Second-Price Autobidding Auctions

• Second-price autobidding auctions are auction environments that integrate automated bidding systems with second-price-like payment rules to meet campaign-level objectives.
• They span various formats—including sponsored search, shopping, and parallel single-item auctions—with uniform bid scaling and shifting bidder objectives from direct value maximization to constraint-based optimization.
• Research shows that delegating bidding to automated systems alters strategic incentives, equilibrium properties, and dynamic behavior, impacting welfare and revenue guarantees.

Second-price autobidding auctions are auction environments in which second-price or second-price-like payment rules operate jointly with automated bidding systems that optimize campaign-level objectives—typically value subject to return-on-investment, return-on-spend, budget, or target-CPA constraints—instead of direct quasilinear utility. The term spans ordinary single-item second-price auctions, parallel second-price auctions with uniform bid scaling, generalized second-price position auctions, and VCG-style second-price mechanisms for sponsored search and sponsored shopping. Across these settings, the central research lesson is that delegation to autobidders changes the relevant strategic interface: truthfulness, efficiency, equilibrium computation, and dynamic behavior must be analyzed at the level of reported constraints, learned multipliers, and platform information, not only at the level of a single direct bid (Deng et al., 2023, Kolumbus et al., 2021, Li et al., 2022, Anagnostides et al., 9 Feb 2026).

1. Auction formats and formal scope

A first formal family is the multi-auction sponsored-search model for generalized second-price auction (GSP). There are $n$ bidders, $m$ auctions, and $s$ slots per auction; bidder $i$ has per-auction value $v_{i,j}$, slot $k$ in auction $j$ has discount factor $d_{j,k}$ with $d_{j,k}\ge d_{j,k+1}$, and realized value from slot $k$ is $m$0. If $m$1 is the bidder with the $m$2-th highest bid in auction $m$3, then $m$4 gets slot $m$5 and pays

$m$6

This is the canonical “second-price-like” autobidding model for multi-slot position auctions, and the discount factors are interpreted as click-probability ratios relative to the top slot (Deng et al., 2023).

A second family is parallel single-item second-price auctions with uniform multiplicative bidding and return-on-spend constraints. An instance is

$m$7

bidder $m$8 chooses a multiplier $m$9, and submits

$s$0

Items are allocated to highest bidders, winners pay the second-highest bid, and the allocation may be fractional. This model is explicitly built around “autobidding equilibrium” rather than ordinary pure Nash equilibrium, with maximal pacing used to handle tie structure and equilibrium existence (Anagnostides et al., 9 Feb 2026).

A third family is sponsored shopping, where allocation is combinatorial because a single advertiser can secure multiple slots simultaneously with different products. There are $s$1 slots with click-through rates $s$2, bidder $s$3 has item values $s$4, and uniform autobidding takes the form

$s$5

In this setting, GSP remains item-ranked, while the second-price benchmark is VCG with bidder-level externality payment

$s$6

The paper proves that this combinatorial setting still admits an autobidding-equilibrium notion for both GSP and VCG (Dütting et al., 25 Feb 2026).

These models are supplemented by repeated separate-item second-price auctions with reserves and uniform bid scaling, stochastic sequential models with payment rule $s$7, and information-design models in which the platform sends calibrated private signals that bidders use directly as bids. This breadth suggests that “second-price autobidding auctions” is not a single mechanism class but a research area organized around the interaction between second-price payment semantics and automated bid control (Aggarwal et al., 2024, Du et al., 23 Jul 2025).

2. Bidder objectives, autobidder policies, and equilibrium notions

The defining behavioral departure from classical auction theory is that autobidders are usually modeled as value maximizers subject to feasibility constraints rather than quasilinear utility maximizers. In the multi-slot GSP model, bidder $s$8 chooses a possibly randomized strategy $s$9 to maximize expected total value subject to expected payment not exceeding expected value; payment enters only as a feasibility constraint. In the parallel second-price model, advertiser $i$0 solves

$i$1

and the paper normalizes to $i$2. This makes autobidding behavior closer to target-CPA or RoS control than to bidder-side surplus maximization (Deng et al., 2023, Anagnostides et al., 9 Feb 2026).

The equilibrium notions are correspondingly specialized. In the parallel second-price model, an exact autobidding equilibrium $i$3 requires: only highest bidders can win, exact second-price payments, full allocation, RoS feasibility,

$i$4

and maximal pacing, meaning that unless $i$5,

$i$6

In sponsored shopping, an autobidding equilibrium $i$7 requires bid-consistent allocation, ROI feasibility $i$8, and maximal autobidding: if the inequality is strict, then $i$9 (Anagnostides et al., 9 Feb 2026, Dütting et al., 25 Feb 2026).

A distinct design line treats the platform as eliciting private constraints rather than values. In that public-values, private-budget/private-ROI model, bidder type is $v_{i,j}$0, cumulative value is

$v_{i,j}$1

and utility is $v_{i,j}$2 if both budget and ROI are respected, and $v_{i,j}$3 otherwise. The central payment identity is not Myerson’s formula but the maximum feasible truthful payment

$v_{i,j}$4

This is a truthful autobidding mechanism with personalized rank scores, similar to GSP in allocation style but not a standard second-price payment rule (Xing et al., 2023).

A stochastic online-learning version further abstracts the bidder to a single learner facing value $v_{i,j}$5, highest competing bid $v_{i,j}$6, and payment function $v_{i,j}$7. In the pure second-price specialization $v_{i,j}$8, the per-round constrained optimization becomes especially simple because the Lagrangian reward is

$v_{i,j}$9

and the safe bid

$k$0

is an optimizer of that reward (Aggarwal et al., 2024).

3. Incentive compatibility beyond the one-shot bid

The most persistent theme in the literature is that classical second-price truthfulness does not survive delegation. In repeated single-item second-price auctions played by regret-minimizing agents, the high-value player’s bids converge to a distribution that is uniform over $k$1, the low-value player’s bids retain full support on $k$2, and the high player still wins while paying an average price strictly below the second price. Because the long-run payment can fall far below the rival’s true value, a user may benefit by exaggerating the value reported to its own learning agent. The paper states this explicitly: a player with true value $k$3 can lose against $k$4 under truthful reporting, yet by misreporting $k$5 to its own agent may win almost always and obtain strictly positive utility on average (Kolumbus et al., 2021).

A related but distinct notion is auto-bidding incentive compatibility (AIC): whether an advertiser can gain by misreporting its budget or target CPA to the autobidder intermediary. In the continuous-query model $k$6, the autobidder solves

$k$7

subject to

$k$8

The main result is that for both budget and target-CPA settings, SPA is not AIC, and the paper extends this to any truthful, scalar invariant, symmetric auction. Thus the advertiser-to-autobidder interface is strategically manipulable even when the underlying auction is classically truthful (Alimohammadi et al., 2023).

The paper on “single-round incentive compatibility” pushes this critique further in platform-run markets with ROI-constrained value-maximizing campaigns. It shows that second price auction exhibits computational hardness, non-monotonicity, instability of bidders’ utilities, and interference in A/B testing, and argues that these failures arise from the runner-up-winner interdependence built into second-price pricing. The winner’s payment being determined by the runner-up is precisely the local feature that gives ordinary single-item incentive compatibility, but in the autobidding world it propagates through campaign-wide constraints and induces nonlocal strategic feedback (Li et al., 2022).

These results collectively separate auction-level incentive compatibility from autobidding-level incentive compatibility. This suggests that the relevant revelation problem in modern advertising markets is often not “bid your value to the mechanism,” but “report your constraint or value proxy to the autobidder,” and the two problems need not have the same answer.

4. Welfare guarantees, price of anarchy, and comparative efficiency

The efficiency literature uses two closely related but not identical conventions. One line defines

$k$9

so larger numbers are better; another defines

$j$0

so smaller numbers are better. This implies that a $j$1 lower bound in the first convention corresponds to a factor-$j$2 guarantee in the second (Deng et al., 2023, Anagnostides et al., 9 Feb 2026).

In the multi-slot GSP autobidding model, the main impossibility result is stark: the price of anarchy can be as bad as $j$3. The paper’s refined theorem ties the lower bound to slot discount geometry, with bottleneck auction $j$4 determined by the smallest ratio $j$5. A simplified corollary is

$j$6

The qualitative conclusion is explicit: the smoother the discount factors are, the better the efficiency guarantee; a severe cliff between the first and second slots can drive efficiency to $j$7. By contrast, VCG retains a $j$8 guarantee in the autobidding world, although that comparison excludes dominated strategies for VCG while GSP can have PoA $j$9 even excluding dominated strategies (Deng et al., 2023).

For the single-slot prior-free benchmark, SPA is the canonical truthful mechanism with PoA $d_{j,k}$0, and deterministic non-truthful mechanisms do no better. The same paper shows that first-price auction has PoA exactly $d_{j,k}$1, that a randomized non-truthful auction $d_{j,k}$2 achieves PoA $d_{j,k}$3 for two bidders, and that the truthful version of the same randomized allocation achieves about $d_{j,k}$4. It also proves that no auction, even randomized and non-truthful, can asymptotically beat factor $d_{j,k}$5 as the number of bidders grows (Liaw et al., 2022).

The sponsored-shopping literature supplies a contrasting multi-item result: autobidding equilibrium always exists for both GSP and VCG, and the Price of Anarchy is exactly $d_{j,k}$6 for both mechanisms. This restores a coarse constant guarantee in a combinatorial environment where a bidder may win multiple slots simultaneously, but it does not restore full efficiency under autobidding (Dütting et al., 25 Feb 2026).

5. Computation, learning, dynamics, and information design

A central algorithmic distinction is between finding some equilibrium and selecting a good one. In parallel second-price autobidding auctions with RoS constraints, autobidding equilibria always exist and computing one is in $d_{j,k}$7. Yet computing an autobidding equilibrium that approximates the welfare-optimal one within a factor $d_{j,k}$8 is NP-hard for every constant $d_{j,k}$9, and deciding whether there exists an autobidding equilibrium with welfare at least a $d_{j,k}\ge d_{j,k+1}$0 fraction of unconstrained optimum is also NP-hard. Revenue is harder still: constant-factor hardness strengthens to $d_{j,k}\ge d_{j,k+1}$1-hardness, and under the Projection Games Conjecture even polynomial-factor approximation is ruled out (Anagnostides et al., 9 Feb 2026).

A separate line studies exact and approximate dynamics rather than static selection. In repeated second-price auctions with uniform bid scaling $d_{j,k}\ge d_{j,k+1}$2, continuous-time updates

$d_{j,k}\ge d_{j,k+1}$3

and discrete-time mirror-descent variants can exhibit quasiperiodicity, Li–Yorke chaos, Devaney chaos, positive Lyapunov exponents, and positive topological entropy. The paper proves exact recovery of the Ricker model under entropic mirror descent and of the logistic map under Euclidean mirror descent, and shows that autobidding systems can approximately simulate Chua’s circuit. The practical implication is that simple second-price autobidding systems may be inherently unpredictable in the long run (Anagnostides et al., 9 Feb 2026).

Against this negative dynamic picture, the stochastic online-learning literature identifies a tractable second-price case. In the full-information setting, there is an algorithm with near-optimal $d_{j,k}\ge d_{j,k+1}$4 regret against the best $d_{j,k}\ge d_{j,k+1}$5-Lipschitz bid function while respecting budget and ROI constraints, and in pure second-price auctions the per-round safe bid

$d_{j,k}\ge d_{j,k+1}$6

actually maximizes the Lagrangian reward. The same paper proves that the dependence on the ROI-slack parameter $d_{j,k}\ge d_{j,k+1}$7 is unavoidable even in second-price auctions (Aggarwal et al., 2024).

Information design introduces another dimension. In a single-impression second-price auction with autobidders who bid calibrated CTR signals, the platform’s problem is

$d_{j,k}\ge d_{j,k+1}$8

Optimal calibrated signaling can be fully characterized, every bid profile in the support can be made multi-maximal, and the conditional second-highest bid takes a three-level form depending on the number of bidders with outcome $d_{j,k}\ge d_{j,k+1}$9. The paper shows that optimal calibrated signaling can extract the full surplus—or even exceed it—depending on a market condition, and gives an FPTAS for the IR-constrained version (Du et al., 23 Jul 2025).

Finally, model quality itself need not be monotone for auction outcomes. Under model refinement $k$0, first-price auctions with uniform bidding guarantee revenue monotonicity for tCPA bidders without budgets, but second-price auctions can break this property. In the single-item SPA/VCG setting, revenue monotonicity fails for MAX-CPA bidders, and for tCPA bidders both revenue and welfare can fail to be monotone. Thus a more refined prediction model can lower second-price revenue, and in tCPA settings can even lower welfare (Badanidiyuru, 29 May 2026).

6. Empirical evidence and mechanism-design implications

Empirical work complements the worst-case theory by comparing second-price-like mechanisms under richer strategy spaces. In a large synthetic position-auction environment with $k$1 advertisers, $k$2 queries, $k$3 slots, and 25 bidding-update rounds, the ranking is unambiguous: for both uniform bid-scaling and non-uniform bid-scaling, FPA is better than GSP and GSP is better than VCG in terms of both welfare and profit. Within the second-price family, VCG behaves exactly as truthful-auction theory predicts—different levels of non-uniform bid-scaling have no effect—whereas GSP is strategically fragile: a higher level of non-uniform bid-scaling leads to lower welfare performance, while profit rises slightly at the highest level (Deng et al., 2023).

Mechanism-design responses diverge accordingly. One response is to design truthful mechanisms directly for public values and private campaign constraints. The proposed family of truthful automated bidding auctions with personalized rank scores uses virtual bids

$k$4

derives item-level critical ROIs from second-ranked scores, computes a bidder-level critical ROI $k$5, and then charges

$k$6

This is similar to GSP in allocation style but not a standard second-price payment rule, and it is proposed precisely because repeated first-price and repeated second-price auctions are not truthful in that public-values, private-constraints model (Xing et al., 2023).

Another response comes from multi-platform competition. In that setting, advertisers strategically report a platform-specific target ROI to each platform’s autobidder, and the platform’s autobidder then uses a uniform bid multiplier. The paper’s main qualitative conclusion is that while first-price auctions are optimal for both revenue and welfare in the absence of competition, this no longer holds in multi-platform settings: there exists a large class of advertiser valuations over impressions such that, from the platform’s perspective, running a second price auction dominates running a first price auction. The key factors are intensity of competition among advertisers, sensitivity of bid landscapes to an auction change, and the relative inefficiency of second-price auctions compared to first-price auctions (Aggarwal et al., 2024).

Taken together, these papers depict second-price autobidding auctions as a heterogeneous but now well-defined research domain. Classical second-price logic remains visible in isolated components—uniform bidding in truthful auctions, second-highest-price payments, VCG externality pricing, or calibrated expected-value bidding—but the surrounding autobidding layer changes almost every global conclusion. Depending on the model, second-price autobidding can be computationally intractable, dynamically chaotic, strategically manipulable, empirically weaker than first-price, or advantageous under platform competition. The unifying lesson is that in the autobidding world, the relevant object of mechanism design is no longer the one-shot auction rule alone, but the combined system of predictions, bidding interface, payment semantics, equilibrium selection, and learning dynamics.