Game Theory

Units

1. 1

   Games and payoffs

   4 lessons

   1. Players, strategies and payoffsDefine a game as a set of players, a strategy set for each player and a payoff for each outcome, state what rationality and common knowledge assume, and label a two-player payoff table with strategies as rows and columns and a payoff pair in each cell.
   2. The normal formWrite a simultaneous-move game in normal form, G = (N, {S_i}, {u_i}) with payoffs u_i(s₁, s₂), and draw the Prisoner's Dilemma as a 2 by 2 payoff matrix with each player's payoff in the cell and the cooperative outcome picked out.
   3. Dominant and dominated strategiesDefine strict and weak dominance, show with a payoff graph and a matrix why a rational player never plays a strictly dominated strategy, and cross out dominated rows and columns step by step to solve a game.
   4. Iterated elimination of dominated strategiesSolve a game where dominance only appears in rounds, striking out strictly dominated strategies of each player in turn on a 3 by 3 matrix until one profile survives, and say why this needs common knowledge of rationality.
2. 2

   Nash equilibrium

   4 lessons

   1. Best responsesDefine the best-response correspondence B_i(s_-i) and find it by underlining the best payoff in each column and each row of a payoff matrix, then read off the cells where both players' underlines coincide.
   2. Pure strategy Nash equilibriumState the definition of a pure strategy Nash equilibrium, that no player gains by deviating alone, u_i(s_i*, s_-i*) ≥ u_i(s_i, s_-i*) for every s_i, and find equilibria by circling mutual best responses in a matrix, including a pure coordination game and Battle of the Sexes.
   3. Nash equilibrium with continuous strategiesSolve a game whose strategies are numbers by setting each player's first-order condition to find the best-response function, using a brief Cournot duopoly with linear demand, and find the Nash equilibrium where the two downward-sloping reaction curves cross.
   4. Coordination, anti-coordination and Hawk-DoveCompare Stag Hunt, Battle of the Sexes and Hawk-Dove side by side, find the equilibria of each with payoff matrices, and see why they have several equilibria, how risk and conflict differ among them, and how a focal point can select one.
3. 3

   Mixed strategies

   4 lessons

   1. Randomizing and expected payoffDefine a mixed strategy as a probability distribution over actions, and compute expected payoffs in Matching Pennies against a pure action and against another mixed strategy.
   2. The indifference principleFinds mixed equilibria by making the opponent indifferent, solving p u(A) + (1 - p) u(B) equal across actions, with a worked 2 by 2 example and the algebra written beside the matrix.
   3. Best-response graphs for mixed playPlots each player's best response as a function of the other's probability, p on one axis and q on the other, with stepped lines whose crossings mark pure and mixed equilibria.
   4. Existence of equilibrium and zero-sum gamesStates Nash's theorem that every finite game has an equilibrium in mixed strategies, and the minimax theorem for zero-sum games, shown with a penalty kick matrix and the value of the game.
4. 4

   Dynamic games and subgame perfection

   4 lessons

   1. Game trees and information setsRepresents sequential play as a game tree with decision nodes, branches and terminal payoffs, and draws dashed information sets for imperfect information, translating trees to and from normal form.
   2. Backward inductionSolves finite perfect-information games by reasoning from the last move, folding up a drawn tree and highlighting the optimal branch at each node, with the entry deterrence game as the example.
   3. Subgame perfect equilibriumDefines subgames and subgame perfect equilibrium, showing how non-credible threats make some Nash equilibria fail, by boxing subgames on a tree and marking the equilibrium path.
   4. Commitment: Stackelberg and ultimatum gamesShows the first-mover advantage in a leader-follower quantity game and in the ultimatum game, using a tree with offers and responses and the follower's reaction curve intersected by the leader's isoprofit curve.
5. 5

   Repeated games

   4 lessons

   1. Finitely repeated gamesExplains why a Prisoner's Dilemma repeated a known number of times unravels by backward induction, drawn as a stage-game matrix repeated along a timeline with defection at every stage.
   2. Infinitely repeated games and discountingIntroduces discount factor delta and present value of payoffs, sum of delta to the t times the stage payoff, drawn as a payoff stream on a timeline with the geometric sum 1 / (1 - delta).
   3. Grim trigger and the folk theoremDerives the condition delta >= (T - R) / (T - P) for cooperation under grim trigger, plotting cooperation payoff against deviation payoff, and states the folk theorem with the shaded set of feasible payoffs.
   4. Tit-for-tat, reputation and cooperationCompares tit-for-tat, grim trigger and always-defect strategies, shown as small state machines with arrows for cooperate and defect, linking to Axelrod's tournaments and real examples of cartels and norms.
6. 6

   Incomplete information and Bayesian games

   4 lessons

   1. Types, beliefs and Bayes' ruleIntroduces private information as types drawn by nature with a common prior, and updating beliefs with Bayes' rule, shown on a probability tree with nature's first move and posterior probabilities computed.
   2. Bayesian Nash equilibriumDefines a Bayesian Nash equilibrium as a strategy for every type maximizing expected payoff, with a worked two-type game drawn as a tree with nature and two payoff matrices.
   3. First-price sealed-bid auctionsSolves bidding with private values uniform on [0, 1], getting the equilibrium bid b(v) = v (n - 1) / n, and plots the bidding function below the 45 degree line for different numbers of bidders.
   4. Second-price auctions and revenue equivalenceShows that bidding your value is a dominant strategy in a second-price auction, and the revenue equivalence theorem, drawn as bid functions and a bar chart of equal expected revenue across formats.
7. 7

   Signaling and screening

   4 lessons

   1. Cheap talk and communicationStudies when costless messages can be informative, using the Crawford-Sobel idea of partition equilibria, drawn as a type interval on a line cut into segments with messages sent for each segment.
   2. Spence job market signalingModels education as a costly signal where high-ability types find it cheaper, with the single crossing condition, drawn as indifference curves of two types and a wage schedule with a separating cutoff.
   3. Pooling, separating and perfect Bayesian equilibriumDefines perfect Bayesian equilibrium with beliefs consistent by Bayes' rule, and classifies separating and pooling outcomes, drawn as a signaling game tree with dashed information sets and beliefs written at each.
   4. Screening and adverse selectionDevelops screening with the used car lemons market and insurance contract menus, drawn as a quality histogram, a price line cutting out the good cars, and a menu of contracts on a coverage and price diagram.
8. 8

   Bargaining and applications

   4 lessons

   1. Alternating offer bargainingSolves the Rubinstein game by backward induction on a timeline, with equilibrium first-mover share (1 - delta) / (1 - delta²) shown as a shrinking pie each round that the players split.
   2. The Nash bargaining solutionDefines the Nash product (u1 - d1)(u2 - d2) maximized over the feasible set, drawn as a utility possibility frontier with disagreement point d and a hyperbola touching the frontier at the solution.
   3. Voting and the median voterAnalyzes two-candidate competition on a policy line with the median voter theorem, and agenda control and Condorcet cycles, drawn as voter ideal points on a line with a median marker and a cycle diagram.
   4. Oligopoly games in briefReviews Bertrand price competition giving price equal to marginal cost against Cournot quantity competition, with reaction curves on one diagram and a repeated-game tie to collusion, as a game theory application.