Kuhn, H. W. (1953). Extensive games and the problem of information. In Kuhn, H. W., & Tucker, A. W. (Eds.), Contributions to the Theory of Games II, pp. 193--216. Princeton University Press. Reprinted in (Kuhn, 1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....each player i makes his own forecast p , where we do not impose pp k i k j = for ij . Thus in place of a single commonly known object ( a p we have distinct beliefs b (a i , p i i ) 8 See Fudenberg and Kreps [10] for a discussion of this point. 9 Given independence, Kuhn s theorem [18] shows that there is no additional loss of generality in restricting attention to expectations that correspond to a single strategy profile p i i , as opposed to a probability distribution over such profiles. 10 In definition 2.1 every strategy belief pair in the belief model is required to ....

H. Kuhn, "Extensive games and the problem of information," Annals of Mathematics Studies no. 28, Princeton University Press, Princeton, NJ (1953).

....in certain ways, as mentioned at the end of this paper. Our construction uses a game tree where alternately the adversary generates a request and the online algorithm serves it. The adversary is not informed about the action of the online algorithm, so the game tree has imperfect information [12]. We consider a finite tree where after some requests, the ratio of online versus optimal offline cost is the payoff to the adversary. This defines a zero sum game, which we solve by linear programming. For a game tree that is sufficiently deep, and restricted to a suitable subset of requests so ....

....reaction by the online player. Each leaf of the tree defines a sequence oe and an online cost ON (oe) depending on the online actions leading to that leaf) with payoff ON (oe) OFF (oe) to the adversary. The restricted information of the adversary in this game tree is modeled by information sets [12]. Here, an information set is a set of nodes where the adversary is to move and which are preceded by the same previous moves of the adversary himself. Hence, the nodes in the set differ only by the preceding moves of the online player, which the adversary cannot observe. An action of the ....

H. W. Kuhn (1953), Extensive games and the problem of information. In: Contributions to the Theory of Games II, eds. H. W. Kuhn and A. W. Tucker, Annals of Mathematics Studies 28, Princeton Univ. Press, Princeton, 193--216.

....in certain ways, as mentioned at the end of this paper. Our construction uses a game tree where alternately the adversary generates a request and the online algorithm serves it. The adversary is not informed about the action of the online algorithm, so the game tree has imperfect information [12]. We consider a nite tree where after some requests, the ratio of online versus optimal o ine cost is the payo to the adversary. This de nes a zero sum game, which we solve by linear programming. For a game tree that is suciently deep, and restricted to a suitable subset of requests so that the ....

....the reaction by the online player. Each leaf of the tree de nes a sequence and an online cost ON ( depending on the online actions leading to that leaf) with payo ON ( OFF ( to the adversary. The restricted information of the adversary in this game tree is modeled by information sets [12]. Here, an information set is a set of nodes where the adversary is to move and which are preceded by the same previous moves of the adversary himself. Hence, the nodes in the set di er only by the preceding moves of the online player, which the adversary cannot observe. An action of the adversary ....

H. W. Kuhn (1953), Extensive games and the problem of information. In: Contributions to the Theory of Games II, eds. H. W. Kuhn and A. W. Tucker, Annals of Mathematics Studies 28, Princeton Univ. Press, Princeton, 193-216.

....at all successor nodes are degenerate. But at such a lowest node the agent who chooses cannot be optimizing, since she is not indifferent between any of the pure outcomes resulting from the various successor nodes. The argument just given is a slight generalization of the one used by Kuhn (1953) to prove his backwards induction theorem characterizing what are now called subgame perfect equilibria of games of perfect information. 1.2. Two Agents, Two Outcomes Assume that there are two players and two outcomes, i.e. n = 2, and Omega is, say, fa; bg. Generically, neither agent is ....

....terminal nodes, and let Omega = Z. Kreps and Wilson (1982) show that for a generic set of u 2 (IR Z ) I the set of distributions on Z ( paths ) induced by sequential equilibria is finite, but they also mention (p. 881) that this remains true if one considers paths induced by Nash equilibria. (Kuhn s (1953) theorem concerning the equivalence of normal form and behavioral strategies implies that this is correct as a statement about the Nash equilibria of the normal form. Their result appears to have considerable epistemological significance insofar as the distribution on terminal nodes is precisely ....

Kuhn, H.W., (1953), "Extensive Games and the Problem of Information," Annals of Mathematical Studies, 28, 193-216.

....(initial node) of the tree and ends at a leaf (terminal node) where each player receives a payoff. The nonterminal nodes are called decision nodes. The player s moves are assigned to the outgoing edges of the decision node. The decision nodes are partitioned into information sets, introduced by Kuhn (1953). All nodes in an information set belong to the same player, and have the same moves. The interpretation is that when a player makes a move, he only knows the information set but not the particular node he is at. Some decision nodes may belong to chance where the next move is made according to a ....

.... tree and with a basis crashing subroutine, as shown by Koller and Megiddo (1996) The best response subroutine in Wilson s (1972) algorithm requires that the players have perfect recall , that is, all nodes in an information set of a player are preceded by the same earlier moves of that player (Kuhn, 1953). For finding all equilibria, Koller and Megiddo (1996) show how to enumerate small supports in a way that can also be applied to extensive games without perfect recall. 4.2. Sequence form The use of pure strategies can be avoided altogether by using sequences of moves instead. The unique path ....

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H. W. Kuhn (1953), Extensive games and the problem of information. In: Contributions to the Theory of Games II, eds. H. W. Kuhn and A. W. Tucker, Annals of Mathematics Studies 28, Princeton Univ. Press, Princeton, pp. 193--216.

....i.e. a vector p = a 1 ; a 2 ; a m ) with a k 2 A denoting the vector of actions taken at round k. A m denotes the set of all possible play paths. S i , with generic element s i , is a set of possible strategies that player i can use for playing the stage game. Following Kuhn s [21] theorem, S i consists of the mixed strategies of the extensive form stage game. g s denotes the probability distribution over play paths induced by a given vector of strategies s = s 1 ; s n ) Let g s Gammai ;r i denote the distribution over play paths obtained when players other than ....

Kuhn, H.[1953],"Extensive Games and the Problem of Information, "in: H. Kuhn and A.W. Tucker,Contributions to the Theory of Games, Vol. II -- Annals of Mathematics Studies 28,Princeton University Press, pp. 193--216

....measure of the computational effort required to solve the decision tree representation as it does not account for other computer operations that are required to solve the problem. 3 . GAME TREE REPRESENTATION In this section, we describe a game tree representation of a decision problem. We use Kuhn s [1953] definition of an extensive form game appropriately modified. In particular, we assume only one player (the decision maker) and we assume perfect recall. A game tree representation of a decision problem consists of the following: i) A rooted tree consisting of nodes and edges. ii) The leaf ....

....(b) states that a node in an information set cannot come after another one in the same information set. Condition (c) states that at each information set, the decision maker remembers past decisions (her choices at earlier decision nodes) Condition (c) is called perfect recall in game theory [Kuhn 1953, Hart 1992] Figure 3 shows a game tree representation of the oil wildcatter s problem. Notice that in the game tree, random variable O (amount of oil) comes before R (test results) Also the decision nodes are partitioned into 5 information sets, I 1 , I 5 . I 1 , for example, encodes the ....

[Article contains additional citation context not shown here]

Kuhn, H. W. (1953), "Extensive games and the problem of information," in H. W. Kuhn and A.

....the search tree, 3. moderate solution lengths, and 4. all moves are reversible. Sokoban is a difficult problem domain for computers, and more challenging than previously studied domains, because of the following reasons: 1. Sokoban has a complex lower bound estimator (O(N 3 ) given N goals [13]) Unfortunately, even this expensive lower bound is not very effective. In other domains, such as the sliding tile puzzles or Rubik s Cube, a table lookup is often sufficient to deliver a high quality lower bound. 2. The branching factor for Sokoban is large and variable (potentially over 100) ....

H.W. Kuhn. Extensive games and the problem of information. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games 2, pages 193--216, Princton, 1953. Princton Univ. Press.

....is done similarly. Moreover, for any mixed policy, the probability space for the state and action processes can be chosen to be the same as the one obtained by some equivalent policy in U . This was established for the more general setting of MDPs with several controllers (stochastic games) by Kuhn (1953), Aumann (1964) and Bernhard (1992) When fi is concentrated on some state x (i.e. fi = ffi x ) we shall use the notation P u x instead of P u fi . Denote p u fi (t; x) P u fi (X t = x) and p u fi (t; x; A) P u fi (X t = x; A t 2 A) A ae A(x) We have for all fi 2 M 1 (X) and ....

H. W. Kuhn (1953), "Extensive games and the problem of information", Ann. Math. Stud. 28, pp. 193-216.

....to view a randomized algorithm as an algorithm that at every step chooses its move from a probability distribution on the set of possible moves. In the theory of multi stage games these two approaches are sometimes called, respectively, mixed strategies and behavior strategies (see, for example, [14]) The de#nitions of cost and competitiveness extend naturally to randomized algorithms, independently of which of the two above de#nitions is being used. If A is a randomized algorithm then cost A ( denotes the expected cost of A on , and inequality (1) remains unchanged. It is quite easy to ....

H. Kuhn, Extensive games and the problem of information, in: Kuhn, H., Tucker, A. (Eds.), Contributions to the Theory of Games, Princeton University Press, 1953, pp. 193--216.

....THREE SHORT PLAYS: STRUCTURAL INFORMATION IN EXTENSIVE GAMES 3 at least not if the other players future actions is at issue in investigating player s information. To start with, we de ne a game in its extensive form (See [Osborne and Rubinstein, 1994] The origin of these de nitions goes back to [Kuhn, 1953], see also [Luce and Rai a, 1957] Let n . An extensive form game G (without chance moves) is a tuple hP; H; s; P j ; SP j i consisting of the following components. P is a set fP 1 : Pn g of players, and H is a set fh 1 : hn g of nite histories h i ; 0 i n, satisfying the ....

Kuhn, H.: (1953) Extensive games and the problem of information, in Kuhn, H. and Tucker, A., (eds.), Contributions to the Theory of Games II, Princeton: Princeton University Press.

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Kuhn, H. W. (1953). Extensive games and the problem of information. In Kuhn, H. W., & Tucker, A. W. (Eds.), Contributions to the Theory of Games II, pp. 193--216. Princeton University Press. Reprinted in (Kuhn, 1997).

No context found.

Kuhn, H. W. (1953). Extensive games and the problem of information. In Kuhn, H. W., & Tucker, A. W. (Eds.), Contributions to the Theory of Games II, pp. 193--216. Princeton University Press. Reprinted in (Kuhn, 1997).

No context found.

Kuhn, Heinrich. "Extensive Games and the Problem of Information." Annals of Mathematics Studies, 28, Princeton, NJ: Princeton University Press. Lambson, Val E. and Probst, Daniel A. "Learning by Matching Patterns." Games and Economic Behavior, 2004, 46 (2), pp. 398-409.

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Kuhn, H. (1953). Extensive games and the problem of information. Annals of Mathematics Studies, 28, 193--216.

No context found.

Harold William Kuhn. Extensive games and the problem of information. In Kenneth J. Arrow and Harold William Kuhn, editors, Contributions to the theory of games, volume 2, pages 193--216. Princeton University Press, 2 edition, 1956.

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H. Kuhn. Extensive games and the problem of information. In H. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, volume 2 of Annals of Mathematics Studies, 28, pages 193--216. Princeton University Press, 1953.

No context found.

H.W. Kuhn, "Extensive games and the problem of information" in Contributions to the Theory of Games II, H.W. Kuhn and A. W. Tucker eds., Annals of Mathematical Studies No. 28, Princeton Univ. Press, 1950.

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H. W. Kuhn (1953), `Extensive games and the problem of information', Ann. Math. Stud., 28, pp. 193-216.

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H. W. Kuhn. Extensive games and the problem of information. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, pages 196--216. Princeton University Press, Princeton, NJ, 1953.

No context found.

H. W. Kuhn. Extensive games and the problem of information. In H.W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games Vol. II, Annals of Mathematics Studies, 28, pages 193 -- 216. Princeton University Press, 1953.

No context found.

H. Kuhn. "Extensive games and the problem of information". In Contributions to the Theory of Games II (H. Kuhn and A. Tucker, Eds.), 193--216, Princeton University Press, Princeton, NJ, 1953.

No context found.

Kuhn, H.W. (1953) Extensive games and the problem of information. In Kuhn, H.W. and Tucker, A.W. Contributions to the Theory of Games Vol I, 193--216. Princeton University Press, Princeton N.J.

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H.W. Kuhn. Extensive games and the problem of information. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games. Annals of Mathematics Study 28, Princeton University Press, 1953. 11

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H. W. Kuhn. Extensive games and the problem of information. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, Vol. II, pages 193--216. Princeton University Press, Princeton, 1953.

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