## 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). |

### Dentro del libro

Resultados 1-5 de 6

Página 4

e Usually, but not always, we draw trees in such a way that the

An example is given in Figure 1.3; all the vertices have been labeled. This tree

has eleven vertices and ten edges. Now let T be a tree. We make a few

definitions.

e Usually, but not always, we draw trees in such a way that the

**root**is at the top.An example is given in Figure 1.3; all the vertices have been labeled. This tree

has eleven vertices and ten edges. Now let T be a tree. We make a few

definitions.

Página 5

The depth of the tree in Figure 1.3 is three because (

greatest length. Some general facts about trees are collected together in the

following: THEOREM 1.1. Let T be a tree. Then we have: (1) No vertex has more

than one ...

The depth of the tree in Figure 1.3 is three because (

**root**, c, g, j) is a path ofgreatest length. Some general facts about trees are collected together in the

following: THEOREM 1.1. Let T be a tree. Then we have: (1) No vertex has more

than one ...

Página 6

The edges of T, are all edges of T which start at a vertex of T. Note that if u is the

tree in Figure 1.3 has vertices c, g, h, i, j, and edges (c, g), (g, i), (g, j), (c, h).

The edges of T, are all edges of T which start at a vertex of T. Note that if u is the

**root**of T, then T = T, and if u is terminal, then T, is a trivial tree. The cutting T of thetree in Figure 1.3 has vertices c, g, h, i, j, and edges (c, g), (g, i), (g, j), (c, h).

Página 14

in the game of Figure 1.5, suppose that P1 uses the choice function c1, where c1(

in the game of Figure 1.5, suppose that P1 uses the choice function c1, where c1(

**root**) = a, c1(d) = k, c1(e) = l. Then the path (**root**, a, e, l) is a (P1, c1)-path. So is (**root**, a, d, k). In fact, these are the only two. Then we have the following: ...Página 17

Alcanzaste el límite de visualización de este libro.

Alcanzaste el límite de visualización de este libro.

### Comentarios de la gente - Escribir un comentario

No encontramos ningún comentario en los lugares habituales.

### 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 |

### Otras ediciones - Ver todas

### 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