Notation & Glossary
\[ \newcommand{\E}{\mathbb{E}} \newcommand{\R}{\mathbb{R}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\BR}{\operatorname{BR}} \newcommand{\eps}{\varepsilon} \newcommand{\given}{\,\vert\,} \newcommand{\argmax}{\operatorname*{arg\,max}} \newcommand{\argmin}{\operatorname*{arg\,min}} \newcommand{\sm}{\setminus} \newcommand{\defeq}{\equiv} \]
A quick reference for the symbols and named concepts used throughout the book.
Symbols
| Symbol | Meaning |
|---|---|
| \(N=\{1,\dots,n\}\) | set of players |
| \(i,\ -i\) | a generic player; “everyone except \(i\)” |
| \(S_i,\ s_i\) | player \(i\)’s strategy space; a strategy |
| \(S_{-i},\ s_{-i}\) | opponents’ strategy space; a profile of opponents’ strategies |
| \(s=(s_i,s_{-i})\) | a strategy profile |
| \(v_i(\cdot)\) | player \(i\)’s payoff function |
| \(\Gamma=(N,\{S_i\},\{v_i\})\) | a normal-form game |
| \(\Delta S_i,\ \sigma_i\) | the simplex of mixed strategies over \(S_i\); a mixed strategy |
| \(\operatorname{supp}\sigma_i\) | the support of \(\sigma_i\) |
| \(\BR_i(s_{-i})\) | player \(i\)’s best-response correspondence |
| \(\E[\cdot]\) | expectation (expected payoff) |
| \(\Prob\) | a probability distribution / common prior |
| \(\succ\) | the precedence relation on nodes of a game tree |
| \(H_i,\ h_i\) | player \(i\)’s collection of information sets; one information set |
| \(A_i(x)\) | actions available to the mover at node \(x\) |
| \(z(s)\) | the terminal node (outcome) induced by profile \(s\) |
| \(\delta\in(0,1)\) | discount factor in repeated/dynamic games |
| \(\Theta_i,\ \theta_i\) | player \(i\)’s type space; a type |
| \(\phi_i(\theta_{-i}\given\theta_i)\) | \(i\)’s posterior belief over others’ types |
| \(\mu\) | a system of beliefs over nodes within information sets |
| \(F,\ f\) | a type/value distribution and its density (auctions) |
Solution concepts at a glance
| Complete information | Incomplete information | |
|---|---|---|
| Static | Nash equilibrium (2 Static Games with Complete Information) | Bayesian Nash equilibrium (4 Static Games with Incomplete Information) |
| Dynamic | Subgame-perfect equilibrium (3 Dynamic Games with Complete Information) | Perfect Bayesian equilibrium (5 Dynamic Games with Incomplete Information) |
Each concept down or across the table is a refinement of the one before it, introduced to rule out predictions that the cruder concept cannot: incredible threats (handled by subgame perfection) and arbitrary off-path beliefs (handled by perfect Bayesian equilibrium and its refinements).
Glossary
Static games, complete information (2 Static Games with Complete Information)
- Normal-form game — players, strategy spaces, payoff functions.
- Strictly dominated / dominant strategy — a strategy that does strictly worse / strictly better than an alternative against every opponent profile.
- Strictly dominant strategy equilibrium — every player plays a dominant strategy (e.g. \((C,C)\) in the Prisoner’s Dilemma).
- Pareto optimal — no other profile makes someone better off without making someone worse off.
- IESDS — iterated elimination of strictly dominated strategies; relies on common knowledge of rationality; order-independent for finite games.
- Best response — a payoff-maximising reply to a fixed opponent profile.
- Nash equilibrium (Definition 2.8) — a profile in which everyone best-responds to everyone else; equivalently, no player has a profitable deviation.
- Mixed strategy — a probability distribution over pure strategies; the indifference principle (Lemma 2.2) characterises mixed best responses. Every finite game has a (possibly mixed) equilibrium (Theorem 2.1).
- Cournot / Bertrand / Hotelling — quantity competition / price competition / spatial (electoral) competition; the last yields the median voter theorem.
Dynamic games, complete information (3 Dynamic Games with Complete Information)
- Extensive form / game tree — moves, information, and payoffs in sequence.
- Information set — decision nodes a player cannot tell apart; singletons mean perfect information.
- Perfect recall — no player forgets what they once knew.
- Pure / mixed / behavioral strategy — a contingent plan over information sets; a randomisation over plans; independent randomisation at each information set (Kuhn’s theorem links the last two).
- Backward induction — solving a finite perfect-information game from its end.
- Subgame; subgame-perfect equilibrium (Definition 3.11) — a profile that is a Nash equilibrium in every subgame; rules out non-credible threats.
- One-shot deviation principle (Theorem 3.6) — to check subgame perfection it suffices to rule out single-stage profitable deviations.
- Repeated game — a stage game played repeatedly; the discount factor \(\delta\) measures patience. Grim-trigger strategies sustain cooperation when players are patient enough (folk-theorem logic).
- Rubinstein bargaining — alternating offers over an infinite horizon; a unique stationary subgame-perfect equilibrium with shares \(\tfrac{1}{1+\delta}\).
Static games, incomplete information (4 Static Games with Incomplete Information)
- Bayesian game (Definition 4.1) — players have private types drawn from a common prior; strategies are type-contingent.
- Bayesian Nash equilibrium (Definition 4.3) — every type best-responds given its posterior beliefs; equivalently, a Nash equilibrium of the Harsanyi extensive-form game with a move by Nature.
- Adverse selection — private information about quality unravels trade (the market for lemons).
- Pivotal voter — in committee voting, a rational juror conditions on being decisive, so voting one’s own signal need not be an equilibrium.
- Purification — a mixed equilibrium reinterpreted as the limit of pure-strategy Bayesian equilibria of nearby games with small private shocks.
- Auctions — second-price/English (truthful bidding weakly dominant); first-price/Dutch (shade your bid); revenue equivalence (standard formats raise the same expected revenue); winner’s curse (winning is bad news about a common value).
Dynamic games, incomplete information (5 Dynamic Games with Incomplete Information)
- System of beliefs (Definition 5.2) — probabilities over the nodes within each information set.
- Perfect Bayesian equilibrium (Definition 5.3) — strategies sequentially rational given beliefs, with on-path beliefs derived by Bayes’ rule.
- Signaling game — an informed player takes a costly action that may credibly reveal type; separating (types choose distinct signals) vs pooling (types choose the same signal) equilibria. Spence’s education model: schooling signals ability even when unproductive, because imitation is too costly for the low type.
- Intuitive criterion — a refinement (Cho–Kreps) that disciplines off-path beliefs and kills implausible pooling equilibria.
- Cheap talk — costless, non-binding messages (Crawford–Sobel); informative communication is possible only when the sender’s and receiver’s interests are close enough (bias \(b<\tfrac14\) for a two-message equilibrium in the canonical uniform-quadratic model).