At an EC conference many years ago I made some remarks on the role of budgets in sponsored search auctions. Unsurprisingly, they fell on deaf ears. I lack the gravitas of a Maskin or a Myerson or I was (am) wrong. Following (Walter Scott’s version of) Robert the Bruce, I try again.

To recap, sponsored search is a form of advertising sold by auction. When a user queries a search engine like Bing (product placement) for ipod, advertisers bid to be featured alongside the search listings. The advertisements appear in a list in a section of the page designated as sponsored. Each position in the list is called a slot. Generally, advertisements that appear in a higher ranked slot garner more attention and clicks from users. Thus, all else equal, merchants generally prefer higher ranked slots to lower ranked slots. Advertisers pay only if a user clicks on their slot. Furthermore, advertises also have the opportunity to submit a budget.

The standard model has N bidders and K < N slots. The probability that someone clicks on slot i is C(i) and is called the click through rate (CTR). Assume that C(1) >  C(2) > ….> C(K), i.e., lower ranked slots have smaller CTR’s. Notice also that the CTR is assumed independent of the identity of advertisers. Furthermore, the CTR’s assumed to be common knowledge.

When a user clicks on a slot occupied by advertiser j, advertiser j earns a benefit of V(j). This benefit does not depend on whether the click came via the top slot, bottom slot or any slot in between. Furthermore, this benefit is the private information of the advertiser. Advertiser j, assigned to slot i, enjoys a benefit V(j)C(i). Thus, each advertiser prefers a higher ranked slot to a lower ranked slot. Advertisers submit bids that represent what they will pay for each click received. Controlling for advertiser quality, higher bids are assigned to higher slots.

The advantage of the standard model is that the standard machinery of auction theory can be brought to bear upon it. There are features of the model one can criticize like CTR’s being common knowledge, CTR’s are advertiser independent and so on. These are well known and attempts to address them have been made in the literature. I would like to direct attention to the investigations that incorporate budget constraints. I argue these are wrong headed.

1) My first argument is by based on reverent authority. Preston McAfee has claimed that budget constraints don’t matter.

2) My second argument is that if advertisers have budgets that bind something is awry in the standard model. Observer that whether a click comes on the top or bottom slot, it is worth the same to the advertiser. Thus, the slot an advertiser is assigned to, only determines the speed at which clicks arrive. If an advertiser does not care about speed, it does not matter which slot they are assigned to. Given a fixed budget, an advertiser will want to buy clicks as cheaply as possible. So, an advertiser would want to be on the bottom slot where prices are presumably the lowest! One can pick apart the assumptions I made (like hard budget constraint, infinite patience). My point is, that when one accounts for the fact that the advertiser is consuming a flow rather than a quantity, it is no longer clear that an advertiser prefers the slot with the highest CTR.

3) My third argument is related to the second. If the search engine is going to keep the advertiser on a slot until their budget is exhausted, then it is the budget that determines how much revenue the search engine makes. The click through rate determines the speed at which the budget is consumed. So, the search engine should be concerned with assigning advertisers to slots so as to maximize the rate at which $’s are generated.

To summarize. Either budgets don’t matter or they do in which case simply inserting them into the standard model is wrong.