#### Hal Varian (Google Chief Economist) – Nash equilibria and bidding in Google Auctions (Feb 2006)

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#### Abstract

Understanding the Dynamics of Google’s Auction System: A Deep Dive into the Nash Equilibrium and Advertising Strategies

In the rapidly evolving world of online advertising, Google’s auction system stands as a pivotal model for digital marketplaces. This article delves into the intricate workings of this system, as explained by Hal Varian, a prominent economist at Google. By dissecting the Nash equilibrium within the context of Google’s auction strategy, the evolution of the auction system, and the strategies employed by advertisers, this piece aims to provide a comprehensive understanding of the mechanisms that drive one of the world’s most influential advertising platforms.

The Genesis and Evolution of Google’s Auction System

Google’s auction system, a cornerstone of its advertising success, underwent significant evolution. Initially conceptualized to auction search results, the model has matured into auctioning ad space. This system, characterized as a second-price auction, allows advertisers to bid for clicks, where the payment is pegged to the bid below the winning bid. This structure aims to ensure that advertisers with higher-value bids gain more prominent positions, enhancing the probability of receiving clicks. Importantly, Varian stresses that advertisers incur costs only when users engage with their ads, creating a synergy between advertisers and publishers.

The origins of Google’s auction system can be traced back to an employee of the Idea Factory who was inspired by Charlie Plott’s auction course. The system evolved from auctioning search results to auctioning ad space while generating algorithmic search results. This transition occurred around the time when GoTo, later known as Overture, commercialized the ad auction model, which Google adopted in 2001. The key feature of Google’s auction is its employment of a second-price auction mechanism, where advertisers pay the bid of the advertiser just below them, not their own bid. Ads are ranked by the product of their bid and predicted click-through rate, not just the bid. This innovative model aligns publisher and advertiser interests by allowing the publisher to sell impressions while advertisers pay for clicks.

The Intricacies of Nash Equilibrium in Auctions

The Nash equilibrium, a fundamental concept in game theory, finds a specific application in Google’s auction system. This equilibrium exists when no bidder can improve their profit by unilaterally changing their bid, assuming others’ bids remain constant. A subset of this, the symmetric Nash equilibrium, involves monotone bids satisfying certain inequalities. Understanding these equilibria is crucial as they dictate the strategic bidding behavior in auctions. The work of Edelman, Ostrovsky, and Schwartz, which resonates with Varian’s findings, further illuminates this concept.

In the context of generalized second price auctions used in Internet advertising, an example with four positions and four bidders illustrates the Nash equilibrium and the conditions for bidding to satisfy this equilibrium. The Nash equilibrium is defined as a set of bids where no bidder can increase their profit by changing their bid, assuming others’ bids remain unchanged. The symmetric Nash equilibrium introduces the concept where bids are monotone and satisfy certain inequalities. It demonstrates that symmetric Nash equilibrium inequalities imply the previously defined Nash equilibrium inequalities and provides a recursive formula to solve for a symmetric Nash equilibrium.

Bidding Strategies in Sponsored Search Auctions

In sponsored search auctions, equilibrium bids diverge from bidders’ true values. Instead, these bids represent a weighted average of the bidder’s value and the bid immediately below them, influenced by relative click-through rates. This complex interplay of bids and values, defined by marginal costs or the incremental cost per click of moving up a position, shapes advertisers’ strategies.

Equilibrium in a Google auction is not achieved by truthful bidding. Advertisers report a weighted average of their value and the bid of the advertiser below them. The weights depend on the relative click-through rates of the advertisers. An excluded bidder may bid their true value, or a reserve price might exist. Advertisers do not have direct knowledge of other bids during the auction. They experiment and adjust their bids to maintain a particular position. Modeling assumes knowledge of other bids, similar to Nash equilibrium. Dynamic exploration of bids could lead to unpredictable behavior.

Advertisers set bids based on the incremental cost-per-click value proposition. Bidding should stop when the incremental cost per click exceeds the value of that click to the advertiser. Finding the equilibrium for the bidding process is not necessary. Market testing through repeated bidding experiments can effectively determine the optimal bid for advertisers. Advertiser awareness of the impact of increased bids on clicks, and evaluating the value of additional clicks against the increased cost is crucial.

Search Engine Management (SEM) and Search Engine Optimization (SEO) companies often assist advertisers in bid optimization. These intermediaries understand advertiser objectives, the bidding model, and ways to optimize for specific goals. Bids can be changed continuously using an Application Programming Interface (API). The system updates bids in real time.

Keyword Bidding and Market Definition

Advertisers strategically bid on keywords to position their ads effectively in search results. They employ specific bidding rules to balance the maximization of click value against budget constraints. Factors like language, geography, and time of day play a crucial role in defining the market for these ads. Additionally, advertisers often engage intermediaries like SEMs or SEOs to optimize their bidding strategies.

Keywords used by advertisers determine who they compete against in the auction. Matching queries to keywords can be specified as broad, exact, phrase, or negative match. Advertisers define the target audience through language and geographical specifications, a practice known as day parting. Google offers auto advertisers the option of ad exclusivity, which prevents multiple ads from the same advertiser from appearing on the same page. Google has a limited number of ad slots, usually 11 per page, with three top ads and eight right-hand side ads. Google ensures that only one ad from a given advertiser appears on a page to avoid a bad user experience.

Landing page quality is crucial to prevent users from clicking on ads that lead to the same destination. Affiliates, like those of Amazon, may have multiple ads for the same product or service. Car manufacturers and their dealers may have separate ads for the same product, representing different business units. Generally, multiple ads leading to the same landing page are avoided, as it results in a negative user experience.

The Economics of Click-Through Rates and Ad Placement

The relationship between click-through rates, ad quality, and bid amounts is central to ad placement and profitability. Ads with higher click-through rates can outperform those with higher bids, underlining the importance of ad quality. This dynamic guides the ranking of advertisers and ultimately determines ad placement.

Real-World Data and Nash Equilibrium in Google’s Auctions

Empirical analysis of Google’s auction system reveals a convex relationship between cost and click-through rate, aligning with Nash equilibrium predictions. The observed cost functions, exhibiting monotone convex characteristics, suggest that the system effectively maintains a balance between clicks and costs, ensuring value consistency for advertisers.

Broader Implications and Additional Insights

The system not only facilitates direct sales but also builds brand value. Cost per acquisition metrics, varying across products, offer insights into the lifetime value of customers versus immediate profits. Varian highlights the alignment between VCG auctions, promoting truthful bidding, and Google’s auction system, where the dominant strategy equilibrium in VCG auctions corresponds with the Nash equilibrium in Google auctions. Beyond auction mechanics, user behavior and external factors like online trends and search behavior significantly influence advertising dynamics.

Advertisers do not have direct knowledge of other bids during the auction and adjust their bids to maintain a particular position. Bid values for each position can be derived from the model, and the total auction revenue is calculated using a specific expression. In equilibrium, an advertiser’s value is trapped between upper and lower bounds determined by marginal costs.

In online advertising auctions, the equilibrium state ensures that the incremental cost per click increases with the click-through rate. This is because a decrease in incremental cost with increasing bid would lead to buying expensive clicks and missing out on cheap ones, which cannot occur in equilibrium.

The other bidders in the auction collectively define a supply curve of clicks. Advertisers can pick their desired position on this supply curve based on their value per click. The incremental cost per click is represented by the slope of the line connecting clicks (horizontal axis) and expenditure (vertical axis). Profit maximization involves finding the point where the iso-profit line (negative of profit with vertical intercept) intersects the supply curve.

Profit maximization can be viewed as a convex optimization problem. By adjusting the iso-profit line, advertisers can find the optimal point that maximizes their profit. The profit-maximizing point implies a range of possible values for the advertiser. This range is determined by the geometry of the supply curve and the iso-profit line. Observing the advertiser’s optimized position reveals the upper and lower bounds of their value range.

Traditional bidding models consider only the per-click value, which may not lead to optimal revenue. Expected revenue considers the product of per-click value and the number of clicks, ensuring the most profitable ad placement. CTR (click-through rate) is influenced by two effects: Position-specific effect (ZI) indicates additional clicks due to more prominent placement. Ad quality effect (EI) represents clicks an ad would receive in an average position. CTR is modeled as the product of ZI and EI, allowing for separate treatment of position and quality effects.

Ad quality (EI) remains consistent regardless of position, ensuring that high-quality ads continue to attract clicks even in lower positions. Bidders are ranked based on EI × BI (ad quality × bid), prioritizing ads with both high quality and bid value. Ad quality can be approximated by the predicted click-through rate, leading to the calculation of cost per impression or price per impression.

Concluding Thoughts

This exploration of Google’s auction system, through the lens of Hal Varian’s insights and the concept of Nash equilibrium, offers a window into the sophisticated mechanisms driving online advertising. As the digital marketplace continues to evolve, understanding these dynamics remains crucial for advertisers, economists, and digital strategists alike.

Varian emphasizes the importance of logistic regression in estimating click-through rates (CTRs) for individual ads. By utilizing historical data and various factors like day of the week, country, and ad creative, advertisers can predict the likelihood of users clicking on their ads, informing their bidding strategies.

Varian introduces the concept of dynamic pricing in Google’s auction system, where the price paid by an advertiser is determined based on the quality of their ad relative to others. This leads to a Nash equilibrium, where no advertiser can improve their position by unilaterally changing their bid. This equilibrium ensures efficient allocation of ad space and maximizes the overall value generated by the auction.

Varian presents empirical evidence to validate the theoretical predictions of the Nash equilibrium in Google’s auction system. By analyzing cost-per

-click (CPC) distributions for different keyword auctions, he demonstrates that the actual CPC function closely resembles an increasing convex function, supporting the theoretical model. This provides empirical confirmation of the Nash equilibrium’s role in shaping advertiser bidding behavior.

To assess the robustness of the Nash equilibrium in Google’s auction data, Varian conducts a perturbation analysis. He examines how much the data needs to be modified to satisfy the Nash equilibrium conditions. The results show that the average absolute deviation required is around 5%, indicating a good fit between the model and the observed data. This suggests that the Nash equilibrium is a reasonable representation of the strategic bidding behavior in Google’s auction system.

Varian acknowledges the importance of learning and experimentation in online advertising. Advertisers often adjust their bids based on observed performance to find their optimal positions in the auction. The deviations from the theoretical Nash equilibrium can be attributed to this learning process, as advertisers experiment to maximize their returns. This highlights the dynamic nature of online advertising and the need for continuous adaptation to changing market conditions.

To further validate the Nash equilibrium model, Varian calculates the implied value per click for advertisers based on their bids and compares it with the actual CPC they pay. He finds that the upper and lower bounds on the implied value per click align well with the actual prices, supporting the model’s predictions. This provides additional evidence for the validity of the Nash equilibrium in explaining advertiser behavior in Google’s auction system.

Advertisers with higher surplus to cost ratios often have sophisticated strategies and understand the relationship between bids, clicks, and keywords. Initiatives are underway to provide more information and guidance to smaller advertisers to help them bid more intelligently. Providing better guidance to advertisers aims to reduce noise in the system, helping advertisers make more informed bidding decisions.

Notes by: WisdomWave