n\}$ the set of consumers, each with a monotone non-decreasing valuation function $v_i: 2^M \to \mathbb{R}_{\geq 0}$. A bundling $B = (B_1, \ldots, B_\ell)$ is a partition of $M$ into $\ell$ bundles. A competitive bundling equilibrium (CBE) consists of:

• a bundling $B$,
• an allocation $(T_1, \ldots, T_n)$ of bundles to consumers ($T_i \subseteq B$, each bundle assigned to one consumer, $\bigcup_i T_i = M$),
• bundle prices $\{p_B\}_{B \in B}$,

such that for every buyer $i$ and alternative subset $T \subseteq B$,

$$v_i\left(\bigcup_{B\in T_i}B\right) - \sum_{B\in T_i} p_B \geq v_i\left(\bigcup_{B\in T}B\right) - \sum_{B\in T}p_B.$$

The CBE revenue is $$\operatorname{Rev}(E) = \sum_{i=1}^n \sum_{B \in T_i} p_B$$. The minimal-revenue equilibrium state is defined by the CBE with the lowest possible $\operatorname{Rev}(E)$ achievable in equilibrium.

Competitive Bundling with Oligopoly

In the presence of single-item competitors, minimal-revenue is defined as the principal’s minimal revenue over all Nash equilibria induced when agile competitors react to the principal’s announced product-bundle menu:

• $m$ heterogeneous goods, a single buyer with additive values $v \sim F = \times_{i=1}^m F_i$, principal menu $p:2^{[m]} \to \mathbb{R}_{\geq 0}$.
• Each rival single-item seller $i$ posts price $q_i \geq 0$ after observing $p$.
• The buyer chooses which bundles to buy from the principal and which items from rivals, maximizing total utility.
• For any menu $p$, the minimal-revenue equilibrium $R_{\min}(p) := \min_{s^* \in NE(p)} R(p,s^*)$ is the principal’s minimum expected revenue across all Nash equilibria $s^*$ of the item-pricing subgame (Babaioff et al., 19 Jun 2024).

2. Fundamental Revenue Guarantee Theorems

CBE Revenue Lower Bounds

In the weighted-uniform matroid (multi-unit demand) and common-matroid value settings, it is guaranteed that there exists a CBE $E$ with

$$\operatorname{Rev}(E) \geq \Omega\left(\frac{\mathrm{OPT}}{\log m}\right),$$

where $\mathrm{OPT}$ is the maximal social welfare across all possible (integral) allocations (Dobzinski et al., 2014). The lower bound is tight: there exist markets for which every CBE achieves only $O(\mathrm{OPT}/\log m)$ revenue.

Oligopoly Bundling Benchmarks

When the principal faces single-item competitors, the minimal equilibrium revenue over all Nash equilibria is upper-bounded by the expected truncated welfare

$$B := \mathbb{E}_{v \sim F} \left[\sum_{i=1}^m \min(v_i, r_i) \right],$$

where $r_i$ is the monopoly (Myerson) price for $F_i$. For every bundle menu $p$ and Nash equilibrium $s^*$:

$R(p, s^*) \leq B,$

hence $R_{\min}(p) \leq B$ (Babaioff et al., 19 Jun 2024). Under (λ,C)-price sensitivity and sufficiently large variance in $B$, offering only the grand bundle with an appropriately chosen price guarantees, at every equilibrium,

$R_{\min}(p) \geq \frac{1}{3} B.$

3. Tightness: Instance Constructions and Zero-Revenue Cases

The lower bounds for minimal-revenue equilibria are sharp in both domains.

Tightness in CBE

• Unit-demand construction: Let $n = m$, $v_i(j) = 1/i$ for all $i, j$. Here, $\mathrm{OPT} = H_m \approx \log m$, but every CBE extracts at most revenue $1$, so minimal revenue is $O(1/\log m)$ fraction of $\mathrm{OPT}$.
• Zero-revenue in standard competitive equilibrium: With $m > n$, all consumers unit-demand and value 1 per item, the unique Walrasian equilibrium sets all prices to zero: revenue is zero.

Tightness in Oligopoly Bundling

If the value distributions violate price-sensitivity or variance suppositions, there exist settings where:

• Some partition-bundle menu yields revenue $\Theta(m)$ in every equilibrium,
• But every grand-bundle pricing yields only $O(1)$ revenue in any equilibrium.

Constructed distributions have extremely rare, high-valued items so grand bundles almost never sell, but partition menus exploit the rare coincidences, demonstrating failure of simple menu designs without suitable distributional regularities (Babaioff et al., 19 Jun 2024).

4. Mechanisms Attaining Minimal-Revenue States

The minimal-revenue equilibrium is always achieved through explicit constructions tied to the market’s structure:

Context	Mechanism type	Features
CBE (matroid/GS classes)	High-demand priced bundling	Bucketing items, reserve pricing, O(m) bundles, at most $\Theta(\log m)$ welfare fraction captured
Oligopoly with rivals	Grand-bundle pricing or partition	Menu design constrained by anti-concentration and price-sensitivity of $F_i$; minimal revenue tight to $B$

Technical Devices

• High-demand priced bundling: Cluster items by value into $k$ bundles, each priced $p$ so at least $k$ consumers strictly demand the bundle, yielding revenue $kp$.
• Extra-consumer solutions: Introduce a hypothetical $(n+1)$-st consumer to enforce a reserve price; derive a full CBE by reallocation.

In all constructions, the inability of any single price point (or menu) to extract more than a $1/\log m$ fraction of OPT, due to variance or tail/heavy-tailed value distribution, underpins the minimal revenue outcome.

5. Economic Interpretation and Impact

Bundling serves as a mechanism to elevate the principal’s revenue floor above zero, which is frequently the outcome in traditional item-pricing competitive equilibrium due to supply-demand mismatch or negligible marginal willingness-to-pay. By collapsing goods into fewer bundles or leveraging buyer complementarities, bundling can forestall undercutting and force positive reserve prices on packages, guaranteeing revenue proportional to the aggregate of truncated values.

In markets where competitive equilibrium revenue is potentially zero, bundling—or its analog in oligopoly menu design—systematically ensures minimal revenue bounded below by $\Omega(\mathrm{OPT}/\log m)$ (Dobzinski et al., 2014), or fraction $1/3$ of the truncated welfare benchmark under distributional restrictions (Babaioff et al., 19 Jun 2024). However, such guarantees are envelope bounds; for structurally adverse distributions, even sophisticated menus can suffer unboundedly poor minimal-revenue performance.

6. Algorithmic and Complexity Considerations

Finding minimal-revenue equilibrium states is generally computationally tractable for the explicit constructions outlined:

• In CBE under gross substitutes, matroid, and multi-unit cases, the requisite bundlings and price settings are effectively constructible.
• In cost-minimization for equilibrium transitions in normal-form games, computing the minimal reward is NP-complete, APX-hard in general, but tractable for constant numbers of strategies or players; additive-approximation algorithms exist for arbitrary game profiles (Huang et al., 2023).

A plausible implication is that, barring pathological value distributions, minimal-revenue states can generally be calculated and operationalized in practical market designs when the gross substitutes structure or appropriate regularity is present.

7. Broader Context and Analytical Significance

Minimal-revenue equilibrium states provide sharp lower bounds for revenue extraction in both combinatorial and competitive pricing environments. They clarify how market design, competition, and valuation structure limit achievable principal revenue, distinguishing between mechanism-induced and purely competitive revenue collapse. These results connect to classical questions in mathematical economics regarding the limits of “market clearance” and the effectiveness of aggregation mechanisms in extracting surplus. Their construction and analysis also inform the design of robust auction menus and market platforms, highlighting the importance of value distribution regularity, bundling, and the mathematical structure of agent preferences in determining revenue guarantees.

These advances collectively frame the paper of minimal-revenue equilibrium states as a fundamental inquiry into the lower boundary of revenue viability in diverse equilibrium concepts and market architectures.