Glossary — #equilibria
Glossary
Compact definitions of the recurring formal terms. Where a tutorial gives the long form, the entry here links to it.
A
Action. The atomic choice available to a player at a decision point. Distinct from strategy (which is a complete plan over all decision points).
Adversarial setting. Any game in which players’ interests are at least partly opposed; the limiting case is zero-sum.
Anonymity (axiom). Property of a solution concept (notably Shapley value): the value depends only on the structure of the characteristic function, not on the labels of players.
ARIMA / AR / MA. Time-series models: autoregressive, moving-average, and combined integrated forms. Used in Time Series & Econometrics.
Auction (sealed-bid). Mechanism in which bidders submit private bids; the seller allocates and prices according to a rule (first-price, second-price/Vickrey, all-pay, …).
B
Backward induction. Method for solving finite extensive-form games with perfect information: solve the last subgame, fold its value into the parent, repeat. Yields a subgame-perfect equilibrium.
Bayesian game. Game of incomplete information in which players have private types. Strategies map type to action; equilibrium concept is Bayesian Nash.
Bayesian Nash equilibrium. Strategy profile \((\sigma_i^*(\theta_i))\) such that each player’s strategy is a best response in expectation over other players’ types, given common-prior beliefs.
Best response. Action (or mixed strategy) that maximises own expected payoff given a belief about others’ play. A Nash equilibrium is a mutual best response.
Bimatrix game. Two-player normal-form game with separate payoff matrices for each player. Special case of an \(n\)-player game; the standard pedagogical setup.
C
Characteristic function \(v: 2^N \to \mathbb{R}\). Maps each coalition \(S \subseteq N\) to the total payoff that coalition can guarantee in a cooperative game.
Cheap talk. Pre-play communication that is non-binding and payoff-irrelevant. May still influence equilibrium selection (Crawford-Sobel).
Coalition. Any subset of players in a cooperative game. The grand coalition is \(N\).
Common knowledge. A fact is common knowledge among a group when everyone knows it, everyone knows that everyone knows it, and so on ad infinitum.
Common prior. Assumption that players’ beliefs about uncertain states derive from a single shared probability distribution (Harsanyi).
Cooperative game theory. Branch in which binding agreements between players are permitted; solution concepts focus on which payoffs (or imputations) the grand coalition can sustain.
Correlated equilibrium. Aumann’s generalisation of Nash: a joint distribution over actions such that conditional best-responses hold. Computable via LP.
D
Dominant strategy. A strategy that yields a weakly better payoff than every alternative, against every possible opponent play. Strictly dominant if strictly better.
DAG (Directed Acyclic Graph). The data structure for representing causal assumptions in Causal Inference.
DiD (Difference-in-Differences). Causal-inference estimator that contrasts treatment and control trajectories before and after intervention.
E
ESS (Evolutionarily Stable Strategy). Strategy that, when adopted by an entire population, cannot be invaded by a small fraction playing an alternative. Concept from Evolutionary Game Theory.
Expected utility. Linear functional \(\mathbb{E}[u(X)] = \sum_x p(x) u(x)\) over a lottery. Foundation of Decision Theory.
Extensive form. Game representation as a tree: nodes = decision points, edges = actions, leaves = payoffs.
F
Fictitious play. Learning dynamic in which each player best-responds to the empirical frequency of opponents’ past actions.
Folk theorem. Family of results stating that in infinitely repeated games with sufficiently patient players, virtually any individually rational payoff vector is sustainable in equilibrium.
G
GARCH. Generalised autoregressive conditional heteroskedasticity — volatility model from Time Series & Econometrics.
Grim trigger. Repeated-game strategy: cooperate until any defection, then defect forever after.
H
Hawk-Dove (Chicken). 2×2 coordination/conflict game with two pure NE plus a symmetric mixed NE. Canonical example of anti-coordination.
I
Imputation. Payoff vector \((x_1, \dots, x_n)\) that is individually rational (\(x_i \ge v(\{i\})\)) and Pareto-efficient (\(\sum x_i = v(N)\)).
Information set. In extensive-form games, the collection of nodes a player cannot distinguish between when choosing. Singleton information sets ⇒ perfect information.
Iterated elimination of strictly dominated strategies (IESDS). Recursive removal of strategies that are strictly dominated; the surviving profiles are the set of rationalisable strategies.
K
Knightian uncertainty. State of belief where probabilities cannot be assigned to outcomes (as opposed to risk, where they can).
M
Markov perfect equilibrium (MPE). Subgame-perfect equilibrium in a Markov game where players’ strategies depend only on the payoff-relevant state.
Mechanism design. Reverse engineering: given a social-choice objective and private types, design a game whose equilibrium implements the objective.
Minimax theorem. In any finite two-player zero-sum game, \(\max_\sigma \min_\tau u(\sigma, \tau) = \min_\tau \max_\sigma u(\sigma, \tau)\) (von Neumann 1928). Mixed strategies suffice.
N
Nash equilibrium. Strategy profile in which every player is best-responding to the others. Exists in mixed strategies for any finite game (Nash 1951).
No-regret learning. Online-learning algorithms whose average regret vs. the best fixed action vanishes. Empirical play converges to coarse correlated equilibrium.
Normal form. Game representation as a payoff matrix/tensor over the product of pure strategy sets.
P
Pareto efficient. Outcome from which no Pareto improvement (making one player better off without making another worse off) is possible.
Perfect Bayesian equilibrium (PBE). Strategy + belief system in an extensive-form game of incomplete information satisfying sequential rationality at every information set and Bayes-consistent belief updating wherever possible.
Potential game. Game admitting a function \(\Phi: S \to \mathbb{R}\) such that any unilateral deviation’s change in payoff equals the change in \(\Phi\). Pure NE exists trivially: argmax \(\Phi\).
Prospect theory. Descriptive theory of choice under risk (Kahneman-Tversky) that explains expected-utility violations via probability weighting and loss aversion.
Q
Quantal Response Equilibrium (QRE). Equilibrium concept relaxing strict best-response with logit-like noisy choice. Used heavily in Experimental Economics.
R
Rationalisability. Set of strategies that survive iterated elimination of strictly dominated strategies. Necessary but weaker than Nash.
Replicator dynamics. ODE describing strategy frequencies in an evolutionary population: \(\dot{x}_i = x_i (f_i(x) - \bar{f}(x))\).
Revenue equivalence. Under symmetric private-value auctions with risk-neutral bidders and standard assumptions, all auction formats yielding the same allocation produce the same expected revenue.
S
Sequential equilibrium. Refinement of PBE that strengthens belief consistency via convergent sequences of completely-mixed strategies.
Shapley value. Unique cooperative solution satisfying efficiency, symmetry, dummy, and additivity: \(\phi_i(v) = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!(n - |S| - 1)!}{n!} (v(S \cup \{i\}) - v(S))\).
Signalling game. Two-stage Bayesian game: an informed sender chooses an action observable by the receiver, who updates beliefs and acts.
Strategy. A complete contingent plan: for every information set at which a player might be called to act, specifies the action.
Subgame. A subtree of an extensive-form game that starts at a single decision node and contains all descendants, with the property that every information set involving those nodes is contained entirely within it.
T
Trembling-hand perfection. Equilibrium refinement: a profile is trembling-hand perfect if it is the limit of equilibria of perturbed games where every action has positive probability.
Type. Privately observed parameter encoding a player’s information in a Bayesian game; the distribution over types is common prior.
U
Utility. Real-valued representation of preferences over outcomes. Expected utility extends this to lotteries via the von Neumann-Morgenstern axioms.
V
Value of a zero-sum game. The unique payoff guaranteed by both players’ minimax strategies; equal to \(\max \min = \min \max\) by von Neumann.
VCG (Vickrey-Clarke-Groves) mechanism. Truthful direct mechanism for quasi-linear environments: charges each agent the externality they impose on others.
Z
Zero-sum game. Two-player game where payoffs sum to zero (or any constant). Equivalent to strictly competitive.