arXiv subjects

Kelly Spendlove

Publications and source records attributed to Kelly Spendlove.

12 recordsLinked to original sources

arXiv2026

Autobidding Equilibria in Sponsored Shopping

As commerce shifts to digital marketplaces, platforms increasingly monetize traffic through Sponsored Shopping auctions. Unlike classic ``Sponsored Search", where an advertiser typically bids for a single link, these settings involve advertisers with broad catalogs of distinct products. In these auctions, a single advertiser can secure multiple slots simultaneously to promote different items within the same query. This creates a fundamental complexity: the allocation is combinatorial, as advertisers simultaneously win a bundle of slots rather than a single position. We study this setting through the lens of autobidding, where value-maximizing agents employ uniform bidding strategies to optimize total value subject to Return-on-Investment (ROI) constraints. We analyze two prevalent auction formats: Generalized Second-Price (GSP) and Vickrey-Clarke-Groves (VCG). Our first main contribution is establishing the universal existence of an Autobidding Equilibrium for both settings. Second, we prove a tight Price of Anarchy (PoA) of 2 for both mechanisms.

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arXiv2026

Tight Inapproximability for Welfare-Maximizing Autobidding Equilibria

We examine the complexity of computing welfare- and revenue-maximizing equilibria in autobidding second-price auctions subject to return-on-spend (RoS) constraints. We show that computing an autobidding equilibrium that approximates the welfare-optimal one within a factor of $2 - \epsilon$ is NP-hard for any constant $\epsilon > 0$. Moreover, deciding whether there exists an autobidding equilibrium that attains a $1/2 + \epsilon$ fraction of the optimal welfare -- unfettered by equilibrium constraints -- is NP-hard for any constant $\epsilon > 0$. This hardness result is tight in view of the fact that the price of anarchy (PoA) is at most $2$, and shows that deciding whether a non-trivial autobidding equilibrium exists -- one that is even marginally better than the worst-case guarantee -- is intractable. For revenue, we establish a stronger logarithmic inapproximability, while under the projection games conjecture, our reduction rules out even a polynomial approximation factor. These results significantly strengthen the APX-hardness of Li and Tang (AAAI '24). Furthermore, we refine our reduction in the presence of ML advice concerning the buyers' valuations, revealing again a close connection between the inapproximability threshold and PoA bounds. Finally, we examine relaxed notions of equilibrium attained by simple learning algorithms, establishing constant inapproximability for both revenue and welfare.

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arXiv2026

Chaos in Autobidding Auctions

As autobidding systems increasingly dominate online advertising auctions, characterizing their long-term dynamical behavior is brought to the fore. In this paper, we examine the dynamics of autobidders who optimize value subject to a return-on-spend (RoS) constraint under uniform bid scaling. Our main set of results show that simple autobidding dynamics can exhibit formally chaotic behavior. This significantly strengthens the recent results of Leme, Piliouras, Schneider, Spendlove, and Zuo (EC '24) that went as far as quasiperiodicity. Our proof proceeds by establishing that autobidding dynamics can simulate -- up to an arbitrarily small error -- a broad class of continuous-time nonlinear dynamical systems. This class contains as a special case Chua's circuit, a classic chaotic system renowned for its iconic double scroll attractor. Our reduction develops several modular gadgets, which we anticipate will find other applications going forward. Moreover, in discrete time, we show that different incarnations of mirror descent can exhibit Li-Yorke chaos, topological transitivity, and sensitivity to initial conditions, connecting along the way those dynamics to classic dynamical systems such as the logistic map and the Ricker population model. Taken together, our results reveal that the long-term behavior of ostensibly simple second-price autobidding auctions can be inherently unpredictable and complex.

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arXiv2024

Auto-bidding and Auctions in Online Advertising: A Survey

In this survey, we summarize recent developments in research fueled by the growing adoption of automated bidding strategies in online advertising. We explore the challenges and opportunities that have arisen as markets embrace this autobidding and cover a range of topics in this area, including bidding algorithms, equilibrium analysis and efficiency of common auction formats, and optimal auction design.

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arXiv2024

Graded differential groups, Cartan-Eilenberg systems and conjectures in Conley index theory

Cartan-Eilenberg systems play an prominent role in the homological algebra of filtered and graded differential groups and (co)chain complexes in particular. We define the concept of Cartan-Eilenberg systems of abelian groups over a poset. Our main result states that a filtered chain isomorphism between free, P-graded differential groups is equivalent to an isomorphism between associated Cartan-Eilenberg systems. An application of this result to the theory of dynamical systems addresses two open conjectures posed by J. Robbin and D. Salamon regarding uniqueness type questions for connection matrices. The main result of this paper also proves that three connection matrix theories in the literature are equivalent in the setting of vector spaces, as well as uniqueness of connection matrices for Morse-Smale gradient systems.

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arXiv2024

Complex Dynamics in Autobidding Systems

It has become the default in markets such as ad auctions for participants to bid in an auction through automated bidding agents (autobidders) which adjust bids over time to satisfy return-over-spend constraints. Despite the prominence of such systems for the internet economy, their resulting dynamical behavior is still not well understood. Although one might hope that such relatively simple systems would typically converge to the equilibria of their underlying auctions, we provide a plethora of results that show the emergence of complex behavior, such as bi-stability, periodic orbits and quasi periodicity. We empirically observe how the market structure (expressed as motifs) qualitatively affects the behavior of the dynamics. We complement it with theoretical results showing that autobidding systems can simulate both linear dynamical systems as well logical boolean gates.

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arXiv2022

The algebra of semi-flows: a tale of two topologies

To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies. We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraization yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows.

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arXiv2022

Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics

Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for $ε$-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to $ε$-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any $ε$ between zero and $0.09$. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics.

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arXiv2021

Morse Theoretic Templates for High Dimensional Homology Computation

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We demonstrate the effectiveness of templates in two settings: first, by computing the homology of certain cubical complexes which are homotopy equivalent to $\mathbb{S}^d$ for $1\leq d \leq 20$, and second, by computing Conley complexes and connection matrices for a collection of examples arising from a Conley-Morse theory on spaces of braids diagrams.

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arXiv2021

A computational framework for connection matrix theory

The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.

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arXiv2021

Exploration-Exploitation in Multi-Agent Competition: Convergence with Bounded Rationality

The interplay between exploration and exploitation in competitive multi-agent learning is still far from being well understood. Motivated by this, we study smooth Q-learning, a prototypical learning model that explicitly captures the balance between game rewards and exploration costs. We show that Q-learning always converges to the unique quantal-response equilibrium (QRE), the standard solution concept for games under bounded rationality, in weighted zero-sum polymatrix games with heterogeneous learning agents using positive exploration rates. Complementing recent results about convergence in weighted potential games, we show that fast convergence of Q-learning in competitive settings is obtained regardless of the number of agents and without any need for parameter fine-tuning. As showcased by our experiments in network zero-sum games, these theoretical results provide the necessary guarantees for an algorithmic approach to the currently open problem of equilibrium selection in competitive multi-agent settings.

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arXiv2020

Homology of configuration spaces of hard squares in a rectangle

We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p \times q$ rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.

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