The Bigger Picture . . . or the Smaller Picture Notes

The Bigger Picture . . . or the Smaller Picture Notes - The...

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The Bigger Picture . . . or the Smaller Picture Adam Brandenburger * Version 01/01/15 1 Introduction In his classic book How to Solve It , 1 the mathematician George P´ olya offered advice on how to tackle problems in mathematics. Among his advice is: If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? In this note we offer some brief thoughts on two components of this advice, one about looking to break off a part of the problem at hand and solve the smaller problem, the other about looking to embed the problem at hand in a bigger problem. 2 The Smaller Picture There are problems that cannot be solved even in principle, regardless of the resources, intellectual or material, that are devoted to them. In 1937, the computer pioneer Alan Turing found a famous such problem, 2 when he proved that there cannot exist an algorithm that, given any computer program together with an input, can decide whether or not the program will halt on that input. Then there are problems which could in principle be solved, but the solutions of which lie beyond today’s technological capabilities. The game of Chess makes a good example. 3 It is straightforward to prove that for Chess, precisely one of the following three statements is true: (i) there is a strategy for White that guarantees White a win every time, regardless of how Black plays; (ii) there is a strategy for Black that guarantees Black a win every time, regardless of how White plays; (iii) there is a strategy for White that guarantees White at least a draw, and the same is true for Black. 4 In principle, it is possible to determine which of these three statements is true, and even to determine the corresponding strategy or strategies, by drawing out the game tree for Chess and working backwards through the tree. But this is only in principle. The reason is that the tree is so large — estimated to contain more than 10 120 different paths through it 5 — that no conceivable machine could analyze it. In practice, Chess is (partially) analyzed by being broken down into parts usually refereed to as the opening, middle game, and endgame. Many endgames, in particular, have been analyzed exhaustively and complete solutions are known for many of them. 6 * Stern School of Business, Polytechnic School of Engineering, Institute for the Interdisciplinary Study of Decision Making, Center for Data Science, New York University, New York, NY 10012, U.S.A., [email protected], adambrandenburger.com
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So far, the problems we have mentioned lie in the mathematical realm.
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