1  Introduction

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Game theory is the study of strategic interaction: situations in which the outcome for each participant depends not only on what they do, but on what everybody else does too. This short chapter says what makes such situations different from the optimisation problems of single-agent economics, what game theory offers in return, and reviews the one tool from decision theory we rely on throughout — expected utility.

1.1 Individual decisions versus strategic interactions

In intermediate microeconomics every problem is an individual decision problem: a single decision-maker chooses to maximise an objective that is completely determined by their own choice. Two familiar examples:

• A consumer choosing a bundle \((x_1, x_2)\) to maximise utility subject to a budget constraint, \[ \max_{x_1, x_2 \ge 0} \; u(x_1, x_2) \quad\text{s.t.}\quad p_1 x_1 + p_2 x_2 \le w . \]
• A competitive firm choosing output \(y\) to maximise profit, \[ \max_{y \ge 0} \; p\,y - c(y). \]

Both are solved with the standard toolkit — first-order conditions, Lagrange multipliers. The defining feature is that nobody else’s behaviour enters the objective.

Most interesting economic situations are not like this. In chess, oligopolistic competition, an auction, or a political campaign, one participant’s payoff depends on the choices of others as well as their own. We call these strategic interactions.

What makes them genuinely harder is that beliefs become circular. Consider rock–paper–scissors:

If I predict that you will play Rock, I should play Paper. But if you anticipate that prediction, you will play Scissors — in which case I should play Rock — in which case you should play Paper — …

There is no end to the regress, and no first-order condition that resolves it. A strategic interaction is not a simple optimisation problem, because the “right” choice for me depends on a choice you are making for symmetric reasons. We need a framework that closes this loop.

The founding text, von Neumann and Morgenstern’s Theory of Games and Economic Behavior (1944), anticipated exactly this difficulty:

“[I]t is unlikely that a mere repetition of the tricks which served us so well in physics will do for the social phenomena too. … [The] process of mathematization is not at all obvious.”

Game theory is the body of mathematics that grew up to meet the problem.

1.2 What game theory provides

Game theory does two things for us.

1. A language. It gives a precise, formal way to describe a strategic interaction — who the players are, what they can do, what they know, and what they care about. Much of the work in any application is learning to write the situation down as a formal model.
2. A prediction. Given that model, it offers a prediction of what rational players will do — a solution concept. Most of this book is about defining, justifying, and computing such solutions.

Throughout, we maintain the standing assumption that players are rational: each player acts so as to maximise their own payoff, given what they know and believe about the others.

1.3 A review of expected utility

Because players will often face uncertainty — over a chance move, or over what an opponent does — we need to say how a rational player evaluates a gamble over outcomes rather than a single sure outcome. The answer is expected utility, and it requires only two definitions.

Definition 1.1 (Simple lottery) A simple lottery over a finite set of outcomes \(X = \{x_1, x_2, \dots, x_n\}\) is a probability distribution \(p = \big(p(x_1), \dots, p(x_n)\big)\), where \(p(x_k) \ge 0\) for every \(k\) and \(\sum_{k=1}^{n} p(x_k) = 1\).

Two illustrations:

• A lottery ticket that pays $0 with probability \(0.99\) and $100 with probability \(0.01\): here \(X = \{0, 100\}\) and \(p(0) = 0.99,\ p(100) = 0.01\).
• Tomorrow’s weather, which is sunny, cloudy, or raining with probabilities \(80\%, 15\%, 5\%\): here \(X = \{s, c, r\}\) and \(p(s) = 0.8,\ p(c) = 0.15,\ p(r) = 0.05\).

A player ranks lotteries by the expectation of their payoff.

Definition 1.2 (Expected payoff) Let \(u(x)\) be the player’s payoff (utility) over outcomes in \(X = \{x_1, \dots, x_n\}\), and let \(p = (p_1, \dots, p_n)\) be a lottery over \(X\). The player’s expected payoff from the lottery \(p\) is \[ \E[u(x)] \defeq p_1 u(x_1) + \cdots + p_n u(x_n) = \sum_{k=1}^{n} p_k\, u(x_k). \]

That is, the expected payoff is the weighted average of the outcome payoffs, with the lottery’s probabilities as weights. A rational player chooses the action whose induced lottery has the highest expected payoff.

Worked example: to carry an umbrella?

Keep the weather lottery \(p(s) = 0.8,\ p(c) = 0.15,\ p(r) = 0.05\), and suppose the agent must choose whether to carry an umbrella (\(u\)) or not (\(n\)). Their payoff depends on the choice and on whether it rains:

Table 1.1: Payoffs from the umbrella decision.

	it rains	it does not rain
carry (\(u\))	\(10\)	\(-1\)
don’t carry (\(n\))	\(-10\)	\(0\)

Both sunny and cloudy count as “does not rain”, so the two actions induce two lotteries over payoffs. Their expected payoffs are \[ \E[u \mid \text{carry}] = 0.8\times(-1) + 0.15\times(-1) + 0.05\times 10 = -0.45, \] \[ \E[u \mid \text{don't carry}] = 0.8\times 0 + 0.15\times 0 + 0.05\times(-10) = -0.5 . \] Since \(-0.45 > -0.5\), carrying the umbrella is optimal. This is still an individual decision problem — nobody else’s choice entered. From the next chapter on, the lotteries a player faces will be generated by the choices of other players, and expected utility is exactly the criterion we use to evaluate them.

1.4 The road ahead

The rest of the book is organised by two binary distinctions that between them classify every game we study.

• Static vs. dynamic. In a static game players move simultaneously (or, more precisely, without observing each other’s moves). In a dynamic game they move in sequence and can condition on what they have seen.
• Complete vs. incomplete information. Under complete information the structure of the game — including every player’s payoffs — is common knowledge. Under incomplete information some players hold private information (their “type”) that others do not observe.

Crossing the two distinctions gives four classes of game, each with its own solution concept, and each concept a refinement that fixes a shortcoming of the last:

The four classes of game and their solution concepts.

	Complete information	Incomplete information
Static	Nash equilibrium (Chapter 2)	Bayesian Nash equilibrium (Chapter 4)
Dynamic	Subgame-perfect equilibrium (Chapter 3)	Perfect Bayesian equilibrium (Chapter 5)