Game Theory

Definition Action At An Information Set formalized

Definition Agent Normal Form staged

Definition Agent Normal Form Of A Bayesian Game staged

Theorem Antisymmetric Matrix Game Has Value Zero proved

Definition Approximate Solution staged

Proposition Approximate Solutions Are Reny Solutions staged

Theorem Aumann-Maschler Cav-u Theorem staged

Example Backward And Forward Induction Can Conflict staged

Definition Backward Induction Value formalized

Definition Balanced Coalitional Game staged

Definition Balanced Collection Of Coalitions staged

Definition Bayesian Equilibrium staged

Theorem Bayesian Equilibrium As Nash Equilibrium staged

Definition Bayesian Game staged

Definition Bayesian Strategy staged

Definition Behavioral Equilibrium staged

Definition Belief System staged

Definition Best Response admitted

Proposition Best Response Face staged

Definition Best-Response Invariance staged

Definition Better Reply Secure Game staged

Theorem Bewley-Kohlberg Asymptotic Value staged

Theorem Big Match Uniform Value staged

Theorem Blackwell B-Set Approachability staged

Theorem Bondareva-Shapley Theorem staged

Lemma Bracket Invariant For Admissible Sequences admitted

Definition Calibrated Strategy staged

Theorem Calibrated Strategy Exists staged

Definition Canonical Correlation Device staged

Lemma Cesàro Payoff Convergence From Robinson Lemma admitted

Theorem Chomp Strategy Stealing staged

Example Common Knowledge Can Filter Nash Equilibria staged

Proposition Compact Mixed Versus Finite-Support Minimax admitted

Theorem Completely Mixed Nash Is Subgame Perfect staged

Proposition Continuous Fictitious Play Closes The Duality Gap admitted

Definition Convex Coalitional Game staged

Definition Convex Game staged

Proposition Convex Game Nash Variational Characterization staged

Theorem Convex Games Have Nonempty Core staged

Definition Core Of A Coalitional Game staged

Theorem Core Payoffs Are Imputations staged

Example Correlated Best Response Need Not Be Independent staged

Definition Correlated Equilibrium staged

Theorem Correlated Equilibrium Exists In Finite Games staged

Theorem Correlated Equilibrium Set Is Convex And Compact staged

Definition Correlated Strategy staged

Proposition Cournot Duopoly Equilibrium From Supermodularity staged

Proposition Cournot Fishermen Equilibria staged

Theorem Degenerate Correlated Equilibrium staged

Definition Derived Game admitted

Definition Determined Game formalized

Theorem Deterministic Blackwell Sequence staged

Proposition Diagonal Matrix Game Value proved

Proposition Directional Derivative Of The Value admitted

Theorem Discounted Folk Theorem staged

Definition Discounted Repeated Game staged

Theorem Discounted Value Is the Unique Shapley Fixed Point staged

Definition Discounted Value of a Zero-Sum Stochastic Game staged

Theorem Dominant Strategy Profile is a Nash Equilibrium admitted

Definition Dominated Strategy staged

Definition Duality Gap In A Matrix Game formalized

Theorem ESS Replicator Lyapunov Function staged

Proposition ESS Uniform Invasion Barrier staged

Definition Empirical Frequency Of Play formalized

Example Entrant Incomplete Information Value Example staged

Definition Epsilon Equilibrium staged

Definition Epsilon Perfect Equilibrium staged

Definition Epsilon Proper Equilibrium staged

Definition Epsilon-Optimal Strategy formalized

Definition Equilibrium Manifold staged

Definition Essential Equilibrium Component staged

Lemma Euclidean Division Pasting In Robinson Induction admitted

Definition Evaluation Game Equilibrium staged

Definition Evolutionarily Stable Strategy staged

Theorem Existence Of Behavioral Equilibrium staged

Theorem Existence Of Perfect Equilibrium staged

Theorem Existence Of Proper Equilibrium staged

Theorem Existence Of Reny Solutions staged

Theorem Existence Of Sequential Equilibrium staged

Theorem Existence Of Subgame-Perfect Equilibrium With Perfect Recall staged

Theorem Existence of Optimal Mixed Strategies (Loomis Foundations) proved

Definition Expected Payoff of a Matrix Game formalized

Definition Extensive Form Perfect Equilibrium staged

Definition Extensive Game With Imperfect Information formalized

Definition Extensive-Game Theorems Catalog staged

Definition External Regret staged

Definition Feasible And Individually Rational Payoffs staged

Definition Fictitious Play formalized

Theorem Fictitious Play Convergence admitted

Theorem Finite Bayesian Equilibrium Exists staged

Definition Finite Extensive Game With Perfect Information formalized

Definition Finite Game Catalog staged

Definition Finite Game-Tree Strategy Induces An Outcome formalized

Example Finite Nash Equilibrium Computation Examples staged

Proposition Finite-Opponent Separation Minimax admitted

Theorem Finitely Repeated Folk Theorem staged

Definition Finitely Repeated Game staged

Theorem Fink Nash Existence in Stochastic Games staged

Theorem First-Order Nash Condition staged

Definition Focal Point staged

Theorem Folk Theorem (Baseline Statement) staged

Definition Forward Induction staged

Definition Game Tree formalized

Theorem General Mixed Minimax Theorem admitted

Definition General Mixed Strategy Space admitted

Definition General Value Operator admitted

Proposition General Value Operator Is Nonexpansive admitted

Theorem Generic Finite Odd Equilibria staged

Proposition Generic Uniqueness Of Subgame-Perfect Equilibrium staged

Definition Hannan Set staged

Definition History And Subgame formalized

Definition Imputation staged

Definition Information Set staged

Proposition Information Set Refinement And Equilibrium staged

Definition Internal Regret staged

Lemma Intersection Lemma admitted

Theorem Isbell Behavioral-To-Mixed Equivalence staged

Definition Iterated Elimination Of Strictly Dominated Strategies staged

Theorem Kakutani From Two-Player Nash Existence staged

Theorem Kohlberg Mertens Structure Theorem staged

Theorem Kuhn Existence Of Pure Subgame-Perfect Equilibrium (No-Chance) proved

Theorem Kuhn Existence Of Pure Subgame-Perfect Equilibrium (With Chance) staged

Theorem Kuhn Mixed-To-Behavioral Equivalence staged

Definition Linear Extensive Game staged

Theorem Local Characterization Of Behavioral Equilibrium staged

Definition Marginal Contribution staged

Lemma Markov Stationary Distribution as Zero-Sum Game Value admitted

Example Matching Pennies formalized

Definition Matrix Game formalized

Definition Maximin and Minimax Values formalized

Lemma Maximin is Bounded by Minimax proved

Definition Mediated Game staged

Definition Mertens Stable Sets staged

Theorem Mertens-Neyman Uniform Value staged

Definition Mixed Behavioral And General Strategies staged

Theorem Mixed Equilibrium In Compact Continuous Games staged

Definition Mixed Extension Of A Matrix Game formalized

Definition Mixed Extension Of A Strategic Game staged

Example Mixed Extension Pathology admitted

Definition Mixed Nash Equilibrium staged

Theorem Mixed Nash Equilibrium of a Matrix Game proved

Definition Mixed Strategy staged

Definition Mixed Strategy Simplex formalized

Proposition Monotone Decreasing Values Limit admitted

Example Monotone Family Assumption Counterexamples admitted

Definition Monotone Smooth Game staged

Definition Monotonic Coalitional Game staged

Theorem Nash Characterization And Uniqueness In Monotone Games staged

Proposition Nash Equilibria Are Rationalizable staged

Definition Nash Equilibrium admitted

Theorem Nash Equilibrium Induces Correlated Equilibrium staged

Theorem Nash Existence For Finite Games proved

Theorem Nash Existence In Better Reply Secure Games staged

Theorem Nash Existence In Compact Quasi-Concave Games staged

Theorem Nash Existence In Convex Games staged

Theorem Nash Existence In Supermodular Games staged

Definition Nash Field staged

Theorem Nash Payoff Uniqueness in Zero-Sum Games proved

Proposition Nash Payoffs Are Individually Rational staged

Definition Nash Wardrop Equilibrium staged

Definition Nature In Extensive Games staged

Theorem No External Regret Converges To Hannan Set staged

Theorem No Internal Regret Converges To Correlated Equilibrium staged

Theorem Noisy Duel With Multiple Bullets admitted

Proposition Noisy One-Bullet Duel Value admitted

Definition Non-Player-Controlled Nonterminal State formalized

Example Noncontractible Feasible Payoff Example staged

Example Normal And Extensive Perfection Can Differ staged

Proposition Normal Form Invariance Of Extensive Representations proved

Definition Normal Form Reduction formalized

Theorem Obedience Condition staged

Proposition One-Sided Compact Convex Minimax admitted

Theorem Optimal Pairs Are Exactly Saddle Points proved

Definition Optimal Strategy Sets formalized

Proposition Optimal Strategy Sets Are Polytopes proved

Lemma Payoff Negation in Zero-Sum Games proved

Definition Payoff Vector staged

Definition Perfect Equilibrium staged

Proposition Perfect Equilibrium And Undominated Equilibrium staged

Definition Perfect Recall staged

Theorem Perron-Frobenius For Positive Matrices proved

Definition Player Guarantee formalized

Example Poker Game Value Example staged

Definition Population Game staged

Definition Positive Correlation Dynamics staged

Definition Potential Game staged

Theorem Potential Is Lyapunov For Replicator Dynamics staged

Theorem Potential Maximizer Is Nash staged

Example Prisoner's Dilemma admitted

Example Proper And Perfect Equilibrium Examples staged

Definition Proper Equilibrium staged

Theorem Proper Equilibrium Induces Sequential Equilibrium staged

Definition Prudent Strategy staged

Definition Pure Strategy With Imperfect Information formalized

Definition Quasi-Concave Game staged

Definition Quasi-Concavity And Semicontinuity admitted

Proposition Rationalizable Set As Largest Best-Response Fixed Set staged

Definition Rationalizable Strategy staged

Definition Reached Information Set staged

Theorem Reached Subgame Nash Restriction staged

Definition Recommendation Strategy staged

Definition Reny Solution staged

Definition Repeated Feasible Individually Rational Payoffs staged

Definition Repeated Game staged

Definition Repeated Game History staged

Definition Repeated Game Payoff Aggregation staged

Definition Repeated Game Strategy staged

Definition Repeated Game With Incomplete Information staged

Definition Repeated Game With Signals staged

Definition Replicator Dynamics staged

Lemma Robinson Admissible Sequence Lemma admitted

Theorem Rubinstein Bargaining Subgame Perfect Equilibrium staged

Definition Saddle Point formalized

Proposition Semi-Algebraic Mixed Nash Set staged

Definition Semi-Reduced Normal Form formalized

Definition Sequential Equilibrium staged

Definition Shapley Axioms staged

Definition Shapley Operator staged

Theorem Shapley Operator Is a γ-Contraction staged

Theorem Shapley Uniqueness Theorem staged

Definition Shapley Value staged

Theorem Shapley Value Is Efficient staged

Theorem Shapley Value Lies In The Core Of A Convex Game staged

Proposition Silent One-Bullet Duel Has No Pure Value admitted

Proposition Silent Versus Noisy One-Bullet Duel Value admitted

Definition Simple Game staged

Definition Simple Perfect-Information Game formalized

Example Sion Boundary Counterexample admitted

Theorem Sion Minimax Theorem admitted

Example Sion-Wolfe Game Has No Mixed Value admitted

Definition Smooth Game staged

Definition Solvable Game staged

Proposition Solvable Games Have A Unique Nash Equilibrium staged

Theorem Stable Game Framework (Hofbauer–Sandholm) staged

Definition Standard Repeated Game staged

Definition Stochastic Game staged

Definition Stochastic Game History staged

Theorem Stochastic Matrix Has An Invariant Distribution proved

Definition Strategic Game admitted

Definition Strategic Stability staged

Definition Strategy In A Perfect-Information Game formalized

Definition Strategy Profile admitted

Definition Strategy Profile And Induced Outcome staged

Definition Strict Dominance admitted

Proposition Strict Domination And Never Best Response (via auxiliary zero-sum game) admitted

Definition Strict Equilibrium staged

Proposition Strict Equilibrium Is Proper staged

Theorem Strictly Competitive Perfect-Information Determinacy proved

Definition Strictly Dominant Strategy admitted

Theorem Strong Complementarity proved

Theorem Subgame Perfect Uniform Folk Theorem staged

Lemma Subgame Reduction For Non-Useful Indices admitted

Proposition Subgame-Perfect Equilibrium Implies Nash Equilibrium proved

Definition Subgame-Perfect Equilibrium In Imperfect-Information Games staged

Definition Subgame-Perfect Equilibrium In Perfect-Information Games formalized

Definition Subroot And Subgame In Imperfect-Information Games formalized

Definition Superadditive Coalitional Game staged

Proposition Supermodular Best Responses Are Lattices staged

Definition Supermodular Game staged

Theorem Support Complementarity proved

Definition Support Of A Mixed Strategy formalized

Definition Symmetric Game staged

Theorem Symmetric Mixed Equilibrium staged

Definition Threat Point staged

Example Three-By-Two Matrix Game Computation formalized

Example Three-Player Minimax Failure formalized

Proposition Three-Player Minority Game Equilibria staged

Definition Transferable-Utility Coalitional Game staged

Example Two-By-Two Value Formula formalized

Definition Uniform Equilibrium staged

Theorem Uniform Folk Theorem staged

Definition Unilateral Deviation admitted

Lemma Useful Window Bound For Admissible Sequences admitted

Theorem Value In Finite Zero-Sum Perfect-Information Games (No-Chance) proved

Theorem Value In Finite Zero-Sum Perfect-Information Games (With Chance) staged

Definition Value Of A Zero-Sum Game formalized

Proposition Value Operator Properties admitted

Theorem Von Neumann Minimax Theorem proved

Proposition Wardrop Variational Characterization staged

Definition Weak Bayesian Perfect Equilibrium staged

Definition Weak Dominance admitted

Proposition Weak Domination And Completely Mixed Tests (via auxiliary zero-sum game) admitted

Definition Weakly Dominant Strategy admitted

Theorem Zermelo Determinacy For Finite Perfect-Information Games proved

Definition Zero-Sum Game formalized