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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Página 7
... Define a cycle in a directed graph to be a path which begins and ends at the same vertex . Prove that trees do not contain cycles . Then prove that paths in trees consist of distinct vertices . ( 9 ) Let G be a directed graph . For a ...
... Define a cycle in a directed graph to be a path which begins and ends at the same vertex . Prove that trees do not contain cycles . Then prove that paths in trees consist of distinct vertices . ( 9 ) Let G be a directed graph . For a ...
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... defined . It is played as follows . The player who owns the root chooses one of the children of the root . If that child is intermediate , the player to whom it belongs chooses one of its children . The game continues in this way until ...
... defined . It is played as follows . The player who owns the root chooses one of the children of the root . If that child is intermediate , the player to whom it belongs chooses one of its children . The game continues in this way until ...
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... defined by a tree the players obviously move consecutively . To remove this apparent difficulty , suppose that a neutral third party is present . Player P1 whispers her move ( that is , the number of fingers she wishes to hold up and ...
... defined by a tree the players obviously move consecutively . To remove this apparent difficulty , suppose that a neutral third party is present . Player P1 whispers her move ( that is , the number of fingers she wishes to hold up and ...
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... defines a concept which is our first approximation to what a strategy should be . DEFINITION 1.2 . Let T be a game tree and let P be one of the players . Define a choice function for P to be a function c , defined on the set of all ...
... defines a concept which is our first approximation to what a strategy should be . DEFINITION 1.2 . Let T be a game tree and let P be one of the players . Define a choice function for P to be a function c , defined on the set of all ...
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... definition of strategy . The first is that choice functions are usually defined at many vertices where they need not be defined , that is , at vertices which can never be reached in the course of the game , given earlier decisions by ...
... definition of strategy . The first is that choice functions are usually defined at many vertices where they need not be defined , that is , at vertices which can never be reached in the course of the game , given earlier decisions by ...
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