About This Course

This book is about decisions whose consequences depend on what other people decide. A firm choosing a price, a bidder choosing a bid, a job applicant choosing how much schooling to buy, and three towns deciding how to split the cost of a shared water plant all face the same difficulty: no one of them controls the outcome, and each knows that the others are reasoning too. Game theory supplies a precise language for such situations and a small number of solution concepts, each of which answers the question “what will happen, or what should be agreed?” under stated assumptions about timing, information, and commitment.

The book is written for a reader working alone who is comfortable with proofs. You need three things from earlier courses. From Discrete Mathematics: sets, functions, quantifiers, induction, and counting. From Probability Theory: expectation, conditional probability, Bayes’ rule, independence, and the distribution function and density of a continuous random variable. From Continuous Optimization: maximizing a concave function of one variable, the meaning of a convex set, and duality for linear programs. Two chapters lean on a theorem from analysis, and it is stated in full where it is used. No economics is assumed; the economic examples are explained from scratch. When a result that another course owns is needed, it is stated with the hypotheses the argument uses and marked once with its library identifier; it is not proved again here.

A reader who finishes the book should be able to carry out six tasks with the book closed. Write a strategic situation as a game, saying exactly who chooses what, when, and knowing what. Find the Nash equilibria of a finite game in pure and in mixed strategies, and prove that a proposed profile is or is not an equilibrium by exhibiting the deviation inequalities. Solve a zero-sum game as a linear program. Solve a game tree by backward induction, decide which Nash equilibria of a dynamic game survive subgame perfection, and compute the discount factor at which cooperation becomes self-enforcing in a repeated game. Solve static games with private information, track beliefs through a signaling game, and say which equilibria rest on unreasonable beliefs. Derive equilibrium bids in the standard auctions, state and use the revelation principle and revenue equivalence, and compute the core and the Shapley value of a small coalitional game.

Notation is fixed throughout. Players are numbered \(1,\dots,n\) and the set of players is \(N\) . A subscript \(i\) refers to player \(i\) and a subscript \(-i\) to everyone else, so that a strategy profile \(s=(s_1,\dots,s_n)\) is also written \((s_i,s_{-i})\) . In a two-player payoff table, player 1 chooses the row and player 2 the column, and each cell lists player 1’s payoff first. Mixed strategies are written with the letter \(\sigma\) , types with \(\theta\) , beliefs with \(\mu\) , and the discount factor with \(\delta\) . A star marks an equilibrium object.

Every chapter ends with exercises that run from recall to proof, and several of them ask you to find the error in a plausible argument, because in this subject the wrong arguments are often more persuasive than the right ones. In the companion volume some exercises are solved in full, others receive a hint, and a few only a final answer to check against. Every equilibrium in the worked examples was recomputed by machine from the payoffs; you should verify each one by hand before reading on, since checking that no player gains by deviating is the single skill the whole subject rests on. The cumulative review at the end mixes the chapters without labels, because deciding which model and which solution concept fit a situation is part of the problem.