Binary Constrained Bidding (BiCB)

• Binary Constrained Bidding (BiCB) is an auction mechanism where bidders communicate using only a few discrete signals, typically binary, to convey their valuation thresholds.
• It employs threshold strategies and asymmetric, dynamic programming methods, leveraging SDP relaxations and branch-and-bound techniques to manage computational complexity.
• BiCB offers practical insights for real-time, low-bandwidth applications like online advertising, ensuring near-optimal welfare despite severe communication constraints.

Binary Constrained Bidding (BiCB) refers to auction and market design settings in which each bidder communicates only a small and discrete set of possible bids, often as few as two, due to communication, computational, or practical constraints. BiCB models are highly relevant in environments with restricted bandwidth, privacy considerations, or real-time decision needs, and have catalyzed theoretical and algorithmic advances in mechanism design, bidding games, and optimization. The foundational property of BiCB is that bidders—whose valuations may be drawn from a continuum—are restricted to coarse “binary” signals (for example, a single bit indicating high or low interest, or a finite menu of bid thresholds), leading to distinctive equilibrium, efficiency, and strategic phenomena.

1. Communication Constraints and Mechanism Design

BiCB is grounded in the paper of auctions under severely bounded communication, where each bidder transmits $t$ bits, allowing only $k = 2^t$ possible bids. In the minimal case ($t = 1$), a bidder has two possible bids, a framework termed “binary constrained bidding” (Blumrosen et al., 2011). Mechanism design under such constraints reveals several striking features:

• Threshold strategies: Optimal mechanisms induce threshold-based bidding, wherein bidders partition their valuation space and report only the interval (threshold) containing their value.
• Priority games: For two bidders, the optimal mechanism (for both welfare and revenue) is a priority game. The higher message wins; ties are broken via fixed priority. Modified variants allow the seller to retain the object under certain conditions (e.g., high reservation value).
• Mutually centered thresholds: Threshold values are determined by conditional expectations of opponents’ values:

$$x_{(i)} = E[v_B\,|\, y_{(i-1)} < v_B < y_{(i)}] \qquad y_{(i)} = E[v_A\,|\, x_{(i-1)} < v_A < x_{(i)}]$$

guaranteeing sharp demarcation of winning and losing regions.

Performance loss from full to binary communication is shown to be mild; with $k = 2$ bids per agent, welfare is about $97\%$ of the unconstrained optimum (e.g., $0.648$ vs $0.667$ for uniform valuations). This loss diminishes rapidly as $k$ increases, specifically as $O(1/k^2)$ (Blumrosen et al., 2011).

2. Asymmetry, Equilibria, and Strategic Complexity

Contrary to intuition, optimal BiCB mechanisms often favor asymmetry:

• Asymmetric priority and payment rules: Even for ex-ante identical bidders, optimal mechanisms assign non-anonymous priority and payment schedules. Welfare for the optimal asymmetric mechanism exceeds that of the best symmetric one (e.g., $0.648$ vs $0.625$ for two bidders, uniform valuations) (Blumrosen et al., 2011).
• Binary bidding games: Extensive-form general-sum games in which each node presents up to two possible actions (binary) have rich equilibrium structure. Every binary bidding game admits a pure subgame-perfect equilibrium (PSPE), organized into a semilattice with a unique Bottom Equilibrium (1311.0913).
  • Bottom Equilibrium exhibits near-tied bids, monotonicity in budget allocation, Pareto-optimality, and Pareto-surjectivity (every Pareto-efficient outcome is attainable)—properties not generically present with richer bidding languages.
  • In combinatorial bargaining, a player with $X\%$ of the "scrip-budget" is guaranteed an allocation she weakly prefers to at least $X\%$ of all feasible allocations (1311.0913).

3. Dynamic and Sequential Auction Settings

Sequential auction environments, including those with complementarities or budget constraints, pose significant computational challenges for optimal bidding strategy determination:

• Quasi-linear dynamic programming: Reformulating the utility as $v(R_S) - \sum_{r \in R_S} z_r$ and treating payments as state-transition costs enables dynamic programming solutions with dramatically reduced state space ($O(2^n)$ rather than $O(m \times 2^n)$ for additive formulations), yielding an $m$-fold speed-up (Hattori et al., 2013).
• Semi-optimal proration for budget constraints: When hard budget caps are present, a two-step procedure computes an unconstrained optimal strategy, then prunes bids to respect the remaining budget via:

$$Z_\text{mar} = Z_\text{opt} \times \frac{Z_\text{bud} - Z_\text{past}}{Z_\text{pre} + Z_\text{opt}}$$

This maintains near-optimal welfare while limiting the risk associated with incompatible bundles (Hattori et al., 2013).

• Sequential vs simultaneous mechanisms: Multi-round mechanisms extract only a linear reduction in required communication per bidder compared to simultaneous models; for $m$ bits of total communication and $n$ bidders, performance matches that of a simultaneous mechanism with $m/n$ bits per bidder (Blumrosen et al., 2011).

4. Computation of Equilibria and Algorithmic Solutions

BiCB frequently leads to decision and allocation problems formulated over binary variables. Advanced solvers and theoretical analyses address the computational tractability:

• Max-Cut transformation and SDP relaxations: Binary quadratic problem (BQP) instances, representative of BiCB mechanisms, can be converted to equivalent Max-Cut problems via exact penalty reformulations and linear augmented variables (Gusmeroli et al., 2020).
• Branch-and-bound algorithms: Tighter bounding via strengthened semidefinite programs (SDPs) with hypermetric inequalities (triangle, pentagonal, heptagonal) enables practical solution of large NP-hard BiCB-type problems. Parallelization (e.g., MPI-based coordinator–worker models) achieves scalability to hundreds of cores (Gusmeroli et al., 2020).
• Polynomial-time algorithms in binary bidding games: Interval-profile properties of binary extensive-form bidding games allow for polynomial-time computation of equilibria, leveraging recursive ascending-auction simulations (1311.0913).

5. Budget Constraints and Welfare Efficiency

The introduction of budget caps to BiCB significantly alters equilibrium outcomes and welfare properties:

• Coupled allocation-pricing complexity: In combinatorial exchanges, budget constraints yield $\Sigma_2^p$-hard allocation and pricing problems, requiring mixed-integer bilevel linear programs (MIBLPs) to compute core or least-core prices (Bichler et al., 2018).
• Liquid welfare and price of anarchy: In simple per-item allocation mechanisms, with agents bidding only “binary” bids for each item and respecting budget caps, the liquid welfare (sum of $\min\{v_i(X_i), c_i\}$ across agents) serves as the efficiency benchmark. Price of anarchy and price of stability are tightly bounded by $2$ in such settings, including under binary or per-item bid constraints (Voudouris, 2020).
• Sequential pruning strategies: Focusing on limited coalition sizes (e.g., restricting blocking coalitions to at most 3 or 5 members) enables computation of stable outcomes within practical runtime, even when the theoretical core may be empty (Bichler et al., 2018).

6. Strategic Behaviors and Bidding Language Design

BiCB settings constrain the expressivity of bids but also restrict strategic manipulations:

• Simple vs complex bidding in combinatorial auctions: Restricting bids to those corresponding directly to genuine interest may be strictly suboptimal. Complex bidding (bids on unrelated bundles) in first-price or VCG-nearest mechanisms enables conditional exploitation of the winner-determination process (Bosshard et al., 2020). In certain cases, optimal complex bidding requires exponentially many XOR bids per agent; restricting bidding languages to binary alternatives trades expressivity for tractability.
• Demand reduction in uniform price auctions: In BiCB models where each bidder submits a single “lumpy” bid for total quantity, strategic demand reduction occurs: bidders may forgo units to achieve a lower equilibrium price, resulting in low-price equilibria distinct from those seen under richer bidding schemes (Yoon, 6 Sep 2024).

7. Practical Applications and Emerging Directions

Binary Constrained Bidding is well suited for contemporary high-frequency, communication-limited decision environments:

• Real-time auto-bidding in live advertising: BiCB algorithms combine LP-derived optimal bid formulas, dual analysis, and future traffic prediction to maximize campaign effects under budget and cost constraints (Yang et al., 8 Aug 2025). Projected gradient descent and lightweight regression models (e.g., LightGBM) enable fast, low-overhead computation; empirical studies demonstrate constraint satisfaction and improved revenue performance.
• Online learning with binary feedback and prediction: Advanced bandit algorithms in first-price auction settings leverage external predictions of the highest competing bid, achieving zero regret when predictions are accurate and bounded sublinear regret otherwise, using the BROAD-OMD mirror descent framework (Tandiary, 18 Jun 2025).

These results confirm that binary constrained bidding mechanisms—with tailored priority and threshold strategies, scalable algorithmic solutions, and interface to prediction and control platforms—provide near-optimal efficiency in practice, despite substantial reductions in expressive power and information requirements. This positions BiCB as a foundational approach in communication-efficient mechanism design, discrete games, and on-line optimization.