Lecture 1: Introduction to Individual Decision-Making

ELI5 / TLDR

This is the opening lecture of an MIT game theory course, and it spends its whole hour on a deliberately small question: how does a single person make a sensible choice? Before you can study how people scheme against each other, you have to nail down what it means for one person to “prefer” one thing to another. The answer turns out to be surprisingly mechanical — a preference is just a description of what you’d pick, and once there’s uncertainty in the mix, the way you choose is to assign probabilities to the things you can’t control, attach a number called utility to each possible outcome, and pick whatever gives you the highest average. The lecture builds this up from coffee orders and walks home, and ends on why a sane person will happily take less money on average just to avoid a bad surprise.

The Full Story

Setting the stage: what is game theory, and why start by ignoring it

The course is about game theory, which the lecturer (Ian Ball) frames with a one-liner borrowed from Nobel laureate Robert Aumann:

It’s interactive decision theory. It’s about people making decisions when their decisions interact.

The word doing all the work there is interact. Plenty of situations involve lots of people choosing things, but most of the time your choice and mine don’t touch. Game theory is specifically about the cases where they do — where the best move for me depends on what you’re about to do, and vice versa. The textbook image is a penalty kick: there’s no reason to shoot left rather than right, except that it depends on which way the keeper is going to dive, who is in turn guessing about you. Economists call this strategic interdependence.

To make the idea concrete, the class plays a game. Everyone secretly picks a number from 1 to 100, and the winner is whoever lands closest to two-thirds of the class average. The trap is that there’s no good answer in isolation. If you think the average will be 50, you should guess 33. But everyone else is thinking that too, so the average will be lower, so you should guess two-thirds of that — and now you’re in a hall of mirrors. One round, the winning guess was 24. Replay it, and the winners drift down toward 14, then lower. (The name for this is the Keynesian beauty contest — Keynes’ idea that picking stocks is less about which company you like and more about guessing which company everyone else will guess everyone else likes.)

So I have to think about what would they think about what I think what they think. And all of a sudden, it gets very, very hard.

Two side-lessons fall out of the game. First, a few students always deliberately guess 100 just to mess up the prediction — which isn’t irrational, it’s just a reminder that the model is only as good as its assumptions about what people actually want. If you assume everyone wants to win and some people want to be a nuisance instead, your forecast breaks. Second, stakes change behavior: nobody plays the trickster when there’s real money on the line.

Having shown why interaction is hard, the lecturer then sets it aside entirely. Today is about a single decision-maker in isolation. You have to understand how one rational person chooses before you can watch two of them collide.

What “rational” actually means (it’s not what you think)

People hear “rational” and picture a cold optimizer, and then object that humans aren’t like that. The lecture quietly redefines the word. In economics, rationality is not about having good goals — it’s about consistency in pursuing whatever goals you happen to have.

Preferences cannot be irrational.

Liking chocolate over vanilla can’t be a mistake; it’s just your taste. But if you genuinely prefer chocolate and then reach for vanilla, that is irrational, because you failed to act on your own preference. Rationality is the link between what you want and what you do, not a judgment on what you want. The lecturer concedes this consistency assumption holds in some settings and not others — your preferences at 20 and at 2 are wildly different — but across a short horizon, today versus tomorrow, it’s a decent enough description to build on.

Preferences, made boringly precise

Start with no uncertainty at all. You’re at a café choosing between coffee, espresso, and tea, and your preference is written:

C ≻ E ≻ T

Read it as “coffee over espresso over tea.” Buried in writing it left-to-right is an assumption called transitivity: if coffee beats espresso and espresso beats tea, then coffee must beat tea. It feels obvious, and the math will quietly rely on it everywhere.

But what is a preference? Philosophers can spend careers here; economists take a deliberately flat, almost tautological view. To prefer coffee to espresso means exactly one thing: when offered the two, you pick coffee. Nothing about neurons or feelings — preference is defined by choice. And it’s a pairwise notion: a preference is the answer to “which of these two?” for every possible pair, from which richer choices can be built up.

Utility: a convenient bookkeeping trick

Those squiggly ≻ signs get unwieldy fast once there are many options. So instead we attach a number to each choice — a utility function:

u(C) = 5, u(E) = 4, u(T) = 1

This “represents” the preferences because the ordering of the numbers matches the ordering of the tastes. Now here’s the subtle part. The lecturer writes a second utility function with totally different numbers but the same ranking, and asks: same preferences or different? Same. Because in this world, only the order of the numbers matters, not their size. This is called an ordinal utility function — “ordinal” as in ordinal numbers, first/second/third. The implication: there are infinitely many number-sets that encode the same preferences, and the units (sometimes jokingly called “utils”) are meaningless. Saying you get “5 utils” from coffee is not a measurement of anything; it’s just a label that happens to sort correctly.

A student raises a sharp point that the lecturer flags as crucial: a choice problem must include every option, and you can pick exactly one. If “drink nothing” is possible, it has to be on the menu with its own utility, or the model misdescribes the situation.

Uncertainty changes everything — now the numbers matter

Here’s the pivot. Real decisions, especially the ones game theory cares about, happen under uncertainty — when you choose, you don’t yet know the consequence. You buy a stock without knowing if it’ll rise. You walk home without knowing if it’ll rain.

The running example: walk home or take the subway (the “T” in Boston), not knowing whether it’ll be sunny or rainy. The payoffs:

	Sunny	Rainy
Walk	7	2
Take T	5	5

When it’s sunny, walking is lovely (7); when it’s rainy, walking is miserable (2); the subway is a steady 5 regardless. With certainty this is trivial — you’d walk in sun, ride in rain. The whole difficulty is the uncertainty.

What do you do? You form beliefs. You check the weather app and it hands you a probability — say, a chance p that it’s sunny, and 1 − p that it rains. Game theory takes a thoroughly Bayesian stance: facing the unknown, you assign probabilities to it. (The lecturer is honest that this is more believable for tomorrow’s weather than for “the odds your great-great-grandchild is short on cash in 200 years,” but it’s the working assumption.)

Then you maximize expected utility: weight each outcome’s utility by its probability and pick the highest average. He flags that this isn’t the only logically possible rule — an optimist could just chase the best-case payoff, a pessimist could avoid the worst case — but game theory folds optimism and pessimism into your beliefs (your p) rather than your decision rule, and assumes expected-utility maximization throughout. The justification lives in an axiomatic appendix; the course just takes it as given.

Crank the handle for walking: with probability p you get 7, with 1 − p you get 2, so expected utility is 7p + 2(1−p) = 2 + 5p. The subway is a flat 5. Walking wins when 2 + 5p ≥ 5, i.e. when p ≥ 3/5. So the rule is intuitive: walk home if the weather app says sun is at least 60% likely. (A student catches the edge case — at exactly p = 3/5 you’re indifferent, the third possibility alongside “strictly prefer A” and “strictly prefer B.”)

The deep consequence: those payoff numbers are no longer just rankings. If 7 were 700, the expectation would shift and you’d walk in almost any forecast. So under uncertainty, the magnitudes matter. This kind of utility — where the actual numbers carry weight — is cardinal (“cardinal” as in the counting numbers, size matters), and the formal name is a Von Neumann–Morgenstern (VNM) utility function. The rest of the course lives in this cardinal world.

The general machinery: outcomes, lotteries, and a tidy notation

The lecture then generalizes, following its house pattern (simple example → abstract model → richer applications). The pieces:

A set of outcomes (or consequences), written Z = {Z₁, …, Zₘ}. Crucially, an outcome must capture everything relevant — in the walk-home problem there are really four outcomes (walk-sunny, walk-rainy, T-sunny, T-rainy), not two.
A small-u VNM utility assigning a number to each outcome.
A lottery: a probability vector (p₁, …, pₘ) over outcomes, with each pᵢ ≥ 0 and the lot summing to 1. The “café-econ” word lottery just means “a thing with probabilistic outcomes” — not literal scratch tickets.

The reframing is that you don’t choose outcomes directly — you choose lotteries over outcomes. Walking is the lottery “sun-walk with probability p, rain-walk with probability 1 − p” (and zero chance of the two subway outcomes — choosing to walk can’t teleport you onto the train). The set of all such lotteries gets the shorthand Δ(Z).

Then big-U expected utility is a function over lotteries: for any lottery p, U(p) = p₁·u(Z₁) + … + pₘ·u(Zₘ). Note the careful split: small-u is defined on outcomes, big-U on lotteries. A student asks the right question — are beliefs (the p’s) just handed to us? For individual decision-making, yes: the uncertainty is exogenous, you form beliefs about nature by introspection. The course flags that interactive game theory will later have to derive beliefs about what other players do, which is much harder.

Money lotteries and why you’ll take the safe bet

The lecture closes on the most important special case: when outcomes are just dollar amounts (Z is the real number line). Two lotteries:

A: $10 with probability 0.99, $0 with probability 0.01. Expected value: $9.90.
B: $1,000 with probability 0.01, $0 with probability 0.99. Expected value: $10.00.

B is worth more on average. Yet most people pick A. Why? Because, as a student put it, A feels safer — it’s near-certain. The resolution is the central insight of the whole lecture: you don’t maximize expected money, you maximize expected utility, and money and utility are not the same thing. When the utility-of-money curve bends over — gaining your second thousand dollars thrills you less than your first — the function is concave, and a person with a concave utility curve is risk averse.

Formally: a risk-averse person always prefers a sure thing equal to a lottery’s average over the lottery itself. Offered “$10 guaranteed” versus “a gamble averaging $10,” they take the guarantee. This dissolves a puzzle that genuinely vexed mathematicians back in the 1700s (the St. Petersburg paradox): why turn down a bet with a higher expected payout? Because your inner yardstick isn’t dollars — it’s the curved utility of those dollars.

You don’t maximize expected money, you maximize expected utility. And some people’s utilities can be concave like this.

Key Takeaways

Game theory = interactive decision theory. Its defining feature is strategic interdependence: the best choice for you depends on what others will do, and they’re reasoning about you in turn.
The 2/3-of-the-average guessing game (Keynesian beauty contest) demonstrates the regress at the heart of strategic reasoning — and why winning guesses spiral toward zero over repeated rounds.
“Rational” means consistent, not wise. Preferences themselves are never irrational; only acting against your own preferences is. Rationality is the bridge between wanting and doing.
A preference is defined by choice. To prefer A to B just means: offered both, you pick A. Economics takes this operational, almost tautological view rather than a psychological one.
Transitivity is the quiet assumption baked into ranking things: A ≻ B and B ≻ C forces A ≻ C.
Ordinal utility: without uncertainty, only the ranking of utility numbers matters. Any numbers in the right order encode the same preferences; the units (“utils”) are meaningless.
Cardinal (VNM) utility: once there’s uncertainty, the magnitudes of utility numbers matter, because you average them. This is the regime the whole course operates in.
Decision-making under uncertainty has two steps: (1) form beliefs — assign probabilities to what you can’t control (a Bayesian move); (2) pick the option with the highest expected utility.
Optimism/pessimism live in your beliefs, not your decision rule. Everyone is assumed to maximize expected utility; your temperament shows up in the probabilities you assign.
A choice problem must list every option exactly once, and you pick one. Forgetting an option (like “do nothing”) misdescribes the problem.
You choose lotteries, not outcomes. Each action induces a probability distribution over outcomes; an outcome must encode everything relevant about the situation.
For an individual, uncertainty is exogenous — beliefs about nature come from introspection. In multi-player games, beliefs about other players must be derived, which is the hard part to come.
Expected value ≠ expected utility. You don’t maximize average dollars; you maximize average utility of dollars.
Risk aversion = a concave utility-of-money curve. A risk-averse person prefers a sure amount to a gamble with the same average payout — which is why most people take $10 guaranteed over a coin-flip-ish bet averaging $10.

Claude’s Take

This is a genuinely good first lecture, and it earns its slowness. Most intros to game theory rush to the prisoner’s dilemma and Nash equilibrium; Ball does the unglamorous but correct thing and spends the whole hour making sure you understand a single decision-maker before any strategy enters. The payoff is that the famously slippery concepts — why utility numbers sometimes don’t matter (ordinal) and sometimes do (cardinal), why “rational” doesn’t mean “smart,” why a sane person takes the worse average bet — all land cleanly because they’re built from café orders and walks home rather than dropped as axioms.

The teaching is honest in a way that’s rare. He repeatedly flags where the model’s assumptions are shaky (forming beliefs about your great-great-grandchild’s finances, treating preferences as stable over decades) instead of pretending the framework is universal. The student questions are unusually sharp and he doesn’t dodge them — the edge case at p = 3/5, the “must every option be on the menu” point, the exogenous-beliefs question all get real answers.

What keeps it from a 9: it’s a lecture, so by design it doesn’t go anywhere yet — this is foundation-pouring, and the interesting stuff (actual games, equilibria) is explicitly deferred to future weeks. The risk-aversion ending is the one moment of real “oh, that’s why” payoff. As the opening brick of a course it’s excellent; as a standalone watch it’s more nutrition than dessert. If you’ve taken any microeconomics, the ordinal/cardinal distinction and VNM utility will be review, though cleanly re-derived. Eight out of ten: rigorous, well-paced, intellectually honest, and the kind of thing where the value compounds only if you watch the lectures that follow.