Game Theory Lecture: Core Concepts, Utility and Uncertainty

Game theory is the study of mathematical models of conflict and cooperation between intelligent, rational decision makers. Roger Myerson defines it as “mathematical models of conflict and cooperation,” while Robert Aumann calls it “interactive decision theory.” The discipline examines how agents make choices when those choices affect one another.

Foundations of Decision Making

Rationality

In game theory, rationality means acting consistently toward one’s objectives. It does not refer to everyday “sensible” behavior, and preferences themselves cannot be irrational.

Strategic Interdependence

A player’s best action often depends on what other players do. This interdependence is illustrated by the Keynesian beauty‑contest experiment, where participants guess a number closest to two‑thirds of the class average. Success requires anticipating others’ guesses and their anticipation of yours, creating an “interactive regress” of reasoning layers.

Scope of Application

The concepts apply to individuals, firms, evolutionary processes, and artificial intelligence, providing a baseline for later interactive models.

The Keynesian Beauty Contest

In the classroom experiment run on the Moblab platform, each student submits a number. The winning guess is 2/3 of the average of all guesses. Empirically, participants tend to lower their numbers as they expect others to do the same, producing a regression of reasoning. Player preferences matter: some participants act mischievously rather than trying to win, and higher stakes reduce such behavior.

Utility Theory

Ordinal Utility

Ordinal utility captures only the ranking of outcomes. Any utility function that preserves the order represents the same preferences, so the actual numerical values are irrelevant.

Cardinal (Von Neumann‑Morgenstern) Utility

Cardinal utility, also called VNM utility, assigns numerical values that matter for decisions under uncertainty. It enables the calculation of expected utility, where each outcome’s utility is weighted by its probability:

[ U(p)=\sum_{i=1}^{m} p_i \, u(Z_i) ]

Here (p_i) is the probability of outcome (Z_i) and (u(Z_i)) is its VNM utility.

Decisions Under Uncertainty

When outcomes are uncertain, agents form beliefs in the form of probabilities and then maximize expected utility. Risk aversion describes a preference for a certain payoff equal to the expected monetary value of a lottery rather than the lottery itself. Mathematically, risk‑averse behavior is modeled by a concave utility function, reflecting diminishing marginal utility of wealth.

Mechanisms & Explanations

• Interactive Regress – Players think about others’ thoughts, who in turn think about the original player, leading to deeper layers of strategic anticipation.
• Expected Utility Calculation – The formula (U(p)=\sum p_i u(Z_i)) aggregates weighted utilities to guide choice under risk.
• Risk Aversion – A concave utility curve captures the desire for certainty over a risky prospect with the same expected value.