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Certainty, Uncertainty, and Risk
The essence of the phenomenon of gambling is decision making. The act of making a decision consists of selecting one course of action, or strategy, from among the set of admissible strategies. A particular decision might indicate the card to be played, a horse to be backed, the fraction of a fortune to be hazarded over a given interval of play…. Decisions can be categorized according to the specific relationship between action and outcome.—Richard Epstein, The Theory of Gambling and Statistical Logic
In The Theory of Gambling and Statistical Logic, mathematician Richard Epstein investigates the mathematics of uncertainty in gambling. His research, however, can be applied to all kinds of games. In his emphasis on decision making and the relationship between action and outcome, Epstein echoes some of our own core ideas. In his book, Epstein identifies three types of decision-outcome relationships, leading to three degrees of uncertainty: uncertainty, risk, and certainty. Each category corresponds to a different kind of decision-outcome relationship and game experience. A game that is completely certain is hardly a game at all, and certainly not much fun to play. It is like flipping a two-headed coin: there is no doubt what the end result will be. Sometimes, certainty is contextual. A game of Tic-Tac-Toe between two people that are completely familiar with the logic of the game play has a certain outcome: the game will always end in a draw. Although the specific decisions of the players aren't certain, the overall result of the game will be. In a game that is completely certain, meaningful play is impossible. Epstein's other two categories describe what we normally think of as uncertainty in games. Risk refers to a situation in which there is some uncertainty but the game's players know the nature of the uncertainty in advance. For example, playing a game of Roulette involves placing bets on the possible outcome of a spin and then spinning the roulette wheel to get a random result. There is some uncertainty in the spin of the wheel, but the percentage chance for a particular result occurring and the resulting loss or gain on the bet can be calculated precisely. Of the thirty-one numbers on the Roulette wheel, 15 are red, 15 are black and one of them (the zero) is neither red nor black. If you bet on red, you have 15 out of 31 chances (or 48.39%) to win and double your bet. In other words, in a game of pure risk, you can be completely certain about the degree of uncertainty in the outcome of the game. Epstein's category of uncertainty describes a situation in which players have no idea about the outcome of the game. For example, imagine that you are a moderately skilled Chess player and you go to an online game site to play a game of Chess with an opponent that you select at random. You have no idea who you are going to play against. It might be a Chess master, who will most likely beat you, or it might be someone learning to play for the first time, who you will most likely beat. There is no way for you to predict the outcome of the game. If, in contrast, you are playing a friend that you have played many times before and you know that you usually win three out of four games, you have a good sense of the outcome. But without knowing your opponent, you can't make that kind of guess. Although games of pure certainty are extremely rare (and not much fun to boot), games of pure risk and games of pure uncertainty are also quite rare. Most games possess some combination of risk and uncertainty. Even though you know something about the general chances of winning against your friend, you certainly don't have absolute mathematical certainty about your chances of winning. And although you know the exact risk each time you make a bet on the Roulette wheel, your overall loss or winnings over an evening of play is much more uncertain.
