Cooperative Advertising Optimization Model

• Cooperative advertising optimization models are frameworks for joint budget allocation and coordinated bidding that integrate entropic regularization and game-theoretic design.
• They employ Sinkhorn-style matrix scaling to enforce bilateral budget and channel capacity constraints, ensuring market equilibrium and robust exploration.
• Empirical evaluations demonstrate significant improvements in conversions, reduced cost-per-conversion, and enhanced revenue in large-scale ad campaigns.

Cooperative advertising optimization models provide rigorous methodologies for joint budget allocation, coordinated bidding, and mechanism design in environments where multiple advertisers seek to maximize conversions, revenue, or overall utility. Such models explicitly address cross-channel dynamics, competitive interactions between campaigns, and strategic multi-agent cooperation within large-scale online advertising platforms. Recent developments integrate entropic regularization, game-theoretic mechanism design, and scalable matrix-scaling solvers to harmonize individual advertiser objectives with global platform goals, yielding substantial improvements in conversion efficiency, cost-per-conversion, and long-term platform profitability (Shen et al., 2023).

1. Mathematical Formulation and Objective

The foundational cooperative advertising optimization model posits $N$ ad campaigns, each with daily budgets $b_i$, distributed over $M$ channels with capacity limits $h_j$. The allocation matrix $P \in \mathbb{R}_+^{N \times M}$ assigns $P_{i,j}$ units of budget from campaign $i$ to channel $j$. The cost-per-conversion (CPC) for each campaign-channel pair is $C_{i,j}$, with total conversions equivalent to minimizing the dollar-weighted CPC:

$$\min_{P \ge 0} \left[ \sum_{i=1}^N \sum_{j=1}^M P_{i,j} C_{i,j} - \varepsilon H(P) \right]$$

Where the entropy regularizer is

$b_i$0

$b_i$1 adjusts the trade-off between pure optimality and allocation smoothness. The bilateral constraints enforce complete budget use and channel saturation:

$b_i$2

This entropic optimal transport formulation ensures market equilibrium, mitigating budget over-concentration and promoting robust exploration across poorly estimated or low-conversion channels (Shen et al., 2023).

2. Advertiser Competition Modeling

Unlike classical local optimization, cooperative models impose global coordination, explicitly capturing market competition by enforcing channel capacity constraints. This simultaneous optimization across all campaigns ensures that no channel is over-allocated, even when multiple advertisers prefer dominant high-performance channels. The formulation achieves a market-maker objective that balances aggregate welfare against individual incentives; campaigns adjust allocation strategies in response to the global optimum, providing a de facto cooperative fixed point. The absence of explicit Nash equilibrium computation underscores the model's divergence from traditional noncooperative game-theoretic methods, with future research encouraged to analyze Price of Anarchy implications (Shen et al., 2023).

3. Sinkhorn-style Matrix Scaling Algorithm

The computational backbone of the cooperative model is a Sinkhorn-type matrix scaling procedure, used to efficiently compute the entropic OT solution at massive scale $b_i$3. The Gibbs kernel is defined as

$b_i$4

Dual variables $b_i$5, $b_i$6 satisfy

$b_i$7

with iterative updates:

$b_i$8

One iterates until both row and column sums of $b_i$9 meet their respective marginal constraints within a small tolerance $M$0. This entropic-regularized formulation guarantees $M$1-strong convexity and rapid convergence (typically 20–50 iterations) robust to numerical instability and data sparsity (Shen et al., 2023).

4. Implementation and Scalability

Per-iteration complexity is $M$2, with practical exploitation of sparsity in $M$3 (reflecting campaign-channel inactivity) and GPU or distributed architectures for parallelization. Estimation of $M$4 is stabilized by mixing sparse conversion observations with model-based pCVR and priors, while $M$5 values are derived from offline, unconstrained auction simulations over a rolling $M$6-day window. Once computed, the resulting allocation matrix $M$7 is deployed as precomputed budget-share weights, ensuring negligible incremental computational cost in live auctions (Shen et al., 2023).

5. Empirical Evaluation and Results

Offline Validation

Tested on 200,000 campaigns across billions of auctions, AdCob yields substantial conversion and cost improvements over baselines such as FCFS and single-advertiser unified allocation:

Algorithm	Revenue	Conversions	CPC
FCFS (baseline)	1.000	1.000	1.000
80% unified allocation	0.965	0.910	1.052
AdCob ($M$8)	1.029	1.191	0.864

AdCob achieves $M$9 conversions, $h_j$0 CPC, and $h_j$1 revenue relative to FCFS, with larger $h_j$2 values yielding more uniform allocations at the expense of CPC (Shen et al., 2023).

Online A/B Bucketing

Live A/B bucketed budget management demonstrates AdCob's robust performance:

Metric	Control (A)	AdCob (B)
Clicks	1.000	0.968 ($h_j$33.2%)
Conversions	1.000	1.246 (+24.6%)
CPC	1.000	0.824 ($h_j$417.6%)

Budget bucketing averts traffic divergence and confirms theoretical gains: higher CVR-concentrated budgets with reduced click-through but substantially improved conversion rates and lower advertising costs (Shen et al., 2023).

6. Impact and Strategic Implications

Cooperative advertising optimization frameworks such as AdCob establish principled methods for allocating budgets in environments with complex cross-channel interdependencies and competitive advertiser incentives. Their market-wide coordination yields improvements in conversion efficiency and CPC without diminishing platform revenue, refuting the conventional assumption that cooperation inevitably reduces auctioneer profit. The entropic-OT approach further provides remarkable scalability and numerical stability, facilitating daily or intraday optimization across massive campaign portfolios. The generalization of local greedy algorithms to joint global optimization has empirically proven benefits for both advertisers and platform operators (Shen et al., 2023).

7. Extensions and Related Models

Research on cooperative advertising optimization continues to expand towards multi-agent, multi-objective environments involving reinforcement learning, mean-field control, and game-theoretic mechanism design. Extensions include joint auctions for brand–store bundles, mean-field delay models for goodwill dynamics, and multi-agent MARL frameworks integrating reward sharing, incentive compatibility, and mixed cooperative–competitive paradigms. These advances marshal the theoretical underpinnings and empirical effectiveness of cooperative strategies across both search and recommendation domains, positioning cooperative optimization as a cornerstone of modern advertising platform science (Zhang et al., 2024, Gozzi et al., 2024, Guan et al., 2021).