Decision Theory for Founders: Choosing Under Uncertainty with Maximum Expected Value

In early stage companies, decisions must often be made without certainty. You do not know if a launch will succeed, whether customers will accept a price increase, or if a campaign will break even. But delaying decisions is itself a risk. This is where I use decision theory — a mathematical framework for making rational choices under uncertainty.

I apply these concepts regularly in client work: launch planning, pricing trials, offer structure, ad spend strategy, and risk modelling. It helps us avoid paralysis, stay probabilistic in our thinking, and act based on structured evaluation instead of instinct alone.

Expected Value: The Rational Default

The most basic and widely used decision model is the expected value (EV) principle. It tells us to choose the option that offers the highest payoff on average, weighted by probability.

If an action ( A ) has outcomes ( x_1, x_2, ..., x_n ) with probabilities ( p_1, p_2, ..., p_n ), then:

EV(A) = sum_{i=1}^{n} p_i cdot x_i

I use this to evaluate pricing strategies, upsell offers, and campaign variations. For example, suppose you are deciding between:

Offer A: 40 percent chance of £20,000 gain, 60 percent chance of £5,000
Offer B: 100 percent chance of £10,000

Then:

EV_A = 0.4 cdot 20000 + 0.6 cdot 5000 = 11000

EV_B = 1.0 cdot 10000 = 10000

Rationally, you choose A — even though it is riskier, the expected return is higher.

Utility Functions: Adjusting for Risk Preferences

But in reality, founders do not always value money linearly. A £10,000 loss hurts more than a £10,000 gain feels good. This is where utility theory comes in.

A utility function ( U(x) ) maps outcomes to subjective value. Risk-averse people have concave utility functions:

U(x) = sqrt{x}, quad U(10000) = 100, quad U(40000) = 200

In that case, even if the expected monetary value of Option A is higher, Option B may offer higher expected utility. This is common when:

Founders have limited cash runway
Failure has existential consequences
Upside is capped, but downside is painful

I help clients define utility functions based on business context, not ego.

Maximin and Minimax: Planning for Worst-Case and Adversarial Scenarios

Sometimes we want to plan pessimistically — especially in irreversible decisions like migration, rebranding or long-term contracts.

The maximin rule chooses the option with the best worst-case outcome:

ext{Maximin: } max left( min_{j} x_{ij} ight)

The minimax regret rule focuses on minimising the maximum regret:

ext{Regret}_{ij} = max_j x_j - x_{ij}

ext{Minimax Regret: } min_i left( max_j ext{Regret}_{ij} ight)

I use this in risk workshops — when planning for worst-case reactions to pricing, platform bans, or brand missteps.

Multi-Stage Decisions and Decision Trees

When decisions unfold over time, I build decision trees that model conditional probabilities.

Example:

Step 1: Launch campaign (success 70 percent)
Step 2: If successful, retarget with upsell (conversion 20 percent)

Each branch has probabilities and payoffs. I compute the EV of each path, then work backwards to the root. This is backward induction, and it gives clients clarity on the whole picture — not just the next step.

How I Use This in Client Work

For a SaaS client choosing between launching in the UK or Germany, I modelled expected value of adoption, adjusted for localisation cost and probability of breaking even in 6 months
For an ecommerce brand debating whether to create a new category page, I built a decision tree with organic traffic probabilities, conversion estimates and content investment
For a founder with limited budget, I helped weigh paid social campaigns versus SEO content strategy by modelling both expected value and cost variance

The result is not just rational choices — but rational confidence. Clients move faster when they understand why they are betting.

Decision Support Is Not Prediction — It Is Framing

None of these tools remove risk. But they frame it. They show what we are assuming. They quantify uncertainty. They prevent wishful thinking.

I use decision theory not to slow teams down, but to help them move deliberately.

If you are making high-leverage decisions with incomplete data, and the stakes are real — I can help you structure the choice, weigh the payoff, and act with confidence, not hope.