Game Theory: Penn State Math 486 Lecture Notes Version 2.1.1 Christopher Griffin « 2010-2021 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License With Major Contributions By: James Fan George Kesidis and Other Contributions By: Arlan Stutler Sarthak Shah Contents List of Figuresv Preface xi 1. Using These Notes xi 2. An Overview of Game Theory xi Chapter 1. Probability Theory and Games Against the House1 1. Probability1 2. Random Variables and Expected Values6 3. Conditional Probability8 4. The Monty Hall Problem 11 Chapter 2. Game Trees and Extensive Form 15 1. Graphs and Trees 15 2. Game Trees with Complete Information and No Chance 18 3. Game Trees with Incomplete Information 22 4. Games of Chance 24 5. Pay-off Functions and Equilibria 26 Chapter 3. Normal and Strategic Form Games and Matrices 37 1. Normal and Strategic Form 37 2. Strategic Form Games 38 3. Review of Basic Matrix Properties 40 4. Special Matrices and Vectors 42 5. Strategy Vectors and Matrix Games 43 Chapter 4. Saddle Points, Mixed Strategies and the Minimax Theorem 45 1. Saddle Points 45 2. Zero-Sum Games without Saddle Points 48 3. Mixed Strategies 50 4. Mixed Strategies in Matrix Games 53 5. Dominated Strategies and Nash Equilibria 54 6. The Minimax Theorem 59 7. Finding Nash Equilibria in Simple Games 64 8. A Note on Nash Equilibria in General 66 Chapter 5. An Introduction to Optimization and the Karush-Kuhn-Tucker Conditions 69 1. A General Maximization Formulation 70 2. Some Geometry for Optimization 72 3. Gradients, Constraints and Optimization 76 4. Convex Sets and Combinations 78 iii iv CONTENTS 5. Convex and Concave Functions 79 6. Karush-Kuhn-Tucker Conditions 80 7. Relating Back to Game Theory 83 Chapter 6. Zero-Sum Matrix Games with Linear Programming 85 1. Linear Programs 85 2. Intuition on the Solution of Linear Programs 86 3. A Linear Program for Zero-Sum Game Players 90 4. Matrix Notation, Slack and Surplus Variables for Linear Programming 93 5. Solving Linear Programs by Computer 95 6. Duality and Optimality Conditions for Zero-Sum Game Linear Programs 98 Chapter 7. Quadratic Programs and General Sum Games 105 1. Introduction to Quadratic Programming 105 2. Solving QP's by Computer 106 3. General Sum Games and Quadratic Programming 106 Chapter 8. Nash's Bargaining Problem and Cooperative Games 115 1. Payoff Regions in Two Player Games 115 2. Collaboration and Multi-criteria Optimization 119 3. Nash's Bargaining Axioms 122 4. Nash's Bargaining Theorem 123 Chapter 9. A Short Introduction to N-Player Cooperative Games 131 1. Motivating Cooperative Games 131 2. Basic Results on Coalition Games 132 3. Division of Payoff to the Coalition 133 4. The Core 134 5. Shapley Values 136 Appendix A. Utility Theory 139 1. Decision Making Under Certainty 139 2. Advanced Decision Making under Uncertainty 146 Bibliography 151 List of Figures 0.1 There are several sub-disciplines within Game Theory. Each one has its own unique sets of problems and applications. We will study Classical Game Theory, which focuses on questions like, \What is my best decision in a given economic scenario, where a reward function provides a way for me to understand how my decision will impact my result." We may also investigate Combinatorial Game Theory, which is interested in games like Chess or Go. If there's time, we'll study Evolutionary Game Theory, which is interesting in its own right. xiii 1.1 An (American) roulette wheel is shown above. A French roulette wheel lacks the 00 pocket. This image was obtained from http://www.math.uah.edu/stat/ games/Roulette.html.7 1.2 Example card counting strategies. This table is adapted from https: //en.wikipedia.org/wiki/Card_counting. 10 1.3 You are sitting at a Black Jack Table. The dealer holds a king and something. You hold a 7 and a King. Do you hit? 10 1.4 The Monty Hall Problem is a multi-stage decision problem whose solution relies on conditional probability. The stages of decision making are shown in the diagram. We assume that the prizes are randomly assigned to the doors. We can't see this step{so we've adorned this decision with a square box. We'll discuss these boxes more when we talk about game trees. You the player must first choose a door. Lastly, you must decide whether or not to switch doors having been shown a door that is incorrect. 12 2.1 Digraphs on 3 Vertices: There are 64 = 26 distinct graphs on three vertices. The increased number of edges graphs is caused by the fact that the edges are now directed. 16 2.2 Two Paths: We illustrate two paths in a digraph on three vertices. 16 2.3 Directed Tree: We illustrate a directed tree. Every directed tree has a unique vertex called the root. The root is connected by a directed path to every other vertex in the directed tree. 17 2.4 Sub Tree: We illustrate a sub-tree. This tree is the collection of all nodes that are descended from a vertex u. 18 2.5 Rock-Paper-Scissors with Perfect Information: Player 1 moves first and holds up a symbol for either rock, paper or scissors. This is illustrated by the three edges leaving the root node, which is assigned to Player 1. Player 2 then holds up a symbol for either rock, paper or scissors. Payoffs are assigned to Player 1 and 2 at terminal nodes. The index of the payoff vector corresponds to the players. 20 v vi LIST OF FIGURES 2.6 New Guinea is located in the south pacific and was a major region of contention during World War II. The northern half was controlled by Japan through 1943, while the southern half was controlled by the Allies. (Image created from Wikipedia (http://en.wikipedia.org/wiki/File:LocationNewGuinea.svg), originally sourced from http://commons.wikimedia.org/wiki/File: LocationPapuaNewGuinea.svg. 20 2.7 The game tree for the Battle of the Bismark Sea. The Japanese could choose to sail either north or south of New Britain. The Americans (Allies) could choose to concentrate their search efforts on either the northern or southern routes. Given this game tree, the Americans would always choose to search the North if they knew the Japanese had chosen to sail on the north side of New Britain; alternatively, they would search the south route, if they knew the Japanese had taken that. Assuming the Americans have perfect intelligence, the Japanese would always choose to sail the northern route as in this instance they would expose themselves to only 2 days of bombing as opposed to 3 with the southern route. 21 2.8 Simple tic-tac-toe: Players in this case try to get two in a row. 22 2.9 The game tree for the Battle of the Bismark Sea with incomplete information. Obviously Kenney could not have known a priori which path the Japanese would choose to sail. He could have reasoned (as they might) that there best plan was to sail north, but he wouldn't really know. We can capture this fact by showing that when Kenney chooses his move, he cannot distinguish between the two intermediate nodes that belong to the Allies. 24 2.10 Poker: The root node of the game tree is controlled by Nature. At this node, a single random card is dealt to Player 1. Player 1 can then decide whether to end the game by folding (and thus receiving a payoff or not) or continuing the game by raising. At this point, Player 2 can then decide whether to call or fold, thus potentially receiving a payoff. 26 2.11 Reduced Red Black Poker: We are told that Player 1 receives a red card. The resulting game tree is substantially simpler. Because the information set on Player 2 controlled nodes indicated a lack of knowledge of Player 1's card, we can see that this sub-game is now a complete information game. 27 2.12 A unique path through the game tree of the Battle of the Bismark Sea. Since each player determines a priori the unique edge he/she will select when confronted with a specific information set, a path through the tree can be determined from these selections. 28 2.13 The probability space constructed from fixed player strategies in a game of chance. The strategy space is constructed from the unique choices determined by the strategy of the players and the independent random events that are determined by the chance moves. 30 2.14 The probability space constructed from fixed player strategies in a game of chance. The strategy space is constructed from the unique choices determined by the strategy of the players and the independent random events that are determined by the chance moves. Note in this example that constructing the LIST OF FIGURES vii probabilities of the various events requires multiplying the probabilities of the chance moves in each path. 31 2.15 Game tree paths derived from the Simple Poker Game as a result of the strategy (Fold, Fold). The probability of each of these paths is 1=2. 32 2.16 The game tree for the Battle of the Bismark Sea. If the Japanese sail north, the best move for the Allies is to search north. If the Japanese sail south, then the best move for the Allies is to search south. The Japanese, observing the payoffs, note that given these best strategies for the Allies, there best course of action is to sail North.