Introduction to Game TheorySpringer Science & Business Media, 2012 M12 6 - 252 páginas 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). |
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... proof . At these places , we have tried to make it clear that there is a gap , and point the reader who wishes to ... proofs . It is also recognized , however , that this is an applied subject and so its computational aspects have not at ...
... proof . At these places , we have tried to make it clear that there is a gap , and point the reader who wishes to ... proofs . It is also recognized , however , that this is an applied subject and so its computational aspects have not at ...
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... PROOF . ( 1 ) Suppose that w is a vertex with two parents u and v . By definition of a tree , there is a path from the root to u . Appending the edge ( u , w ) to this path produces a path from the root to w . Similarly , appending the ...
... PROOF . ( 1 ) Suppose that w is a vertex with two parents u and v . By definition of a tree , there is a path from the root to u . Appending the edge ( u , w ) to this path produces a path from the root to w . Similarly , appending the ...
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... PROOF . First , u has no edges of Tu going into it because the starting vertex of such an edge could not be a descendant of u . Second , let v be a vertex of Tu . Then v is a descendant of u and so there is a path in Tu from u to v ...
... PROOF . First , u has no edges of Tu going into it because the starting vertex of such an edge could not be a descendant of u . Second , let v be a vertex of Tu . Then v is a descendant of u and so there is a path in Tu from u to v ...
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Contenido
Linear Programming | 65 |
Solving Matrix Games 99 | 98 |
NonZeroSum Games | 115 |
NPerson Cooperative Games | 149 |
GamePlaying Programs | 185 |
Appendix Solutions | 201 |
Bibliography | 223 |
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alpha-beta pruning arbitration pair Axiom basic solution bi-matrix chance moves characteristic function form choice function choice subtree choose coefficient column player compute constant-sum constraints cooperative defined definition denoted directed graph dual basic form edges entry equation equilibrium N-tuple equilibrium pairs example Exercise expected payoff feasible tableau feasible vector game in characteristic game theory game tree grand coalition imputation inequality inessential joint strategy labeled mancala matrix game maximin values maximize maximum minimax theorem nonbasic variable noncooperative nonnegative normal form objective function optimal mixed strategies optimal strategy P₁ Pareto optimal path payoff matrix payoff pair payoff region pivot 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 subject to x1 supergame Suppose symmetric terminal vertex theorem vc(M verify vertices zero zero-sum