Game Theory

Players, actions, payoffs and the rationality assumptions; why optimisation against a responding opponent is a different problem.

Payoff matrices, strategy profiles, and reading a bimatrix game correctly.

Assumes: What Game Theory Studies

Strict and weak dominance, iterated elimination, and best-response correspondences.

Assumes: Normal-Form Games

The definition, finding pure-strategy equilibria by inspection, multiplicity, and what the concept does and does not predict.

Assumes: Dominance and Best Response

Randomising over actions, expected payoffs, the indifference principle, and computing mixed equilibria by hand.

Assumes: Nash Equilibrium · Expectation

Why every finite game has at least one equilibrium, via Brouwer and Kakutani fixed-point arguments.

Assumes: Mixed Strategies

Security levels, the value of a game, and von Neumann's theorem that maximin equals minimax in mixed strategies.

Assumes: Mixed Strategies

Solving a game by LP, and how strong duality *is* the minimax theorem — two results that turn out to be the same statement.

Assumes: Zero-Sum Games and the Minimax Theorem · Linear Programming Duality

When individually rational play is collectively disastrous; public goods, tragedy of the commons and free riding.

Assumes: Nash Equilibrium

Multiple equilibria, focal points, risk versus payoff dominance, and the stag hunt.

Assumes: The Prisoner's Dilemma and Social Dilemmas

Game trees, sequential moves, information sets, and the difference between perfect and imperfect information.

Assumes: Nash Equilibrium

Solving trees from the leaves up, subgame-perfect equilibrium, credible versus empty threats, and the centipede paradox.

Assumes: Extensive-Form Games

Finite versus infinite horizons, discounting, trigger and tit-for-tat strategies, and the folk theorem.

Assumes: Backward Induction and Subgame Perfection

Types, beliefs and Bayesian Nash equilibrium when players do not know each other's payoffs.

Assumes: Repeated Games and Cooperation · Bayes' Theorem

Costly signals, separating and pooling equilibria, adverse selection and the market-for-lemons result.

Assumes: Bayesian Games and Incomplete Information

Designing the rules rather than playing the game: incentive compatibility, the revelation principle and VCG mechanisms.

Assumes: Signalling and Screening

First-price, second-price, English and Dutch auctions; why second-price is truthful, and revenue equivalence.

Assumes: Mechanism Design

Coalitions, characteristic functions, superadditivity, and which allocations are stable.

Assumes: Nash Equilibrium

Four axioms that pin down a unique fair allocation, the permutation formula, and a full worked computation — the foundation of SHAP.

Assumes: Cooperative Games and the Core · Counting, Permutations and Combinations

Populations instead of rational agents: evolutionarily stable strategies and replicator dynamics.

Assumes: Repeated Games and Cooperation

Allowing a shared signal, why the set is larger than Nash and easier to compute, and the traffic-light example.

Assumes: Mixed Strategies

Regret matching and multiplicative weights, and the result that no-regret play converges to correlated equilibrium.

Assumes: Correlated Equilibrium · Stochastic Optimisation

Stochastic games as the multi-agent generalisation of MDPs, minimax-Q, self-play, and why convergence guarantees mostly disappear.

Assumes: Zero-Sum Games and the Minimax Theorem · No-Regret Learning and Equilibrium

GAN training as a zero-sum minimax problem, adversarial examples as a constrained attacker's best response, and robust optimisation.

Assumes: Zero-Sum Games and the Minimax Theorem · Projected Gradient and Constrained Descent

Incentives in recommendation and ad auctions, multi-agent LLM debate and negotiation, collusion risk, and designing rules agents cannot exploit.

Assumes: Auction Theory · Markov Games and Multi-Agent Learning