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Let’s Make A Deal: Bayesian Probability
INTRODUCTION
One of the aspects of Bayesian Inference is, which can be applied to marketing, is that one makes decisions and inferences in which evidence or observations are used to update probabilities or to infer that a probability is true.
THE PRICE IS RIGHT
In the Grand Prize part of Let’s Make a Deal (LMAD), the contestant has 3 doors. Behind one door is the grand prize and behind the other two doors, there is nothing. Once the contestant picks the door, the host asks the contestant if they want to switch. The probability of the same game is based on the contestant having a strategy for switching versus not switching. Assume door #1 has the Grand Prize.
ODDS BY NOT SWITCHING:
Probability of Picking Door 1: Win (1/3).
Probability of Picking Door 2: Lose (1/3).
Probability of Picking Door 3: Lose (1/3).
Odds of Winning: 1/3.
Odds of Losing: 2/3 = (1/3) + (1/3)
In this instance, the contestant has a 1/3 chance of winning by ‘not switching’. Anybody who has watched TPIR knows that the host never opens the door with the Grand Prize and will ALWAYS open a losing door. By having this piece of evidence or information it allows one to use Bayesian Inference to determine the correct odds. In order to do that, we have to provide additional basic statistical information.
Expected Value is the probability of an even happening multipled by the value of an event. It also requires Conditional Probability or the odds of event B happening given event A having just happened. So after picking a door, what is the next step to maximize probability of winning.
ODDS BY SWITCHING:
Information: The host will always open a losing door.
1. Pick Door 1 (winner), Doors 2 or 3 are shown, player switches and loses with 100% probability. Given the contestant a 1/3 chance of picking the correct door and 100% chance of losing by switching, the odds of losing in this scenario are 1/3.
2. Pick Door 2, Door 3 is opened (host knowing it is a loser), player switches and wins with 100% probability. Given the contestant picked a losing door, was shown a losing door, switching increases the probability of winning to 100%. 1/3 chance of picking Door 2 and 100% of winning by switching or 1/3 chance of winning.
3. Pick Door 3, Door 2 is opened (host knows it was a loser), player switches and wins with 100% probability. Given that the contestant picked a losing door, was shown another losing door, switching increases probability of winning to 100%. 1/3 chance of picking Door 3 and 100% chance of winning by switching or 1/3 * 100% = 1/3 chance of winning.
So by switching there is a 2/3 chance of winning.
By using the knowledge that the host will never open a door with the Grand Prize, one is able to form an logical argument for switching. Knowing this information that the host will never open a winning door, the Contestant has a 1/3 chance of winning by never switching and based on the the contestant has a 2/3 chance of winning by switching.
DERIVING RELATIVE AD RANK
In Google AdWords, you are paying .01 the Ad Rank of the ad in the position below you divided by your Quality score + .01. Currently, an advertiser does not know their quality score, but it is my understanding that Google will release this number in the near future. For now we do not have access to our our Q Score.
By using your Actual CPC with no knowledge of your own Q Score, you can estimate the ratio:
#1: Ad Rank (A) = Max CPC (A) * Q Score (A).
#2: Actual CPC (A) = (Ad Rank(B)/ Q Score(A)) + .01
Where: Q Score (A) = Ad Rank(B) / (Actual CPC (A) - .01).
Substituting Q Score (A) into equation #1: Ad Rank (A) = (Max CPC(A)) * (AdRank (B) / Actual CPC(A) + .01).
We can derive that the Ratio of AdRank (B) / AdRank (A) = (Actual CPC(A) - .01) / (Max CPC(A))
or AdRank(A) / AdRank(B) = (Max CPC(A) )/ (Actual CPC(A) -.01) = k.
If you are competing in a market vertical whereby your competitor has a comparable profit structure and assumptions on Max CPC’s can be made, this type of analysis can help you determine how much better or worse your Q score is relative to your competition. Essentially, if I did the math properly, one should be able to derive relative Ad Rank and therefore for comparable industries a relative quality score. Once Google releases an actual metric for their Quality Score I believe that there will be many more ways to get information about how to optimize a campaign for Quality Score Metrics.
I encourage any readers well-versed in math to see if there are any inconsistencies in my analysis as I did a lot of the derivations in my head instead of using a computer and I will try to revise them or walk through step-by-step. The purpose of this article is show how inferences can be drawn and decisions based on using incomplete information to make decisions.
CONCLUSION
Once Google releases an estimate of your Q Score, a smart advertiser should be able to derive their competitors Max CPC and Q Score through a similar analysis as described above. Alternatively, with comparable Q scores, one should be able to determine their competition’s Max CPC.
Hey, please check my math. This was an off the cuff quick & dirty analysis and may not be perfect, so I invite any comments, questions, corrections, or concerns for me to address. I did leave steps out in deriving formulas.
4 Comments
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I don’t see what the game or Bayesian Probability has to do with Google Quality Scores, but the game you’re talking about isn’t “The Price is Right”. It’s Monte Hall’s “Let’s Make a Deal”.
Michael, Thanks or bringing that to my attention. IT IS Monty Hall’s Let’s Make A Deal and has been corrected. Bayesian Probability is used to make decisions based upon information. The information one appears is able to glean from the analysis, under various assumptions, is your Ad Rank relative to the position below you and if you make an assumption with regard to Max CPC I believe you can estimate Quality Scores.
The assumption that your competitor has a similar max cpc is really the weakpoint in the argument (or that you can estimate their CPC). I’ve rarely seen two competitors so economically aligned that they bid the same. Heck, I rarely see competitors bidding the same across PPC engines.
Hi Dave,
Thanks for the feedback.
I see many scenarios where that is not true - one of them being the PPC arbitrage space for Ringtone Affiliates where their CPA is approximately the same and the bids are very tightly clustered.
In other spaces, where the economic value of a lead is somewhat known such as the mortgage and debt space, this also holds true.
At the end of the day, if you assume that people are running their business efficiently (not always a great assumption) or that they have the value attached to a CPL, they should have comparable bids. In a lot of mathematical proofs you have to make some simplifying assumptions to demonstrate concepts.