Solving the Auction of Google Ads
When advertised websites come up on a Google search result, the ordering of these suggestions is far from random. Having an ad in a high spot in the ordering is infinitely beneficial for an advertiser because of the increased click-through rates it brings.
Auction theory fuels many real-world concepts, including the seemingly unrelated market of search engine optimization. Google, surprisingly to me, uses an auction-style algorithm for its paid advertisement spots. Google uses this system because of the competitive landscape of having ads on such valuable virtual space. Basically, entities that are trying to get an ad under a specific keyword are able to place bids on the positioning of their ad, with the winning bid being in the best position. For example, Expedia would probably want to be the first ad that comes up under the keyword “hotel”, and since they have the wealth to do so, they probably would place a large bid in hopes of getting the best position.
It turns out the mathematical solution to this situation is very similar to a combination of auction theory in a second-price sealed bid auction and a two-sided matching model. Hal R. Varian quantifies this auction process in game theory terms in his essay “Position Auctions” by assigning advertisers to agents (a=1,…,A) and positions to slots (s1,…,S). An agent’s value for a slot s is equivalent to the expected profit they gain per click of the advertisement, both of which are represented as v(a)*x(s), or the agent’s value * the position of slot s. An agent’s bid is represented as b(a), but similarly to a second-price closed auction, the agent only pays the bid of the agent below him (b(s+1)). The profit agent a makes if their bid for slot s is successful is (v(a)-b(s+1))*x(s). The situation is different than that of a regular second-price sealed auction because we cannot assume that every agent wants the first slot, as they may not have the budget. This means there is also a matching component involved. The Nash Equilbrium in which each agent prefers the slot they are in to any other agent’s slot is found when (v(s)-p(s))*x(s)>=(v(s)-p(t))*x(t) for t>s and (v(s)-p(s))*x(s)>=(v(s)-p(t-1))*x(t) for t<s when t is another slot option and p is the profit of the agent getting assigned a specific slot.
Source: https://people.ischool.berkeley.edu/~hal/Papers/2006/position.pdf