# Envy, Regret, and Social Welfare Loss

## TL;DR

This paper introduces a novel metric, IC-Envy, to evaluate incentive compatibility in auctions, showing it can be computed efficiently and used to bound social welfare loss and predict incentive issues.

## Contribution

It proposes IC-Envy as a new, efficient metric for incentive compatibility, establishing theoretical bounds and demonstrating its predictive power beyond existing methods.

## Key findings

- IC-Envy is greater than or equal to IC-Regret in position auctions.
- IC-Envy can be used to bound social welfare loss due to misreports.
- Using IC-Envy as a feature improves prediction of IC-Regret in various auction environments.

## Abstract

Incentive compatibility (IC) is one of the most fundamental properties of an auction mechanism, including those used for online advertising. Recent methods by Feng et al. and Lahaie et al. show that counterfactual runs of the auction mechanism with different bids can be used to determine whether an auction is IC. In this paper we show that a similar result can be obtained by looking at the advertisers' envy, which can be computed with one single execution of the auction. We introduce two metrics to evaluate the incentive-compatibility of an auction: IC-Regret and IC-Envy. For position auction environments, we show that for a large class of pricing schemes (which includes e.g. VCG and GSP), IC-Envy $\ge$ IC-Regret (and IC-Envy = IC-Regret when bids are distinct). We consider non-separable discounts in the Ad Types environment of Colini-Baldeschi et al. where we show that for a generalization of GSP also IC-Envy $\ge$ IC-Regret. Our final theoretical result is that in all these settings IC-Envy be used to bound the loss in social welfare due advertiser misreports. Finally, we show that IC-Envy is useful as a feature to predict IC-Regret in auction environments beyond the ones for which we show theoretical results. In particular, using IC-Envy yields better results than training models using only price and value features.