D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1--8, 1956.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....up a (1 ffl) factor, we are able to achieve this additional guarantee, and it is in this respect that our algorithm outperforms the Ellipsoid algorithm. Note that our goals follow a classic line of research in game theory and machine learning, namely that of minimizing regret in repeated games [5, 10, 14, 9, 2]. Freund and Schapire [9] point out that the Weighted Majority (WM) algorithm of Littlestone and Warmuth [14] can be used for repeated play in any 2 player zero sum game, and will perform nearly as well as the best fixed strategy in hindsight. However, the WM algorithm involves placing weights on ....

D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1--8, 1956.

....is an algorithm for finding the value of a zero sum game, and so for solving a linear program. For a detailed treatment see [15] A short time after Hannan announced his result, David Blackwell [2] showed how Hannan s theorem could be obtained as a corollary of his approachability theorem, [3]. To use the theorem one needs to define an auxiliary game with vector valued payoffs and a target set. If the row player chooses strategy i and the column player chooses strategy j, the payoff is an n 1 vector with a 1 in the j position, a ij in the (n 1) st position and zeros everywhere ....

....which the vector payoffs reside there is a convex set G, called the target set. R s goal is to play the game so as to force A T to approach G arbitrarily closely almost surely. If R can succeed at approaching G, the set G is said to be approachable. In the case when G is a convex set, Blackwell [3] gave a necessary and sufficient condition for a convex target set to be approachable. To describe the condition let A T 62 G be the current average payoff and g the point in G closest to A T . Let l be the plane perpendicular to the line joining A T and g that touches G. Such a plane can be ....

Blackwell, D., `An analog of the minimax theorem for vector payoffs', Pacific Journal of Mathematics, 6, 1-8, 1956.

....n = 1, B = f Gamma1; 1g and ffi = 0:1. Any norm will do (since we are working in just one dimension) and the loss function is L(s) ae 0 if s 2 [2; 7] 1 otherwise. We will return to this example later in the paper. Drifting games bear a certain resemblence to the kind of games studied in Blackwell s 1956 celebrated approachability theory. However, it is unclear what the exact relationship is between these two types of games and whether one type is a special case of the other. paper.tex; 31 07 2000; 21:37; p.4 Drifting Games 5 3. Relation to boosting In this section, we describe how the ....

Blackwell, D.: 1956, `An analog of the minimax theorem for vector payoffs'. Pacific Journal of Mathematics 6(1), 1--8.

.... 1g and ffi = 0:1. Any norm will do (since we are working in just one dimension) and the loss function is 4 R. E. SCHAPIRE L(s) ae 0 if s 2 [2; 7] 1 otherwise. We will return to this example later in the paper. Drifting games bear a certain resemblence to the kind of games studied in Blackwell s (1956) celebrated approachability theory. However, it is unclear what the exact relationship is between these two types of games and whether one type is a special case of the other. 3. Relation to boosting In this section, we describe how the general game of drift relates directly to boosting. In the ....

Blackwell, D. (1956). An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6(1), 1--8.

....2 rather than Theorem 1. The details of the algorithm about which this corollary applies are largely unimportant and could, in principle, be applied to any algorithm with similar properties. Indeed, algorithms for this problem with similar properties were derived by Hannan [13] 1 Blackwell [1] and Foster and Vohra [6, 5, 4] Also, Fudenberg and Levine [10] independently proposed an algorithm equivalent to LW and proved a slightly weaker version of Corollary 2. As a simple first corollary, we see that the loss of LW can never exceed the value of the game M by more than D T . Corollary ....

David Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6(1):1--8, Spring 1956.

....of the investment problem to overcome this difficulty. The side information aspect of the present universal investment problem is new, but several prior works have considered the problem with no side information. Cover and Gluss [13] show that the approachability excludability theorem of Blackwell [11, 12] can be used to define an investment scheme with universal properties if the price relatives are restricted to a finite set. Larson [14] shows that variants of the investment scheme suggested by the compound Bayes technique have exponential growth rates arbitrarily close (but not equal) to that of ....

D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1--8, 1956.

....No useful analysis can be done in a model where those changes are completely arbitrary. 4. Repeated games with complete information, or more generally, multistage games and stochastic games have been extensively studied in game theory and economics. A very partial list includes: Shapley, 1953; Blackwell, 1956; Luce Raiffa, 1957) and more recently (Fudenberg Tirole, 1991; Mertens, Sorin, Zamir, 1995) and the evolving literature on learning (e.g. Fudenberg Levine 1997) The incomplete information setup in which the player is ignorant about the game being played was inspired Dynamic ....

....group structure. That is, our approach to decision making is qualitative (or ordinal) This distinguishes our work from previous work on non Bayesian repeated games, which used the probabilistic safety level criterion as a basic solution concept for the one shot game 15 . These works, including (Blackwell, 1956; Hannan, 1957; Banos, 1968; Megiddo, 1980) and more recently (Auer, Cesa Bianchi, Freund, Schapire, 1995; Hart Mas Colell, 1997) used several versions of long run solution concepts, all based on some optimization of the average of the utility values over time. That is, in all of these ....

Blackwell, D. (1956). An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematic, 6, 1--8.

....significantly smaller than the game value. Overcoming these difficulties in the one shot game is hopeless. In repeated play, however, one can hope to learn to play well against the particular opponent that is being faced. Algorithms of this type were first proposed by Hannan [20] and Blackwell [3], and later algorithms were proposed by Foster and Vohra [15, 13, 12] These algorithms have the property that the loss of the row player in repeated play is guaranteed to come close to the minimum loss achievable with respect to the sequence of plays taken by the column player. In this paper, we ....

David Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6(1):1--8, Spring 1956.

....by Nature. Remarkably however, the work on universal prediction, forecasting, and data compression has shown that in many cases the learning algorithm can achieve surprisingly small regret, i.e. it can make predictions almost as well as if it knew ahead of time which expert s advice to take [5, 4, 19, 20, 11, 33, 27, 14, 35, 15, 30, 6, 7, 37, 2, 17, 31, 28]. In particular, it has been shown that the Lempel Ziv algorithm universally achieves quite a small regret when compared to the best finite state predictor [15] A more general analysis of universal prediction is given in [31] for a comparison class that is a smooth parametric family, and in more ....

D. Blackwell, "An analog of the minimax theorem for vector payoffs," Pacific Journal of Mathematics, vol. 6, pp. 1--8, 1956.

.... that one can remove all statistical assumptions about the data and still achieve extremely tight bounds (see Freund [18] This problem and many variations and extensions have been addressed in a number of different communities, under names such as the sequential compound decision problem [32] [4], universal prediction [16] universal coding [33] universal portfolios [13] and prediction of individual sequences ; the notion of the competitiveness is also called the min max regret of an algorithm. A web page uniting some of these communities and with a discussion of this general ....

....we then would get another opportunity to probabilistically select an expert to use and so forth. Freund and Schapire show that extensions of the randomized Weighted Majority Algorithm discussed above can be made to fit nicely into this scenario [19] see also the classic work of Blackwell [4]) Another scenario fitting this framework would be a case where each expert is a page replacement algorithm, and an operating system needs to decide which algorithm to use. Periodically the operating system computes losses for the various algorithms that it could have used and based on this ....

D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific J. Math., 6:1--8, 1956.

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D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1--8, 1956.

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Blackwell, D.: An Analog of the Minimax Theorem for Vector Payoffs. Pacific J. Math., 6, pp 1-8, 1956.

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D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6:1--8, 1956.

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Blackwell, D. (1956b): "An Analog of the Minimax Theorem for Vector Payoffs," Pacific Journal of Mathematics, VI, 1-8.

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