1
Mark a bimatrix
Read a two-player table, mark each player's best response to each move of the other, and name the cells where both marks fall.
Course · Philosophy
Games and the Social Contract
Take a social-contract argument and write it as a game. Read a payoff table, find every cell where no one gains by moving alone, see why the cell can be worse for everyone than another, and compute exactly what changes it: a repeated game, a sovereign's penalty, a fee, a change in who meets whom.
Nash: no player gains by moving alone
dilemma: each defects, both do worse
convention: stable because others conform
GRIM holds if δ ≥ (T − R) / (T − P)
commons: the gain is mine, the cost is shared
replicator: shares follow relative payoff
What this course is about
- Subject
- Philosophy
- Lessons
- 11
- Format
- Interactive
Overview
What you will be able to do
By the end you can take a situation of interdependent choice, write it as a table or a pair of payoff expressions, and compute what follows from it.
2
Show that a game is a dilemma
Establish that a strategy strictly dominates, locate the equilibrium it leads to, and show that another cell pays every player more.
3
Compute an equilibrium and a threshold
Solve a two-by-two game for its mixed equilibrium exactly, and read the same number as the belief at which a risky choice is worth taking.
4
Derive the discount factor
Compute the continuation probability above which cooperation is stable against a punishing partner, and state why a known last round removes it.
5
Model the state of nature and the commons
Write Hobbes's argument, the herders' commons and the public-goods game as games, find each equilibrium and optimum, and show what a penalty or a fee changes.
6
Run the replicator step
Update a population's share of cooperators generation by generation in exact fractions, and say where it ends up.
Lessons
11 lessons
One-shot games come first, then the same dilemma repeated, then many players at once.
Strategic Form and Best ResponsesA game is a table of payoffs, and a best response is the reply that pays most against one particular move.Not marked
The Prisoner's DilemmaDefecting is the better move whatever the other does, and two players who follow it both do worse.Not marked
Nash Equilibrium in Pure and Mixed StrategiesAn equilibrium is a pair of strategies, each a best response to the other, and where no cell qualifies the pair may randomise.Not marked
Coordination, Conventions and the Stag HuntWhen several equilibria pay, what makes people land on one is a shared expectation, and the belief it takes can be computed.Not marked
Repeated Games and ReciprocityPlayed again and again, the dilemma rewards strategies that answer in kind, and a tournament shows which ones score.Not marked
The Shadow of the FutureCooperation is stable when the future counts for enough, and the amount can be computed from the four payoffs.Not marked
Hume's Farmers and ConventionTwo farmers who would each gain from the other's help refuse in one season and help over many, with no promise between them.Not marked
Hobbes and the State of NatureWithout a sovereign, two rational and uncertain people attack each other, and a penalty on attacking changes the equilibrium.Not marked
The Tragedy of the CommonsEach herder gains from one more animal while the cost falls on all, so every herder adds and all are left worse off.Not marked
Public Goods and Free-RidingA contribution that benefits everyone returns little to the contributor, so each does better by letting the others pay.Not marked
The Evolution of CooperationIn a population, a type spreads when it out-earns the other type, and what spreads need not be good for the group.Not marked
Recommended background
Decision and Rationality, for the idea that a payoff stands for what an outcome is worth and that a player prefers the larger expected one, and nothing else.
Practice suggestions
Work forward. Each lesson uses the table-reading of “Strategic Form and Best Responses” and the idea of an equilibrium from “Nash Equilibrium in Pure and Mixed Strategies”. The repeated games in “Repeated Games and Reciprocity” make no sense without the one-shot dilemma in “The Prisoner's Dilemma”.
Change the payoffs. Every lab here lets you retype the table, and most of the course is a claim that one number matters or does not: the temptation in the dilemma, the crash in chicken, the hare in the stag hunt, the fee on a commons. Make the claim false and watch which tile says so.
Keep the payoffs and the people apart. A game is a table of numbers, and which numbers describe a real pair of hunters, drivers or states is a judgment the lab cannot make. The arguments on this course are conditional: if the payoffs are these, then this follows.
Read the quizzes for the reason, not the answer. Each wrong choice is the answer someone gives from one of the course's standing misconceptions, and the explanation says which one.
What it does not cover
Games played in turns. Every game here is in strategic form, with the players choosing at once. Trees, backward induction as a general method, subgame perfection and credible threats are not built; the only backward argument is the unravelling of a known last round.
Games of incomplete information. The players know the payoffs and each other's strategies. Bayesian games, signalling, and reputation built on doubt about a partner's type are left out.
Cooperative game theory, bargaining and mechanism design. Shapley values, the Nash bargaining solution and the design of rules to produce a chosen outcome are different subjects; the course stops where a sovereign's penalty changes the payoffs.
Proofs. Nash's theorem that a finite game has a mixed equilibrium is stated and the equilibria are computed, but the theorem is not proved, and the replicator step is given as a rule rather than derived from a model of selection.
The history of social contract theory as history. Hobbes and Hume appear as arguments, written as games and tested; what they said, to whom, and why is not here, and nor is the experimental evidence on how real people play these games.