Game Theory
Module 1 · Lecture 1: Rationality and Rational Choice · following Asya Magazinnik's MIT course.
This course begins with a simple question: how can we explain a political outcome? The lecture notes start from causal mechanisms: an explanation of how one situation produces another. A downturn may precede an incumbent's defeat, but the theory asks what actors, goals, opportunities, incentives, and constraints connect one event to the other.
It then builds one layer at a time. First we study an individual choosing from a set of options. Then we add other players, whose choices affect the outcome. After that we add uncertainty, sequence, repetition, private information, signals, and collective voting rules.
Why start with rationality?
Before there is a game, there is an actor choosing from a set of possibilities. A preference relation records how that actor ranks alternatives. The lecture notes call it rational when it is complete - the actor can compare any two alternatives - and transitive - if x is preferred to y and y to z, then x is preferred to z.
A choice structure records the feasible sets an actor may face and what the actor chooses from each set. A preference relation rationalizes the choices when the chosen options are exactly the best available options under that relation.
Problem Set 1 uses a two-round runoff election to show how a voting rule can violate WARP. Follow the alternatives available at each round, identify what is chosen, and locate the pair that is ordered inconsistently.
Magazinnik presents game theory as a graduate-level introduction to formal theoretical analysis. The local archive preserves the lecture notes and problem sets; the original course remains on MIT OpenCourseWare ↗.
Games in Strategic Form and Nash Equilibrium
Lecture 2 moves from individual preferences to a static game of complete information. Static means the players choose without observing one another's current choice. Complete information means everyone knows the players, the available actions, and the payoffs attached to every combination.
A normal-form game names the players, lists each player's strategies, and records the payoff attached to every strategy profile. A strategy profile is one complete combination of choices, one for each player.
The Prisoner's Dilemma
Two suspects choose whether to remain silent or confess. Strict dominance means that one strategy gives a higher payoff than another no matter what the other player does. Here, confessing strictly dominates silence: whatever the other suspect does, confession gives that suspect a better payoff. The Nash equilibrium is therefore mutual confession, even though both would prefer mutual silence.
| Clyde: silent | Clyde: confess | |
|---|---|---|
| Bonnie: silent | -1, -1 | -9, 0 |
| Bonnie: confess | 0, -9 | -6, -6 |
The source then applies the same method to campaign contributions and lobbying, the war of attrition, and the median voter theorem. The point is not that every political situation is a prisoner's dilemma; it is that the model makes the strategic dependence explicit.
For each player in the matrix, compare the payoff from silence with confession in each column or row. Identify the best responses. Where do they intersect? Then explain why the equilibrium can be stable without being collectively best.
Lecture 2 and Problem Set 2 are preserved locally. Problem Set 2 asks learners to eliminate dominated strategies, find pure Nash equilibria, and formalize distributive politics and auctions.
Check your understanding
What makes a strategy profile a Nash equilibrium?
Mixed Strategy Nash Equilibrium
Lecture 3 begins with a gap in the previous lesson: pure strategies map actions to certain outcomes, but many decisions lead to risky outcomes. A lottery is a probability distribution over consequences. Expected utility gives us a way to compare those lotteries.
A mixed strategy is a probability distribution over actions. The point is not that a player is confused. The player randomizes because the probability itself changes what the other player wants to do.
Public goods
Problem Set 3 asks about a public good that appears when at least one of N people contributes a cost c. In a symmetric mixed equilibrium, everyone contributes with the same probability. The calculation is useful because it exposes a political tension: as the group becomes larger, the probability that someone contributes can fall.
Let q be the probability that one person contributes. Compare the payoff from contributing with the payoff from not contributing. Set them equal to find the symmetric mixed equilibrium. Then compute the probability that at least one person contributes and explain what happens as N grows.
Lecture 3 develops lotteries, expected utility, mixed-strategy Nash equilibrium, public-goods provision, and defense against terrorism. Problem Set 3 also applies the logic to voter turnout.
Check your understanding
Why can mixing be part of an equilibrium?
Extensive Form Games with Complete Information
In a static game, players choose once and the timing disappears into the matrix. In an extensive-form game, players move at different points. A strategy is therefore a complete contingent plan: what a player would do at every history, including histories that never occur.
The lecture distinguishes perfect information, where everyone sees the history so far, from imperfect information, where a player does not. Backward induction solves a perfect-information game by starting at the final decision and working backward.
Incumbent versus challenger
Problem Set 4 gives the challenger the first move: enter or stay out. The incumbent then chooses high or low campaign spending. The payoffs and winning probabilities make the tree concrete. Draw the tree first, solve the final decision, then work backward to the entry decision.
Trace the players' moves in order. At each final node, identify the incumbent's preferred action. Then ask whether the challenger would enter, given the incumbent's rational continuation choice. Explain the difference between a Nash equilibrium and a subgame-perfect equilibrium here.
Lecture 4 and Problem Set 4 are full texts preserved locally. The lecture's examples include colonial control, the tragedy of the commons, mixed and behavioral strategies, and Grossman and Helpman's “Protection for Sale.”
Check your understanding
What does subgame perfection add?
Repeated Games
Lecture 5 studies repeated games: the same stage game is played again and again. Repetition does not change the one-shot payoffs, but it changes what a player can condition on. A future reward or punishment may make cooperation possible even when the one-shot equilibrium is inefficient.
The lecture compares finitely repeated games with infinitely repeated games, then introduces discounting, grim trigger, tit-for-tat, intermediate punishments, and the folk theorem. More equilibria also means more difficulty making a precise prediction.
Rubinstein bargaining
Problem Set 5 turns repetition into a bargaining problem. Two politicians divide one million dollars. One proposes, the other accepts or rejects; after rejection, the available amount shrinks by δ and the other proposes. To solve it, compare accepting now with rejecting and receiving the continuation value.
Start with the final offer. What would the second politician accept? Move one step earlier and calculate what the first politician can offer while keeping the second indifferent. State where δ enters the result and what happens as the surplus becomes smaller.
Lecture 5 and Problem Set 5 are preserved locally. The set also covers infinite investigation, individual rationality, feasible payoffs, and sustaining the cooperative outcome (9, 9).
Static Games of Incomplete Information
Complete information means everyone knows the players, available actions, and payoffs. A Bayesian game relaxes that assumption. Players may have different types, and a type can affect both preferences and actions.
A strategy in a Bayesian game is a plan for every possible type, not just the action chosen by the type that happens to exist. Equilibrium uses expected payoffs and beliefs about other players' types.
Bayesian politics in the source
The lecture uses jury voting, public goods, electoral competition, and Fearon's rationalist explanation for war. Problem Set 6 gives a politician, Bob, who may be innocent or guilty, and a journalist, Alice, who chooses where to look for him. The type changes Bob's preferences, while Alice must reason from her prior belief.
Construct two payoff matrices for Bob: one when θ = 0 and one when θ = 1. For each possible action by Alice, identify Bob's best response. Then calculate Alice's expected payoff using her prior probability. Do not collapse the two types into one average player.
Lecture 6 and Problem Set 6 are full texts preserved locally. The source also names Gibbons, Chapter 3, and includes a majority-rule bargaining extension.
Check your understanding
What is a strategy in a Bayesian game?
Dynamic Games of Incomplete Information
Lecture 7 combines two difficulties: players move at different times, and some players know more than others. The model must therefore describe not only actions, but also what each player believes after observing an action.
A Perfect Bayesian equilibrium combines sequential rationality with consistent beliefs. The Beer-Quiche game shows how an action can both deter a challenger and reveal something about the actor's type.
Separating and pooling
In a separating outcome, different types choose different actions. In a pooling outcome, types choose the same action. The distinction is not a label added after the fact: it changes what an observer can learn.
Identify the informed player, the possible types, the observed action, and the observer's response. Then ask whether the signal separates types or pools them, and whether the proposed beliefs are consistent with the actions.
The source covers Perfect Bayesian Equilibrium, sequential rationality, weak consistency of beliefs, the Beer-Quiche game, and advice from experts. Problem Set 7 applies Bayesian reasoning to representative democracy and trade restrictions.
Check your understanding
What happens in a separating outcome?
Social Choice
Social choice asks how individual preferences aggregate into the preferences or decisions of a group. The problem is not simply to find what each person wants. It is to specify the rule that turns those wants into a collective outcome.
Lecture 8 uses Andrew, Bonnie, and Chuck, who must choose between the Museum of Fine Arts, Walden Pond, and a Red Sox game. Pairwise majority votes can cycle: one option can beat a second, the second can beat a third, and the third can beat the first.
Sincere and strategic voting
The source example then asks whether Chuck should vote sincerely for his first choice or vote strategically to affect the pairwise contests. The answer depends on the backup rule. A voting system is not fully described by the phrase “majority rule”; we also need to know how alternatives are compared and what happens when no option wins.
Write the three voters' rankings in a table. Compare each pair of alternatives by majority vote. Then identify whether a voter can change the outcome by voting for something other than their first choice.
The lecture introduces Condorcet, Borda, Lewis Carroll, Arrow, Riker, Sen, and Duncan Black, then develops strategic voting, majority rule, the median voter theorem, and value restriction.
Check your understanding
Why does the voting rule matter?
Additional Topics
The final lecture asks what changes when the standard assumptions of game theory become too strong. A player may have limited computational power, limited foresight, limited knowledge of the game, or little willingness to use a complicated strategy.
Bounded rationality does not mean abandoning models. It means modelling the limits of the decision-maker. The lecture also introduces algorithmic game theory, evolutionary game theory, and agent-based modelling.
From the course to your next model
The MIT course has moved from preferences to strategic form, from pure to mixed strategies, from static to dynamic games, from complete to incomplete information, and finally to collective choice and bounded rationality.
Choose one political situation. Name the players, their actions, what each knows, the timing, the payoff logic, and the rule that turns individual actions into an outcome. Then identify the assumption you would relax first: complete information, unlimited foresight, or sincere voting.
Read further
- Full text preserved locally: Magazinnik's nine lecture PDFs and seven problem sets.
- Named textbook: Robert Gibbons, Game Theory for Applied Economists.
- Named applications: Fearon's “Rationalist Explanations for War,” Maskin and Tirole's “The Politician and the Judge,” and Grossman and Helpman's “Protection for Sale.”
This page is a readable synthesis of the preserved MIT course materials, not a replacement for the original lectures, proofs, or problem-set solutions. Works named by the course are distinguished from documents preserved in the local archive.