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Nash Incentive-compatible Online Mechanism Learning via Weakly Differentially Private Online Learning
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Proposes a Nash incentive-compatible online mechanism learner that mixes Hedge (which is weakly DP) with a penalizing commitment mechanism, achieving O(T^{(1+h)/2}) regret for application objectives like facility location and VCG reserve pricing.
Problem
Online mechanism design where agents are non-myopic and strategic across rounds, so per-round incentive compatibility does not ensure incentive compatibility over the full learning process, while the mechanism must learn an objective-maximizing mechanism without knowing type distributions.
Approach
Randomly mix, each round, between a Hedge recommendation (shown to be a weak eta-DP sequence per round) and a strictly DSIC commitment mechanism that penalizes misreports; a backward-induction argument shows truth-telling is a Nash equilibrium whenever the commitment penalty dominates the agents' long-sightedness times the DP gain.
Key results
- Hedge-based NICOM achieves regret O((log|Pi| + beta^{-1}) sqrt(alpha_T T)), which is O(T^{(1+h)/2}) when long-sightedness alpha_T is O(T^h).
- Lemma: Hedge is a weak 4eta-DP sequence, so the exponential mechanism's DP guarantee suffices for NIC.
- Applications: facility location regret O((k log m + m^2) T^{(1+h)/2}); VCG with bidder-specific reserves regret O((n log m + m^2) T^{(1+h)/2}).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Combine a weakly differentially private recommendation sequence with a commitment mechanism | Online mechanism learning · Adversarial online mechanism learning | Nash incentive compatibility: \frac{\lambda}{4\eta}\beta(\picom)\geq\alpha(A) | Helped |
| Use Hedge as the weakly differentially private recommendation sequence | Hedge NICOM · Adversarial online mechanism learning | weak differential privacy parameter: weak $4\eta$-DP sequence | Helped |
| Use Hedge with a commitment mechanism and suitable parameters | Hedge NICOM · Adversarial online mechanism learning | regret: O\big((\log|\Pi|+\beta(\picom)^{-1})\cdot\sqrt{\alpha_T(\A)\,T}\big) | Helped |
| Use the commitment mechanism with penalty gap at least $m^{-2}$ | Online facility location NICOM · Online facility location | regret: O\!\rbra{ (k\log m + m^2)\cdot T^{\frac{1+h}{2}}} | Helped |
| Use the commitment mechanism with penalty gap at least $(2m)^{-2}$ | VCG NICOM · VCG with bidder-specific reserve prices | regret: O\!\rbra{(n\log m + m^2)\cdot T^{\frac{1+h}{2}}} | Helped |
| Use the resource-allocation commitment mechanism with penalty gap at least $(2nk)^{-1}$ | Resource allocation NICOM · Online optimal resource allocation | regret: O\Big( (n\log k + nk)\cdot T^{\frac{1+h}{2}} \Big) | Helped |
Limitations
- Requires exact commitment mechanisms, which assume discrete agent types; continuous types need approximate IC extension.
- Regret grows with long-sightedness alpha_T; becomes vacuous when agents are linearly long-sighted (h=1).
- Mechanism class must be finite or discretized; individual rationality not explicitly analyzed.
Experimental setup
Metrics: regret penalty gap
Baselines: strongly DP online learning for auctions (Myerson-style online auction) dynamic mechanism design strategic regret posted-price algorithms
Builds on: Hedge (exponential weights) exponential mechanism commitment mechanism (Kearns/Roth-style DP mechanism design) VCG mechanism with reserve prices dynamic mechanism design
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