## Introduction to Game TheoryThe 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). |

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Thelma Louise (1,-1) (-1,1) (-5,5) (5,-5) FIGURE 1.1. Matching Coins. 1.1. Trees A

directed graph is a finite set of points, called

directed line segments, called edges, between some pairs of distinct

can ...

Thelma Louise (1,-1) (-1,1) (-5,5) (5,-5) FIGURE 1.1. Matching Coins. 1.1. Trees A

directed graph is a finite set of points, called

**vertices**, together with a set ofdirected line segments, called edges, between some pairs of distinct

**vertices**. Wecan ...

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D. Now let T be a tree and let u be any vertex of T. The cutting of T determined by

u, denoted T., is defined as follows: The

descendants of u. The edges of T, are all edges of T which start at a vertex of T.

D. Now let T be a tree and let u be any vertex of T. The cutting of T determined by

u, denoted T., is defined as follows: The

**vertices**of T, are u itself plus all of thedescendants of u. The edges of T, are all edges of T which start at a vertex of T.

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For example, in the tree of Figure 1.3, consider the set of terminal

f, j}. Then the subtree determined by U is the union of the three paths (root, a, d), (

root, b, f), (root, c, g, j). Exercises (1) Sketch all cuttings of the tree in Figure 1.4.

For example, in the tree of Figure 1.3, consider the set of terminal

**vertices**U = {d,f, j}. Then the subtree determined by U is the union of the three paths (root, a, d), (

root, b, f), (root, c, g, j). Exercises (1) Sketch all cuttings of the tree in Figure 1.4.

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In terms of the tree, it also means that Louise does not know which of her two

move (that is, decide which coin to hold) based on which of these two

she has ...

In terms of the tree, it also means that Louise does not know which of her two

**vertices**she is at. It is therefore impossible, under the rules, for her to plan hermove (that is, decide which coin to hold) based on which of these two

**vertices**she has ...

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Thus, the choice subtree determined by P and c has as its set of

according to the choice function c. It is clear that a choice subtree is a subtree (by

...

Thus, the choice subtree determined by P and c has as its set of

**vertices**all the**vertices**of T which can be reached in the course of the game, given that P playsaccording to the choice function c. It is clear that a choice subtree is a subtree (by

...

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### Contenido

1 | |

TwoPerson ZeroSum Games 35 | 34 |

Linear Programming | 65 |

Solving Matrix Games 99 | 98 |

NonZeroSum Games | 115 |

NPerson Cooperative Games | 149 |

GamePlaying Programs | 185 |

Appendix Solutions | 201 |

Two cuttings | 202 |

Equilibrium pair | 219 |

Bibliography | 223 |

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### Términos y frases comunes

alpha-beta pruning arbitration pair Axiom basic solution bi-matrix chance moves characteristic function form choice function choose coefficient column player compute constant-sum constraints cooperative core defined definition denoted directed graph dominated rows dual basic form edges entry equation equilibrium N-tuple equilibrium pairs example Exercise expected payoff feasible tableau feasible vector game in characteristic game shown game tree grand coalition imputation inequality inessential joint strategy labeled m x n matrix game maximin values maximum minimax theorem moves left nonbasic variable noncooperative nonnegative normal form objective function optimal mixed strategies optimal strategy Pareto optimal path payoff matrix payoff pair payoff region pivot player and column player plays plays according primal problem Prisoner's Dilemma probability proof Prove pure strategies root row player saddle point Shapley value shown in Figure simplex algorithm Solve stable set strategically equivalent supergame Suppose symmetric terminal vertex Ti(S ur(M ve(M verify vertices vr(M zero zero-sum