Types of Auction

Auctions come in a wide variety of flavours. The most common are: English auctions, Dutch auctions, Sealed first-bid auctions, Sealed second-bid auctions, and All-pay auctions.

English: An english auction is the traditional auction that most people recognise. It involves multiple bidders all able to bid and outbid each other, progressively raising the price to beat the others. When one bidder is left, he/she wins the item with the bid that nobody else was willing to beat.

Dutch: In a dutch auction, the auctioneer progressively lowers the price until the first bidder decides to buy. This is similar to several of the cable tv channels selling items by progressively dropping the price. The people who want the item do not want to bid until they believe they are at risk of not winning the item, but they do not want to leave it too late and lose.

Sealed first-price: In this auction each party makes just one bid, and the highest bidder wins the item, paying what they bid.

Sealed second-price: In a second-price auction each bidder makes a single bid, and the highest bidder wins the item, paying the next highest bidder’s bid. This is the format of most internet auctions like eBay and Google Adwords.

All-pay: In an all-pay auction each bidder pays up front, then the highest bidder wins the item. Nobody gets their bid returned. This isn’t a common design of auction for public use, but it closely resembles the dynamics of systems like political bribery or corporate hospitality - everybody will make significant entertainment outlays in the hope of winning the client, but the bidder who puts on the most lavish display has shown themselves to be the most keen and gets the business.

As different (and in some cases counter-intuitive) as these may seem, the strategy in all (other than the all-pay, in which optimal behaviour is a little different) is the same. Each party will bid the true value to them of winning the item. I’ll go through each in turn and describe why this is true.

English: In the English auction, the bidder is progressively bidding against others. The bidder will not stop early, before their value is met. Although this would save them from paying more money and might make them more cash-rich overall, they had already decided what the item was worth and would have been better off continuing to bid, forgoing the cash in favour of winning. Conversely, the bidder will not bid past their value. As soon as the bidding goes beyond the level the bidder values the item at, every extra unit of currency they hypothetically bid is wasted money for no additional value from the purchase. So the bidder is best off by bidding up to their item. If the bidding stops and the price has not yet reached their value, then they have effectively “won” by having the item value, plus residual cash.

Dutch: In the dutch auction, the bidders are all waiting for the price to reach an acceptable value. Everybody’s acceptable value may be different. For example, when bidding on a piece of land the value to a developer planning to build a discount store may be very different to that of a developer planning to build a casino. Their projected profits from the investment are different, so they will value the land differently. But nobody knows anybody else’s value, because nobody bids until the winning bid takes place. So it is not in a bidder’s favour to bid early, because they will pay more than their valuation and receive back the same land value but lose cash. And again, it is not in the bidder’s interest to wait past their value, because although they could return with the value and the extra cash, the risk of losing the item that was already at or below the value placed on it mean that the bidder is likely to come away with nothing.

Sealed first-price: In this auction the bidders do not have any idea about anybody else’s bids. The same motivations apply as in the english auction.

Sealed second-price: This one’s tricky. Because of the fact that the winning bidder will not be paying their bid, it is easy to think that the optimal strategy would be to bid slightly higher than your true value, knowing that you won’t end up paying that amount. However this still does not produce the best results for the bidder. The key is that it is impossible to know the second-place bidder’s value that they are willing to pay. So if the second bidder is well below the first bidder’s value, then the second bidder has lost nothing by bidding their true value - they won’t pay any higher given the second bidder’s bid doesn’t change. If the second bidder is only slightly below the first bidder’s value, then the first bidder would lose the item if they bid lower, even though they would be paying at or below the value they placed on it. Conversely, if the bidder were to bid above their true value then they are in an identical situation if the second bidder is below the first bidder’s true value - they pay the second bidder’s price regardless. However if the second bidder is above the first bidder’s true value then the first bidder ends up paying above their true value and loses out. So the optimal strategy is still to bid the true value.

All-pay: In an all-pay auction the strategies change somewhat. Risk neutrality levels change the optimal situations, and strategies do not generally arise from pure equilibria. This is beyond the scope of this article.

Google’s Auction

The dynamics of Google’s auction broadly follow a second price auction. However the auction occurs not for how much you pay, but for what rank you appear in. Because the winning rank does not determine who gets clicked on and therefore who pays, it needs to behave a little differently.

Google’s Quality Score feature is what allows them to have some control over the ranking process. It is made up mostly of click-through rate, ad relevance and landing page relevance. This is an area that is constantly in flux and is not fully available to the public to understand. Debates and investigations continue to try to ascertain the true algorithm as closely as possible, but broadly speaking the most important of the three public factors is click-through rate. Quality score is assigned a value from 1 to 10.

Rank is then decided by the auction of QS x Bid. The highest rank sits first on the page, the second highest rank sits second on the page, and so on. So far this doesn’t seem much like an auction - people have bid, but nobody has paid for anything.

To determine how much people pay, Google applies the second-price philosophy of only paying the next highest bid. But instead of bid, it determines that you must pay enough to equal the next highest rank. So if your QS was 8 and your bid 3, then your rank would be 24. If the next highest rank is 22, then the amount that you would pay to reach that rank would be 2.75.

This system means that when your ad is clicked on you pay the amount that your rank on the page requires you to pay to stay appearing above the next person. You can never pay more than your bid, and the amount you pay less depends on how good the rank of the next highest ranked ad is.

In this situation it is not just conceivable but indeed likely that a higher ranked ad would pay less if clicked than a lower ranked ad would. All that would be required would be for the quality scores to be appreciably difficult. Knowing this, how is your bidding strategy affected?

What You Pay

In Google’s auction, the end result is that the proportion of your bid that you pay depends on how far your rank is above that of the ad below yours. The reasoning goes like this: regardless of your quality score and the other bidders’ quality scores and ranks, to beat the ad below you to reach your position on the page you have to have a higher rank. And regardless of your quality score (which we assume doesn’t change based on your position) your payment will then be discounted from your bid in order to beat match the ranking of the next ad and keep you in that position, but paying the “second price” (which is based on rankings, not bids).

So if the ad positioned below yours has a rank of 90% of yours, you will have to pay 90% of your bid if you get clicked on. Similarly if the ad below yours has a ranking of 50% of yours, you will pay 50% of your bid if you get clicked on.

What does this mean? It means that we can have situations like the one below:

Position	Rank	Quality Score	Bid	Pay
1	28	8	3.50	3.38
2	27	?	?	?
3	14	?	?	?

Suppose your ad is showing in position 1, and you drop below the ad beneath you to become position 2. The new table looks like this:

Position	Rank	Quality Score	Bid	Pay
1	27	?	?	?
2	26	8	3.25	1.75
3	14	?	?	?

That’s a difference of 48% in cost per click for a drop of 7% in bid!

As you can see, when in position 1 the ad is paying 96% of the bid in cost per click, and the rank of the second ad is 96% of the rank of the first ad. By dropping our bids so that we are in a position where there is a bigger gap in ranking between our ad and the next highest ranked ad, we can see that we are now paying just 54% of our bid, and that the next highest rank is 54% of ours.

Strategies For Pay Per Click

So what do we do because of this? Well obviously we can’t see the rank of the other bidders in the auction. So we don’t know beforehand which position or what bid will allow us to take advantage of this best. But we can learn some things, if we’re willing to make some assumptions:

1. A higher position leads to a better click-through rate. This assumption is not always true, but it is true enough of the time to be safe for this analysis.
2. A higher position leads to more overall impressions. In advertising areas with a lot of competition, there are more adverts that can be shown on the first page. Google rotates the ads for various reasons like repeated searches by a user, standard vs accelerated ad serving, and simple random factors so ads lower down the page are rotated off more often.
3. Quality score stays stable across positions. This one is a bit more dodgy. Google claim that they take into account the effect of position on click-through-rate when using it to determine quality score, but the system is imperfect. Imperfect, but improving.

As a result of this we can assume that we will get more clicks if we show in a higher position. But we can’t always afford to run that far up the page. What we want is a position that gives us the highest position to cost-per-click ratio (ignoring other factors like budgetary limits and conversion rate differences in different positions).

But we saw earlier that in a regular second-price auction we want to bid our true value. Has that changed now that we have some influence over the proportion of that we end up paying? Now that we know we can find more efficient positions? Simply put, no.

In the case above where the owner of the ad dropped their bid by 7% and ended up paying 48% less, they could just have well have dropped their bid by 47% and wound up in the same position on the page, paying the same. There is no significant benefit to us in running an ad with a cost per click much lower than the bid. If we assume quality score stays stable then we can say that we could drop the bid and nothing would result. Except in the event of competitors changing bids. Because this is a sealed auction, we do not know what competitors are doing. If the position three ad from the table above were to raise their bid enough to increase their rank to 25, then in the case shown our ad will still be in second position - but now paying 3.13. However if we had reduced our rank down to 15 (to keep us just above position three) then we would now be in third position and paying an entirely different proportion of our new bid (based on the position four ad).

Let’s take a look then at the strategies open to us:

1. We bid above our true value, but in a position where our actual cost per click is roughly equal to our true value. In this circumstance if a competitor lowers his/her rank, we pay less and we gain. However if a competitor raises his/her rank, we pay above our true value and we lose overall.
2. We bid below our true value. In this strategy there is no situation whereby we gain. We are not getting enough clicks because our position is too low. We are paying below the value we have placed on getting each click and we lose overall.
3. We bid our true value. By doing this we are in a position where if competitors move up or down the page below us our cost per click will fluctuate, but as long as they stay below our rank we will still gain overall. We will be paying at maximum the value we placed on each click, and in some cases we will pay less. If competitors move above our rank we may lose clicks, but only clicks that would otherwise have been more expensive.

The third strategy is the only one where we can win, but not lose.

So when you’re looking at how much to bid per click on Adwords, do still think about positions, efficiencies, budget limitations, and trends. But don’t forget that the Google system is clever, but not obtuse. You still get the value you are willing to pay, so prepare to pay your real value.

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