Optimistic Auto-bidding Incentive Compatibility

• OAIC is a refinement of incentive compatibility in auto-bidding, where the best-case equilibrium outcome under truthful reporting is compared to outcomes from any downward deviation.
• It relaxes the stringent AIC criterion by evaluating the highest equilibrium payoff rather than the worst-case scenario, accommodating advertisers with varying risk attitudes.
• OAIC influences auction design by promoting mechanisms, such as second-price auctions, where truthful reporting guarantees an equilibrium outcome that is at least as favorable as any misreport.

Optimistic Auto-bidding Incentive Compatibility (OAIC) is a refinement of incentive compatibility for auto-bidding environments in which advertisers delegate bidding to agents subject to high-level constraints and the induced auction game may admit multiple equilibria. In this setting, truthful reporting cannot be evaluated by a single deterministic outcome; instead, each report induces a set of possible equilibrium allocations. OAIC evaluates truthfulness through the advertiser’s most favorable equilibrium outcome: a mechanism is OAIC if the best-case equilibrium under truthful reporting is at least as good as the best-case equilibrium under any permitted misreport. The notion was introduced together with Risk-Averse Auto-bidding Incentive Compatibility (RAIC) in order to relax the much stronger Auto-bidding Incentive Compatibility (AIC) criterion and to model advertisers with different attitudes toward equilibrium uncertainty (Liaw et al., 22 Aug 2025).

1. Formal definition in the multi-equilibrium auto-bidding model

The OAIC framework of "Risk-Averse and Optimistic Advertiser Incentive Compatibility in Auto-bidding" studies a set of advertisers $A$ and a set of queries $Q$. Advertiser $i\in A$ has a true tCPA target $T_i$ and values $v_{i,j}$ for each query $j\in Q$. After advertisers report targets $(T_i')_{i\in A}$, their auto-bidders play a second-price auction on each query and reach an equilibrium (Liaw et al., 22 Aug 2025).

For a fixed advertiser $i$, the report $T_i'$ induces the equilibrium allocation set

$\Pi(T_i')^{Eq},$

defined as the set of all allocation vectors $Q$0 that can arise in some auto-bidder equilibrium when advertiser $Q$1 reports $Q$2 and the other advertisers fix their reports. The advertiser’s value under allocation, often called total “lifted welfare,” is

$Q$3

The mechanism is Optimistic Auto-bidding Incentive Compatible if, for every advertiser $Q$4 and every misreport $Q$5,

$Q$6

In words, the best-case equilibrium payoff when advertiser $Q$7 tells the truth is at least as good as the best-case payoff under any downward deviation of its reported target (Liaw et al., 22 Aug 2025).

This definition is explicitly report-relative and equilibrium-set-relative. It does not collapse the strategic problem to a unique bid vector or a unique allocation rule. Instead, it treats equilibrium multiplicity as the primitive obstacle and asks whether truth-telling remains optimal when the advertiser evaluates each report by its most favorable equilibrium realization.

2. Position relative to AIC and RAIC

The central motivation for OAIC is that the original AIC notion of Alimohammadi et al. is very stringent. Under AIC, the worst equilibrium under truth-telling must dominate the best equilibrium under any deviation. Formally, AIC requires

$Q$8

whereas RAIC requires

$Q$9

OAIC instead compares best-case to best-case and is therefore the most permissive of the three notions (Liaw et al., 22 Aug 2025).

Notion	Equilibrium comparison
AIC	truthful worst-case vs. deviation best-case
RAIC	truthful worst-case vs. deviation worst-case
OAIC	truthful best-case vs. deviation best-case

The distinction is not merely technical. The 2025 formulation states that RAIC and OAIC allow a clearer modeling of ordinal preferences for advertisers with differing attitudes towards equilibrium uncertainty (Liaw et al., 22 Aug 2025). An advertiser who is effectively pessimistic about equilibrium selection is better represented by RAIC; an advertiser who evaluates reports by the best equilibrium consistent with them is better represented by OAIC. The original AIC, by contrast, precludes having ordinal preferences on the possible constraints that the advertiser can report and often fails even in standard auction formats.

A notable consequence of this refinement is the reevaluation of second-price auctions. Alimohammadi et al. show that both First-Price Auction and Second-Price Auction are not AIC, yet the refined analysis establishes that SPA satisfies both RAIC and OAIC in the tCPA auto-bidding setting (Liaw et al., 22 Aug 2025). This separates failure of the strongest robustness notion from failure of more behaviorally targeted notions.

3. OAIC in single-slot second-price auctions with non-uniform bidding

A principal positive result is Theorem 3.2: in a single-slot second-price auction (SPA) with arbitrary per-query bids, subject to undominated bids $i\in A$0, SPA is OAIC for tCPA auto-bidders (Liaw et al., 22 Aug 2025).

The proof idea is constructive. Fix advertiser 1’s true target $i\in A$1 and an arbitrary misreport $i\in A$2. Let $i\in A$3 be the best-case equilibrium allocation under the misreport. The argument then constructs a new bid profile consistent with the true target $i\in A$4. For every query lost by advertiser 1 in $i\in A$5, the winner’s bid is set to $i\in A$6; for each query won by advertiser 1 in $i\in A$7 but not profitable under $i\in A$8, the other bidder outbids advertiser 1 by bidding $i\in A$9. This yields an equilibrium $T_i$0 in which advertiser 1 wins at least the same set of queries as in $T_i$1, and hence

$T_i$2

The result establishes OAIC because the best truthful equilibrium weakly dominates the best equilibrium achievable by any downward deviation (Liaw et al., 22 Aug 2025).

The significance of the theorem is methodological as much as substantive. It shows that equilibrium multiplicity need not force the analyst into worst-case reasoning. Under SPA, the existence of multiple equilibria is compatible with a strong optimistic truthfulness guarantee, provided the comparison class is best-case equilibrium payoff rather than worst-case equilibrium payoff.

4. Uniform bidding and the two-advertiser prefix characterization

The second main OAIC theorem concerns uniform bidding. The setup restricts attention to two advertisers, each using a uniform multiplier $T_i$3 with bids

$T_i$4

Queries are sorted so that

$T_i$5

and every equilibrium corresponds to a prefix allocation $T_i$6 in which advertiser 1 wins queries $T_i$7 (Liaw et al., 22 Aug 2025).

Under this restriction, Theorem 4.3 states that SPA remains OAIC. The proof proceeds in two stages. First, it characterizes precisely when each prefix $T_i$8 can arise as an equilibrium for reported targets $T_i$9: existence holds if and only if a set of monotone inequalities in $v_{i,j}$0, $v_{i,j}$1, and prefix sums of $v_{i,j}$2 is satisfied. Second, it shows monotonicity with respect to advertiser 1’s target: if $v_{i,j}$3 is feasible under $v_{i,j}$4, then under the larger true target $v_{i,j}$5 one can find some $v_{i,j}$6 with $v_{i,j}$7 that is still feasible. Consequently, the maximum truthful lifted welfare, corresponding to the largest feasible prefix, exceeds the maximum lifted welfare obtainable under the lower reported target (Liaw et al., 22 Aug 2025).

This two-advertiser prefix structure is analytically important because it converts equilibrium feasibility into an ordered combinatorial object. A plausible implication is that OAIC becomes easier to verify when the equilibrium set admits a lattice-like or monotone representation. The paper itself, however, stops at the two-advertiser uniform-bidding case and identifies extension beyond that regime as open.

5. Illustrative contrast with AIC and the role of equilibrium selection

The paper gives a two-query, two-bidder example to illustrate the difference between OAIC and AIC: bidder 1 has values $v_{i,j}$8, bidder 2 has values $v_{i,j}$9, bidder 1’s true tCPA is $j\in Q$0, and the deviation considered is $j\in Q$1. The example is used to show that OAIC can hold even when the original AIC fails (Liaw et al., 22 Aug 2025).

The conceptual point is straightforward. OAIC asks whether the advertiser can do strictly better, in its most favorable equilibrium, by misreporting. AIC asks a much harsher question: whether even the advertiser’s least favorable truthful equilibrium still dominates the best deviation equilibrium. Because these are distinct comparisons, truthful reporting may survive the optimistic benchmark while failing the worst-case benchmark.

This also clarifies a common misconception. OAIC is not a statement that all truthful equilibria are superior to all deviating equilibria. Nor does it imply that misreporting is never beneficial in some realized equilibrium. The 2025 paper is explicit that OAIC only says that no misreport can strictly improve the very best equilibrium beyond the best truthful equilibrium; it does not prohibit an advertiser from benefiting in some intermediate equilibria by misreporting (Liaw et al., 22 Aug 2025).

The dependence on equilibrium selection is therefore intrinsic, not incidental. OAIC presumes that the advertiser evaluates reports optimistically by the top element of the equilibrium payoff set. If the platform, tie-breaking rule, or learning dynamics selects a different equilibrium, the OAIC guarantee may not predict realized behavior.

6. Limits, design implications, and related research directions

OAIC assumes an optimistic equilibrium selection and offers no guarantee on what happens if the system actually picks a pessimistic equilibrium. Highly risk-averse advertisers may prefer RAIC instead. In addition, both RAIC and OAIC rely on knowing the full set of equilibria $j\in Q$2, which can be complex in richer auction formats. The paper therefore notes a practical design direction: designers may want to engineer single-equilibrium auctions or tie-breaking rules to align actual outcomes with the optimistic equilibrium, thereby realizing the OAIC guarantee. It also identifies open extensions, including analysis with more than two bidders under uniform bidding and settings with budget constraints without tCPA (Liaw et al., 22 Aug 2025).

OAIC also sits within a broader literature on truthful reporting under auto-bidding and financial constraints. Ni and Tang show that, under very mild conditions on the mechanism environment, incentive compatibility for an ex-ante constrained player is characterized by an auto-bidding form: the bidder chooses

$j\in Q$3

for a fixed multiplier $j\in Q$4, and every IC interim rule can be implemented exactly in this form (Ni et al., 2022). Their exposition further discusses an optimistic auto-bidding setting in which the bidder uses an estimated multiplier $j\in Q$5; to preserve optimistic IC, the path of estimates must remain at or above the true Lagrange multiplier $j\in Q$6, with $j\in Q$7 held fixed and the convexity and gradient identities holding round by round (Ni et al., 2022). This suggests a second, distinct use of “optimistic” in the literature: not equilibrium-selection optimism over a set of allocations, but optimism in the choice of multiplier governing ex-ante constrained bidding.

A separate adjacent development is the market-clearing approach for value maximizers under private budgets and private Return-on-Spend constraints. That work designs a market-clearing mechanism that is incentive-compatible with respect to financial constraints and presents a decentralized online algorithm with sublinear regret guarantees. It does not itself develop an “optimistic” variant, but states that its online algorithm is compatible with standard optimistic OCO extensions and that replacing RDA by Optimistic Mirror Descent could potentially yield $j\in Q$8 regret rather than $j\in Q$9 (Liu et al., 22 Feb 2026). In this sense, OAIC can be situated within a wider transition in auction theory from quasi-linear utility maximization to value maximization under delegated auto-bidding and constraint-reporting regimes.

Taken together, these results place OAIC as a precise equilibrium-sensitive truthfulness notion tailored to modern auto-bidding systems. Its main contribution is not to eliminate strategic ambiguity, but to index incentive compatibility to an advertiser’s attitude toward equilibrium uncertainty and thereby recover positive guarantees for SPA in environments where the original AIC criterion is too strong (Liaw et al., 22 Aug 2025).