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The mathematical theory of games has as its purpose the analysis of a wide range of competitive situations. These include most of the recreations which people usually call "games" such as chess, poker, bridge, backgam mon, baseball, and so forth, but also contests between companies, military forces, and nations. For the purposes of developing the theory, all these competitive situations are called games. The analysis of games has two goals. First, there is the descriptive goal of understanding why the parties ("players") in competitive situations behave as they do. The second is the more practical goal of being able to advise the players of the game as to the best way to play. The first goal is especially relevant when the game is on a large scale, has many players, and has complicated rules. The economy and international politics are good examples. In the ideal, the pursuit of the second goal would allow us to describe to each player a strategy which guarantees that he or she does as well as possible. As we shall see, this goal is too ambitious. In many games, the phrase "as well as possible" is hard to define. In other games, it can be defined and there is a clear-cut "solution" (that is, best way of playing).
Content Level »Lower undergraduate
Keywords »Prisoner's dilemma - calculus - game theory
Table of contents
1. Games in Extensive Form.- 1.1. Trees.- 1.2. Game Trees.- 1.2.1. Information Sets.- 1.3. Choice Functions and Strategies.- 1.3.1. Choice Subtrees.- 1.4. Games with Chance Moves.- 1.4.1. A Theorem on Payoffs.- 1.5. Equilibrium N-tuples of Strategies.- 1.6. Normal Forms.- 2. Two-Person Zero-Sum Games.- 2.1. Saddle Points.- 2.2. Mixed Strategies.- 2.2.1. Row Values and Column Values.- 2.2.2. Dominated Rows and Columns.- 2.3. Small Games.- 2.3.1. 2 × n and m × 2 Games.- 2.4. Symmetric Games.- 2.4.1. Solving Symmetric Games.- 3. Linear Programming.- 3.1. Primal and Dual Problems.- 3.1.1. Primal Problems and Their Duals.- 3.2. Basic Forms and Pivots.- 3.2.1. Pivots.- 3.2.2. Dual Basic Forms.- 3.3. The Simplex Algorithm.- 3.3.1. Tableaus.- 3.3.2. The Simplex Algorithm.- 3.4. Avoiding Cycles and Achieving Feasibility.- 3.4.1. Degeneracy and Cycles.- 3.4.2. The Initial Feasible Tableau.- 3.5. Duality.- 3.5.1. The Dual Simplex Algorithm.- 3.5.2. The Duality Theorem.- 4. Solving Matrix Games.- 4.1. The Minimax Theorem.- 4.2. Some Examples.- 4.2.1. Scissors-Paper-Stone.- 4.2.2. Three-Finger Morra.- 4.2.3. Colonel Blotto’s Game.- 4.2.4. Simple Poker.- 5. Non-Zero-Sum Games.- 5.1. Noncooperative Games.- 5.1.1. Mixed Strategies.- 5.1.2. Maximin Values.- 5.1.3. Equilibrium N-tuples of Mixed Strategies.- 5.1.4. A Graphical Method for Computing Equilibrium Pairs.- 5.2. Solution Concepts for Noncooperative Games.- 5.2.1. Battle of the Buddies.- 5.2.2. Prisoner’s Dilemma.- 5.2.3. Another Game.- 5.2.4. Supergames.- 5.3. Cooperative Games.- 5.3.1. Nash Bargaining Axioms.- 5.3.2. Convex Sets.- 5.3.3. Nash’s Theorem.- 5.3.4. Computing Arbitration Pairs.- 5.3.5. Remarks.- 6. N-Person Cooperative Games.- 6.1. Coalitions.- 6.1.1. The Characteristic Function.- 6.1.2. Essential and Inessential Games.- 6.2. Imputations.- 6.2.1. Dominance of Imputations.- 6.2.2. The Core.- 6.2.3. Constant-Sum Games.- 6.2.4. A Voting Game.- 6.3. Strategic Equivalence.- 6.3.1. Equivalence and Imputations.- 6.3.2. (0,1)-Reduced Form.- 6.3.3. Classification of Small Games.- 6.4. Two Solution Concepts.- 6.4.1. Stable Sets of Imputations.- 6.4.2. Shapley Values.- 7. Game-Playing Programs.- 7.1. Three Algorithms.- 7.1.1. The Naive Algorithm.- 7.1.2. The Branch and Bound Algorithm.- 7.1.3. The Alpha-Beta Pruning Algorithm.- 7.2. Evaluation Functions.- 7.2.1. Depth-Limited Subgames.- 7.2.2. Mancala.- 7.2.3. Nine-Men’s Morris.- Appendix. Solutions.