Risk-Averse Auto-bidding Incentive Compatibility

• Risk-Averse Auto-bidding Incentive Compatibility is defined as a design framework that ensures truthful advertiser reporting amid risk preferences and equilibrium ambiguity.
• It integrates classical auction theory with concave utility models and ex-ante constraints to address risk-averse behavior in auto-bidding environments.
• Practical applications include optimizing first- and second-price auctions with tCPA constraints in dynamic digital advertising markets.

Searching arXiv for the cited work on risk aversion, auctions, and auto-bidding to ground the article in the literature. arXiv search query: "Risk-Averse Auto-bidding Incentive Compatibility auto-bidding risk aversion first-price second-price" Risk-Averse Auto-bidding Incentive Compatibility (RAIC) denotes incentive requirements for auction systems in which advertisers delegate bidding to platform-controlled auto-bidders and are not adequately described by the risk-neutral, single-auction model. In recent work, the term appears in two closely related formal senses. One is an equilibrium-selection notion for advertisers facing multiple auto-bidder equilibria, in which truthful reporting is evaluated by comparing least favorable or most favorable equilibrium outcomes. The other is a runtime notion for sparse-click advertising systems, in which truthful reporting remains optimal only if realized cost-per-acquisition tracks the reported target throughout execution. More broadly, RAIC is grounded in older results on concave utility, pay-your-bid semantics, ex-ante constrained mechanism design, and ROI- or CPA-constrained auto-bidding (Liaw et al., 22 Aug 2025, Wang et al., 2024).

1. Risk aversion and incentive compatibility before auto-bidding

The prehistory of RAIC lies in mechanism design with non-linear utility for money. One tractable specialization is capacitated utility,

$u_C(w)=\min\{w,C\},$

where small $C$ corresponds to severe risk aversion, large $C$ to mild risk aversion, and $C=\infty$ to risk neutrality. In that model, optimal BIC revenue mechanisms can be restricted to two-priced auctions; for one-priced or pay-your-bid mechanisms, Theorem 5.1 gives a payment identity; and with common capacity and i.i.d. regular values, the first-price auction achieves a 5-approximation to the optimal revenue for capacitated agents (Fu et al., 2013).

A second foundational result shows that risk aversion need not force a redesign of randomized allocation rules. For any truthful-in-expectation mechanism $(A,p)$, one can keep the allocation rule $A$ unchanged and define

$p'_i(v)=v_i(A(v))-\mathbb{E}[v_i(A(v))-p_i(v)].$

Under truthful reporting, the agent’s payoff becomes deterministic, expected payments are preserved, and the transformed mechanism is dominant-strategy incentive compatible for arbitrary unknown non-decreasing concave utilities. The same logic extends to Bayesian settings (Dughmi et al., 2012).

Algorithmic mechanism design for risk-averse agents also supplies explicit approximation guarantees. In multi-unit auctions with a risk-averse seller, a polynomial-time deterministic SPM achieves $(1-1/e-\varepsilon)$ of the expected utility of the optimal DSIC mechanism. When buyers as well as the seller are risk-averse, truthful-in-expectation mechanisms achieve $(1-1/e)^2 y(k)$ of the optimal BIC benchmark, with $y(k)=1-1/e^k$ (Bhalgat et al., 2011).

Comparative statics sharpen the behavioral content. In first-price auctions, under the conditions that winning is never worse than the outside option and winning with a low bid is preferable to winning only with a high bid, greater risk aversion makes high bids more appealing. In second-price auctions with a known outside option, bidding more increases risk exposure conditional on winning, so greater risk aversion favors lower bids. The paper translates these bid-level forces into corresponding equilibrium comparative statics (Pease et al., 10 Mar 2026).

2. Ex-ante constraints, auto-bidder reports, and the breakdown of classical IC

Modern auto-bidding shifts the strategic object from per-auction bids to campaign-level reports. In a general ex-ante constrained Bayesian model, expected utility is

$C$0

subject to an ex-ante feasibility condition

$C$1

Under compactness and continuity, IC is fully characterized by a taxation principle: either truthful types maximize $C$2, or there exists $C$3 such that

$C$4

with the ex-ante constraint binding when $C$5. The paper’s conclusion is that IC mechanisms are fully characterized by auto-bidding mechanisms (Ni et al., 2022).

In ROI-constrained auto-bidding, the report space is itself a performance constraint. For a tCPA advertiser with constant conversion value $C$6, a cumulative delivery mechanism $C$7 is IC and IR iff

$C$8

For tROI, the delivered ROI must equal the reported target $C$9, and utility must be non-increasing in $C$0. The same framework introduces a double-layer framework together with Algorithm 1, Interior Controlling Strategy, and Exterior Controlling Strategy (Li et al., 2020).

These positive characterizations do not imply that canonical auction formats are auto-bidding incentive compatible. In the continuous query model, both FPA and SPA are not AIC for budget and tCPA advertisers, and any truthful, scalar invariant and symmetric auction is not AIC; by contrast, FPA is AIC when auto-bidders are constrained to use a uniform bidding policy, and with two advertisers FPA and SPA are auction equivalent (Alimohammadi et al., 2023). In ROI-constrained second-price markets, related failures appear as PPAD-hardness of finding an $C$1-approximate auto-bidding equilibrium, APX-hardness of optimizing revenue or welfare over equilibria, non-monotonicity, instability of bidders' utilities, and interference in A/B testing. The structural source is runner-up–winner interdependence (Li et al., 2022).

3. Formal RAIC notions

The literature contains more than one AIC baseline. One compares equilibrium sets under alternative constraint reports; another, developed for OCPC with tCPA bidders, is defined ex ante in expected utility. RAIC refines the first baseline, whereas TIC strengthens the second at runtime.

Notion	Comparison object	Representative condition
AIC (equilibrium-set)	Worst truthful vs best misreport	$C$2
RAIC	Worst truthful vs worst misreport	$C$3
OAIC	Best truthful vs best misreport	$C$4
AIC (ex ante, OCPC)	Expected utility	$C$5
TIC	Runtime realized CPA	$C$6

In the equilibrium-selection formulation, the advertiser reports a tCPA $C$7, the auto-bidder subgame generates a set $C$8 of equilibrium allocations, and welfare is

$C$9

RAIC requires that, for any $C=\infty$0, the least favorable equilibrium under truthful reporting dominate the least favorable equilibrium under the misreport; OAIC replaces minima by maxima. The same paper shows that SPA satisfies both RAIC and OAIC, and that SPA also satisfies both notions for two advertisers under uniform bidding (Liaw et al., 22 Aug 2025).

In the runtime formulation, a bidder is called risk-averse when she adjusts her bids or withdraws from the auction after observing deviations between actual and expected outcomes during the auction process. In OCPC with tCPA bidders, AIC is defined ex ante by expected utility, but TIC is stronger: $C=\infty$1 The paper also defines $C=\infty$2-TIC through the multiplicative band

$C=\infty$3

This makes RAIC a runtime property rather than only an ex ante one (Wang et al., 2024).

4. Mechanism classes and guarantees

Among simple mechanisms, pay-your-bid and one-price formats recur because they admit explicit risk-averse incentive and approximation statements. For capacitated agents, any auction can be replaced by a two-priced auction with no lower revenue. In arbitrary downward-closed environments with asymmetric capacities and regular distributions, there exists a one-priced BIC mechanism whose revenue is at least one third of optimal two-priced revenue. For common capacity and i.i.d. regular values, the first-price auction’s BNE revenue is at least one fifth of the optimal capacitated BIC revenue. The one-price payment identity is built from

$C=\infty$4

and a recursively defined $C=\infty$5, which determines equilibrium payments and bids from the allocation rule (Fu et al., 2013).

In the tCPA second-price setting with equilibrium multiplicity, SPA enjoys a different kind of guarantee: truthful reporting is protected at the level of equilibrium-set comparisons rather than revenue approximation. Specifically, SPA is RAIC and OAIC under non-uniform bidding, and it remains RAIC and OAIC under uniform bidding when there are two advertisers (Liaw et al., 22 Aug 2025).

OCPC mechanisms introduce a runtime template. CFP uses a monotone ranking score for allocation and the payment rule

$C=\infty$6

which yields

$C=\infty$7

and therefore AIC and IR. DFP keeps the allocation rule but solves an online payment problem so that

$C=\infty$8

thereby obtaining TIC. To realize this payment rule online, the paper uses a PPO-based RL algorithm. On 31 days of logs from a large search advertising platform, DFP yields $C=\infty$9 upper $(A,p)$0, lower $(A,p)$1, mean $(A,p)$2, whereas CFP yields upper $(A,p)$3, lower $(A,p)$4, mean $(A,p)$5 (Wang et al., 2024).

5. Market dynamics, approximation, and strategic stability

RAIC is also connected to market-level auto-bidding dynamics. In a linear Fisher market with multiple-item sellers, buyers allocate seller-level budgets according to the proportional rule

$(A,p)$6

while each seller’s sub-market clears through a first-price pacing equilibrium. The resulting proportional dynamics converges to the competitive equilibrium, and the average Eisenberg–Gale objective converges at an ergodic $(A,p)$7 rate (Li et al., 2024).

The incentive picture in that market is asymmetric. Buyers who equalize bang-per-bucks across sellers obtain at least half of the optimal utility, so the proportional rule is an approximate best response. Sellers, by contrast, can use additive boosts to implement any feasible allocation in their sub-market; a deviating seller can gain up to 5 times equilibrium revenue; and the seller-side deviation game admits a unique pure Nash equilibrium. Even then, Nash social welfare at that equilibrium is at least a $(A,p)$8-fraction of the competitive-equilibrium NSW (Li et al., 2024).

This suggests that RAIC cannot be reduced to advertiser-side report truthfulness alone. In auto-bidding markets with delegated bidding, seller-side manipulation, pacing dynamics, and market competitiveness remain part of the incentive problem.

6. Limitations and open directions

Existing RAIC results are model-specific. Capacitated-auction analyses focus on single-item, downward-closed, or position environments; use capacitated utility rather than CARA or CRRA; assume ex post IR; and omit explicit budgets, pacing, and multi-period campaign risk (Fu et al., 2013). The generic truthful-in-expectation transformation for concave utilities assumes agents can both pay and receive money, with no budget constraints or liquidity issues, so direct application to one-directional ad-payment systems is limited (Dughmi et al., 2012).

The runtime TIC program is developed for OCPC with tCPA bidders, assumes the platform has CTR and CVR estimates, and realizes DFP through RL because exact conversion information is not available online. The theory targets sparse-click deviations between ex ante expectations and realized CPA, but the implemented payment rule is an approximation learned from data (Wang et al., 2024).

The equilibrium-selection RAIC/OAIC results are currently specific to SPA with tCPA constraints. They cover non-uniform bidding with any number of advertisers and uniform bidding with two advertisers; extending uniform-bidding RAIC/OAIC to more than two advertisers is left open. The same paper also notes that budget-only constraints in SPA do not yield a meaningful RAIC/OAIC notion (Liaw et al., 22 Aug 2025).

Broader theory papers point to additional gaps. The ex-ante constrained characterization is explicitly risk-neutral, so a generalized taxation principle for concave or coherent risk objectives remains undeveloped (Ni et al., 2022). ROI/CPA models in auto-bidding treat risk primarily through hard feasibility constraints rather than concave utility (Li et al., 2020). Proportional-dynamics analyses likewise assume linear utilities and state that new potential functions and fairness notions would be needed once utilities become concave (Li et al., 2024).

A plausible synthesis is that RAIC names a family of design objectives rather than a single theorem. Across the literature, the common target is to make truthful constraint reports, or the prescribed auto-bidding behavior induced by those reports, stable for advertisers who care about downside risk, equilibrium ambiguity, or realized deviations from expectation. Under that broader interpretation, RAIC spans at least four layers: concave utility over money, campaign-level ex-ante constraints, equilibrium-set comparisons under multiplicity, and runtime alignment between reported and realized performance.