After the New York Times announced its metered paywall last week there has been a lot of empty blather. Standing out from all the noise are two very good analyses. The first was by Felix Salmon for Reuters, analyzing a consumers decision of whether or not to pay. The second one was by Jonathan Stray on Nieman Lab, showing the effect of several different variables on revenue.

This stuff is right up my alley, and I’m currently working on a senior thesis in the field and so I’ll try to extend Salmon’s analysis a little bit. Later on, I’ll take on Stray’s model as well.

##### Salmon's Analysis

Let’s say a reader in a given period reads $N$ articles from the New York Times. Then suppose the New York Times sets the paywall after a consumer has read some $% $ articles. In order to read the $n+1th$ article, the reader must pay a fee of $F$. If $v$ is the value the reader gets from each article, then he will only pay the fee if $v(N-n) > F$. This is a good simple model synopsis.

##### Article Values are Different

Let $n,N,F$ be as before. The first issue that jumps out is that the value of any given article is not constant. The value of articles over a period varies, so let’s arrange them in order of value from highest to lowest.

Let $\{v_i\}_{i=1..\infty}$ be a monotonically decreasing sequence of article values for our reader, with $v_i = 1 \:\forall\: i>N$. Then the reader gets value,

The reader would clearly choose to read the articles he values most first, and after that only pay the subscription if the rest of the articles he has yet to read are still valuable enough. Only if $\sum_{i=n}^{N}{v_i} > F$ will the reader pay the fee.

But this is not quite right either. There’s no way for a reader to know ahead of time which articles are most valuable to him.

##### Predicting future value

Now, instead of ordering the values of articles from highest to lowest, let’s say that the value of articles our reader reads are drawn independently from a probability distribution. Let the value of articles be a random variable $V \sim N\left ( \mu,\: \sigma^2 \right )$ with a normal distribution and $\mu_x$ the average value of an article. $V_1, V_2, V_3,\cdots$ are the value of the first article read, second article read, etc.

Let the period of time for which the reader pays be represented as $\left [ 0,1 \right ]$, and the moment when the reader has read $n$ free articles and must choose whether or not to pay the fee be at time $t\in \left [ 0,1 \right ]$. Assume the reader reads articles at some constant rate $r$ throughout the entire period. Then $t= \frac{n}{r}$.

Now the reader must predict what the value of articles he will read will be to determine whether or not he should pay the fee. Up to point $t$, he has gotten value $\sum_{i=1}^{n}{V_i}$ and average value per article of $\overline{V}= \frac{\sum_{i=1}^{n}{V_i}}{n}$. $\overline{V}$ is also the sample mean of the distribution.

##### Result

Our reader will choose to pay the fee if $( 1-t ) r \frac{\sum_{i=1}^{n}{V_i}}{n} > F$. As $r$ goes up, so does $F$ and as $n$ goes up, $F$ goes down.

There are some interesting suggestions from this. When the New York Times imposes the paywall, they should carefully monitor the rate at which people read its articles. Those that have a low rate would be ideally suited for targeted discounts. Also, since readers make their predictions based on past articles they’ve read, the ideal time to convert non-paying readers is right after a reader reads a series of good articles. If the Times can be subtle about dialing up and down $n$, then they can exploit variance in article value to increase sales.

##### Further work

This analysis is of course still incomplete. Problems I still see with it.

• Knowing that you’ll only get a limited amount of articles for free will change a reader’s behavior. If they’re still uncertain about whether or not paying the fee will be worth it, they will more carefully pick which articles they read before time t. This will bias $\overline{V}$ upwards, but push $r$ downwards. At time $t$, there will also be a back-log of articles that would have been read but weren’t influencing the decision of whether to pay $F$ or not.

• How will the reader decide whether or not to read an article before time $t$? He’ll have to depend on the headline and a summary if available to make a prediction. Before actually reading the article, the reader will predict some value $V_{i}'$ and after reading the article realize some value $V_i$. This average spread $\frac{\sum_{i=1}^{m}{V_i-V_{i}'}}{m}$ will likely affect predictions of future value.

• As is, the model says decreasing $n$ and increasing $F$ leaves the reader’s decision of whether to buy unchanged. But as $n\rightarrow 0$ this becomes a strict paywall, which the gut says people would be less willing to pay for. Another factor in the reader’s decision of whether or not to pay is their confidence about their decision. The larger $n$ is the more confident they will be about their value prediction since the sample mean’s standard deviation will fall, as $\overline{V} \sim N\left ( \mu,\: \frac{\sigma^2}{n} \right )$.

• Paywalls, as described by the New York Times and as currently implemented by the Financial Times and WSJ, are easily bypassed. This can be done either by spoofing the referrer header, or by clearing cookies. This avoidance could also be modeled in in some way.

• Letting people in for free if they come via social media or links from other sites screws everything up. I think this may turn out to be such a huge gaping hole in the paywall that they severely restrict it, but if they don’t there are several ways it can be modeled. You could divide articles between different distributions of those that are primarily found through social media and those that aren’t. The reader would choose whether or not to pay based on the value of those that aren’t. Alternately, an article’s ability to be found through social media could just affect its $V_i$.

• Print subscribers get free access as well. In Salmon’s post he looks at $P-F$, the difference between print subscriber’s fee and online subscribers. If this is less than the value of getting the print paper then the reader will choose the print subscription.

• What if users can choose between a short period, and a longer period with a discount? What does the renewal decision look like?

There are undoubtedly more things that can be done with this model. One of the most obvious is to try and figure out what $n$ and $F$ should be set to.

##### Finding good values for F and n

Since it’s reader’s will not have the same distribution for $V$ it would be theoretically ideal to pick values for $n$ and $F$ individually for every reader. Realistically, the New York Times probably shouldn’t be that opaque about their pricing as it would cause confusion and a negative reaction among readers.

If forced to pick a single price, it would be necessary to find the average value of articles for all readers. That’s what Stray did with his paywall simulation. However, part of the reason that simulation has such wild swings in revenue from relatively small changes is because many of the variables are dependent on each other. For example, the percentage of people who pay for a subscription does not stay constant when $n$ or $F$ change.

I’ll tackle this issue more in my next post.

###### Special Bonus! A pricing algorithm for the FT

This part might still be a bit half baked, but working backwards from the consumer’s decision, it seems possible to figure out a demand curve for each individual piece of content if enough data is available. Since the Financial Times already has a metered subscription plan, if they’ve been good about collecting user data they should have what’s necessary to do this. Here’s an outline of the method.

It requires some change of notation from the above.

Let $a_i \in A \;\forall i\in\mathbb{N}$ be an article, and $x_i \in X \;\forall i\in\mathbb{N}$ be a reader. We will now represent the value of an article to a reader as a mapping $V: A\times X \mapsto \mathbb{R}$ with $V(a_i,x_i)$ representing to the value of article $a_i$ to reader $x_i$. The functions $F(x_i)$ and $r(x_i)$ replace $F$ and $r$ as the fee and rate for reader $x_i$. $n$ is as before.

Define the set $R(x_i)$ such that $a_i \in R(x_i)$ iff $x_i$ reads $a_i$ before deciding whether or not to buy.

So our former equation $\left ( 1-t \right ) r \frac{\sum_{i=1}^{n}{V_i}}{n} > F$ becomes $\left ( r(x_i)-n \right ) \frac{\sum_{a_i \in R(x_i)}{V(a_i,x_i)}}{n} > F(x_i)$.

Rearranging, we get $\frac{\sum_{a_i \in R(x_i)}{V(a_i,x_i)}}{n} > \frac{F(x_i)}{r(x_i)-n}$.

The left side of the above equation is the average value of an article that a reader reads before making the buying decision. So if $x_i$ does buy a subscription, we then know that the average value was at least the right side.

Now that we have an estimate of a given readers average value for content we want to estimate that value across all readers. For any given piece of content, some fixed $a_i$, to determine its value we sum the average value for content of all readers who read $a_i$ before purchasing, and then divide by the total number of readers (who aren’t already subscribers) who’ve read $a_i$.

Define, $% \frac{F(x_i)}{r(x_i)-n}\\ 0 &,\:if\: \frac{\sum_{a_i \in R(x_i)}{V(a_i,x_i)}}{n} \leq \frac{F(x_i)}{r(x_i)-n}\end{Bmatrix} %]]>$ .

Equivalently, $% $ .

This function $\overline{V(x_i)}$ is an estimator of the average $x_i$ has for an article.

Now define the set $S(a_i)$ such that $x_i \in S(a_i)$ iff $x_i$ reads $a_i$ before deciding whether or not to buy a subscription. This set is all non-subscribing readers that read article $a_i$ in the current period, whether or not they’ve ultimately paid for a subscription by the end of the period or not.

If we take $\overline{V(x_i)}$ for each $x_i$ in the set $S(a_i)$, we have a distribution of estimated values for article $a_i$. That might look something like this. Finally, to come up with a set value for a specific piece of content, we sum over the entire set and divide by the number of readers.

With this value, you can now derive a demand curve for the entire site. Or you can dynamically set prices based on what articles a reader has viewed before hitting the paywall.

Exciting stuff, if actually implemented.

If you think I’ve screwed up the math in some way, or if anything isn’t clear, please please let me know. The thoughts in this post are still very much a work in progress.