GemNet: Menu-Based, Strategy-Proof Multi-Bidder Auctions Through Deep Learning

TL;DR

GemNet learns auctions with better revenue than affine maximization methods, achieves exact SP whereas previous general multi-bidder methods are approximately SP, and offers greatly enhanced interpretability.

Abstract

Automated mechanism design (AMD) uses computational methods for mechanism design. Differentiable economics is a form of AMD that uses deep learning to learn mechanism designs and has enabled strong progress in AMD in recent years. Nevertheless, a major open problem has been to learn multi-bidder, general, and fully strategy-proof (SP) auctions. We introduce GEneral Menu-based NETwork (GemNet), which significantly extends the menu-based approach of the single-bidder RochetNet (Dütting et al., 2024) to the multi-bidder setting. The challenge in achieving SP is to learn bidder-independent menus that are feasible, so that the optimal menu choices for each bidder do not over-allocate items when taken together (we call this menu compatibility). GemNet penalizes the failure of menu compatibility during training, and transforms learned menus after training through price changes, by considering a set of discretized bidder values and reasoning about Lipschitz smoothness to guarantee menu compatibility on the entire value space. This approach is general, leaving trained menus that already satisfy menu compatibility undisturbed and reducing to RochetNet for a single bidder. Mixed-integer linear programs are used for menu transforms, and through a number of optimizations enabled by deep learning, including adaptive grids and methods to skip menu elements, we scale to large auction design problems. GemNet learns auctions with better revenue than affine maximization methods, achieves exact SP whereas previous general multi-bidder methods are approximately SP, and offers greatly enhanced interpretability.

Figures (10)

• Figure 1: GemNet is the first differentiable economics method that is generally expressive, truthful (or strategy-proof, DSIC), and supports multiple bidders (and items).
• Figure 2: Illustrating our method for additive valuations. (a) An example of incompatible menus where the utility-maximizing menu elements over-allocate the first item. (b) Prices are adjusted so that the feasible menu element, $[0,1]$ in this case, has greater utility than infeasible elements like $[1,1]$. Menus after price adjustment are compatible at grid points. (c) Extending menu compatibility to off-grid values by introducing safety margins. In this example, the utility of an infeasible bundle $[1,1]$ changes from $1.3 to $2.7 between two grid points. Here, the safety margin represents the gap between the utilities of feasible and infeasible menu elements at grid points preventing their utility ranges from overlapping.
• Figure 3: Learned mechanisms in the setting with two additive bidders, two items, and i.i.d. uniform values on $[0,1]$. Bidder 2's values are set at $(0, 0.6)$, and the x- and y-axis in each subplot is Bidder 1's value for Item 1 and 2. Rows 1 and 2 show the probability of Bidder 1 getting Item 1 and Item 2, respectively, and row 3 shows the price for Bidder 1. Columns represent different methods. Dotted lines in the second column distinguish the pre- and post-transformation mechanisms. Compared to RegretNet, GemNet exhibits a clear decision boundary and thereby improves the interpretability of designed mechanisms. AMenuNet has a large set of types in the left-top region where Bidder 1 receives no items and makes no payments. In contrast, GemNet increases revenue by allocating Item 1 to Bidder 1 within this region, highlighting the improved expressive capacity enabled by our method.
• Figure 4: Three items, additive bidders, i.i.d. uniform values on $[0,1]$. The value of a particular bidder for each of three items varies in each cube, and annotation $G_{\{S\}}$ means the bidder gets items in set $S$ in a region. (a) One bidder. The optimal allocation structure for the 1-bidder case giannakopoulos2014duality. An optimal analytical solution is not known for two or more bidders. (b-d) Three bidders. Fix the values of Bidders 2 and 3 to $v_2=v_3=(0.2,0.2,0.2)$, showing the allocation for Bidder 1 learned by AMenuNet duan2023scalable, RegretNet duetting2023optimal, and GemNet. AMenuNet learns a sub-optimal mechanism, e.g., Bidder 1 gets Item 2 even when its value is 0 (when $v_1(2)=0$, $v_1(1)>0.5$, $v_1(3)>0.2$). It is interesting that GemNet obtains an allocation rule conforming to the optimal structure in the 1-bidder case. (e-f) Three bidders.GemNet adapts the allocation for Bidder 1, while maintaining the highlevel structure, for different values of Bidders 2 and 3.
• Figure 5: Slicing the GemNet mechanism in the auction setting with 2 additive bidders, 2 items, and uniform values on $[0,1]$. 2D Plots: We vary Bidder 1's values, $v_1(1)$ and $v_1(2)$ in each subplot on the x- and y-axis respectively, varying Bidder 2's values, $v_2(1)$ and $v_2(2)$, across each subplot. 3D Plots: We first vary $v_1(1)$, $v_1(2)$, $v_2(1)$ in each subplot on x-, y-, z-axis, respectively, varying $v_2(2)$ across each subplot. We then vary $v_1(1)$, $v_1(2)$, $v_2(2)$ in each subplot, on x-, y-, z-axis, respectively, varying $v_2(1)$ across each subplot.

Theorems & Definitions (7)

• Theorem 1: SP auctions via menus (necessary) hammond1979straightforward
• Definition 2: Menu compatibility
• Theorem 3: SP auctions via menus (sufficient) hammond1979straightforward
• Theorem 4: 2-Bidder Menu Compatibility
• Theorem 5: General $n$-Bidder Menu Compatibility
• Theorem 6: Exact Strategy-Proofness
• Lemma 7: Menu Element Utility Bounds