Tag Archives: marketing

A problem well stated is a problem half-solved

Lately, I’ve been reflecting on how we can prevent “things from getting worse” in a variety of settings — at work, within our families, and even more broadly in society. I’ve come to believe that what truly matters is our ability to course correct: to adjust and improve situations once we understand that they are deteriorating.

The difficulty, of course, lies precisely in that word — understand. Recognizing that things are getting worse is often far from easy. In many cases, it is hard even to articulate clearly what we mean when we say that something is “not going in the right direction.” And in most environments, it can be genuinely difficult to speak up and say, “Something is not right here.”

This is what brings to mind the quote often attributed to Charles Kettering, former head of research at General Motors: “A problem well stated is a problem half solved.” There is considerable wisdom in that statement, particularly when dealing with complex issues and when trying to mobilize groups of people toward meaningful course correction.

Why do I say this? Fundamentally because, in my experience, when problems or opportunities are not well stated, a host of negative dynamics tend to emerge.

  • People begin to adapt to problems instead of solving them — a powerful driver of many organizational and social failures.
  • A lack of clarity makes it difficult for competent individuals to take the lead.
  • Those who have adapted successfully to a flawed situation often resist change, even when the overall impact is negative.
  • Morale and energy decline as conditions worsen and collaboration becomes harder.

Within the limits of this post, I want to share some thoughts on what can help prevent this outcome. In particular, I want to highlight one family of tools that humans have developed to state problems with exceptional clarity: quantitative models. To be clear, the core point is not about mathematics per se, but about fostering clarity of language and transparency in order to enable course correction. Quantitative models are simply a particularly powerful way to achieve that.

Quantitative models: well-stating problems the hard way

Quantitative models are not always available — after all, they require measurable quantities — but when they are, they are remarkably effective. They force assumptions to be explicit, make key trade-offs visible, and provide a shared and precise language that greatly facilitates collaboration. It is no coincidence that the extraordinary progress of physics, chemistry, and engineering from the 16th century onward coincided with the widespread adoption of mathematical modeling.

In some cases, such as physics, we are even able to state incredibly complex problems with great precision without fully understanding them. Quantum mechanics is a striking example: we can formulate models that answer factual questions with astonishing accuracy, even when their interpretation in plain language remains deeply contested.

Another major benefit of mathematical models is that they uncover relationships that would otherwise remain hidden. A simple example illustrates this.

Suppose you run a sales effort in which you provide services at a loss, hoping that a fraction of prospects will eventually convert to paid customers. You would like to understand how to balance this investment in order to maximize profit. Assume that you are sophisticated enough to have an estimate of the probability of conversion for different prospect profiles.

A natural question arises: up to what probability of conversion should we be willing to provide services at a loss?

At a high level, the answer is intuitive: the gains from converted prospects must offset the losses from those who do not convert. In quantitative terms, this can be written as:

Conv\% \cdot (ConvValue-Investment) \ge (1-Conv\%) \cdot Investment

Solving this inequality at breakeven yields the minimum conversion probability required for profitability:

Conv\% = \frac{Investment}{ConvValue}

This expression immediately clarifies several things. Profitability depends only on the investment per prospect and the value of a converted customer. And if the investment exceeds the conversion value, there is simply no viable business.

A less obvious insight concerns sensitivity. The conversion threshold depends on investment and conversion value in exactly opposite relative terms: increasing investment by 10% raises the required conversion rate by 10%, while increasing customer value by 10% lowers the threshold by 10%. This kind of elasticity-based reasoning is extremely hard to see without writing the problem down explicitly.

Of course, this model is simplified. In practice, conversion rates often depend on investment — for example, offering a richer free trial may increase the likelihood of conversion. At first glance, this seems to make the problem much harder: the conversion rate depends on investment, but investment decisions depend on the conversion rate.

Yet writing this down actually simplifies the situation. If conversion probability is a function of Conv\%(Investment), profitability requires

Conv\%(Investment) \ge \frac{Investment}{ConvValue}​

Rather than a fixed threshold, we now have a relationship defining a region of profitability. Far from being an obstacle, this opens the door to optimization: by segmenting prospects by expected value, we can refine investment levels and improve outcomes.

This is a general principle in quantitative modeling: relationships between variables may complicate the mathematics, but they expand the space of possible strategies.

From thresholds to overall profits

So far, the discussion has focused on whether to pursue a given prospect segment. But what about overall profitability once we act?

If conversion rates are not easily influenced by investment, total profits can be written as:

Profits = MarketSize \cdot ( P(Converting)\cdot(ConvValue - Investment) - P(Not\ Converting)\cdot Investment )

Suppose the baseline conversion rate is 30% and the long-term value of a converted customer is three times the investment. Plugging in the numbers yields a loss: on average, each prospect served generates a loss equal to 10% of the investment.

At this point, several strategic levers are available: improve the product without raising costs, improve it while raising costs but increasing conversion, reduce free-trial costs, or develop targeting models to focus on prospects more likely to convert.

How should a team — say, a group of founders with very different backgrounds — decide which lever to prioritize? Disagreement is inevitable, and mistakes will be made. This is precisely why course correction matters, and why developing a precise language around the problem is so important.

Consider targeting. Suppose we segment the market into two equal-sized groups: Segment A with a 40% conversion rate, and Segment B with a 20% conversion rate. Targeting only Segment A yields positive profits — a substantial improvement driven by a very rough segmentation (equivalent to a two-bin scorecard with a Gini of roughly 23%). See below:

SegmentAProfits=\frac{1}{2}\cdot MarketSize\cdot(40\%\cdot(3\cdot Investment-Investment)-60\%\cdot Investment)=\frac{1}{2}\cdot MarketSize\cdot(20\%\cdot Investment)

With further work, we could address questions such as: how valuable is improving targeting further? How does that compare with reducing free-trial costs or increasing customer lifetime value? Quantitative models allow us to ask — and answer — these questions systematically.

Clarity, knowledge, and course correction

One might object that quantitative models are difficult for many people to understand, and therefore limit broad participation in decision-making. This is a fair concern. But clarity is never free. Whether expressed mathematically or otherwise, precision requires effort.

Course correction depends on acquiring and applying new knowledge, and conversations about knowledge are rarely easy. We cannot hope to improve conversion through product enhancements without learning what users value most — and learning often requires time, attention, and risk. As Feynman put it, we must “pay attention” at the very least.

Recognizing knowledge, applying it, and revising beliefs accordingly is hard, even for experts. A well-known anecdote from Einstein’s career illustrates this. After developing general relativity, Einstein initially concluded — incorrectly — that gravitational waves did not exist. His paper was rejected due to a mistake, which he initially resisted. Yet within a year, through discussion and correction, he recognized the error and published a revision.

Even giants stumble. Progress depends not on being right the first time, but often on being willing — and able — to correct course.

Recommended book on how powerful quantiative models can be:

The Art of Planning, DeepSeek and Politics

A while back I discussed in a post some of the nuances of data driven decision making when we need to answer a straight question at a point in time e.g. : “shall we do this now?”.

That was a case presented as if our decision “now” would have no relationsip to the future, meaning that it would have no impact on our future decisions on the same problem.

Luckily, often, we do know that our decisions have an impact on the future, but the issue here is slightly different, we are looking not only on impact in the future but an impact on future decisions.

This is the problem of sequential decisioning. Instead of “shall we do this now?” answers the questions:

“Shall we adopt this strategy/policy over time?”

In other words it tries to solve a dynamical problem, so that the sequence of decisions made could not have been different no matter what happened. When you have an optimal solution to this problem, whatever decision is made, at any point in time, is a decision that one would never regret.

I will answer three questions here:

  • What is an example of such a problem?
  • Is there a way to solve such problems?
  • What is the relationship with DeepSeek and politics?

Sequential decisioning problem example – Customers Enagegement Management

A typical example could be that of working on marketing incentives on a customer base. The problem is that of deriving a policy for optimal marketing incentives depending on some measures of engagement in the customer base.

Why is it sequential? Because we have a feedback loop between what we plan to do, the marketing incentive, and the behaviour of our customers that triggers customer incentives: engagement.

Whenever we have such loops we cannot solve the problem at one point in time and regardless of the future.

The solution could look something like: “Invest $X in customer rewards whenever the following engagement KPIs, email open rate, web visits, product interaction etc.etc. drop below certain levels.

It is important to note that we need something to aim for, e.g. maximise return on marketing incentives.

Does a solution to such problems exist?

The good news is that yes, solutions exist, but we have been able to deal only with such cases that would be amenable to some mathematical/statistical modelling.

We can 100% find a solution If we have a good model that answers the following:

  • How does a given marketing incentive influence customers enagement on average?
  • Given a certain level of customer engagement, what is the expected value of the relationship with the customer?
  • How does the engagement of our customer evolves overtime, in absence of any incentive?

So we do need some KPIs that give us a good sense of what level of cusotmer engagement we have, and we need to have a sense of what is the expected value of the relationship with the customer given a particular set of KPIs. To be clear what we need is something true on average, and at least over a certain period of time.

Example, we should be able to say that, on average, a customer using our products daily for the past month will deliver X value over 2 years.

We also need to be able to say that a given marketing incentive, on average, increase customers daily enagement by X% and costs some defined amount of $

We also need to be able to say something like: our customer engagement rate tends to decrease X% every few months, again on average.

The above modelling of engagement and customer value is not something that most businesses would find difficutl today. Engagement rates, or attrition rates overtime, are easy to get, as well as results of marketing campaigns on engagement rates. Harder to get is a reliable estimate of lifetime value given current engagement metrics, but we can solve for a shorter time horizon in such cases.

Richar Bellman is the mathematician credited for having solved such problems in the most general way.

His equation, the Bellman equation, is all you need after you have a mathematical model of your problem.

The equation, in the form I’d like to present it is the one below, where V is the value of the optimal policy pi*:

It says the following:

The optimal strategy, depending on parameters a (e.g. how $ spent on incentives), is that which maximizes the instant rewards now, as well as the expected future rewards (future rewards conditional on the policy), no matter what the variables of the probelm are at that point in time (e.g. customer engagement and marketing budget left).

It is a beautiful and intuitive mathematical statement which is familiar to most humans:

The best course of action is that which balances instant gratification with delayed gratification.

We all work this way, and this is arguably what makes us human.

To be clear this problem can be, and it is solved, in a large variety of sequential decisioning problems.

The Bellman equation is the basic tool for all planning optimizations in supply management, portfolio management, power grid management and more…

Politics?

Well, in some sense politics is the art of collective, at least representative, policy making. And here a problem obviously arises: How to dial well instant versus delayed gratification when the views of the collective might differ? What if the collective doesn’t even agree on the various aspects of the problem at hand? (e.g. the KPIs, the expected rewards etc.etc.).

A key aspect of this should be the following: LEARNING.

The illustration below should be of guidance:

There has to be a feedback loop as the Bellman equation gives conditions for optimality, but often the solution can only be found iteratively. Futhermore, we also need to be open minded on revising the overall model of the problem, if we gather evidence that the conditions for optimality do not seem to come about.

So we have multiple challenges, which are true for politics but also for business management. We need:

  • Consensus on how to frame the problem
  • A shared vision on the balance between instant versus delayed rewards (or my reward versus your reward)
  • A framework for adapting as we proceed

The above is what I would call as being “data driven”- basically “reality” driven, but it is hard to get there. We all obviously use data when we have it, but the real challenge is to operate in such a way that the data will be there when needed.

How about DeepSeek?

If you have followed thus far you might have understood that solving a sequential problem basically involves algorithms to learn the dynamics between a policy maker action (e.g. marketing $ spent) and the long term feedback of the policy. This is the principle behing “reinforcement learning” in the AI world, as one want the learning to be reinforced by incentives for moving in the right direction. This was considered as a very promising framework for AI in the early days, but it is indeed an approach that requires time, and a fair bit of imagination as well as data, and was not pursued in recent years… until DeepSeek stormed the AI world.

A key element of DeepSeek successful R1 generative AI model was that it leveraged reinforecement learning, which forced it to develop advanced reasoning at a lower cost.

This was achieved through very clever redesigning of the transformer architecture, but I won’t go through it (as I have not yet understood it well myself).

As usual let me point you to a great book on the above topics.

My favorite is “Economics Dynamics” by John Stachursky. Great resource on sequential problem solving under uncertainty. Pretty theoretical, but that’s inevitable here.