Draft, December 2013 CONTINUOUS TIME GAME THEORY: AN INFINITESIMAL APPROACH MAXWELL B. STINCHCOMBE 1 Contents 1. Introduction and Overview . 6 1.1. Equilibrium Refinement via Infinitesimals. 6 1.2. Compact and Continuous Normal Form Games . 7 1.2.1. Limit games and limit equilibria . 8 1.2.2. The need for exhaustiveness . 8 1.2.3. Respecting weak dominance . 8 1.3. Extensive Form Games with Infinite Action Sets . 8 1.4. Control Problems and Differential Games . 8 1.4.1. An Easy Control Problem . 8 1.4.2. A Less Easy Control Problem. 9 1.4.3. Differential Games. 10 1.5. Continuous Time Stochastic Processes and Diffuse Monitoring . 10 1.5.1. Brownian monitoring . 10 1.5.2. Poisson monitoring . 10 1.5.3. Levy processes and monitoring . 11 1.6. Continuous Time Game Theory with Sharp Monitoring . 11 1.7. Large Population Games. 11 2. Constructing Infinitesimals and the Tools to Use Them . 12 2.1. A Purely Finitely Additive Point Mass. 12 2.2. The Equivalence Classes . 13 2.3. Normal Form Equilibrium Refinement . 13 2.3.1. Notation . 13 2.3.2. Perfection . 14 2.3.3. Properness . 14 2.3.4. p-Stability . 14 2.4. The Basic Results . 14 2.5. Compact and Continuous Games . 15 3. Extensive Form Games . 15 3.1. Decision Theory with Full Support Probabilities . 15 3.2. Dynamic Decisions, the Basic Model . 15 3.3. Bridge Crossing . 16 3.4. Heirarchies of Beliefs . 16 3.5. Extensive Form Equilibrium Refinement . 17 3.5.1. Perfect, proper, and stable equilibrium outcomes. 17 3.5.2. Iterative dominance arguments . 17 4. Continuous Time Control and Games . 17 4.1. Control Theory . 17 4.1.1. When it Works Wonderfully . 17 4.1.2. When it Works Less Well . 17 4.2. Games with Instantaneous Monitoring . 18 4.2.1. (R; ≤) is Totally Ordered but Not Well-Ordered . 18 4.2.2. Implications . 18 4.3. Games Played on a Near Interval . 19 2 4.3.1. Actions, Histories, Strategies, and Outcomes . 20 4.3.2. Safety in Continuously Repeated Games . 21 4.3.3. Some Examples. 22 4.3.4. Revisiting Cournot vs. Bertrand . 23 4.3.5. Preemption Games . 23 4.3.6. Wars of Attrition . 25 4.4. Brownian Monitoring . 28 4.5. Poisson Monitoring . 28 4.6. Continuous Time Martingales. 28 4.7. It^o's Lemma . 28 5. Standard and Nonstandard Superstructures . 28 5.1. Purely Finitely Additive Point Masses . 28 ∗ 5.2. The equivalence relation ∼µ and X .................................. 29 5.3. Superstructures. 29 5.4. Defining V (∗S) inductively . 29 5.5. Internal Sets for Stochastic Processes . 30 5.6. Some External Sets . 30 5.7. Statements and the Transfer Principle . 31 6. Some Real Analysis. 32 6.1. Closed Sets and Closure . 32 6.1.1. The standard part mapping. 32 6.1.2. Closedness of Refined Sets of Equilibria. 32 6.2. Continuity and Uniform Continuity . 32 6.2.1. C(X; R), X compact . 32 6.2.2. Near continuity. 32 6.2.3. The Riemann-Stieltjes Integral. 32 6.2.4. Some near interval control theory . 32 6.3. Theorem of the Maximum . 32 6.3.1. In Control Theory . 32 6.3.2. Single person problems . 32 6.3.3. Limit games and limit equilibria . 32 6.4. Compactness . 32 6.4.1. Existence of optima . 32 6.4.2. Existence of equilibria . 32 6.4.3. Existence of extened equilibrium outcomes. 33 6.4.4. Compact sets of probabilities on R................................ 33 6.5. Probabilities on Metric Spaces . 33 6.5.1. Loeb Measures . ..