arXiv subjects

Benjamin Heymann

Publications and source records attributed to Benjamin Heymann.

32 records · Page 2Linked to original sources

arXiv2024

Conditional Separation as a Binary Relation. A Coq Assisted Proof

The concept of d-separation holds a pivotal role in causality theory, serving as a fundamental tool for deriving conditional independence properties from causal graphs. Pearl defined the d-separation of two subsets conditionally on a third one. In this study, we present a novel perspective by showing i) how the d-separation can be extended beyond acyclic graphs, possibly infinite, and ii) how it can be expressed and characterized as a binary relation between vertices. Compared to the typical perspectives in causality theory, our equivalence opens the door to more compact and computational proofing techniques, because the language of binary relations is well adapted to equational reasoning. Additionally, and of independent interest, the proofs of the results presented in this paper are checked with the Coq proof assistant.

cs.DM↗

arXiv2023

Maximizing the Success Probability of Policy Allocations in Online Systems

The effectiveness of advertising in e-commerce largely depends on the ability of merchants to bid on and win impressions for their targeted users. The bidding procedure is highly complex due to various factors such as market competition, user behavior, and the diverse objectives of advertisers. In this paper we consider the problem at the level of user timelines instead of individual bid requests, manipulating full policies (i.e. pre-defined bidding strategies) and not bid values. In order to optimally allocate policies to users, typical multiple treatments allocation methods solve knapsack-like problems which aim at maximizing an expected value under constraints. In the industrial contexts such as online advertising, we argue that optimizing for the probability of success is a more suited objective than expected value maximization, and we introduce the SuccessProbaMax algorithm that aims at finding the policy allocation which is the most likely to outperform a fixed reference policy. Finally, we conduct comprehensive experiments both on synthetic and real-world data to evaluate its performance. The results demonstrate that our proposed algorithm outperforms conventional expected-value maximization algorithms in terms of success rate.

cs.IR↗

arXiv2023

Some bidding games converging to their unique pure equilibrium

We introduce a class of Bayesian bidding games for which we prove that the set of pure Nash equilibria is a (non-empty) sublattice and we give a sufficient condition for uniqueness that is often verified in the context of markets with inelastic demand. We propose a dynamic that converges to the extrema of the equilibrium set and derive a scheme to compute the extreme Nash equilibria.

cs.GT↗

arXiv2023

Fixed point label attribution for real-time bidding

Problem definition: Most of the display advertising inventory is sold through real-time auctions. The participants of these auctions are typically bidders (Google, Criteo, RTB House, Trade Desk for instance) who participate on behalf of advertisers. In order to estimate the value of each display opportunity, they usually train advanced machine learning algorithms using historical data. In the labeled training set, the inputs are vectors of features representing each display opportunity and the labels are the generated rewards. In practice, the rewards are given by the advertiser and are tied to whether or not a particular user converts. Consequently, the rewards are aggregated at the user level and never observed at the display level. A fundamental task that has, to the best of our knowledge, been overlooked is to account for this mismatch and split, or attribute, the rewards at the right granularity level before training a learning algorithm. We call this the label attribution problem. Methodology/results: In this paper, we develop an approach to the label attribution problem, which is both theoretically justified and practical. In particular, we develop a fixed point algorithm that allows for large scale implementation and showcase our solution using a large scale publicly available dataset from Criteo, a large Demand Side Platform. We dub our approach the Fixed Point Label Attribution (FiPLA) Algorithm. Managerial implications: There is often a hidden leap of faith when transforming the advertiser's signal into display labelling. DSP providers should be careful when building their machine learning pipeline and carefully solve the label attribution step.

cs.GT↗

arXiv2023

Repeated Bidding with Dynamic Value

We consider a repeated auction where the buyer's utility for an item depends on the time that elapsed since his last purchase. We present an algorithm to build the optimal bidding policy, and then, because optimal might be impractical, we discuss the cost for the buyer of limiting himself to shading policies.

econ.TH↗

arXiv2022

Offline Evaluation of Reward-Optimizing Recommender Systems: The Case of Simulation

Both in academic and industry-based research, online evaluation methods are seen as the golden standard for interactive applications like recommendation systems. Naturally, the reason for this is that we can directly measure utility metrics that rely on interventions, being the recommendations that are being shown to users. Nevertheless, online evaluation methods are costly for a number of reasons, and a clear need remains for reliable offline evaluation procedures. In industry, offline metrics are often used as a first-line evaluation to generate promising candidate models to evaluate online. In academic work, limited access to online systems makes offline metrics the de facto approach to validating novel methods. Two classes of offline metrics exist: proxy-based methods, and counterfactual methods. The first class is often poorly correlated with the online metrics we care about, and the latter class only provides theoretical guarantees under assumptions that cannot be fulfilled in real-world environments. Here, we make the case that simulation-based comparisons provide ways forward beyond offline metrics, and argue that they are a preferable means of evaluation.

cs.IR↗

arXiv2022

Kuhn's Equivalence Theorem for Games in Product Form

We propose an alternative to the tree representation of extensive form games. Games in product form represent information with $σ$-fields over a product set, and do not require an explicit description of the play temporality, as opposed to extensive form games on trees. This representation encompasses games with a continuum of actions, randomness and players, as well as games for which the play order cannot be determined in advance. We adapt and prove Kuhn's theorem-regarding equivalence between mixed and behavioral strategies under perfect recall-for games in product form with continuous action sets.

cs.GT↗

arXiv2021

Thresholding at the monopoly price: an agnostic way to improve bidding strategies in revenue-maximizing auctions

We address the problem of improving bidders' strategies in prior-dependent revenue-maximizing auctions and introduce a simple and generic method to design novel bidding strategies if the seller uses past bids to optimize her mechanism. We propose a simple and agnostic strategy, independent of the distribution of the competition, that is robust to mechanism changes and local (as opposed to global) optimization of e.g. reserve prices by the seller. This strategy guarantees an increase in utility compared to the truthful strategy for any distribution of the competition. In textbook-style examples, for instance with uniform [0,1] value distributions and two bidders, this no-side-information and mechanism-independent strategy yields an enormous 57% increase in buyer utility for lazy second price auctions with monopoly reserves. When the bidder knows the distribution of the highest bid of the competition, we show how to optimize the tradeoff between reducing the reserve price and beating the competition. Our formulation enables to study some important robustness properties of the strategies, showing their impact even when the seller is using a data-driven approach to set the reserve prices. In this sample-size setting, we prove under what conditions, thresholding bidding strategies can still improve the buyer's utility. The gist of our approach is to see optimal auctions in practice as a Stackelberg game where the buyer is the leader, as he is the first one to move (here bid) when the seller is the follower as she has no prior information on the bidder.

cs.GT↗

arXiv2021

A heuristic for estimating Nash equilibria in first-price auctions with correlated values

Our paper concerns the computation of Nash equilibria of first-price auctions with correlated values. While there exist several equilibrium computation methods for auctions with independent values, the correlation of the bidders' values introduces significant complications that render existing methods unsatisfactory in practice. Our contribution is a step towards filling this gap: inspired by the seminal fictitious play process of Brown and Robinson, we present a learning heuristic-that we call fictitious bidding (FB)-for estimating Bayes-Nash equilibria of first-price auctions with correlated values, and we assess the performance of this heuristic on several relevant examples.

cs.GT↗

arXiv2021

Causal Inference Theory with Information Dependency Models

Inferring the potential consequences of an unobserved event is a fundamental scientific question. To this end, Pearl's celebrated do-calculus provides a set of inference rules to derive an interventional probability from an observational one. In this framework, the primitive causal relations are encoded as functional dependencies in a Structural Causal Model (SCM), which are generally mapped into a Directed Acyclic Graph (DAG) in the absence of cycles. In this paper, by contrast, we capture causality without reference to graphs or functional dependencies, but with information fields and Witsenhausen's intrinsic model. The three rules of do-calculus reduce to a unique sufficient condition for conditional independence, the topological separation, which presents interesting theoretical and practical advantages over the d-separation. With this unique rule, we can deal with systems that cannot be represented with DAGs, for instance systems with cycles and/or 'spurious' edges. We treat an example that cannot be handled-to the extent of our knowledge-with the tools of the current literature. We also explain why, in the presence of cycles, the theory of causal inference might require different tools, depending on whether the random variables are discrete or continuous.

cs.DM↗

arXiv2021

Topological Conditional Separation

Pearl's d-separation is a foundational notion to study conditional independence between random variables. We define the topological conditional separation and we show that it is equivalent to the d-separation, extended beyond acyclic graphs, be they finite or infinite.

cs.DM↗

arXiv2020

Kuhn's Equivalence Theorem for Games in Intrinsic Form

We state and prove Kuhn's equivalence theorem for a new representation of games, the intrinsic form. First, we introduce games in intrinsic form where information is represented by $σ$-fields over a product set. For this purpose, we adapt to games the intrinsic representation that Witsenhausen introduced in control theory. Those intrinsic games do not require an explicit description of the play temporality, as opposed to extensive form games on trees. Second, we prove, for this new and more general representation of games, that behavioral and mixed strategies are equivalent under perfect recall (Kuhn's theorem). As the intrinsic form replaces the tree structure with a product structure, the handling of information is easier. This makes the intrinsic form a new valuable tool for the analysis of games with information.

math.OC↗

arXiv2019

Optimal auctions for networked markets with externalities

Motivated by the problem of market power in electricity markets, we introduced in previous works a mechanism for simplified markets of two agents with linear cost. In standard procurement auctions, the market power resulting from the quadratic transmission losses allows the producers to bid above their true values, which are their production cost. The mechanism proposed in the previous paper optimally reduces the producers' margin to the society's benefit. In this paper, we extend those results to a more general market made of a finite number of agents with piecewise linear cost functions, which makes the problem more difficult, but simultaneously more realistic. We show that the methodology works for a large class of externalities. We also provide an algorithm to solve the principal allocation problem. Our contribution provides a benchmark to assess the sub-optimality of the mechanisms used in practice.

econ.TH↗

arXiv2019

Cost Per Action Constrained Auctions

A standard result from auction theory is that bidding truthfully in a second price auction is a weakly dominant strategy. The result, however, does not apply in the presence of Cost Per Action (CPA) constraints. Such constraints exist, for instance, in digital advertising, as some buyer may try to maximize the total number of clicks while keeping the empirical Cost Per Click (CPC) below a threshold. More generally the CPA constraint implies that the buyer has a maximal average cost per unit of value in mind. We discuss how such constraints change some traditional results from auction theory. Following the usual textbook narrative on auction theory, we focus specifically on the symmetric setting, We formalize the notion of CPA constrained auctions and derive a Nash equilibrium for second price auctions. We then extend this result to combinations of first and second price auctions. Further, we expose a revenue equivalence property and show that the seller's revenue-maximizing reserve price is zero. In practice, CPA-constrained buyers may target an empirical CPA on a given time horizon, as the auction is repeated many times. Thus his bidding behavior depends on past realization. We show that the resulting buyer dynamic optimization problem can be formalized with stochastic control tools and solved numerically with available solvers.

cs.GT↗