The Tools of Decision-Making: Expected Value, Decision Trees, and Models

By Pritesh Yadav June 22, 2026 16 min read —

In the previous chapter you learned the big idea behind decision science: a good decision is not the same thing as a good outcome. You make decisions forward, with limited information and real uncertainty; outcomes get judged backward, once luck has had its say. This chapter is about the working tools that help you make that forward-looking choice as well as possible. These are the everyday instruments of the trade — the things you can actually pick up and use on Monday morning.

We'll build them up in order, simplest first:

Expected Value (EV) — how to put a single number on a risky choice.
Utility — why that number isn't the whole story, and how to fix it.
Decision trees — how to map a choice that unfolds in stages.
Mental models — the reusable thinking tools you carry between problems.

None of these requires hard math. They require a little arithmetic and the willingness to write your guesses down instead of keeping them in your head.

29.1 First, three flavors of "not knowing"

Every tool in this chapter is a tool for deciding when you don't know what will happen. But "not knowing" comes in different strengths, and it matters which one you're facing.

Certainty: You know exactly what will happen. Press B4 on the vending machine, get the chips.
Risk: You don't know the outcome, but you do know the odds. A roulette wheel: you can't predict the spin, but you know each number's chance.
Uncertainty (also called Knightian uncertainty, after economist Frank Knight): You don't know the outcome and you don't even know the odds. Launching a brand-new product into a market nobody has measured — you're guessing the probabilities themselves.

Most of real life is uncertainty wearing a risk costume. The tools below want clean probabilities, but you'll usually be feeding them estimates. That's fine — and it's the point. Writing down "I think there's about a 60% chance" is far better than pretending you know nothing, and far more honest than pretending you know everything.

Key takeaway: Risk means the odds are known; uncertainty means they're not. These tools force you to name your odds even when you have to estimate them — and a written estimate you can check later beats a vague feeling every time.

29.2 Expected Value: putting one number on a gamble

Expected value is the most useful single idea in this chapter, and it's genuinely simple.

Expected Value (EV): The average payoff you'd get if you could repeat a choice many, many times. You compute it by multiplying each possible outcome by its probability, then adding those up.

The formula in plain words: EV = (outcome 1 × its chance) + (outcome 2 × its chance) + …

Example — the lottery ticket. A ticket costs €2. It gives you a 1-in-10-million chance of winning €5,000,000. What's the average payoff?
€5,000,000 × (1 / 10,000,000) = €0.50.
So the average ticket is worth about 50 cents — and you're paying €2 for it. Over millions of tickets, you'd lose three-quarters of your money. The lottery is a beautifully reliable way to convert €2 into 50 cents.

Flip that around and you understand why casinos and insurance companies are so profitable: they arrange to be on the positive side of the EV. They don't win every bet — the casino pays out jackpots, the insurer pays out claims — but on average, across thousands of bets, the math runs in their favor. They don't need luck. They need volume and a positive EV.

A bet whose EV is in your favor is called +EV (or "EV-positive"). A great mental shift is to ask of any risky choice: "Is this +EV?" — is it worth taking on average, even though it might lose this particular time?

Common mistake: Judging a +EV bet by whether it happened to win. If you took a coin-flip that paid €30 on heads and lost €10 on tails, that's a great bet (EV = €10). If it lands tails and you lose, the bet was still smart. Confusing the outcome with the decision is called resulting, and it's the costliest habit in decision-making. The flip side is just as dangerous: a reckless bet that happens to pay off was still a bad decision.

29.3 Why EV alone can fool you: utility

Pure expected value treats every euro the same. But money — and most things — don't feel the same in every situation.

Analogy: €1,000,000 means everything to someone who's broke and almost nothing to a billionaire. Same dollars, wildly different personal value. The satisfaction a euro brings depends on how much you already have.

Utility: The personal value or satisfaction an outcome gives you — not its raw dollar amount.
Expected Utility: Expected value, but using utility instead of dollars: you weight how much each outcome is worth to you by its probability.

Consider a clean choice:

Option A: a guaranteed €1,000,000.
Option B: a 50/50 coin flip for €2,000,000 or nothing.

The expected value of both is identical: €1,000,000. Yet almost everyone takes the sure million. Are they being irrational? No — they're being sensible about utility. The jump from €0 to €1M changes your life. The jump from €1M to €2M is nice but adds far less than the first million did. This is diminishing marginal utility: each extra unit of money matters less than the one before. Because the second million is "worth less" to you, gambling your sure first million to chase it is a bad trade in utility terms — even though it's an even trade in dollar terms. Preferring the sure thing here is called risk aversion, and for most people it's perfectly rational.

This idea is old. In 1738, mathematician Daniel Bernoulli introduced utility to solve a puzzle called the St. Petersburg paradox — a coin-flip game whose expected value is mathematically infinite, yet which no sane person would pay more than a few coins to play. The resolution: people don't value money linearly; they value its usefulness, which grows ever more slowly. Once you measure choices in utility rather than dollars, the paradox dissolves.

Key takeaway: Use raw expected value for repeated, low-stakes bets where the amounts are small relative to your wealth. For big, one-shot, life-changing stakes, switch to expected utility — ask how much each outcome is worth to you, not just how many dollars it is. The €1M-or-coin-flip choice shows why a "lower-EV" sure thing can be the smarter call.

29.4 Decision trees: mapping a choice that unfolds in stages

Expected value handles a single gamble. But real decisions branch: you choose, then chance happens, then maybe you choose again. A decision tree is a diagram that lays the whole thing out so you can compute the EV of each path and pick the best opening move.

Analogy: A decision tree is a choose-your-own-adventure book where, instead of flipping to read what happens, you've written the odds on each fork and the prize at the end of each path. Then you trace backward to find which first move gives the best average ending.

The grammar of a decision tree has three symbols:

Squares = decision nodes — you choose here.
Circles = chance nodes — the world rolls the dice here; you write the probabilities.
Payoffs = the value at the tip of each branch.

You solve a tree by folding back (also called rolling back): start at the tips, work leftward, and at each chance node compute its expected value. At each decision node, pick the branch with the highest value. Whatever survives back at the start is your best move.

Example — should you take the new job offer? You have a safe current job worth, let's say, a steady satisfaction we'll score as 50. A startup offers you a role. You estimate:

60% chance it goes well → payoff 100 (great pay, growth, equity).
40% chance it folds within a year → payoff 10 (back to job-hunting).

EV of the startup branch = (100 × 0.60) + (10 × 0.40) = 60 + 4 = 64.
EV of staying = 50.
On these numbers, the startup is the higher-EV move (64 vs 50). But notice two things: the answer depends entirely on the numbers you put in, and you might still prefer staying if a payoff of 10 would be ruinous for you (that's utility and risk aversion talking). The tree doesn't decide for you — it makes your assumptions visible so you can argue about the right things.

              ┌─ goes well (60%) ───────► payoff 100
   [DECISION] │
   take job?──○  (chance node, EV = 64)
      │       └─ folds (40%) ───────────► payoff  10
      │
      └─ stay put ───────────────────────► payoff  50

   Square □ = you choose   Circle ○ = chance decides
   Fold back: EV(startup)=0.6·100 + 0.4·10 = 64  >  50
   → Best opening move: take the job.

The real power of a tree shows up when decisions have multiple stages — choose a strategy, see how the market reacts, then choose again. Drawing it forces you to plan your later moves before you're emotionally committed, and to notice paths you'd otherwise forget.

Best practice: Write the probabilities and payoffs down, even when you're estimating them. The tree's main gift isn't the final number — it's that it drags your hidden assumptions into the open where you (and your team) can challenge them. A tree built on honest "about 60%" guesses beats a confident decision built on a gut feeling you never examined.

29.5 Mental models: the reusable thinking tools

EV and decision trees are specific tools. But experts also carry a kit of more general thinking tools they reach for across every domain. These are mental models.

Mental model: A reusable concept — borrowed from any field — that helps you make sense of a situation. Examples: supply and demand, compound interest, base rates, second-order effects.

Analogy: Investor Charlie Munger put it best — to a person with only a hammer, every problem looks like a nail. If your one tool is "trust my gut," you'll hammer every decision with it, even the ones that need a screwdriver. A full toolbox — a "latticework of mental models" — lets you pick the right tool for the situation in front of you.

Three mental models are so useful for everyday decisions that they're worth installing right now.

Base rates (the "outside view")

A base rate is the background frequency of something before you look at the specific case in front of you.

Example: Before deciding your startup will surely succeed, note that roughly 90% of startups fail. That 90% is the base rate — the "outside view." Your detailed, exciting plan is the "inside view." Both matter, but beginners almost always over-trust the inside view and ignore the base rate entirely. The fix is an explicit order of operations: start from the base rate, then adjust for what's special about your case — never skip straight to the special case.

This is exactly how the best forecasters work. Researcher Philip Tetlock's "superforecasters" — ordinary, trained people who out-predicted intelligence analysts in the Good Judgment Project — explicitly begin every forecast from the base rate, then update gradually as specific evidence comes in.

Second-order thinking

Second-order thinking means asking "and then what?" — tracing the consequences of the consequences, not just the first, obvious effect.

Analogy: A weak chess player grabs the free pawn — that's first-order thinking ("this feels good now"). A strong player asks what the free pawn sets in motion three moves later, and notices it leads to checkmate. Investors like Howard Marks and Ray Dalio built careers on this: the first-order move that everyone can see is rarely where the advantage is.

Example: A shop cuts prices to win customers. First-order: more sales this week. Second-order: the rival next door cuts prices too, both of you now earn thin margins, and you've started a price war neither can easily exit. The first-order move felt smart; the second-order consequence was a trap.

Reversibility: one-way vs. two-way doors

Not all decisions deserve the same care. A simple model from Jeff Bezos: ask whether a decision is a one-way door (hard or impossible to undo) or a two-way door (easily reversed).

One-way doors — selling your house, quitting to start a company, a big irreversible spend. Go slow, gather information, use a decision tree, run a premortem.
Two-way doors — trying a new supplier for one order, changing a button color, testing a price for a week. Decide fast and learn from the result. Agonizing over reversible choices wastes your scarcest resource: your decision-making attention.

Common mistake: Treating every decision as a one-way door. Beginners often deliberate for weeks over choices they could reverse in an afternoon, then rush the genuinely irreversible ones. Match the speed of the decision to the cost of being wrong: slow for one-way doors, fast for two-way doors.

29.6 Putting the tools together: a worked decision

Watch how the tools stack. Suppose you're a small print-shop owner deciding whether to buy a second printing machine for €40,000.

Frame the uncertainty. You don't know future demand — this is uncertainty, so your probabilities will be estimates. Name them anyway.
Start from the base rate. How often does buying extra capacity pay off for shops your size? Suppose, from talking to peers, it's roughly 50/50. That's your outside view before optimism kicks in.
Adjust with the inside view. You have a new big client almost confirmed, so you nudge the "demand is high" chance up to, say, 65%.
Build the tree and compute EV. If demand is high (65%), the machine earns +€90,000 over its life; if low (35%), you've sunk €40,000 for little return, a net of −€30,000.
EV = (90,000 × 0.65) + (−30,000 × 0.35) = 58,500 − 10,500 = +€48,000.
Check utility, not just dollars. A +€48,000 average looks great — but if losing €40,000 would bankrupt you, that downside is far more painful than its dollar size suggests. Risk aversion is rational when a loss could wipe you out. Maybe you lease the machine instead to cap the downside.
Apply reversibility. Buying outright is closer to a one-way door; leasing turns it into a two-way door. Choosing the lease keeps the upside while making the decision reversible.

Notice that no single tool decided this. EV pointed at "yes." Utility and reversibility refined how to say yes. That's the realistic picture: these tools don't replace judgment — they organize it.

Tool	What it answers	Best used when…
Expected Value	What's the average payoff?	Repeated or small-stakes bets; comparing options on one number.
Expected Utility	What's it worth to me?	Big, one-shot stakes where a loss could hurt badly.
Decision Tree	What's my best move across stages?	Choices that unfold over time, with chance steps in between.
Base rate	What usually happens?	Always — start here before the specifics.
Second-order thinking	And then what?	Any move others will react to, or that has delayed effects.
Reversibility test	How carefully should I decide?	To set the speed of any decision.

29.7 The limits of the tools (and why that's okay)

It's tempting to think that with enough tools you can calculate your way to certainty. You can't, for three honest reasons.

Your inputs are guesses. A decision tree built on numbers you invented is only as good as those numbers — "garbage in, garbage out." The tree doesn't manufacture truth; it makes your guesses explicit and checkable. That's still a huge gain, but it's not magic.

More information isn't always better. Past a certain point, extra data adds noise and false confidence rather than accuracy. You feel more certain without actually being more right. Economist Herbert Simon called the realistic alternative bounded rationality: we have limited time and brainpower, so we satisfice — pick the first option that's good enough — rather than chase the mathematically perfect one. For most decisions, satisficing is the wise move, not a failure.

Common mistake: Believing that gathering more information always improves a decision. Beyond a point it just delays the call and inflates your confidence. Decide what facts would actually change your mind before you go hunting; if a piece of data wouldn't change your choice, you don't need it.

Outcomes still belong to luck. Even a perfectly built +EV decision can lose this one time. The tools improve your process, and the process is the only thing you actually control. The outcome is process plus luck — and you can't control luck.

Key takeaway: The process is the product. You can't control whether any single decision turns out well — you can only control the quality of how you decided. EV, utility, decision trees, and mental models are tools for raising that quality. Use them to think clearly, then accept that good decisions sometimes lose. Over many decisions, a sound process wins.

29.8 Chapter recap

Expected Value turns a gamble into one number: sum of (outcome × probability). Ask "is this +EV?" of any risky choice.
Utility fixes EV's blind spot — a euro's value depends on your situation (diminishing marginal utility), which is why risk aversion can be perfectly rational on big, one-shot stakes.
Decision trees map staged choices with squares (you choose), circles (chance decides), and payoffs; solve by folding back. Their real gift is exposing hidden assumptions.
Mental models — especially base rates first, then adjust, second-order thinking, and the reversibility test — are the reusable tools you carry between every kind of decision.
None of these replaces judgment or beats luck. They improve your process, which is the only thing you control — and over the long run, a good process is what wins.

In the next chapters we turn from "how to be right" toward "why we're so often wrong" — the predictable biases that pull real human decisions away from these clean tools — and then to the prescriptive habits that close the gap.

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