An agentic market is a decentralized serving-time market where AI agents bid to claim each user prompt. An agent's bid is based on its expected value for the prompt, which is the expectation of the reward for its response minus its inference costs. First-price auctions (FPAs), in which the highest bidder pays its bid and receives the prompt, are a natural way to allocate prompts in such a market, since they are simple and transparent to run, and do not require central knowledge of private capabilities and costs. However, FPAs are not truthful: participants may bid less than their true value in the hope of a larger profit, and the prompt may then be allocated to someone other than the bidder who values it the most. This welfare loss is quantified by the $\textit{price-of-anarchy}$ (PoA). In this paper, we argue that agentic markets are likely to possess features that reduce the price-of-anarchy. The first is a $\textit{competitive floor}$, which occurs when multiple bidders usually value the prompt above some baseline share of the optimal welfare (e.g., because the best models are often distilled into cheaper alternatives, which perform similarly). We show that competition among them then limits the welfare loss, which can be driven to zero as the floor rises. The second feature is $\textit{near-independence}$ of values after conditioning on the prompt, which occurs when most of the predictable variation in the expected value is determined by the difficulty of the prompt. As the value distribution approaches conditional independence, we can derive an efficiency guarantee that approaches a factor of $1 - 1/e^2$ of the optimal welfare. Thus, both features give rise to welfare lower bounds that are significantly better than the well-known general case of $1 - 1/e$. We validate these results empirically using auctions with strategic bidders simulated on the EmbedLLM prompt routing dataset.
This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings. This theory complements traditional economic techniques, which generally focus on exact and optimal solutions and are accordingly limited to relatively stylized settings. We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information and deterministic strategies; extension theorems, which extend such guarantees to r…
We consider auctions in which greedy algorithms, paired with first-price or critical-price payment rules, are used to resolve multi-parameter combinatorial allocation problems. We study the price of anarchy for social welfare in such auctions. We show for a variety of equilibrium concepts, including Bayes-Nash equilibrium and correlated equilibrium, the resulting price of anarchy bound is close to the approximation factor of the underlying greedy algorithm.
We study the efficiency of sequential multiunit auctions with two-buyers and complete information. For general valuation functions, we show that the price of anarchy is exactly $1/T$ for auctions with $T$ items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by $1-1/e\simeq 0.632$. This bound is asymptotically tight as the number of items sold tends to infinity.
We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e-1) for first-price auctions with fractionally subadditive valuations which reveals the worst-case price dist…
We study the price of anarchy of the first-price auction in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders). We show that with autobidders only, the price of anarchy of the first-price auction is $1/2$, and with both kinds of bidders, the price of anarchy degrades to about $0.457$ (the precise number is given by an optimization). These results complement the recent result by Jin and Lu [2022] showing that the price of anarchy of the first-price auction with traditional bidders only is $1 - 1/e^2$. We furt…
The Generalized Second Price auction (GSP) has been widely used by search engines to sell ad slots. Previous studies have shown that the pure Price Of Anarchy (POA) of GSP is 1.25 when there are two ad slots and 1.259 when three ad slots. For the cases with more than three ad slots, however, only some untight upper bounds of the pure POA were obtained. In this work, we improve previous results in two aspects: (1) We prove that the pure POA for GSP is 1.259 when there are four ad slots, and (2) We show that the pure POA for GSP with more than four ad slots is also 1.259 given the bidders are ra…
Game-theoretic models relevant for computer science applications usually feature a large number of players. The goal of this paper is to develop an analytical framework for bounding the price of anarchy in such models. We demonstrate the wide applicability of our framework through instantiations for several well-studied models, including simultaneous single-item auctions, greedy combinatorial auctions, and routing games. In all cases, we identify conditions under which the POA of large games is much better than that of worst-case instances. Our results also give new senses in which simple auct…
This paper develops tools for welfare and revenue analyses of Bayes-Nash equilibria in asymmetric auctions with single-dimensional agents. We employ these tools to derive price of anarchy results for social welfare and revenue. Our approach separates the standard smoothness framework into two distinct parts, isolating the analysis common to any auction from the analysis specific to a given auction. The first part relates a bidder's contribution to welfare in equilibrium to their contribution to welfare in the optimal auction using the price the bidder faces for additional allocation. Intuitive…
Incorporating budget constraints into the analysis of auctions has become increasingly important, as they model practical settings more accurately. The social welfare function, which is the standard measure of efficiency in auctions, is inadequate for settings with budgets, since there may be a large disconnect between the value a bidder derives from obtaining an item and what can be liquidated from her. The Liquid Welfare objective function has been suggested as a natural alternative for settings with budgets. Simple auctions, like simultaneous item auctions, are evaluated by their performanc…
We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi-…
We initiate the study of the social welfare loss caused by corrupt auctioneers, both in single-item and multi-unit auctions. In our model, the auctioneer may collude with the winning bidders by letting them lower their bids in exchange for a (possibly bidder-dependent) fraction $γ$ of the surplus. We consider different corruption schemes. In the most basic one, all winning bidders lower their bid to the highest losing bid. We show that this setting is equivalent to a $γ$-hybrid auction in which the payments are a convex combination of first-price and the second-price payments. More generally, …
We expand the literature on the price of anarchy (PoA) of simultaneous item auctions by considering settings with correlated values; we do this via the fundamental economic model of interdependent values (IDV). It is well-known that in multi-item settings with private values, correlated values can lead to bad PoA, which can be polynomially large in the number of agents $n$. In the more general model of IDV, we show that the PoA can be polynomially large even in single-item settings. On the positive side, we identify a natural condition on information dispersion in the market, termed $γ$-hetero…
We study the price of anarchy of mechanisms in the presence of risk-averse agents. Previous work has focused on agents with quasilinear utilities, possibly with a budget. Our model subsumes this as a special case but also captures that agents might be less sensitive to payments than in the risk-neutral model. We show that many positive price-of-anarchy results proved in the smoothness framework continue to hold in the more general risk-averse setting. A sufficient condition is that agents can never end up with negative quasilinear utility after playing an undominated strategy. This is true, e.…
In many natural settings agents participate in multiple different auctions that are not simultaneous. In such auctions, future opportunities affect strategic considerations of the players. The goal of this paper is to develop a quantitative understanding of outcomes of such sequential auctions. In earlier work (Paes Leme et al. 2012) we initiated the study of the price of anarchy in sequential auctions. We considered sequential first price auctions in the full information model, where players are aware of all future opportunities, as well as the valuation of all players. In this paper, we stud…
We study the efficiency of simple auctions in the presence of complements. [DMSW15] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of $O(\log m)$ for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinemen…
The Generalized Second Price (GSP) auction is the primary auction used for monetizing the use of the Internet. It is well-known that truthtelling is not a dominant strategy in this auction and that inefficient equilibria can arise. In this paper we study the space of equilibria in GSP, and quantify the efficiency loss that can arise in equilibria under a wide range of sources of uncertainty, as well as in the full information setting. The traditional Bayesian game models uncertainty in the valuations (types) of the participants. The Generalized Second Price (GSP) auction gives rise to a furthe…
Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanism…
In this work we consider selling items using a sequential first price auction mechanism. We generalize the assumption of conservative bidding to extensive form games (henceforth optimistic conservative bidding), and show that for both linear and unit demand valuations, the only pure subgame perfect equilibrium where buyers are bidding in an optimistic conservative manner is the minimal Walrasian equilibrium. In addition, we show examples where without the requirement of conservative bidding, subgame perfect equilibria can admit a variety of unlikely predictions, including high price of anarchy…
We study equilibria in two-buyer sequential second-price (or first-price) auctions for identical goods. Buyers have weakly decreasing incremental values, and we make a behavioural no-overbidding assumption: the buyers do not bid above their incremental values. Structurally, we show equilibria are intrinsically linked to a greedy bidding strategy. We then prove three results. First, any equilibrium consists of three phases: a competitive phase, a competition reduction phase and a monopsony phase. In particular, there is a time after which one buyer exhibits monopsonistic behaviours. Second, the…
The design of the best economic mechanism for Sponsored Search Auctions (SSAs) is a central task in computational mechanism design/game theory. Two open questions concern the adoption of user models more accurate than that one currently used and the choice between Generalized Second Price auction (GSP) and Vickrey-Clark-Groves mechanism (VCG). In this paper, we provide some contributions to answer these questions. We study Price of Anarchy (PoA) and Price of Stability (PoS) over social welfare and auctioneer's revenue of GSP w.r.t. the VCG when the users follow the famous cascade model. Furthe…
We study cardinal auctions for selling multiple copies of a good, in which bidders specify not only their bid or how much they are ready to pay for the good, but also a cardinality constraint on the number of copies that will be sold via the auction. We perform first known Price of Anarchy type analyses with detailed comparison of the classical Vickrey-Clarke-Groves (VCG) auction and one based on minimum pay property (MPP) which is similar to Generalized Second Price auction commonly used in sponsored search. Without cardinality constraints, MPP has the same efficiency (total value to bidders)…
We study the efficiency of simple combinatorial auctions for the allocation of a set of items to a set of agents, with private subadditive valuation functions and budget constraints. The class we consider includes all auctions that allocate each item independently to the agent that submits the highest bid for it, and requests a payment that depends on the bids of all agents only for this item. Two well-known examples of this class are the simultaneous first and second price auctions. We focus on the pure equilibria of the induced strategic games, and using the liquid welfare as our efficiency …
We study the social efficiency of several well-known mechanisms for the allocation of a set of available (advertising) positions to a set of competing budget-constrained users (advertisers). Specifically, we focus on the Generalized Second Price auction (GSP), the Vickrey-Clarke-Groves mechanism (VCG) and the Expressive Generalized First Price auction (EGFP). Using the liquid welfare as our efficiency benchmark, we prove a tight bound of 2 on the liquid price of anarchy and stability of these mechanisms for pure Nash equilibria.
We study two standard multi-unit auction formats for allocating multiple units of a single good to multi-demand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to be paid per unit. Variants of both formats find applications ranging from the allocation of state bonds to investors, to online sales over the internet, facilitated by popular online brokers. For these formats, we consider two bidding interfaces: (i) standard bidding, which is most prevalent in the scientific l…
This paper establishes the Price of Stability (PoS) for First Price Auctions, for all equilibrium concepts that have been studied in the literature: Bayes Nash Equilibrium $\subsetneq$ Bayes Correlated Equilibrium $\subsetneq$ Bayes Coarse Correlated Equilibrium} $\bullet$ Bayes Nash Equilibrium: For independent valuations, the tight PoS is $1 - 1/ e^{2} \approx 0.8647$, matching the counterpart Price of Anarchy (PoA) bound \cite{JL22}. For correlated valuations, the tight $\PoS$ is $1 - 1 / e \approx 0.6321$, matching the counterpart PoA bound \cite{ST13,S14}. This result indicates that, in t…
The Generalized Second Price auction is the primary method by which sponsered search advertisements are sold. We study the performance of this auction under various equilibrium concepts. In particular, we demonstrate that the Bayesian Price of Anarchy is at most $2(1-1/e)^{-1} \approx 3.16$, significantly improving upon previously known bounds. Our techniques are intuitively straightforward and extend in a number of ways. For one, our result extends to a bound on the performance of GSP at coarse correlated equilibria, which captures (for example) a repeated-auction setting in which agents appl…
Auto-bidding is an area of increasing importance in the domain of online advertising. We study the problem of designing auctions in an auto-bidding setting with the goal of maximizing welfare at system equilibrium. Previous results showed that the price of anarchy (PoA) under VCG is 2 and also that this is tight even with two bidders. This raises an interesting question as to whether VCG yields the best efficiency in this setting, or whether the PoA can be improved upon. We present a prior-free randomized auction in which the PoA is approx. 1.896 for the case of two bidders, proving that one c…
Auto-bidding is now widely adopted as an interface between advertisers and internet advertising as it allows advertisers to specify high-level goals, such as maximizing value subject to a value-per-spend constraint. Prior research has mostly focused on auctions which are truthful (such as SPA) since uniform bidding is optimal in such auctions, which makes it manageable to reason about equilibria. A tantalizing question is whether one can obtain more efficient outcomes by leaving the realm of truthful auctions. This is the first paper to study non-truthful auctions in the prior-free auto-biddin…
The internet advertising market is a multi-billion dollar industry, in which advertisers buy thousands of ad placements every day by repeatedly participating in auctions. An important and ubiquitous feature of these auctions is the presence of campaign budgets, which specify the maximum amount the advertisers are willing to pay over a specified time period. In this paper, we present a new model to study the equilibrium bidding strategies in standard auctions, a large class of auctions that includes first- and second-price auctions, for advertisers who satisfy budget constraints on average. Our…
The literature on the Price of Anarchy (PoA) of simple auctions employs a no-overbidding assumption but has completely overlooked the no-underbidding phenomenon, which is evident in empirical studies on variants of the second price auction. In this work, we provide a theoretical foundation for the no-underbidding phenomenon. We study the PoA of simultaneous 2nd price auctions (S2PA) under a new natural condition of {\em no underbidding}, meaning that agents never bid on items less than their marginal values. We establish improved (mostly tight) bounds on the PoA of S2PA under no underbidding f…
We introduce a new class of games, called social contribution games (SCGs), where each player's individual cost is equal to the cost he induces on society because of his presence. Our results reveal that SCGs constitute useful abstractions of altruistic games when it comes to the analysis of the robust price of anarchy. We first show that SCGs are altruism-independently smooth, i.e., the robust price of anarchy of these games remains the same under arbitrary altruistic extensions. We then devise a general reduction technique that enables us to reduce the problem of establishing smoothness for …
Internet ad auctions have evolved from a few lines of text to richer informational layouts that include images, sitelinks, videos, etc. Ads in these new formats occupy varying amounts of space, and an advertiser can provide multiple formats, only one of which can be shown. The seller is now faced with a multi-parameter mechanism design problem. Computing an efficient allocation is computationally intractable, and therefore the standard Vickrey-Clarke-Groves (VCG) auction, while truthful and welfare-optimal, is impractical. In this paper, we tackle a fundamental problem in the design of modern …
Most of the economic reports forecast that almost half of the worldwide market value unlocked by AI over the next decade (up to 6 trillion USD per year) will be in marketing&sales. In particular, AI will enable the optimization of more and more intricate economic settings, in which multiple different activities need to be jointly automated. This is the case of, e.g., Google Hotel Ads and Tripadvisor, where auctions are used to display ads of similar products or services together with their prices. As in classical ad auctions, the ads are ranked depending on the advertisers' bids, whereas, diff…
Maximal (also miner) extractable value, or MEV, usually refers to the value that privileged players can extract by strategically ordering, censoring, and placing transactions in a blockchain. Each blockchain network, which we refer to as a domain, has its own consensus, ordering, and block-creation mechanisms, which gives rise to different optimal strategies to extract MEV. The strategic behaviour of rational players, known as searchers, lead to MEV games that have different impacts and externalities in each domain. Several ordering mechanisms, which determine the inclusion and position of tra…
As is well known, many classes of markets have efficient equilibria, but this depends on agents being non-strategic, i.e. that they declare their true demands when offered goods at particular prices, or in other words, that they are price-takers. An important question is how much the equilibria degrade in the face of strategic behavior, i.e. what is the Price of Anarchy (PoA) of the market viewed as a mechanism? Often, PoA bounds are modest constants such as 4/3 or 2. Nonetheless, in practice a guarantee that no more than 25% or 50% of the economic value is lost may be unappealing. This paper …
