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

Introduction: Making Choices Without Perfect Information

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.

Practical Example: Choosing Between Two Offers

Suppose you are deciding between:

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

Then:

$$EV_A = 0.4 \times 20000 + 0.6 \times 5000 = £11,000$$ $$EV_B = 1.0 \times 10000 = £10,000$$

Rationally, you choose A, even though it is riskier, because 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:

$$U(x) = \sqrt{x}$$

So:

$$U(10000) = 100, \quad U(40000) = 200$$

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.

Practical Example: SaaS Pricing Decision

A SaaS founder is choosing between:

• Aggressive pricing: 30% chance of 2x revenue, 70% chance of 20% churn spike
• Conservative pricing: 90% chance of steady 10% growth

For a well funded company, the aggressive option might have higher expected value. For a bootstrapped company with 4 months runway, the conservative option has higher expected utility because survival matters more than upside.

Maximin and Minimax: Planning for Worst Cases

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:

$$\text{Maximin: } \max_i \left( \min_j x_{ij} \right)$$

The minimax regret rule focuses on minimising the maximum regret:

$$\text{Regret}_{ij} = \max_j x_j - x_{ij}$$ $$\text{Minimax Regret: } \min_i \left( \max_j \text{Regret}_{ij} \right)$$

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

Practical Example: eCommerce Platform Migration

An eCommerce brand is choosing between staying on Shopify or migrating to a headless stack:

Option	Best Case	Worst Case
Stay on Shopify	+15% efficiency	0% change
Migrate to Headless	+40% efficiency	-20% (failed migration)

Using maximin, you stay on Shopify (worst case is 0%). Using expected value, you might migrate. The right choice depends on risk tolerance and reversibility.

Multi Stage Decisions and Decision Trees

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

Practical Example: Campaign Launch Tree

• Step 1: Launch campaign (70% chance of success)
• Step 2: If successful, retarget with upsell (20% conversion)

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.

$$EV_{path} = P(success) \times P(upsell|success) \times Value_{upsell}$$

For a £50,000 upsell value:

$$EV = 0.70 \times 0.20 \times 50000 = £7,000$$

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.

Final Thought: Act With Confidence, Not Hope

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.