Decision theory for founders: choosing under uncertainty with expected value

Every decision is a bet, and waiting is one of the bets

A founder never gets to decide with the facts in hand. You raise a price before you know how customers will take it. You sign a lease before you know whether the orders will come. You hire before the work is certain. And putting the decision off is not a safe harbour, it is simply another option with its own payoffs, usually the one nobody bothered to price.

Decision theory deals with exactly this: choosing when the outcome depends on things you cannot control or fully know. It does not predict the future. It makes you write down what you are assuming, so the assumptions can be argued with, and then shows which option they favour.

Throughout, I use one invented business. Corrib Roast is a small coffee roaster in Galway, run by Sinéad, selling beans online and by subscription. Nobody real, every number made up so I can show every step.

Write the decision down as a payoff table

Before any maths, a decision needs three lists. The acts are the options you can choose between. The states are the things that could turn out to be true, which you do not control. The outcomes are what you end up with for each act in each state. Acts in rows, states in columns: a payoff table.

Two disciplines matter. The states must cover everything and must not overlap, so exactly one comes true. And "do nothing" is always a row. Leaving it out is how a business ends up comparing two expensive plans with each other and never with keeping the money.

Worked example: Sinéad's wholesale question

Cafés around Galway keep asking Sinéad whether they can buy her beans by the kilo. She sees three options for the coming year:

• Carry on as now, online only.
• Launch a small wholesale range on her current roaster, which needs €8,000 for a grinder, bulk packaging and samples.
• Lease a larger roaster and go after wholesale properly, a commitment of €30,000 over the year.

What happens depends on one thing she cannot control: how many cafés actually sign up and keep ordering. She simplifies that into three states, strong, middling and weak demand. Then she estimates the extra profit each option would bring over the year in each state, after its costs:

Option	Strong demand	Middling demand	Weak demand
Carry on as now	€0	€0	€0
Small wholesale range	€20,000	€5,000	minus €8,000
Larger roaster	€45,000	€5,000	minus €30,000

The roaster earns most when demand is strong. In a weak year it is the worst by a distance, because the lease is paid either way. That tension is the whole decision.

Expected value: weight each outcome by its chance

The first question to ask of a payoff table is what each option is worth on average, if you could make the same choice many times over. That long run average is the expected value: multiply each outcome by its probability and add the results.¹

$$E[a] = \sum_{j=1}^{n} p_j \, x_{aj}$$

Here $E[a]$ is the expected value of act $a$, the sum $\sum_{j=1}^{n}$ runs over the $n$ states of the world, $p_j$ is the probability that state $j$ comes true, and $x_{aj}$ is the outcome of act $a$ if state $j$ comes true. The probabilities $p_j$ must add up to 1, because exactly one state happens. The idea is set out from scratch in Foundations: expected value, and the probabilities themselves in Foundations: probability.

Choosing the highest expected value is the default rule in decision analysis, often called maximising expected monetary value.² It is right whenever the stakes are small next to the business and the decision comes round often, because then the averages really do play out.

Worked example: the expected value of each of Sinéad's options

Sinéad talks to a dozen cafés around Galway and Clare and looks at how her online subscriptions grew. Her best estimate is a 30% chance of strong demand, 50% middling and 20% weak, so $p_{\text{strong}} = 0.3$, $p_{\text{middling}} = 0.5$ and $p_{\text{weak}} = 0.2$. They add up to 1, as they must.

Carrying on as now is worth €0 in every state, so its expected value is €0.

For the small wholesale range, $E[\text{range}] = 0.3 \times 20{,}000 + 0.5 \times 5{,}000 + 0.2 \times (-8{,}000) = 6{,}000 + 2{,}500 - 1{,}600 = 6{,}900$.

For the larger roaster, $E[\text{roaster}] = 0.3 \times 45{,}000 + 0.5 \times 5{,}000 + 0.2 \times (-30{,}000) = 13{,}500 + 2{,}500 - 6{,}000 = 10{,}000$.

In both lines the probabilities are the $p_j$ and the amounts are the outcomes $x_{aj}$ from the table. On expected value the larger roaster wins, €10,000 against €6,900, with doing nothing a distant third at €0.

With a hundred roasteries, the roaster would be the right call. She has one. That is where expected value starts to need help.

Where the probabilities come from

Expected value is only as good as the probabilities you feed it, and in business nobody hands them to you.

Frank Knight drew the line a century ago. Some uncertainty can be reduced to a measurable probability by grouping many similar cases, which he called risk. Some cannot, because there are no similar cases to group, and that is uncertainty in the strict sense.³ A conversion rate measured on 40,000 visitors is risk. Whether a new product category will take off is closer to uncertainty.

In my experience the honest approach is three sources, each labelled. Base rates first: how often similar launches have worked. Then evidence specific to this case. Then judgement, stated as judgement. The result is still an estimate, but one someone can challenge, which beats a gut feeling nobody can.

Worked example: building the 30%

Of the 12 cafés Sinéad spoke to, a good few were keen and four said they would switch suppliers within three months if the price was right. Her online subscriptions grew strongly in their first year, so the coffee sells. Against that, cafés change roasters rarely, and "I would switch" over a flat white is not a signed order. She settles on 30% strong, 50% middling and 20% weak, and writes that reasoning next to the table, so that in six months she can see which part she got wrong.

Utility: when the bad outcome could sink you

Suppose Sinéad's business has €40,000 of cash and reserves. In the weak year the roaster loses €30,000 and leaves €10,000: the business survives, just, with no cushion. Expected value treats a €30,000 loss and a €30,000 gain as mirror images. To the business they are not.

Daniel Bernoulli made this point in 1738. He argued that the value of an extra euro depends on how many euro you already have, and proposed that the usefulness of wealth grows with its logarithm. Each extra euro adds a little less than the one before. Economists call this diminishing marginal utility.⁴ In 1944 John von Neumann and Oskar Morgenstern gave the idea its modern footing, showing that a decision maker who follows a few consistent rules acts as though maximising the expected value of a utility function rather than of money itself.⁵

$$EU(a) = \sum_{j=1}^{n} p_j \, U\!\left(W_0 + x_{aj}\right), \qquad U(w) = \ln w$$

Here $EU(a)$ is the expected utility of act $a$, $p_j$ is again the probability of state $j$, $W_0$ is the wealth you start with, $x_{aj}$ is the gain or loss from act $a$ in state $j$, so $W_0 + x_{aj}$ is where you end up, and $U$ is the utility function that turns an amount of money into how much it is worth to you. Bernoulli's choice is $U(w) = \ln w$, the natural logarithm, which rises steeply at first and flattens out. The logarithm is explained in Foundations: logarithms, and the weighting is the same as in Foundations: expected value.

Utility numbers mean little alone, so I turn them back into money. The certainty equivalent is the sure amount that would be worth exactly as much to you as the gamble.

$$CE(a) = e^{EU(a)} - W_0$$

Here $CE(a)$ is the certainty equivalent of act $a$, $e$ is the base of the natural logarithm (about 2.718), so $e^{EU(a)}$ undoes the logarithm and turns expected utility back into an amount of wealth, and $W_0$ is subtracted so the result is a gain over where you started. The gap between the expected value and the certainty equivalent is the risk premium: what the uncertainty costs you. The groundwork is in Foundations: logarithms.

Worked example: Sinéad's options in utility

Sinéad starts with $W_0 = 40{,}000$. For each option I add the outcome to €40,000, take the natural logarithm, and weight by the probabilities.

Carrying on leaves her at €40,000 in every state, so its expected utility is $\ln 40{,}000 = 10.5966$ and its certainty equivalent is €0.

The wholesale range ends at €60,000, €45,000 or €32,000, so $EU(\text{range}) = 0.3 \times 11.0021 + 0.5 \times 10.7144 + 0.2 \times 10.3735 = 10.7325$.

The roaster ends at €85,000, €45,000 or €10,000, so $EU(\text{roaster}) = 0.3 \times 11.3504 + 0.5 \times 10.7144 + 0.2 \times 9.2103 = 10.6044$.

Each figure inside the sums is $\ln(W_0 + x_{aj})$, for example $\ln 60{,}000 = 11.0021$, weighted by its $p_j$.

Turned back into money, the wholesale range has a certainty equivalent of $e^{10.7325} - 40{,}000$, about €5,800. The larger roaster has $e^{10.6044} - 40{,}000$, about €300.

So the ranking flips. The roaster has the higher expected value, but its risk premium is about €9,700, because the weak year bites so deep into a €40,000 business. The range gives up €3,100 of expected value and carries a risk premium of only about €1,100. To a business this size, it is the better bet.

Run the same sum for a rival with €400,000 in the bank and the roaster wins comfortably. Same table, same probabilities, different answer. That is the point: the right choice depends on what the business can absorb.

Loss aversion is not the same as being careful

You will often hear that founders are cautious because "a €10,000 loss hurts more than a €10,000 gain feels good". That runs two different ideas together, and the difference matters in practice.

Utility, the Bernoulli kind, is about final wealth. It is a rational reason to turn down a bet with a positive expected value, and it fades when the stakes are small next to what you own.

Loss aversion is something else. Daniel Kahneman and Amos Tversky showed in 1979 that people judge outcomes as gains and losses from a reference point, usually where they stand today, rather than as final wealth, and that losses weigh more heavily than gains of the same size.⁶ Their later work named loss aversion as one of the two principles behind the shape of people's choices.⁷ It applies even to sums that could not hurt the business.

A simple way to write it, keeping only the asymmetry:

$$v(x) = \begin{cases} x & \text{if } x \ge 0 \\ \lambda x & \text{if } x < 0 \end{cases}$$

Here $v(x)$ is how an outcome feels, $x$ is the gain or loss measured from today, and $\lambda$ is the loss aversion weight, a number above 1 that makes a loss count for more than a gain of the same size. The full model also curves both sides, left out here. The underlying weighting is still Foundations: expected value.

Worked example: the small bet Sinéad keeps saying no to

Sinéad is offered a sponsored slot in a West of Ireland food newsletter. From what she has seen, it is a coin flip: half the time it brings in €3,000 of profit, half the time it loses the €2,000 fee. The expected value is $0.5 \times 3{,}000 + 0.5 \times (-2{,}000) = 500$, a positive €500.

With an illustrative loss weight of $\lambda = 2$, my own assumption for the example, it feels like $0.5 \times 3{,}000 + 0.5 \times 2 \times (-2{,}000) = 1{,}500 - 2{,}000 = -500$. A gamble worth plus €500 feels like minus €500, so she says no.

Now suppose a slot like this comes up 20 times a year. Taking every one has an expected total of $20 \times 500 = €10{,}000$. She ends the year behind only if seven or fewer of the 20 come up good, and with a fair coin that happens about 13% of the time (a binomial probability, covered in Foundations: distributions). And none of the individual €2,000 losses comes close to threatening the business.

That is the useful test. If the bad outcome would genuinely damage the business, caution is utility, and it is right. If it is merely unpleasant and the bet comes round often, the reluctance is loss aversion, and it is quietly expensive. More of the psychology is in how customers actually decide.

When you cannot put a number on the odds

Sometimes no honest probability is available. A platform might change its rules, a supplier might fold. You can list the states, but any percentage next to them would be invented. For those cases there are rules that use only the payoff table.

Abraham Wald built his statistical decision theory around the minimax criterion: judge each option by how badly it can go, and pick the one whose worst case is least bad.⁸ When the table shows profits rather than losses, the same idea is called maximin.

$$a^{*} = \arg\max_{a} \; \min_{j} \; x_{aj}$$

Here $x_{aj}$ is the outcome of act $a$ in state $j$, $\min_{j}$ finds the worst outcome of each act across all states, $\max_{a}$ picks the largest of those worst outcomes, and $\arg$ means we want the act $a^{*}$ that achieves it, not the number itself. No probability appears anywhere. I read the table here just as in Foundations: expected value, only without the weights.

Maximin is very cautious. It would never let you open a shop. Leonard Savage proposed a less fearful alternative, minimax regret.⁸ Regret is the gap between what you got and what you could have got had you known which state would come true.

$$r_{aj} = \max_{b} \, x_{bj} - x_{aj}, \qquad a^{*} = \arg\min_{a} \; \max_{j} \; r_{aj}$$

Here $r_{aj}$ is the regret of act $a$ in state $j$, $\max_{b} x_{bj}$ is the best outcome any act $b$ achieves in that same state $j$, and $x_{aj}$ is what act $a$ actually achieved. Then $\max_{j}$ finds each act's largest regret, and $\arg\min_{a}$ picks the act $a^{*}$ whose largest regret is smallest. The same table, read column by column instead of row by row. It is the table from Foundations: expected value again, without the weights.

Worked example: the three rules on Sinéad's table

Maximin first. The worst outcomes are €0 for carrying on, minus €8,000 for the wholesale range and minus €30,000 for the larger roaster. The largest of those is €0, so maximin says carry on as now. Its mirror image, maximax, looks only at best cases (€0, €20,000, €45,000) and says buy the roaster.²

Now minimax regret. In the strong column the best outcome is €45,000, so the regrets are €45,000 for carrying on, €25,000 for the range and €0 for the roaster. In the middling column the best is €5,000, so the regrets are €5,000, €0 and €0. In the weak column the best is €0, so the regrets are €0, €8,000 and €30,000.

Each option's largest regret is €45,000 for carrying on, €25,000 for the wholesale range and €30,000 for the roaster. The smallest of those is €25,000, so minimax regret picks the wholesale range.

Along the bottom are Sinéad's three options. The height of each bar is that option's largest regret in euro, the most she could kick herself for in any state. Minimax regret picks the lowest bar. Carrying on looks safe, yet it carries the biggest regret, because a strong year spent doing nothing leaves €45,000 on the table.

Three rules, three answers. That is not a flaw. Each rule encodes a different attitude, and side by side they show how much your answer depends on it. When every rule points the same way, I stop analysing and act.

Decisions in stages: trees, and solving them backwards

Real decisions rarely happen in one go. Often you can take a small step, learn something, then commit. A decision tree draws that sequence: a square where you choose, a circle where chance decides, outcomes at the tips.

You solve a tree from the tips back to the root. At each chance node you take the expected value of the branches. At each decision node you keep the best branch and prune the rest. In the textbook phrase, you work from right to left, computing expectations as you go.² Game theorists call the same procedure backward induction. If a competitor is making choices at some of those nodes, you are in game theory, which I cover in game theory for pricing and positioning.

The step that makes trees powerful is updating the probabilities after you learn something. That is Bayes' rule.

$$P(s_j \mid \text{signal}) = \frac{P(\text{signal} \mid s_j) \, P(s_j)}{P(\text{signal})}$$

Here $P(s_j)$ is your probability of state $s_j$ before you learn anything (the prior), $P(\text{signal} \mid s_j)$ is how likely the signal is if that state is true, $P(\text{signal})$ is how likely the signal is overall, found by adding $P(\text{signal} \mid s_j) \, P(s_j)$ across every state, and $P(s_j \mid \text{signal})$ is the updated probability of the state once you have seen the signal (the posterior). Bayes' rule is worked through in Foundations: probability.

Worked example: a three month pilot with five cafés

Sinéad has a fourth option. Before choosing, she could supply five Galway cafés for three months at a cost of €1,500 in free beans, samples and her own time. At the end, the pilot will either look busy (the cafés reorder and ask for more) or quiet.

The pilot is no crystal ball. She judges that it looks busy 80% of the time if demand is truly strong, 50% if middling and 10% if weak.

First, how likely is a busy pilot overall? $P(\text{busy}) = 0.8 \times 0.3 + 0.5 \times 0.5 + 0.1 \times 0.2 = 0.24 + 0.25 + 0.02 = 0.51$. So a quiet pilot has probability 0.49.

After a busy pilot, Bayes' rule gives strong $0.24 / 0.51 = 0.471$, middling $0.25 / 0.51 = 0.490$ and weak $0.02 / 0.51 = 0.039$. With those probabilities the larger roaster is worth $0.471 \times 45{,}000 + 0.490 \times 5{,}000 - 0.039 \times 30{,}000$, about €22,450, against about €11,550 for the range. She would buy the roaster.

After a quiet pilot, the posteriors are strong $0.06 / 0.49 = 0.122$, middling $0.25 / 0.49 = 0.510$ and weak $0.18 / 0.49 = 0.367$. Now the roaster is worth about minus €2,960, the range about €2,060 and carrying on €0. She would launch the range.

Folding back to the root: $0.51 \times 22{,}450 + 0.49 \times 2{,}060$, which is €11,450 plus €1,010, gives about €12,460 before the pilot's cost, or €10,960 after it. Deciding now, the best she could do was the roaster's €10,000. The pilot route is worth about €960 more, and it changes the decision from a single bet into a plan: roaster if busy, range if quiet.

The diagram reads from the top. The first box is the pilot, the diamond is the moment chance reveals whether it looked busy, the next two boxes are what Sinéad does on each answer, and the last box is the value of the whole plan when everything is folded back.

What it is worth to know more

How much should you spend finding out before you decide? Research, a test campaign, a prototype. Decision theory gives a clean ceiling: the expected value of perfect information.

Imagine someone who could tell you, for free, which state will come true. The expected value of always choosing the best act for that state, minus the best you can do without it, is what perfect foresight is worth. No real test can be worth more, so it caps what you should ever pay for information.⁹

$$\text{EVPI} = \sum_{j=1}^{n} p_j \, \max_{a} \, x_{aj} \; - \; \max_{a} \sum_{j=1}^{n} p_j \, x_{aj}$$

Here $\text{EVPI}$ is the expected value of perfect information, $p_j$ is the probability of state $j$, $\max_{a} x_{aj}$ is the best outcome available in state $j$ if you knew it was coming, so the first sum is the expected value with perfect foresight, and $\max_{a} \sum_{j} p_j \, x_{aj}$ is the highest expected value you can reach without it. Both halves are expected values in the sense of Foundations: expected value.

Worked example: the most Sinéad should spend on finding out

With perfect foresight Sinéad would buy the roaster in a strong year (€45,000), take either wholesale option in a middling year (€5,000), and carry on as now in a weak year (€0). The expected value of that is $0.3 \times 45{,}000 + 0.5 \times 5{,}000 + 0.2 \times 0 = 13{,}500 + 2{,}500 + 0 = 16{,}000$.

Without information, the best expected value was the roaster's €10,000. So $\text{EVPI} = 16{,}000 - 10{,}000 = 6{,}000$.

No survey, consultant or pilot about café demand can be worth more than €6,000 to her. The real pilot, which is imperfect, turned out to be worth €12,460 minus €10,000, about €2,460, comfortably more than its €1,500 cost. A pilot costing €3,000 would have lost money on average, even though it would still have taught her something. Learning is not free, and this is the sum that says when it pays. If the information you want comes from a test, how much data a test needs is the natural next question.

Find the number that flips the answer

Once a decision is on paper, the most useful question is "what would have to be different for the answer to change". That is sensitivity analysis, and it shows which estimate deserves the most scrutiny.

Worked example: how sure does Sinéad need to be?

Keep the chance of strong demand at 30% and let the chance of a weak year vary, with middling taking up the rest. The roaster differs from the wholesale range by plus €25,000 in a strong year, nothing in a middling year and minus €22,000 in a weak year. So the roaster leads by $0.3 \times 25{,}000 - p_{\text{weak}} \times 22{,}000$, which is zero when $p_{\text{weak}} = 7{,}500 / 22{,}000 = 0.34$.

Along the bottom is the probability Sinéad gives to a weak year, from 0% to 50%. Up the side is the expected value of each option in euro. The roaster's line is steep because its weak year is so costly, and it crosses below the range at about 34%. If Sinéad thinks a weak year is more likely than one in three, the range wins even on plain expected value, before utility has a say.

That tells her where to spend her doubt: on whether a weak year is one in five or one in three. The pilot is good at telling those apart.

How I put it together with a founder

None of this needs software. A spreadsheet and an honest hour will do grand.

The diagram runs from the top. The diamond is the honest fork: with defensible probabilities, use expected value and utility, without them the rules that need only the table. It ends in a written decision with a review date, because the review is how the next probabilities get better.

Worked example: Sinéad's one page

Sinéad's page holds the table, the probabilities with their reasoning, the expected values (€0, €6,900, €10,000), the certainty equivalents (€0, about €5,800, about €300), the €6,000 value of perfect information, the pilot plan at about €10,960 after its cost, and the 34% break even. Her decision: run the pilot. Roaster if busy, range if quiet. Review in three months.

In my experience the page matters more than any number on it. It turns an argument about gut feelings into an argument about one probability, which is a much shorter argument.

What decision theory does not do

It does not remove risk. Sinéad can follow every step and still have a weak year. A good decision can have a bad outcome, and a reckless one can get lucky. Judge a decision by what you knew when you made it.

It is also no better than its inputs. As the MIT course notes put it, the results are no better than the model and the assessments behind it.² That is why the reasoning sits next to every number.

What it does is frame the risk. It shows what you are assuming, prices what you do not know, and stops the most confident voice in the room winning by default. If you want the wider habit of thinking in ranges rather than single numbers, I wrote about why averages lie, and for spreading a budget across uncertain options there is optimisation under uncertainty.

Footnotes

1. Barbara Illowsky and Susan Dean, Introductory Statistics 2e, section 4.2 Mean or Expected Value and Standard Deviation, OpenStax, Rice University, Houston, 2023, https://openstax.org/books/introductory-statistics-2e/pages/4-2-mean-or-expected-value-and-standard-deviation

2. George E. Apostolakis, DA 1: The multistage decision model, lecture slides for ESD.72 Engineering Risk Benefit Analysis, MIT OpenCourseWare, 2007, https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/resources/da1/

3. Frank H. Knight, Risk, Uncertainty, and Profit, Part III, The Meaning of Risk and Uncertainty, Houghton Mifflin, 1921, via the Library of Economics and Liberty, https://www.econlib.org/library/Knight/knRUP6.html

4. Martin Peterson, The St. Petersburg Paradox, Stanford Encyclopedia of Philosophy, 2019, revised 2023, https://plato.stanford.edu/entries/paradox-stpetersburg/

5. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, sixtieth anniversary edition, Princeton University Press, 1944 and 2004, https://press.princeton.edu/books/paperback/9780691130613/theory-of-games-and-economic-behavior

6. Daniel Kahneman and Amos Tversky, Prospect Theory: An Analysis of Decision under Risk, Econometrica, volume 47, issue 2, pages 263 to 292, Econometric Society, 1979, https://www.econometricsociety.org/publications/econometrica/1979/03/01/prospect-theory-analysis-decision-under-risk

7. Amos Tversky and Daniel Kahneman, Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, volume 5, issue 4, pages 297 to 323, 1992, https://ideas.repec.org/a/kap/jrisku/v5y1992i4p297-323.html

8. Charles F. Manski, Statistical Decision Theory for Treatment Choice, lecture slides, Northwestern University, Centre for Microdata Methods and Practice (cemmap), https://www.cemmap.ac.uk/wp-content/legacy/uploads/cemmap/resources%20%28slides%29/Manski%20Tetenov/Session%201-2.pdf

9. George E. Apostolakis, DA 2: The value of perfect information, lecture slides for ESD.72 Engineering Risk Benefit Analysis, MIT OpenCourseWare, 2007, https://ocw.mit.edu/courses/esd-72-engineering-risk-benefit-analysis-spring-2007/resources/da2/