Autobidder's Dilemma: Why More Sophisticated Autobidders Lead to Worse Auction Efficiency
Authors: Yuan Deng
Jieming Mao
Vahab Mirrokni
Hanrui Zhang
Song Zuo
Venue: NeurIPS 2024 poster (2024)
Keywords: ad auctions
autobidding
non-uniform bidding
price of anarchy
Primary area: algorithmic_game_theory
TL;DR: We show that automated first-price auctions become more inefficient with more sophisticated autobidders.

Abstract: The recent increasing adoption of autobidding has inspired the growing interest in analyzing the performance of classic mechanism with value-maximizing autobidders both theoretically and empirically. It is known that optimal welfare can be obtained in first-price auctions if autobidders are restricted to uniform bid-scaling and the price of anarchy is $2$ when non-uniform bid-scaling strategies are allowed. 

In this paper, we provide a fine-grained price of anarchy analysis for non-uniform bid-scaling strategies in first-price auctions, demonstrating the reason why more powerful (individual) non-uniform bid-scaling strategies may lead to worse (aggregated) performance in social welfare. Our theoretical results match recent empirical findings that a higher level of non-uniform bid-scaling leads to lower welfare performance in first-price auctions.

Autobidder's Dilemma: Why More Sophisticated Autobidders Lead to Worse Auction Efficiency
Yuan Deng Google Research dengyuan@google.. com
Jieming Mao Google Research mao jm@google com
Vahab Mirrokni Google Research mirrokni@google..om
Hanrui Zhang
Song Zuo
Chinese University of Hong Kong hanruiccse. cuhk. edu.hk hk
Google Research szuo@google. com
Abstract
The recent increasing adoption of autobidding has inspired the growing interest in analyzing the performance of classic mechanism with galue-maximizing auto- bidders both theoretically and empirically. It is known that optimal welfare can be obtained in first-price auctions if autobidders are restricted uniform bid-scaling and the price of anarchy is when non-uniform bid-scaling strategies are allowed.
In this paper, we provide fine-grained price of anarchy analysis for non-uniform bid-scaling strategies in first-price auctions, demonstrating the reason why more powerful (individual) mon-uniform bid-scaling strategies may lead worse (ag- gregated) performance in social welfare. Our theoretical results match recent empirical findings that higher level of non-uniform bid-scaling leads lower welfare performance in first-price auctions.
Introduction
The online advertising market has witnessed increasing adoption of autobidding in recent years, which shifts the bidding behavior model of advertisers from the classic militt-maximizzin bidding value-maximizing bidding [Aggarwal al.) 2019 Balseiro al.) 2021b). Unlike the classic utility maximizers who maximize their quasi-linear utility given by the difference between value and payment, value maximizers maximize the total value subject to a return-on-spend (RoS) con- straint Balseiro et al., 2021b]
The shift in the bidding behavior model has motivated growing body of literature auction design with maximizing autobidders. Notably, has been shown that the price of anarchy (PoA) in second-price auctions [Aggarwal al. 2019 and first-price auctions [Deng et al. 2022 Liaw et al. 2023) are both 2. However, (1) these PoA results measure the welfare performance in the worst-case scenario (i.e., the worst-case equilibrium of the worst-case instance): moreover, (2) these PoA results assume the bidding agents can find the optimal bidding strategies in response to other bidders so that their bidding profile forms an equilibrium, while computing optimal bidding strategies and/or finding equilibria could be computationally infeasible [Aggarwal et al./ 2023 Chen et al. 202
and Tang 2024| Paes Leme al. 2024). On the other hand, Balseiro et al. 202 demonstrate
that optimal welfare can be obtained in first-price auctions if autobidders are restricted to the simple uniform bid-scaling strategy (i.e., always bid @v when the bidder's value is with a universal bid multiplier 0). But, uniform bid-scaling is not always an optimal strategy for autobidders in first-price auctions as non-uniform bidding scaling, in which autobidder may apply different bid multipliers in different auctions, may result in better bidding performance.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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To circumvent the above mentioned limitations of PoA results and bridge the gap between uniform bid bid-scaling and non-uniform bid-scaling, Deng et al.) [024] propose hierarchical structure over ad auction instances in which the auctions are partitioned to different categories following a multi-layer laminar structure. Autobidders are required to adopt the same bid multipliers for auctions within the same slice but can use different bid multipliers across different slices. Such a multi-layer structure enables the comparison between non- uniform bid-scaling strategies of different degrees of freedom and the empirical results from [Deng et al. 20224 show that a higher level of non-uniform bid-scaling may lead to lower aggregated welfare performance in first-price auctions. In this paper, we take a theoretical approach to non-uniform bid-scaling in first-price marketplaces, aiming to (1) provide formal understanding of the phenomenon observed by Deng et al. (2024), and (2) establish principles that can guide the design of marketplaces where non- non-uniform bid-scaling is involved.
1.1 Our Results
Our main technical result is a fine-grained analysis of the price of anarchy of automated first-price marketplaces when autohidders are capable of non-uniform bidding. We adopt the high-level model introduced by Deng al. (2024]: Roughly speaking. the entire market is divided into number of slices, and autobidders may choose one bid multiplier independently for each slice. Here, intuitively. the granularity of the slices measures autobidders' capability of non-uniform bidding. Through refined analysis, we present a parameterized price of anarchy bound, which connects the efficiency of an automated marketplace to the power of autobidders measured by their capability of non-uniform bidding. as well as the "balancedness" of slices. Here, the balancedness of slice roughly measures the smallest "market share" across all bidders. Qualitatively, our results suggest:
First-price markets are more efficient when autobidders are less powerful. This is reminis- cent of the prisoner's dilemma, where both prisoners end up in a strictly worse situation, when each of them chooses an action that is superior to the alternative, regardless of the other prisoner's 's choice. In automated marketplaces, fixing others' bids, more powerful autobidder always achieves (weakly) better payoff. And yet. the interplay of such "better" autobidders could lead to worse auction efficiency. This provides theoretical explanation for the phenomenon observed by Deng et al. [0224].
First-price markets are more efficient with more balanced slices. In particular, when the granularity of non- non-uniform bidding is fixed, one can improve the efficiency of the market by creating slices that are more balanced. Intuitively, this introduces more intense competition within each slice, resulting in better auction efficiency. Such an insight also be applied to marketplaces with the multi-channel setting (Deng et al., 2023a (Susan et al., [2023]. where advertisers can procure ad impressions simulraneously on multiple channels with different bidding strategies. In the multi-channel setting, more balanced channels could lead to better auction efficiency across all channels.
1.2 Further Related Work
The price of anarchy in first- frice auctions with quasi-linear utility maximizers has been extensively studied (Roughgarden et al. 2017]. For Bayesian settings, the price of anarchy is at least 1/2 for subadditive valuations [Feldman et al., 20133 and the price of anarchy is at least 1 1/e for submodular valuations (Syrgkanis and Tardos] 2013). When values are independently distributed, Hoy et al. [2018) improved the price of anarchy to R 0.743. Recently, JJin and Lu (2022) resolve the standitanding open problem and show that the price of anarchy in first-price auctions with independently distributed values is exactly 1 - 1/ee2.
Our work is also closely related to the recent growing body of research AAggrral et al. 2019 Deng
al. 202T Balseiro et al. 2021a Mchta 2022 Deng al., 2023b Liaw et al.) [2024] which study the price of anarchy (PoA) with value-maximin
bidders in several classic auction mechanisms like second-price auctions, (randomized) first-price auctions, and generalized second price auctions. In particular, it has been shown the price of anarchy of first-price auctions is exactly 1/2 [Liaw et al. 2023 Deng et al. 2022]. In the multi-channel setting, Deng et al. [2023a] study the problem of multi-channel bidding where an advertiser aims to maximize their total value across and analyze the effectiveness of levers of return-on-sper -o-send and budget. [Susan et al. (2023) develop multi-channel
bidding algorithms when channels adopt auction rules that may or may not be incentive-compatible

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under the presence of budget constraints. More recently, Feng et al. [2023] investigate the PoA of running first-price auctions with strategic budget-constrained autobidders, in which advertisers may manipulate their reported budget constraints for the autobidders. For more comprehensive overview of auctions and autobidding, see, e.g., [ggarwal al. 2024].
Preliminaries
The multi-auction model. Following prior works [Aggarwal et al. 2019 Balseiro et al. 2021b], we consider the following model where multiple bidders participale in multiple auctions simul- taneously. There are bidders (generally indexed by i) and m auctions (generally indexed by j). In each auction j. each bidder has value Mt.j. and places bid b.j- In this paper, we as- sume these bids are placed through autobidders using non- uniform bidding, in which the bids are subject to certain restrictions to be discussed below. For brevity, we let V; = (V%,1.... U3,m) (.ig....;UU.... and v v--j
1i (Ui,...,, Un), v- (V1...., 1;-1, Ditl,...,, Un). Vj
(o1.)...,, Vi+ (h.j) We use b in a similar way.
Each bidder i's allocation Tı, and payment Pt,j in each auction j is determined by the auction rule and all bidders' bids b, in auction j. We focus on first-price auctions: In each auction j. the bidder with the highest bid" wins, receives the whole item and pays the bid. All other bidders receive nothing and pay O. Formally, for each auction j. let (i) argmax, bi.j- The allocation Tæ and payment pi.j of each bidder is given by
-f f1 Jii = (0.
1fi=101
biij ifi= 0.
otherwise
and
Pij
=
otherwise
We will omit the dependency of visj and púj on b for simplicity. For each auction j. we also identify a "rightful winner" rw(j) = argmax 0i who has the highest value and therefore should win in auction j in the socially optimal allocation
Slices and partially non-uniform bidding. We capture the power of autobidders beyond uniform bidding with a fine-graie -he-igraned slice- bicce-baed model (Deng et al. [024].. Intuitively, we assume that the m auctions are partitioned into slices. Within each slice, the autobidder must bid uniformly, while across slices, it can use different bidding multipliers. Formally, there are slices (generally indexed by k). which together form partition of the m auctions. Each slice 1: is described by the set S₃ S [m] of (indices of) auctions it contains. We always have S₃ n Sw 0 for all l + k' and Up. Sx [m]. For each auction j E Sk, we let slice(j) = k. The bids b, of each bidder is generated in the following way: For each slice k, the autobidder chooses a bid multiplier 0i, such that for each auction j Sp. bijj = v45" 0j We use f in similar way to v and b.
Constrained value maximizers. We adopt the value maximization model prevalent in the autobid- ding literature [Aggarwal et al. 2019 Balseiro et al. (0216]. The objective of a value maximizer is to maximize the total value that they win in all auctions, subject to the constraint that the overall return on spend (RoS) is larger than a given target, by choosing the optimal bid multipliers for all s slices conditioned on other bidders' bids. In addition, we make the following regularity assumption about bidders' strategies: On each slice k. each bidder never underbids (i.c., never chooses a multiplier 0g 1) unless wins in all auctions j Sş where the rightful winner (i.e., where rw(s) = i). The assumption rules out brittle and pathological equilibrium behavior, which allows us to focus on the relation between the power of autobidders and auction efficiency. Intuitively, if bidder i is forced to underbid on slice k. then i must be overbidding on some other slices (because the overall RoS constraint must be binding - see below for the formal definition). In such cases, the assumption says that autobidders are conservative, in the sense that bidder would try to secure i's share in the socially optimal allocation on each slice first, before bidding more aggressively and trying to acquire other bidders' share on other slices. We also remark that such regularity assumptions are used in prior work (e.g.. [Deng et al. 202T) to derive meaningful efficiency bounds for other auction formats.
assume ties are broken arbitrarily and consistently throughout the paper.
TTie are broken in the same way as in winner determination.
3

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Formally, let val., Fii"" Vi,z be i's value in auction j. Let
val;
valuy
and
P
Pi.j
jE[m]
jE(m]
Moreover, for each bidder and slice k. fixing f i let a (1-k = min (1, 0}, where
0 = min Ft.s = 1, Vjj E Sy where rw(j) = i}.
Let
(,( 1 1,1), 00) x...
(..,,)
Then, fixing 0-; (and therefore i), each bidder solves the following optimization problem:
max
val:
0,
s.t. val. 2 Pc-
We say 0, (0-;) 0; achieves the maximum of the above optimization problem given 0 ---- Note that here, we are assuming that bidder i's target RoS is 1. This is without loss of generality, because scaling each bidder's s values by their target RoS preserves the auction outcomes and the liquid welfare, the latter being the standard measure of efficiency with constrained value maximizers.
Equilibria and price of anarchy. We consider stable auction outcomes based on Nash equilibria, where all bidders best responding to each other. Formally, we say strategy profile 0 is an cquilibrium, iff for each bidder i, 0; E BR,(O_.). We measure efficiency by considering the standard notion of the Price of Anarchy (PoA), which is the ratio between the liquid welfare the worst equilibrium and the optimal liquid welfare. Formally, the PoA of a particular instance (n,m, m, s, {Sk}. v) in the multi-aue aultiiaaution model is defined follows:
Eie(n) val:
PoA(n, m, s, (Skh. v)
inf equilibrium
Eje(m) maX Vad
Note that the above definition of PoA with respect to single market instance. later sections, we will adopt specific parametrization of non-uniformity, and consider the worst-case PoA for any fixed non-uniformity parameters. This allows us to clearly depict the relation between non-uniformity and efficiency of auction outcomes. We also remark that this approach refines the standard PoA analysis, which normally establishes a single worst-case PoA bound over all instances.
Fine-Grained PoA of First-Price Auctions with Non-Uniform autobidders
In this section, we present and prove the main result this paper: a fine-grained PoA bound capturing the effect of non-uniform bidding on auction efficiency. As we will discuss later, our result implies that more sophisticated (i.e., more non-uniform) autobidders generally lead to worse auction efficiency.
3.1 The Balancedness Parametrization
We first introduce our parametrization of non-uniformity. Fix market instance (n, m, (8a). v). Definition 1 (Market share). The market share of a bidder i in set of auctions 5 S [m] is i's contribution to the optimal liquid welfare S, as fraction of the optimal liquid welfare. Formally,
Ejestmo()== Vi.j
share,,s
Σje(m) maxyels) Dr.
Definition 2 (Balancedness). The balancedness of set of auctions S S [m] is the minimum market share of all bidders in as a fraction of the sum of all bidders' market shares in S (i.c., the contribution of S to the optimal liquid welfare). Formally,
bals = minge]n] share,
Σ Cie(n) share,,s

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Without loss of generality, we assume (unless otherwise specified) that slices are numbered such that bal; weakly decreasing in k, i.e., for each E 1]. balk+11 S balg. Higher balancedness generally means that the contribution to the optimal welfare is more equally distributed among bidders. We will pay special attention to the market share in, and the balancedness of, a whole slice, where S = Sy for some slice k. Abusing notation, we let share;, = share,,s, for each bidder and slice k. and share, >ie(n) s and balş- = bals, for each slice k.
We discuss the intuition behind these definitions below. Generally speaking, inefficiency in first-price auctions with value maximizers often manifests in the following way:
A bidder wins in large fraction of auctions where that bidder is the rightful winner, without much competition. This means the bidder wins by bidding far below their true value in these auctions, which gives the bidder a significant amount of buyer surplus.
While the bidder (being value maximizer) does not intrinsically care about buyer surplus, it allows the bidder to bid more aggressively (i.e., far above their true value) in other auctions, and win extra auctions in which they are not the rightful winners, which makes the auction outcome less efficient.
For autobidders using slice-based non-uniform bidding strategies. since each bidder must bid uni- formly (i.e., equally aggressively) within each slice, intuitively, more balanced slices make it harder for the above phenomenon to happen. In particular, the first bullet point, on balanced slice, all bidders are bidding seriously because they want to defend their market share, which leads to higher competition and lower buyer surplus. In particular, this means they are more likely to use higher bid multipliers. In the second bullet point, by bidding aggressively above their true values, the bidder also paying much more than they "should" in auctions where they are the rightful winner, which depletes their buyer surplus without hurting the efficiency (because they are the rightful winner in these auctions). We will prove below that this is in fact what happens.
In order to measure the balancedness of a market instance, we will aggregate the balancedness across slices via the balancedness quantile function defined as follows.
Definition 3 (Balancedness quantile function). Fix any market instance (n, m, {S*),, v). The balancedness quantile function F of the market instance is a function such that for any given t E [0, 1]. F(t) is the infimum value b such that least a fraction of the entire market has balancedness most b. That is,
F(t) inf b
Σ balss
share; 2t 2t J
The balancedness quantile function can also be viewed as the inverse of the "cumulative distribution function" of the unbalancedness in the entire market. Note that F(t) [0. 1/2] since ball, E [0, 1/2] whenever n 2 2. Our actual bound will depend on the following parameter related to F(t).
Definition 4 (Unbalancedness of market instance). Fix any market instance (n, m, s, { v). Let be the function such that for any E [0, 1/2) and u E IR,
a(b, u) 1 26 + Vb- (1 11-b) b) u. Moreover, let 33 be the function such that for any w E [0, 1J. B(w) is the unique number such that,
L. a (F df = w.
The unbalancedness unbal(n, m, s, {Sa), v) of the market instance is defined such that unbal(n,, m, s, (S+).
1/ (1 (1+a(F()),(())) V
max #E
dt.
We establish the following intuitive properties of the unbalancedness, which will help make the conceptual messages of our main result (to be discussed the next subsection) clearer.
Proposition 1. The unbalancedness parameter has the following properties:
5

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F(I)
2
F(2)
F(G)
Figure A graphical example the balancedness quantile function. Each grey rectangle corre- sponds to slice k, where the height is bal, and the width share,. Observe that the height of each rectangle is at most bal, S 1/2, and the widths of all rectangles sum to Eke(s) share,
unbal(n, m, (Sk), v) weakly increases when balg decreases for some slice k [s].
unbal(n,, m, {Sk], v) weakly increases when slice is subdivided, i.e.,
unbal(n, m, 8, {Sk). S unbal(n, m, 8 + 1, (S)),
where there exists 1." E [s]. such that S = for all l E [k]] and S5+1 U. = =S
unbal n, m, 8, {Sa}, v) E [2/5, for each market instance (n, m, 8, {Sk}, v) where n 2 and 2 2.
Proof. Note that the unbalancedness depends only on the balancedness quantile function. We first argue that the unbalancedness weakly increases when the balancedness quantile function pointwise weakly decreases. Observe that unbal(-) can be equivalently defined in the following way: For each [0, 1], let
(111+((0) - vI1 a (01))))()))))))))))))))))) X(1) dl,
(w)
max XEA(w))
where A(w) is the of mappings from [0, w| to [0, oo), satisfying for each À A(w),
X(t) di
w.
Note that this maximum guaranteed to exist Then
unbal(n,, m, s, {Sa), v)
max 7w))
we 0,1]
This alternative definition is equivalent Definition because for fixed w. the maximizing À of (A) must satisfy: For each t, to E [0, w]
0(1 + A(4)) v(1 x(D))² + 44(())A()) a(((t))
0(1 + A((2) (1-
a(X(t2))
Herre we not assume slices are ordered such that balancedness weakly increasing.
'To quickly see why this is the case, observe that without loss of generality À is piecewise constant with at most + pieces, because F' is piecewise constant with most pieces.

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This must be true because otherwise one can adjust À locally and achieve larger value of the integral.
Given the above, one can show (see the proof of Lemma Appendix [Aror a detailed argument) that the maximizing À must satisfy: 'here exists some such that for each 7. E [0, w). X(t) = a F()), u), where a is defined in Definition The unique choice of u that satisfies the constraint on À is then (ww), and the alternative definition then reduces to Definition
Now to argue that unbal weakly increases when J' pointwise weakly decreases, we only need to show that for each w E [O, 11, 7(w) weakly increases when I' pointwise weakly decreases. The latter further reduces to: Fixing any satisfying the conditions above, the integral
L (111-000 v(1 X(1)) ( 4F(L)) X(t) di
weakly increases when pointwise weakly decreases. This is true because the integrand increases when Ft)) decreases.
Now we come back to the properties to be proved. The first property becomes almost obvious, because when baly weakly decreases for some slice k, F must pointwise weakly decrease, and therefore unbal
must weakly increase. As for the third property, we consider extreme cases of F: When F(t)
for each & E [0, 1]. in Definition 4 a ((), u) is always 1, and unbal(n, m, 8, {Sa J v) 1. This the largest unbalancedness possible. When F(f) 1/2 for each [0, 1]. (F)))) u) u/2, 3(w) 2/w - 2, and one can show unbal(n, m, s, { v) = 2/5 (achieved when w = 3/5). This the smallest unbalancedness possible.
The second property is a bit trickier. To see why this is true, consider the parameters w and À that achieve unbal(n, m, s, { v) in the alternative definition. Moreover, without loss of generality (recall that À is without loss of generality piecewise constant), suppose À is weakly decreasing on [kE(ke-1j] shares. ExE(k-] sharess). ( Let share', and bal', be the market share and balancedness of each slice
the new market instance. Without loss of generality, suppose balk S bal,+1 Define G' to be the "partially unordered" balancedness quantile function of the new market instance (n, m, s + {S v) that preserves the ordering of the slices before subdivision, i.c.,
F(t), if < Ekepk- -1] share; or >> "xelk--sharee, balk-. if -1 share, < < Share,, bal
G(L)
otherwise
Intuitively, G is obtained by splitting slice k* "in place" from F. One can show that
unbal(n, m, 8 + (,()),, 0) 1." (1+ (1+1()) - v (1) 4G(1) x(t) dt.
This is essentially because the alternative definition is oblivious to "ordering" of the balancedness quantile function. So,
unbal(n, m, + (8)),e) v) - unbal(n, m, (Sk). v)
Exe(ary share,
(V(1- (()= A(1) - V(1- - x(1)²² + 4G(1) ((1)))A()))))))))))))))
A(1) de.
-1e share
We only need to argue that the right hand side is at least O. Since À is weakly decreasing, F is constant, and G is weakly increasing (because balk-- S bal'+ 1). in the worst case, is constant on the interval of interest here (say A(t) <>> 0). Then, the right hand side becomes
share
V(1 8)2 + 4balk.
share V (1= 4balk- E -shree, V(L e)2 4bal's l.
Here we have
share'.- + share', = share and balk-- share'-- + bal, share' S balg. shareg,,
because the two new slices are obtained by subdividing the old slice k. For brevity, let
= share b = share' c= share x = bal'.. y bal' $11 1 2 = balge.
7

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We only need to show
cv 8]2 + Az6 2 av 8)2 4xl + bV (1- 0)2 4yl,
where
a a+b=c and by ce.
Applying Schwar,, we have
(1- 4r 6)2 4yl s Va b) (a(()(()a11111)))) 0)2 4xl) b((1 6)2 4yf))
v(u+b)3( 8)2 (a bb) + 4by))
I Vo 222 -c
= =c 81- + + 4zt.
This finishes the proof.
3.2
Efficiency under the Balancedness Parametrization
Now we are ready to prove our main result, which links auction efficiency with unbalancedness when autobidders perform partially non-uniform bidding.
Theorem (PoA of FPA with partially non-uniform bidding). For any E [2/5, 1}. we have
1 PoA(n, m, (, = 2
inf (n, (W)E,
where
I [n,, m, 8, (Sk), v) (n, m, (Sx). w) S t).
The theorem is a corollary of Lemma T and Lemma 2 which are stated and proved in Appendix A Below discuss the conceptual implications of Theorem
Better autobidders lead to worse efficiency. Theorem and Proposition I together establish possibly counterintuitive relation between the capability of autobidders and the efficiency of the auction outcome: Better optimized autobidders generally are capable of implementing more sophisticated non-uniform bidding strategies, which correspond to establishing finer partitions of the entire market into slices. However, by the second bullet of Proposition T any refinement of the slices can only lead to higher unbalancedness, which, by Theorem [T] leads worse auction efficiency. In the extreme case where autobidders are capable of bidding optimally in each individual auction (which, in our model, corresponds to the case in which each auction forms its own slice with balancedness 0), the unbalancedness of the market is 1, and therefore, Theorem states that the PoA of such market is 1/2, matching the bound established in prior work (Liaw et al. 2023 Deng et al. 2022). In fact, this is precisely the phenomenon observed empirically by Deng et al. 2024 Our results therefore provide theoretical explanation for their empirical findings. Together with empirical results from [Deng et al.) 2024], our theoretical results further suggest that it is unlikely to achieve better auction efficiency (especially in first-price marketplaces) by using autobidders that are (more) capable of non-uniform bidding in practice.
The importance of balanced channels slices. In situations where there are naturally multiple channels or slices (e.g., different ad exchanges operated by different companies, or by different organizations within a single company) [Deng et al., 2023a Susan al., 2023), our results also highlight the importance of keeping each channel slice balanced. In particular, the first bullet of Proposition [ suggests that the balancedness of one slice can be improved without hurting the balancedness of the other slices, then the unbalancedness of the entire market decreases, and Theorem guarantees better efficiency overall. In practice, our results suggest that is better off for different channels / slices to coordinate to achieve higher overall efficiency.
[
Even perfectly balanced slices introduce inefficiency. Proposition I states that whenever there are at least 2 slices in the market instance, the unbalancedness of the market instance is at least 2/5. even if all slices are perfectly balanced. Theorem | then implies that the efficiency such market instances is at most 4/5, which suggests that efficiency loss is inevitable whenever the market consists

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of multiple slices. In contrast, when autobidders perform uniform bidding over the entire market (i.e.. when there is only slice), known that first-price marketplaces achieve full efficiency (Balseiro er al. 2021a]. In other words, this indicates that one may expect to see phase transition in market efficiency when moving away from uniform bidding to (even highly restrictive) non-uniform bidding. We also remark that for second-price marketplaces, the POA is 2 even with perfectly balanced slices, or a single slice. This suggests that first-price marketplaces generally achieve higher efficiency compared to second-price ones.
Acknowledgments and Disclosure of Funding
We thank anonymous reviewers for their helpful feedback.
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Yuan Deng. Jieming Mao, Vahab Mirrokni, Hanrui Zhang, and Song Zuo. Efficiency of the first-price auction in the autobidding world. arXiv preprini arXiv:2208. 10650, 2022.
Yuan Deng, Negin Golrezaei, Patrick Jaillet, Jason Cheuk Nam Liang, and Vahab Mirrokni. Multi-channel autobidding with budget and ROI constraints. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 7617-7644, PMIR, 23-29 Jul 2023a.
Yuan Deng, Mohammad Mahdian, Jieming Mao, Vahab Mirrokni, Hanrui Zhang, and Song Zuo. Efficiency of the generalized accond-price auction value maximizers. arXiv preprint arXiv:2310.03105, 2023b.
Yuan Deng, Jieming Mao, Vahab Mirrokni, Yifeng Teng, and Song Zuo. Non-uniform bid-scaling and equilibria for diffcrent auctions: An empirical study. In Proceedings of the ACM on Web Conference 2024, page 256-266, 2024.
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A Proof of Theorem 1
Lemma 1. For each market instance (n, n,,m,,.. (Sk), v).
1
PoA(n, m, 8, [Sk), v) 2 1 unball(n, m, [Sa]. v).
2
Proof. Fix a market instance (n, m, {Sk): J, v), together with an equilibrium given by bid multipliers 0. Without loss of generality, suppose the optimal liquid welfare is 1, i.e.,
Σ
max Biy =
(E[n]
JE(mJ
For each bidder i [n]] and auction [m]]. let the "loss of efficiency" loss;.j caused by in j be:
losss., = Tijj (Prw(j).j - Vt.j).
Moreover, abusing notation, for each bidder E [v]] and each slice hs E [s], let
lossi,N
Σ
lossi. and loss:
Σ
lossi,
jESu
iE/m
We only need to show
Σ
V Fii 2 uubbal(n, m, {Sk]. v),
ME(M)
which is equivalent
loss Σ un, m, 8, [5k], v)-
2
kE[s]
To upper bound the total loss of efficiency, we first establish a tradeoff between the total loss on slice and the total "overpayment" on the same slice. This will involve upper bounding the loss and lower bounding the overpayment. Fix slice & and partition auctions on k into 3 sets:
Ag: auctions where the rightful winner wins with bid multiplier smaller than 1, i.e.,
A = [ E S Frw(s))s = 1 and ( < 1};
By: auctions where the rightful winner wins with bid multiplier larger than equal to 1, i.e.,
B₁ Lj E Sy (rw(s) 1 and ( 1};
Cj: auctions where the rightful winner does not win, i.c.,
C (j Sk Frw(f).s 0).
For brevity, we define the "reserved" share resy to be the market share set As, i.e., resg A,,re < share:.
Consider any bidder i. First observe that for each auction j Sp. loss,,; 0 only if the following happen simultaneously:
Bidder i must outbid the rightful winner in auction j. i.c., Pijj bb,s brw()).s-
Since the rightful winner rw(j) is not winning j. we must have Orl(s), 2 1, which implies bni 2 Urw(j.ii
Given the above properties, for each auction j S₁ where lossi > O. we have Zij. 1 and 0c Us.j = bj.j 2 b Z Urw())) As a result,
loss,y Unw()) V4. E (nw(). - U/w()))/0 0ek
Unw()) S As+ Urw
0,0

---

where '" [n] is the bidder i" who has the largest bid multiplier on slice k. Recall that Ck is the set of auctions in Sg where E; joss;.; > Since for any auction j E Sş where rw(j) 1" 2e we must have
sharec, I share; re5p share, A S (1 - balk) share; res;-
Summing the upper bound for loss., over i [n] and j E Sk, and plugging in the above inequality involving sharec,, we get
loss
Σ
joss
, Vrw(f)
1
JECx
0z . -1 sharecc,
0i.,k
01 kk -1
((1 balk)) share, resk)-
This is our upper bound the total loss, which will be useful later.
Now we lower bound the total overpayment. Note that Vi, implies bi, = Pi.j. So whenever loss;.j 0, based the observations in the two bullet points above, must have
loss..j S Fi.j (brw()) va.)) Tif (bi. va,)) 24, (pi. v.y) Pi.j Zi,j Va.j.
In words, i can only let
much loss of efficiency in j as the amount overpays in j. To this end,
suri.j = Fij Vig - Pt.j be the surplus gets from j. We have: for each i [m] and j E [m].
lossi,j I max(0, sur;,
Summing over E [n] and j E Ck. we get
Σ
max(0, s Z
Σ
loss = lossk
RE
vE(M).JELx
This is one of the two inequalities that we will use to lower bound the total overpayment.
On the other hand, consider the bidder i" who has the largest bid multiplier on slice k. Recall that for any auction S& where rw(s) = i", Ze 1. This means i* overpays in auctions where i" is the rightful winner without causing any loss of efficiency. In fact, the amount of overpayment here can be bounded
sur
Σ
(0-- - 1) v
jESy:nu(j)=x
(0s. 1) --araerre,
2 (0,., l 1) share, balg.
So, combining this inequality with the one above, we get
Σ
max sure,) (0:- -1) share; bal: loss
This is the desired lower bound on the total overpayment.
Now let us put the bounds for the loss and that for the overpayment together into tradeoff between the two guantities. For brevity, let
sur & =
max (0, )-
12

---

Recall the upper bound we have on the total loss:
loss; I 0g -1
((1 - ((1-bale) shareg res;)
01 0zr,k
This gives
0 (1 balx) ) shares. - ress
balx) share; res; loss
Plugging this into the lower bound we have the total overpayment, get
loss;:
sur (1 balı) -share, resk loss; sharey bal: + loss;:
loss 2 (share: resk + sur lossk ((1 balk) share - resk) surg 2
The latter quadratic inequality lossk, which our case implies
loss: S 2 share; resk + surk (share: resk - sur )a 4share baly surg
The above tradeoff is useful because together with additional constraint on {resk) and {sur }. gives an upper bound on Eg lossæ, which is the quantity we want to bound. The additional constraint is from bidders' RoS constraints. For each bidder i, the RoS constraint requires that [ja(m) suraj 2 O, which implies
sur 0.
On the other hand, for each auction j. sur means
only if Aslice(s) (i.e., (rwls), 1). This
Σ
max(0, sura,) s
Σ
resk.
[E[M]]E[M]
kE[e]
kE|a
As result,
Σ sur
Σ
(0 sur
Σ kE(+)
res
kE
iE(m]JE(m]
This is the additional constraint need. Now the problem of upper bounding ΣeE|s loss; reduces to solving the following optimization problem (replacing all variables with shorthands: for resh. Mh for sur and for loss;:):
max
Zk
s.t.
Vk E [s]
25 S 2
share; I6 + yk (sharee - 26. - 32 + 4share; baly. yk
Σ U/ E Σ 54
kE(#)
kEs
Vk [s] 0 3k share,
Vk E [s] yk 20
The rest of the proof is devoted to solving this optimization problem. First we make on Zg. When optimality is achieved:
structural observations. Let fa (Z 3k) be the right hand side of the upper bound
fM Ms) for all k E [s], and ExE(s) = EmE(s) Sk. The latter is simply because fI Mke is strictly increasing in yk.
For each E [s], if I 2 (1 bala) shareg, then 0 and Fy share This is because in such cases, fa (x*, yy) S O, and fa(#,, yr) when f/s = So we would save our budget for (yee } on slice k and spend it elsewhere. For similar reasons, we would max out Ty create more budget for (. In particular, this property implies that æg share, and yk 0 cannot happen simultanously.
13

---

With the above observations enforced as constraints (which does not change the optimal objective value). we further relax the problem. In particular, for each slice we replace the constraint that Ik S (I uk) with I (T ya), where
9& (xx, 2h)
shaaees 2k + y/ ((sarre: Ik 11)² + 4(shares 22) baly y/k)
This relaxation essentially means we imagine slices are arbitrarily divisible, and we can divide each slice into two with the same balancedness, whose market shares sum that of the one before division. As we will see, the relaxation has no cost under our parametrization, because the hard market instances are precisely the ones that divide slices in the way that achieves the maximum objective value in the relaxed problem. We consider this relaxed problem with the additional constraints (in the bullet points) from now
For this relaxed problem, we make more structural observations when optimality is achieved:
There is at most one slice where 0 < I < share,. This is because fixing Exe(s) Tik the most efficient way to distribute this sum into slices is greedy. That is, we first find the last slice which has the minimum balancedness bal, and increase gg. If we reach the limit, i.e., Te share,, then move to the next slice minimum balancedness and repeat this, etc., until we reach the desired sum. One may check that this procedure in fact induces the sub-optimization problem over { and { with the largest optimal objective value given Σxe(s] L*k-
among the remaining ones with the
For any k. k [s] where min Vk } O, we must have
Ogk. (2k. yk) agk (ow. Mk?)
Oyk
Oyk
because otherwise we can locally adjust y/ and yke (while keeping the sum unchanged) and get strictly larger objective value.
Putting the above observations together, we conclude that the optimal solution to the optimization problem must have the following structure: There integer s' E [s]. positive real number C' E [0, 1]. and "effective" market shares (share')), such that:
share's sharey, for each is E [s' 1], share', I sharey', and
share's C.
kE[s
= share; share', for each k: [s].
There exists a real number D > such that dha(us) D for each E [s/]. where dye
hu (v())) = 1 share', + V (share', yu)² + 4share', baly 1/
In particular, fixing C, yk/ /share,, should only depend on balk.
For each k 8', yk 0, and
Σ Uk = 1 - C.
KE[A]]
14

---

Below we solve the relaxed problem given all simplifying observations above. The key step is to pin down the dependency yv. /haree, on bale given C. fact, for each &t [s'],
dhe(y))
3D, Ak E [s']
dyk
(yh share,)) + 2sharej. balk
aD, vk E [s').
v (yk- - share )2 + 4share,, balx y)
(Mi- --aare))))))) )² + 4share's balı (yI= share()) + 4share 12 bal =
aD, Ak E [s)]
(yk - share,,)2 + 4share' bal: yk
share bal: (1 bbla))
3D, vk E [s)]
D
(yh- share,) + 4share', balı yk
(yk- share, )² + 4share's balı y/
3D, Ak [s'].
D
share 2 balı (1 - balk)
3D, vk E [s'), yk share (11 2balk) v /bale (1 balx) (D
3D, Vk [s'], yk share, ((1 2bals) v /all (1 balx) -D
Moreover, observe that the sign in the above expression must be the same for all & E [s'] in order for the derivatives to be the same, which means
3D R, vk [s'], Us share'. ( ((1-2bale) /bale (1 (1-balx) D)
Abusing notation, let
yk (D) = sharej. (11 (11-bbal)) 2balk) + v bal (1 - balx) D)
Note that here D can be either positive or negative.
Given the above observation, when optimality is achieved, the solution is essentially parametrized by the parameter C. In particular, s' and (share') } are uniquely determined by C, and the parameter D is unique given C In fact, since yı (D) is monotone for each & [s'], there is unique D such that
Σ (D( 1 KE|
So we may alternatively write (C² yk (D(C)) for the unique choice of yk given C. Below link the maximum value of the objective to the unbalancedness of the market instance and conclude the proof.
Recall that
a(b, u) = 2b + Vb. (1 b) u.
So
yk (C) = share a(balk, D(C)).
Note that here share', depends on C, and omit this dependency for brevity. Given the choice of D(C), since the slices are indexed such that bal, is weakly increasing, we must have
fo D(C)
= Σ abbal, D(C)) share'. kE[e
Σ - kE|'
where F(I) is the balancedness quantile function of the market instance. In other words, the choice of
D given C' is precisely D(C) ((C) defined in Definition Then the objective be bounded
"Strictly speaking, this is not when bale for all & [s']. However, in such case, the choice of D does not matter, and one must choose C to satisfy constraint on Eee(e yk. the rest the proof, we omit further discussion as this is relatively straightforward corner and simply assume D is unique given C.
15

---

as
he (3kk)
kE|a]
Σ
(share', + yk
(sharej yx.) + 4share', balk yk
kE[s]
(1 (1+ ya ya/sharee V -0--//sharee 4balı yn y /sarea
Σ
sharet
kE[a]
Σ share (1 (1aalbbal.:3000 - ((C))) + 4bal; (oobalk,,(C)) kE[Tl]
a1all
2 / (1
de
/-" (11)(((0)))))) - -1 -,0 di $ 2 (0
max
unbal(n, {St), v).
2
This concludes the proof of the lemma.
Lemma 2. For each t E [2/5, 1], there exists a market instance (n, m, {Skh, which satisfies s, {Sk} v) = and PoA(n, m, {S = t/2.
unbal(n, m, m,
Proof. Recall that the proof of Proposition establishes that unbal( (F) (abusing notation here since unbal depends only on F) weakly increases whenever F pointwise weakly decreases. Moreover, unbal(FF) is clearly continuous in F (where
any natural norm for the space where F resides, e.g., L1). These properties together ensures that for any E [2/5, I]. there exists some F such that unbal(F) = (- in fact, there exists constant function F such that unbal(F)
i.e., there exists E E [0, 1/2] such that the function F where F(x) = for all & E [0, 1] satisfies unbal(F) = t. So. we only need show that given any constant function where (1) F(z) is constantly f for some [0, 1/2] and (2) unbal(F)) = t. there exists market instance (n, m, s, [8+}, v) whose balancedness quantile function is and PoA(n, m, s, {S*), v)
t/2.
We construct such a market instance with n = 3 bidders, m = 6 auctions, and = slices where each slice contains precisely auctions. Let S (1. S2 {3. 4), and S3 {5, 6). Intuitively, we will construct valuations together with an equilibrium, where (1) on S,, bidder 1 steals from bidder (2) S2, bidder steals from bidder 3, and (3) S3, both bidder 1 and bidder win their market share without any competition. Below we choose the exact valuations that implement this plan.
Let w [0, 1] be a parameter to be optimized later. The valuations we choose depend on f and Below we will only specify the non-zero part of the valuations. Consider S3 first: We let U1.5 = f- w) and U2,6 = (1- f)- (1-w). On slice S,, we let 01 = f ww U3,2 -f f- (1-1) w. and 012 be such that
M3.2
1)
(11,1 +12))
w).
U122
In particular, the choice of 1122 guarantees that when bidder pays precisely the minimum amount to "steal" the entire market share of bidder 3 on S,, bidder I's overall buyer surplus Similarly, on slice Sz, we let U2 = f f) 03,4 (1- f)² w. and U2 be such that
U3.4
1):
(22,3 + 22.4) = f) ww)
U/
In particular, the choice V2,4 guarantees that when bidder pays precisely the minimum amount to "steal" the entire market share of bidder 3 on S2, bidder 2's overall buyer surplus is 0.
16

---

Now observe that the following bidding strategies form an equilibrium: 81,1 = 2/,2//1,22 02,2 03,1/U2,4 and all other bid multipliers are 1 0 This means
01,1 + V1,2 + 12 + 12 + V1.5 + U2,6
PoA(n, m, (Sk), v) v
share; + share + shares
share; - 11 (1 w)) + share; - (12(1111)))) - f)
w)) + share
Recall that we still have the freedom to choose w E [0. 1]. So we only need to show
ma (1 w)) h2((1 - f) (1-w)) unbal(n, m, [Sk). v).
In fact, we show that for each E [0, 1],
1-" (1+26)) - V(7))) - dr. xEA(w)
(f (1- () w))
max
Here, h and h2 are as defined in the proof of Lemmal alternative definition of unbal established in the proof of Proposition
[ This implies the above equation given the
|
To see why this equation
holds, observe that the right hand side is equal to
(ar (1 w)) h2((1 x) (1-w)). z
Moreover, since bal bale f. the optimizer must satisfy 2 3r/share1 (1 -s)//haree, which means x = f. This concludes the proof the lemma.
'Here assume particular tiebreaking rule. The proof works without the assumption since one can perturb the valuations avoid tiebreaking.
17

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Reviews:
[{"review_id": "inWFQmnsV6", "reviewer": "Reviewer_DAvE", "summary": "This paper studies the efficiency, measured by the price of anarchy, of a multi-round single-item first-price auction involving multiple bidders. The auctions are segmented into multiple slices, within each of which all bidders utilize uniform bidding strategies. While the bidding parameters of each bidder must remain consistent within each slice, they can vary across different slices.\n\nThe authors define the unbalanceness of the multi-round auction based on all the valuations of the bidders. Based on such unbalanceness, they show that the price of anarchy of the multi-round auction is $1 - \\frac{1}{2}t$ for $t \\in [2/5, 1]$, where $t$ is the upper bound of the unbalanceness. As the unbalancedness increases with more subdivisions of the slices, the authors conclude that more sophisticated auto-bidders lead to less efficient auction outcomes.", "questions": "None", "limitations": "See the weaknesses.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The fine-grained slice-based model is a powerful tool for analyzing the equilibrium outcomes of more sophisticated autobidders compared to simple uniform-bidding strategies.\n2. The theoretical results are insightful, characterizing the relationship between the price of anarchy and the sophistication level of autobidders, using unbalancedness as the key intermediary.", "weaknesses": "There remains a gap between realistic repeated first-price auctions and the fine-grained slice-based modeling. In reality, budget-constrained autobidders dynamically adapt their bidding strategies (such as through pacing) based on auction outcomes. However, in the paper's model, the authors restrict bidding strategies to static uniform bidding within each slice. Despite this, I still find the theoretical results valuable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful and encouraging comments.\n\n**Practicality of (per-slice) uniform bidding**: while autobidders in reality are presumably more sophisticated, there is evidence that the advertising industry finds per-slice uniform bidding a reasonable approximation. In particular, the recent paper \"Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study\" from Google Research considers a form of non-uniform bidding strategies essentially identical to per-slice uniform bidding.  In their paper, they \"define partitions to the queries\" where \"for each partition d, a non-uniform bid-scaling strategy chooses one bid multiplier\", and study \"how the simulation results vary with the granularity of the partition\".  Our results can be viewed as a theoretical explanation of their empirical findings.  As such, we'd argue that conceptually, our results have similar practical implications.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pGaJm3X9wv", "reviewer": "Reviewer_TSHw", "summary": "This paper investigates why non-uniform bidding in first-price auctions causes inefficiency while uniform bidding does not. The authors propose a new model that partitions auctions into slices. Bidders are allowed to bid differently across different slices but need to bid uniformly in each slice. They characterize the price of anarchy of the game using the unbalancedness defined in the paper. The characterization shows that even just a partition of 2 slices will lead to inefficiency, and the balance across slices will improve the efficiency of the market.", "questions": "1. Can the results extend to cases where bidders have different slice partitions of auctions? \n2. Can you provide more intuitions on the definitions of unbalancedness, or how it is derived?", "limitations": "None.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The properties of the unbalancedness function and its relations with the PoA is interesting within the studied model. The fine-grained analysis between settings with the greatest bidding power and the least provides a reasonable explanation for the inefficiency caused by non-uniform bidding in first-price auctions.", "weaknesses": "1. The model assumption that all bidders have the same auction partitions is not compatible with the empirical results it claims to explain, as bidders may have different slices of auctions. While multi-channel bidding seems a reasonable application of the model, different users may participate in different sets of channels. Moreover, there is no central authority that can adjust the balancedness across channels, which limits practical applications of the insight. \n2. The definition of unbalancedness lacks intuition.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful and detailed comments.\n\n**Bidders have different slice partitions**: first we'd like to note that our model is consistent with prior work on non-uniform bid scaling, and in particular, our results can be viewed as a theoretical explanation for the recent empirical study \"Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study\" from Google Research -- they also consider environments where all bidders share the same partitions.  In light of this, we believe this model already captures the essence of non-uniform bidding in reality.  On the other hand, we do agree that a more general model with different partitions would be even better.  In such a model, one can probably extend our results in the following way:\n- Refine the definition of (un)balancedness.  In particular, the reasonable definition of balancedness of a slice should exclude bidders who don't participate in auctions of this slice.  In addition, the definition of unbalancedness should take into consideration the \"adjacency\" between bidders, where two bidders are adjacent if they are both active in some slice.  The resulting definition would probably be based on a \"balancedness graph\", which degenerates to our definition when the graph is complete.\n- Then, when establishing the bounds, one would follow the same high-level plan of forcing a worst-case equilibrium, but now under more constraints.  More specifically, one would probably need to match slices that \"generate\" surplus to those that \"consume\" surplus, subject to the constraint that the surplus of each bidder flowing out of a slice cannot exceed that bidder's market share.  One would need to find the worst-case matching that forces the worst welfare.\n\nOne important consideration here is the tradeoff between the complexity of the model and the cleanliness / readability of the results.  The above plan for extending our results would probably result in a far more complicated bound that depends on more parameters of the market.  Given this, we'd like to argue that our approach achieves a reasonable balance on the spectrum.\n\n**Intuitions on unbalancedness**: the definition \"naturally\" arises in the process of trying to force a worst-case equilibrium.  The idea is that low welfare in auctions with autobidders is typically because in some auctions, competition is extremely low, so the winner gets surplus almost for free.  These bidders who have high surplus then spend the surplus in other auctions to compete with the \"rightful winner\", which ultimately hurts the welfare.  One observation here is that when the latter phenomenon happens, the surplus \"burnt\" to beat the rightful winner consists of two parts: the part that one overpays for auctions where they are the rightful winner, and the part one overpays to beat the rightful winner.  While the latter amount is always the same as the loss of welfare, the former does not contribute to this loss.  So, to cause loss more efficiently, one would like to minimize the former amount.  Fixing the total surplus burnt, this is done by burning more surplus on slices that are less balanced -- hence the definition of balancedness.\n\nNow, another observation is that the two phenomena can hardly happen simultaneously in one slice, because the first requires a low bid multiplier, and the second requires a high one.  So, to force the worst-case welfare, intuitively one would greedily allocate the most unbalanced slices to the second phenomenon and ensure that the rest of the slices generate just enough surplus for burning.  This corresponds to an intricate optimization problem over a set of functions, which turns out to be controlled by the balancedness of each slice roughly in the way that the unbalancedness is defined.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "P0XVBWwfhJ", "reviewer": "Reviewer_DtQM", "summary": "The paper considers the efficiency of auto bidders in first price auctions that use different shading factors on different slices of the items. Claims to show that the improved efficiency of the multi-slice system yields worse social welfare in equilibrium.", "questions": "Please help me identify the model considered. what is the assumption about other bidders? Do they also use slices? and are they using the same slices? and what is the model of  the value of arriving items? Independent Bayesian? worst case?", "limitations": "No negative impact", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "The topic of auto-bidding and efficiency of the results is extremely interesting.", "weaknesses": "The paper fails to clearly identify the model studied.\n\nbased on the rebuttal and the other reviewers comments, i now understand the model. Like reviewer TSHw I am not convinced at the reasonableness of this special case but will increase my score a bit.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments.\n\nIn words: all bidders are symmetric and use the same slices, which are given exogenously.  Other than slices, we use the standard multi-item model, where essentially all items arrive at once at the very beginning, with the values given exogenously and publicly known (whether the values are public does not affect the behavior of the bidders, since they are responding to the *bids* of other bidders).  Each bidder picks a bid multiplier for each slice, so each bidder's strategy is a vector of bid multipliers.  We then consider equilibrium strategy profiles in this perfect-information environment, where each bidder's strategy must maximize their own total value subject to the ROI constraint, given all other bidders' strategies.  Our measure of efficiency is the price of anarchy, i.e., the welfare under the worst equilibrium, divided by the first-best optimal welfare.  We hope the above informal explanation can help clarify the reviewer's questions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MMv8DQQexu", "reviewer": "Reviewer_uz4Y", "summary": "The paper provide a fine-grained price of anarchy analysis for autobidders with non-uniform bid-scaling strategies in first price auctions, showing that first price auctions are more efficient when autobidders are less powerful and are more efficient with more balanced slices.", "questions": "NA", "limitations": "NA", "rating": 5, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Concrete theoretical analysis for PoA of first price auctions with non-uniform autobidders.\n2. It is interesting to find the definition of balancedness of auctions, which can be regarded as a kind of power of auctioneers, are highly related to the efficiency of auctions w.r.t. autobidders.", "weaknesses": "1. The main conclusions of the paper are indeed unsurprising, although it is good to know with a theoretical proof. This does weaken the contribution of this paper.\n2. The framework of flice-based model is kind of restrictive. It is hard to imagine that bidders in real-world auctions bid uniformly in each slides partitioned by the auctioneer.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful and detailed comments.\n\n**Slice-based model**: first we'd like to note that our model is consistent with prior work on non-uniform bid scaling, e.g., \"Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study\" by researchers at Google.  In light of this, we believe this model already captures the essence of non-uniform bidding in reality.  For example, our messages might inform a platform's decision of merging two channels or (not) splitting a channel into several, whenever practically feasible.  On the other hand, another important consideration here is the tradeoff between the practicality of the model and the cleanliness of the messages.  In principle, one could certainly consider more general / sophisticated models and try to derive similar results there (e.g., we sketch one such possibility in our response to Reviewer TSHw), but doing so would probably make the messages much more obscure.  To this end, we'd like to argue that our approach achieves a reasonable balance on the spectrum.\n\n**Conclusions / messages of this paper**: We position our paper as a paper providing a theoretical explanation of the empirical findings from \"Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study\". The conclusion might seem unsurprising provided the previously known empirical results but we’d like to argue that it is indeed technically non-trivial to develop such a theoretical explanation. Our result is also more interpretable and reliable (in the sense that there is a proof) than empirical ones regarding the same phenomenon.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Reviewers more or less agreed that the general topic is interesting and the results appear to be non-trivial and mathematically interesting. On the negative side, all reviewers raised concerns on the practicality of the results due to possibly unnatural assumptions: static uniform bidding within each slice and all bidders have the same auction partitions. After the rebuttal/discussions, all reviewers remained unconvinced, yet the general consensus is that the interesting results outweigh the strong assumptions. After all, the paper is a solid and non-trivial piece of work. It may not be the last paper in this line of research but may inspire future work. That being said, having more discussions on the motivation and the assumptions would make the paper and future work stronger. (Generally, I think "it was assumed in a previous paper by X group" is not a very convincing argument). We hope the authors find the reviews helpful. Thanks for submitting to NeurIPS!
Decision: Accept (poster)