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Probability and Guesswork
Another important connection between games and information theory is that information is a measure of uncertainty. We know that meaningful play in games requires some degree of uncertainty on the macro-level. If the outcome of a game is known in advance, there really isn't a game at all. If one player is trying to win a simple game by correctly guessing whether a penny is heads-up or tails-down, there are two possible messages the penny can generate (heads and tails). However, if both players know that it is a two-headed coin, the amount of uncertainty drops to zero. Suddenly there is no game to be played, just as there is no information to be generated by the coin. We know this from the example of J.L.: if you already know his favorite color, your answer contains no information. We can also make a connection on the micro-level of uncertainty. Uncertainty, information, and probability all share a set of formal relationships. Think about a die roll as an information theory signal. If one die is rolled, there is less information (and uncertainty of the outcome) than if two are rolled, where the set of possible outcomes is greater.Two die rolls—more possibilities and more uncertainty—equals more information. Because guessing games rely on uncertainty, they make good information theory case studies.
Take a game such as Mastermind. Each turn, one player attempts to guess the correct answer. By receiving coded feedback in the form of black and white pegs, her opponent tells her how correct or incorrect her guess was. The goal of the game is to arrive at the correct answer within a limited number of turns. Understood as an information theory system, Mastermind is a wonderful structure which puts information at play. With each guess, the guesser narrows down the possible answers (decreasing uncertainty), carving out a single guess from a range of all possible guesses. By narrowing the degree of uncertainty, the guesser navigates the space of all possible answers, hopefully arriving at the correct answer before it is too late. Because the answer remains fixed, as long as the guesser doesn't make any logical mistakes, each guess is progressively more certain. Imagine for a moment a variation on Mastermind, in which the guesser's opponent rearranges the solution pegs each time. In this game, a non-winning guess would not reduce the amount of uncertainty, because the system is "reset" every turn. Like a gambler betting on a single Roulette number over and over again, the amount of uncertainty would remain fixed and would not be reduced by successive guesses. It is even possible to think of cultural information in information theory terms. Take the classic guessing game Twenty Questions, which Bertalanffly uses as an example in General Systems Theory.[8] In this simple game, one player has a piece of information, the answer, and the other player tries to guess that piece of information. Each turn, the guesser can ask a yes or no question, and the goal is to arrive at the correct answer within 20 questions."Is it a vegetable?" "No." "Is it an animal?" "Yes." "Is it a mammal?" "Yes." "Is it a rodent?" "No." As with Mastermind, the answer remains fixed as the guesser tries to narrow down the possibilities through clever queries. In this case, however, the answer does not reside within an abstract mathematical system, but instead is an object that exists in the world, in language, in the shared culture of the two players. It is striking that the vast realm of possible solutions can be reduced so quickly and so efficiently through a series of yes/no guesses—usually, fewer than twenty! If meaning is irrelevant to information theory's concept of "information," how is it possible to use information theory to look at a game such as Twenty Questions, a game that relies on culture for its content? Because information is an abstract formal quantity, it can be applied to any form of communication. In the case of Twenty Questions, it is not the linguistic, historical, or cultural relationships that interest information theory, just the mathematical ones. If we really wanted to reduce the game to its formal essence, we could plot the mathematical interrelationships between any possible Twenty Questions query and every possible yes or no answer, using a nonsense language instead of speech. Although this would empty out the cultural content from the game, the informational structure would remain the same. The beauty of Twenty Questions is that such a reduction does occur on some level as the game is played. The game suspends language in a formal web of informational connections. Each guess is a deductive gesture that narrows the focus of the guesser's pursuit until the uncertainty of culture resolves itself into a single coherent answer that, finally, marks the end of the game. [8]Ludwig von Bertalanffy, General Systems Theory Foundations (New York:George Braziller, 1968), p. 42.
