# Online Learning for Measuring Incentive Compatibility in Ad Auctions

## TL;DR

This paper develops online algorithms to accurately measure incentive compatibility regret in ad auctions, providing theoretical guarantees and validation through simulations, addressing both known valuation and worst-case scenarios.

## Contribution

It introduces Regret-UCB algorithms for online IC regret estimation in auction mechanisms, with improved error bounds and extensions for stronger IC regret assessment.

## Key findings

- Error bounds decrease as 1/T with more auctions per step
- Algorithms effectively estimate IC regret in GSP auctions
- Extensions improve estimation accuracy for stronger IC regret

## Abstract

In this paper we investigate the problem of measuring end-to-end Incentive Compatibility (IC) regret given black-box access to an auction mechanism. Our goal is to 1) compute an estimate for IC regret in an auction, 2) provide a measure of certainty around the estimate of IC regret, and 3) minimize the time it takes to arrive at an accurate estimate. We consider two main problems, with different informational assumptions: In the \emph{advertiser problem} the goal is to measure IC regret for some known valuation $v$, while in the more general \emph{demand-side platform (DSP) problem} we wish to determine the worst-case IC regret over all possible valuations. The problems are naturally phrased in an online learning model and we design $Regret-UCB$ algorithms for both problems. We give an online learning algorithm where for the advertiser problem the error of determining IC shrinks as $O\Big(\frac{|B|}{T}\cdot\Big(\frac{\ln T}{n} + \sqrt{\frac{\ln T}{n}}\Big)\Big)$ (where $B$ is the finite set of bids, $T$ is the number of time steps, and $n$ is number of auctions per time step), and for the DSP problem it shrinks as $O\Big(\frac{|B|}{T}\cdot\Big( \frac{|B|\ln T}{n} + \sqrt{\frac{|B|\ln T}{n}}\Big)\Big)$. For the DSP problem, we also consider stronger IC regret estimation and extend our $Regret-UCB$ algorithm to achieve better IC regret error. We validate the theoretical results using simulations with Generalized Second Price (GSP) auctions, which are known to not be incentive compatible and thus have strictly positive IC regret.