Before analyzing games we must be precise about what a “payoff” is and what “rational” means. This chapter builds the decision-theoretic foundation on which all of game theory rests. Readers eager for games may skim it and return later; but the ideas here—especially expected utility—are used constantly from Chapter [ch:mixed] onward.
Preferences and utility
Decision theory starts not from numbers but from preferences. Let \(X\) be a set of outcomes. A decision-maker’s preferences are captured by a binary relation \(\succsim\) on \(X\), where \(x \succsim y\) reads “\(x\) is at least as good as \(y\).” We write \(x \succ y\) (strict preference) when \(x \succsim y\) but not \(y \succsim x\), and \(x \sim y\) (indifference) when both hold.
Rational preference relationrationalpref A preference relation \(\succsim\) on \(X\) is rational if it is
complete: for all \(x,y \in X\), either \(x \succsim y\) or \(y \succsim x\) (or both); and
transitive: for all \(x,y,z \in X\), if \(x \succsim y\) and \(y \succsim z\) then \(x \succsim z\).
Completeness says the decision-maker can always rank alternatives; transitivity rules out preference cycles. Neither is innocuous—real people violate both in experiments—but together they permit a dramatic simplification:
Utility representationutilrep If \(X\) is finite (or countable) and \(\succsim\) is rational, then there exists a utility function \(u : X \to \R\) such that \[x \succsim y \iff u(x) \ge u(y).\] Moreover, \(u\) is unique only up to strictly increasing transformations: if \(f\) is strictly increasing, \(f \circ u\) represents the same preferences.
Proof. Enumerate \(X = \{x_1, x_2, \dots\}\) and set \(u(x) = \#\{y \in X : x \succsim y\}\) when \(X\) is finite (the countable case requires a small extra construction). Completeness and transitivity guarantee \(x \succsim y \iff u(x) \geq u(y)\): transitivity implies that the set of outcomes weakly worse than \(y\) is contained in the set weakly worse than \(x\) whenever \(x \succsim y\), giving \(u(x)\ge u(y)\); completeness gives the converse direction. ◻
Theorem [thm:utilrep] delivers ordinal utility: only the ranking matters, not magnitudes. Saying “\(u(x)=10\) and \(u(y)=5\), so \(x\) is twice as good” is meaningless at this stage. Cardinal meaning arrives only with risk, below.
Choice under risk: lotteries
Games force players to evaluate risky prospects: if my opponent randomizes, my action induces a probability distribution over outcomes. Let \(X = \{x_1, \dots, x_n\}\) be a finite outcome set. A lottery is a probability distribution \(p = (p_1, \dots, p_n)\) with \(p_i \ge 0\) and \(\sum_i p_i = 1\). Let \(\Delta(X)\) denote the set of all lotteries over \(X\). The decision-maker now has preferences \(\succsim\) over \(\Delta(X)\), not just over \(X\).
von Neumann–Morgenstern axiomsvnmaxioms A preference relation \(\succsim\) on \(\Delta(X)\) satisfies the vNM axioms if it is rational (complete, transitive) and additionally:
Continuity: for all \(p, q, r\) with \(p \succ q \succ r\), there exist \(\alpha, \beta \in (0,1)\) such that \(\alpha p + (1-\alpha) r \succ q \succ \beta p + (1-\beta) r\).
Independence: for all \(p, q, r\) and \(\alpha \in (0,1]\), \[p \succsim q \iff \alpha p + (1-\alpha) r \;\succsim\; \alpha q + (1-\alpha) r.\]
Independence is the crucial (and most debated) axiom: mixing both lotteries with a common third lottery should not reverse the ranking.
Expected utility (von Neumann–Morgenstern, 1944)vnm \(\succsim\) on \(\Delta(X)\) satisfies the vNM axioms if and only if there exists \(u : X \to \R\) (a Bernoulli utility function) such that for all lotteries \(p,q\): \[p \succsim q \iff \sum_{i=1}^n p_i\, u(x_i) \;\ge\; \sum_{i=1}^n q_i\, u(x_i).\] Moreover \(u\) is unique up to positive affine transformations: \(v = a u + b\) with \(a>0\) represents the same preferences.
Theorem [thm:vnm] is the license for everything we do later: when we write a payoff matrix and compute expected payoffs against a randomizing opponent (Chapter [ch:mixed]), we are implicitly assuming players are vNM expected-utility maximizers. The affine-uniqueness clause tells us which operations on payoffs are harmless: adding a constant to all of a player’s payoffs, or rescaling them by a positive factor, changes nothing strategically.
Risk attitudes
Suppose outcomes are money amounts. A decision-maker with Bernoulli utility \(u\) facing a lottery with random prize \(W\) compares \(\E[u(W)]\) with \(u(\E[W])\).
Risk aversionriskaversion A decision-maker is risk-averse if she prefers the expected value of any lottery for sure to the lottery itself: \(u(\E[W]) \ge \E[u(W)]\) for all \(W\). By Jensen’s inequality this holds iff \(u\) is concave. She is risk-neutral iff \(u\) is affine, and risk-loving iff \(u\) is convex.
The St. Petersburg puzzlestpetersburg A fair coin is flipped until the first heads; if this happens on flip \(k\) you win \(2^k\) dollars. The expected prize is \(\sum_{k=1}^\infty 2^{-k} \cdot 2^k = \sum_k 1 = \infty\), yet nobody would pay more than a modest amount to play. Daniel Bernoulli’s 1738 resolution: people maximize expected utility, not expected money. With \(u(x) = \log_2 x\), the certainty equivalent is finite and small: \(\E[u] = \sum_{k\ge 1} 2^{-k} \cdot k = 2\), i.e. the gamble is worth \(2^2 = 4\) dollars. This is the historical origin of expected utility theory.
Beliefs, best responses, and rationality in games
In a game, a player’s optimal action depends on what the others do—which she does not observe in a simultaneous-move game. The modern view: a rational player forms a belief (a probability distribution over opponents’ strategy choices) and chooses a best response to that belief.
Best response to a beliefbrbelief Let player \(i\) have belief \(\mu\) over the opponents’ action profiles \(A_{-i}\). Action \(a_i\) is a best response to \(\mu\) if \[\sum_{a_{-i}} \mu(a_{-i})\, u_i(a_i, a_{-i}) \;\ge\; \sum_{a_{-i}} \mu(a_{-i})\, u_i(a_i', a_{-i}) \quad \text{for every } a_i' \in A_i .\] A player is rational if she plays a best response to some belief.
Rationality alone predicts little: almost any action is a best response to some belief. Predictive power comes from assumptions about what players believe about each other:
Common knowledgecommonknowledge An event \(E\) is common knowledge if everyone knows \(E\), everyone knows that everyone knows \(E\), everyone knows that everyone knows that everyone knows \(E\), and so on ad infinitum.
The classical assumption of game theory is common knowledge of rationality (CKR): all players are rational, all know this, all know that all know it, etc. In Chapter [ch:normalform] we will see exactly what CKR implies (iterated elimination of dominated strategies / rationalizability), and in Chapter [ch:nash] what additional assumptions Nash equilibrium requires.
The email/coordinated-attack puzzleemailgame Two generals must attack simultaneously to win. General 1 sees the enemy is weak and emails General 2: “attack at dawn.” Messages fail with small probability, so General 2 replies to confirm, General 1 confirms the confirmation, ad infinitum. After any finite number of confirmations, “attack at dawn” is mutually known to high order but never common knowledge—and one can show coordinated attack cannot be guaranteed. The example (Rubinstein’s email game) shows common knowledge is a genuinely stronger and more delicate notion than “everybody knows.”
Exercises
Transitivity and money pumpsmoneypump A decision-maker has cyclic preferences: \(x \succ y\), \(y \succ z\), \(z \succ x\). Show that if she is willing to pay a small amount \(\varepsilon > 0\) to swap any held item for a strictly preferred one, a trader holding \(x, y, z\) can extract unbounded money from her. (This “money pump” is the classic pragmatic defense of transitivity.)
Affine invarianceaffine Player 1’s payoffs in some game are given by \(u_1\). Show that replacing \(u_1\) by \(v_1 = 3u_1 + 7\) leaves her set of best responses to every belief unchanged. Then give an example showing that a nonlinear increasing transformation (say \(u \mapsto u^3\) with payoffs that can be negative) can change best responses against a randomizing opponent even though it preserves the ordinal ranking of pure outcomes.
Certainty equivalentscertequiv An agent has Bernoulli utility \(u(x) = \sqrt{x}\) and faces a lottery paying \(100\) with probability \(1/2\) and \(0\) otherwise. Compute the expected value, the expected utility, the certainty equivalent, and the risk premium.
Allais paradoxallais Consider lotteries over prizes \(\{0, 1\text{M}, 5\text{M}\}\) (millions): \(A = (0, 1, 0)\) (1M for sure); \(B = (0.01, 0.89, 0.10)\); \(C = (0.89, 0.11, 0)\); \(D = (0.90, 0, 0.10)\). Most people report \(A \succ B\) and \(D \succ C\). Show these two rankings jointly violate the independence axiom.