MenuNet: Neural Mechanism Design

• MenuNet is a unified neural network framework for designing exactly incentive-compatible, revenue-optimal multi-item auction mechanisms.
• It employs a modular two-network architecture, where the Mechanism Network generates allocation-payment menus and the Buyer Network enforces choice-based incentive compatibility via softmax.
• The framework achieves provable optimality and scalability, outperforming traditional LP solvers by recovering near-exact revenue with significantly reduced computation times.

MenuNet is a unified neural network–based framework for the automated design of exactly incentive-compatible, revenue-optimal (or near-optimal) multi-item mechanisms. It addresses key limitations in prior approaches to mechanism design—specifically, representational constraints, lack of exactness in truthfulness or optimality, and domain dependence—by leveraging a modular neural architecture that enforces incentive compatibility (IC) as a consequence of rational buyer choice. MenuNet applies to a range of multi-item auction contexts, including scenarios for which the theoretical optimum was previously unknown, and is additionally the first neural network mechanism design framework to discover and certify optimal multi-item auctions with provable guarantees (Shen et al., 2018).

1. Architecture and Mechanism Representation

MenuNet embodies a two-network architecture:

• The Mechanism Network receives a constant 1-dimensional input and outputs a menu of $k$ items, each specified by an allocation–payment pair $(x_j, p_j)$. Allocation vectors $X \in [0,1]^{m \times k}$ denote item probabilities across menus for $m$ items, with the final column as the mandatory exit option $(x_k = 0)$. The payment vector $p \in \mathbb{R}^k_{\geq 0}$ is constrained by $p_k = 0$.
• The Buyer Network takes the complete menu $\{(x_j, p_j)\}_{j=1}^k$ as input and, for every discretized buyer type $v = (v_1, ..., v_m)$ on a Cartesian grid $V$, produces a distribution $s(v) \in \Delta^k$ over menu items.

For additive, independent valuations, buyer types are represented as value tensors $\mathscr{V}_i$, multiplied (broadcasted) with mechanisms to yield allocation tensors, payments are broadcasted, and utilities $$\mathscr{U}(v, j) = \sum_{i=1}^m \mathscr{X}_i(v, j) - \mathscr{P}(v, j)$$ are calculated per menu per buyer type. A softmax with small temperature $\tau$ yields near-argmax selection behavior.

2. Incentive Compatibility and Individual Rationality

In classical direct mechanisms, incentive compatibility (IC) requires that each buyer type $v$ obtains maximal utility by reporting truthfully, i.e.,

$$v \in \arg\max_{v'} u(x(v'), p(v'); v), \quad u(x, p; v) = v \cdot x - p$$

with individual rationality (IR) satisfied if $u(x(v), p(v); v) \geq 0$. MenuNet operationalizes a finite menu $\{(x_j, p_j)\}$; rational buyers select the menu item maximizing $u(x_j, p_j; v)$. Thus, for every type $v$, the buyer choice $s(v) = \arg\max_j \{ v \cdot x_j - p_j \}$ is incentive compatible and individually rational by construction, enforced precisely via the softmax-based buyer network.

3. Revenue Maximization Objective and Loss Construction

Seller revenue under MenuNet is expressed as

$$\operatorname{Rev} = \mathbb{E}_{v \sim F}[p(s(v))] = \sum_{v \in V} \Pr[v] \sum_{j=1}^k p_j s_j(v)$$

with training targeted on maximizing revenue, equivalently minimizing the loss

$$\operatorname{Loss} = -\operatorname{Rev} = -\sum_{v \in V} \Pr[v]\; p^T s(v)$$

No explicit penalty terms for IC enforcement are required; the construction guarantees incentive compatibility.

4. Training Methodology

The training protocol comprises:

• Discretization of each value dimension into $d_i$ points, forming grid $V$.
• Mechanism network parameterization ($\theta$) and buyer network configuration, with the buyer network either parameter-free (in the known-utility case) or empirically inferred from behavioral data.
• Batched forward-propagation of constant input, producing $(X(\theta), p(\theta))$, utility tensor calculation, softmax-based choice distribution generation, and loss computation.
• Backpropagation of $\nabla_\theta\operatorname{Loss}$ and optimizer-based parameter updates (Adam, SGD).
• Use of fine grid discretizations ($N \sim 100$ per dimension) and high softmax temperature for near-exact argmax behavior.
• Only standard regularization (weight decay, early stopping) is applied.

5. Provable Optimality Guarantees

MenuNet provides certifiable optimality in several canonical settings:

• Uniform Triangle Distribution: For values $v = (v_1, v_2)$ uniform on $T_c = \{ (v_1, v_2) \geq 0 : v_1/c + v_2 \leq 1 \}$, $c \geq 1$.
  • If $c > 4/3$, the three-item menu is optimal, achieving revenue $\frac{2}{27}(4 + c + \sqrt{c(c-1)})$.
  • If $1 \leq c \leq 4/3$, the optimal menu is a two-item posted price, yielding revenue $\sqrt{c/3}$.
  • Duality-based proofs confirm optimality via optimal transport couplings aligned with menu region selection.
• Uniform Square with Restricted Menu Size: For $v \sim U[0,1]^2$, the true optimum for menus restricted to three items is the asymmetric solution (items: $(0,0), (1,0), (1,1)$ with payments $0, \frac{2}{3}, \frac{5}{6}$), attaining revenue $59/108 \approx 0.5463$, strictly exceeding the symmetric menu benchmark ($2\sqrt{6}/9 \approx 0.5443$).

Menu parameterization and region partitioning, supported by symmetry reductions and monotonicity results, permit tractable revenue maximization and unique identification of optima.

6. Experimental Evaluation and Empirical Performance

MenuNet’s scalability and efficacy are established through systematic experimentation across diverse auction settings:

• Tested distributions include $U[0,1]^2$, $U[0,1] \times [0, c]$ (various $c$), correlated triangles, unit-demand allocations, combinatorial value functions, and deterministic binary allocations.
• Benchmarking utilizes closed-form optima (Myerson, Daskalakis et al.) and LP solvers on discretized grids ($N \leq 30$).
• Metrics span absolute revenue, optimality ratio ($\operatorname{Rev}/\operatorname{OptRev}$), runtime, and convergence iterations.

Distribution	MenuNet Revenue	True Optimum	Optimality Ratio
$U[0,1]^2$	0.5491989	$\frac{12+2\sqrt{2}}{27} \approx 0.5492$	$\geq 99.9996\%$
$U[0,1] \times [0,1.5]$	0.6838542	$\frac{15+2\sqrt{3}}{27}$	$\geq 99.9997\%$
$U[0,1]^2$, menu size $\leq 3$	0.5462947	$59/108 \approx 0.546296$	$\geq 99.9997\%$

Off-the-shelf Python LPs require approximately 50 minutes for $N=30$ grids, whereas MenuNet completes $N=200$ discretizations in under five minutes. Iteration time ranges $1$–$30$ ms, with typical convergence (error $< 10^{-4}$) within $O(10^3)$ iterations. In every setting, MenuNet recovers exactly IC mechanisms with revenue matching theoretical optima up to $10^{-4}$ relative error.

7. Domain Agnosticism and Scalability

MenuNet operates independently of specific auction domain knowledge; its neural architecture is generic across settings. Scalability derives from linear complexity in grid size, contrasting with the quadratic expense of LP solvers, and MenuNet is not restricted by mechanism representation, truthfulness constraints, or environment dependence. This enables the automated discovery and certification of optimal multi-item mechanisms in instances where existing theoretical analysis has proven challenging or incomplete (Shen et al., 2018).

A plausible implication is that the methodology may extend readily to other mechanism domains, provided buyer utility and incentive compatibility can be encoded similarly.