Inside
MFG-RegretNet.
Turn private preferences into coordinated contributions. Explore how population context, neural contracts, and incentive constraints work together.
Explore the method ↓A mean-field game studies how clients respond to a fixed mechanism. MFG-RegretNet learns the mechanism itself through allocation, payment, and regret penalties. These are different optimization loops.
Many reports. A compact view of the market.
Each client has a private cost and a privacy capacity. Their reports form an empirical population; a feature mean supplies context to the network.
Population influence shrinks with N. This coordinate changes by Δb/N. The same report can still directly change its own allocation and payment; small population influence alone does not imply small regret.
State distribution, report distribution, and network input
The game models a type/state law μs. A reporting policy π maps those types into reports, producing a report law μb and features m = E[φ(b)]. The deployed AHN retains the full bid profile and a repeated mean-bid vector, together with B. The mean is additional context, not a proven sufficient statistic. This teaching example varies just the cost coordinate.
An equilibrium has to agree with itself.
Hold the mechanism fixed. Clients choose their best reports for a given population. Those choices must reproduce the population used to make them.
Make the feedback loop move.
Three equally likely types prefer an action near their type plus a population effect. Increasing λ makes that feedback stronger.
Each step best-responds, then moves the field 60% toward the mean response. For this toy, λ < 1 makes the map a contraction.
Nash means no profitable unilateral change. A mean-field equilibrium combines optimal responses with population consistency. Transferring it to a finite network also needs sampling and representation error control. Equilibrium reports need not be truthful.
Where HJB, Fokker–Planck, and the neural network fit
The theoretical game
HJB describes an individual's value and optimal response given the population; Fokker–Planck describes the evolving state law. Their consistency defines a dynamic MFE. Under the manuscript's R5 assumption, type dynamics are exogenous, so a bid does not control that state evolution and the response reduces to one-round utility maximization.
The deployed mechanism
MFG-RegretNet learns allocation and payment from bids and context. It does not numerically solve the HJB–FP system. The interactive quadratic-utility game above is a separate, explicitly defined example of fixed-point reasoning, not a paper experiment or the proposed auction.
One input. Two learned decisions.
The network decides how much privacy to procure and how much to offer. A budget projection constrains the total proposed payment for every input.
p̄ᵢ = pᵢ / max(1, Σⱼpⱼ / B)Preserves nonnegative payments and caps their sum.Teach the mechanism to resist misreports.
Optimize utility differences, not a bid-dependent critic. Strong deviation search exposes profitable reports. The largest gain found is a lower bound on true regret; a formal upper bound needs a valid certificate and independent evaluation.
Compensation meets a local decision.
A participant weighs the offered payment against its true privacy cost. It releases an update only if the offer also passes its capacity and cumulative-ledger checks.
Participation and truthfulness are separate. Nonnegative realized utility supports voluntary participation. Regret control addresses whether a different report could earn more. Refusal alone does not establish incentive compatibility.
What should an “optimal” allocation optimize?
More privacy can reduce noise, but unequal contributions can distort the aggregate. A finite teaching example makes this trade-off visible under a fixed financial budget.
Balance representation and noise.
Each ε is in {0, 0.25, 0.5, 0.75, 1}. Costs are [1, 2, 3]; target data shares π are [0.5, 0.3, 0.2]. The budget pays known cost ∑viεi.
Optimal here means best among the displayed grid candidates. The real procurement goal is FL utility subject to incentives, participation, budget, and privacy. This known-cost oracle isolates allocation; it does not implement truthful payments or measure model accuracy.
How this relates to the learned objective
The manuscript's evaluated neural configuration uses a purchased-privacy surrogate with regret and participation penalties. Jκ above comes from the P2 objective development and separates weight mismatch from the noise contribution. A learned network only approximates its chosen training objective; lower surrogate loss alone does not establish globally optimal FL accuracy or superiority to a baseline. Zero procurement has infinite objective in this example.
From a private bid to a shared detector.
Collect reports → form population context → infer allocations and payments → validate locally → train and release → aggregate and settle.