Episode 19: Subjective Probability (Part 2)

I want to take you back to maybe 6th grade math? Ratios! A ratio just so you remember is for example, the number of shots made by a player in a basketball game, say 9 for 14, or 9/14 (a nice efficient game).

But ratios really mess people up; quick, what’s better 71/331 or 42/199? It’s not easy to solve.

A “paper with the best name nominee”, “Six of One, Half Dozen of the Other” by Burson and Larrick set out to find the weird human behavior that arises when people are confronted by ratios.

The biggest part of a ratio that messes people up is when comparing two equal ratios as they change. It’s a variation of the Subjective Probability issue we’ve talked about previously. People misjudge the value of proportional increases.

Here’s a simple example to illustrate the point Burson and Larrick were trying to accomplish. Would you rather increase your score from 80 points to 100? Or 4 points to 5? Their hypothesis is that people like the 80 points to 100 increase more, because again, increasing 20 points is better than increasing 1. Of course, the ratios are the same so it’s an equivalent relative increase in both situations.

In Study 1 Burson and Larrick had subjects evaluate cell phone plans in the first scenario, and a movie plan in the second. Here’s the original tables so you can see how it was all set up:

Let me explain this to you. Start with condition 1 in Table 1. As (I hope) you can see, you should notice that both Plan A, and Plan B are slightly different; one is cheaper, but the other has more value.  In Condition 2, the plans are EXACTLY THE SAME. This is very important. The only thing that has changed is the scale of the ratios. One is by a factor of 10x, the other, price per year vs. price per month. Again. They are exactly the same.

The same happens in Table 2 with the movies. Plan A is cheaper, but Plan B has more movies. It’s a reasonable tradeoff. In Condition 2, the ONLY thing that is different is that the number of movies is expressed per year instead of per week.

There should be no preference for one plan over another. Preference for Plan A should be the same whether it’s the price per month or the price per year, right? It’s all the same.

Well framing is everything. For cell phones, Plan B (the cheaper plan) was preferred 53% to 31% when it had a lower price per year. Most likely because the difference in price looks bigger ($60 instead of $5/month).

Meanwhile Plan A (higher quality plan) was preferred 69% to 23% when It had many more dropped calls… per 1000 instead of per 100.

For the movie plan there was the same result. The only variable that changed was number of movies per week vs. per year (the price stayed monthly). People preferred Plan A (the cheaper plan) 57% to 33% when the number of movies was given per week, because the difference between 7 and 9 is small.

But people preferred Plan B (the higher quality plan) 56% to 38% when the number of new movies was given yearly.

The bottom line from Study 1: framing is important, and people will think that bigger numerical differences create a relatively bigger movement, even when the ratio is exactly the same. This is a tried and true marketing technique: “For only $3 a day you could have premium life insurance” is used instead of “For only $1,080 a year you could have premium life insurance”.

The other classic example: “Only 4 easy payments of $22.95”.

To sum up in a slightly different way: As attribute expansion increases, preference also increases.

Burson and Larrick didn’t stop there. Study 2 re-examined the issue by asking participants what they would be willing to pay.

Participants were again exposed to different movie plans. They were given what an “average plan” costs, and how many new movies they get per week, as well as the “target plan” (aka, the researchers’ target), that only gave the number of movies per week. The price was empty, and subjects were asked to fill it in with what they would be willing to pay.

For example, in Condition 1, the average plan gave you 9 new movies per week for a price of $12/month. If you were to only get 7 movies per week what would you be willing to pay? The average by the way was $9.20 which feels fairly reasonable.

A quick note. This technique is pretty standard for behavioral economists. What we call the “willingness to pay” is a great way to measure how attractive an option is. If the willingness to pay goes up, then the offer must have become more attractive.

There were four Conditions in Study 2.

Two gave the number of new movies per week (per my example in Condition 1). One had a target plan with fewer new movies per week than the average plan, and one had a target plan with more new movies per week than the average plan.

Condition 3 and 4 were identical to Conditions 1 and 2, except the numbers of new movies were given in years not weeks.

Again. The plans and their costs are identical. The ONLY thing that has changed in Condition 3 and 4 is that the number of new movies is now being expressed on a yearly basis.

The goal was to see if there is a difference in what people are willing to pay. The plans are the same. People should pay the same for the same number of movies, whether they’re given per week or per year. It’s the same number of movies! The price is even the same for goodness sake.


This graph is a little hard to read. The first two dots on the left are the plans given with movies per week (Conditions 1 and 2). The dot at the bottom left of $9.20 is the plan we alluded to before with fewer movies than the average plan in Condition 1. The dot on the top left of $11.55 is the plan with more movies than the average plan.  Obviously people should pay more for the plan with more movies compared to the average and that’s what they do.

What gets very interesting is when you take the EXACT same plans, and just expand them to the number of movies per year, which is what the dots on the right are. They should be the same price! It’s silly to expect people to pay less, or more, for the same number of movies but that’s exactly what happens.

The average willingness to pay for the lower movie plan drops to $8.83 when expressed in movies per year, and the willingness to pay for the higher movie plan bumps up to $13.82.

These are considerable movements. While I would not assume you can achieve this level of change in your application or organization, the researchers here were able to get about a 5% drop in relative value for cheap plans if expressed annually, or about a 20% increase in value when expressed annually.

I will quote the paper’s final conclusions:

“Attribute expansion inflated the perceived difference between two alternatives on that attribute, and thereby increased its weight relative to the other attributes.”

So big takeaways:

When you’re comparing your product to the “average” competitor and your product is better than average in a category, make that interval of time as big as possible to maximize the number, and therefore the benefit.

A great example is what student loan companies do. I get letters in the mail from the SoFi’s in the world that say you could save $40,000 today! That number is huge! Of course they get that by comparing your early payoff in 10 years with minimum average payments you’d make for federal student loans over 30. They’ve stretched the window for savings as far as possible to maximize the benefit, and it certainly makes a huge impression.

If you or your customer is comparing your product to the average and your product is worse than average in a category, make that interval of time as small as possible to minimize the number, and therefore the difference of the negative attribute.

If your product is $2880 per year, and your competitor’s product is $2520, don’t use annual prices. Instead say “they” are $7 per day but have no features. Your product is only $8 per day, only one extra dollar, but has this whole list of expanded features!

We’ll talk a lot more about segmentation later. But this is another great example of how framing and segmenting work. Give it a try. It’s all about the numbers.

Burson, K. A., Larrick, R. P., & Lynch, J. G. (2009). Six of One, Half Dozen of the Other. Psychological Science20(9), 1074-1078. doi:10.1111/j.1467-9280.2009.02394.x