## 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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Página 3

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 of

directed line segments, called

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 of

directed line segments, called

**edges**, between some pairs of distinct vertices. Wecan ...

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The

root of T, then T = T, and if u is terminal, then T, is a trivial tree. The cutting T of the

tree in Figure 1.3 has vertices c, g, h, i, j, and

The

**edges**of T, are all**edges**of T which start at a vertex of T. Note that if u is theroot of T, then T = T, and if u is terminal, then T, is a trivial tree. The cutting T of the

tree in Figure 1.3 has vertices c, g, h, i, j, and

**edges**(c, g), (g, i), (g, j), (c, h). Página 7

(11) Let T be a tree; let e be the number of

vertices. Prove that e = w — 1. (12) Let T be a nontrivial tree. Prove that there is a

nonterminal vertex such that all of its children are terminal. (13) Let T be a tree.

(11) Let T be a tree; let e be the number of

**edges**in T; and let v be the number ofvertices. Prove that e = w — 1. (12) Let T be a nontrivial tree. Prove that there is a

nonterminal vertex such that all of its children are terminal. (13) Let T be a tree.

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In this game, there are never more than three

designate these

example, if Pi's first move is to vertex b, then we would designate that

In this game, there are never more than three

**edges**out of a vertex. It is natural todesignate these

**edges**as L (for “left”), R (for “right”), and M (for “middle”). Forexample, if Pi's first move is to vertex b, then we would designate that

**edge**(and ... Página 14

Its

choice subtree S. If the game has reached a vertex u belonging to P, then u is in

S and only one of its children is in S. P's move is the

the ...

Its

**edges**are (root, a) ... for example, that player P is playing according to achoice subtree S. If the game has reached a vertex u belonging to P, then u is in

S and only one of its children is in S. P's move is the

**edge**from u to that child. Onthe ...

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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