7.2: Randomness

Randomness is usually defined as a lack of a definite pattern or predictability in the outcomes of an observable process. We will discuss a more mathematical and precise definition of randomness later in this chapter. For now, we will simply discuss how randomness is an important part of our lives. Randomness is encountered throughout our human experience. Even the simple games played by small children are usually based on some element of a random outcome to make the games more interesting. Certainly, no one would be interested in playing a game that always proceeded in the exact same way. Any game based on spinners, dice, or a shuffled deck of cards is based on the idea that the game is made more interesting by not being able to predict the outcome. Even in games like chess and checkers, where there is no obvious random device, the enjoyment of the game is enhanced by the fact that no one can always predict what move a player may make. Even professional chess players who have met the same opponent many times never know exactly how that opponent will react in a certain situation.

Consider the popular childhood (and adulthood) game rock-paper-scissors. In this game two participants simultaneously form one of three possible hand shapes, which are named rock, paper, and scissors. The winner of the game is then determined by a set of rules that determine which hand shapes beat other hand shapes: scissors beat paper, paper beats rock, and rock beats scissors. Due to the circular (also called “non-transitive”) nature of the relationship between the hand shapes, there is not one choice that has any advantage over the others. See Figure \(\PageIndex{1}\). Therefore, the essential nature of gameplay is based on attempting to guess what shape the opposing player is going to choose. This is where randomness and strategy enter the game. An important aspect of strategy is that the game is usually replicated three times, with the majority winner being declared winner of the set of games.

Figure \(\PageIndex{1}\): The non-transitive nature of how the hand shapes are related in the game rock-paper-scissors (public domain image).

When playing another individual in a game of rock-paper-scissors there are several strategies that can be used. You could, for example, always choose the same hand sign, such as rock. Overall, this is not a good strategy, particularly if known to the opponent, who could easily beat you every time by simply always choosing paper. Another option is to always choose the hand sign randomly. If the hand sign is chosen randomly so that each possible sign is equally likely to be chosen, then this strategy is decent. The advantage of this type of strategy is that your opponent cannot use any strategy to try to beat you because they cannot predict what you are going to do and, when using this strategy, you are just as likely to either win or lose. The last strategy is to use any behavioral information you may have about the tendencies of your opponent. This may be from previous games, or from what you observe in the first one or two plays of a set of games. If this strategy can be used effectively, your chance of winning can be increased—but this also depends on whether the other player is using a similar strategy. For example, an opponent may anticipate your strategy and make a counterintuitive play to nullify the information you have about them.

If your opponent is using either of the two latter strategies, you might not be able to predict what they will do on any play of the game. From your viewpoint, this is an example of how an element of randomness is introduced into the game. Because you cannot predict with certainty what your opponent will do, their play can be considered as random, and your strategy must take this into account. Further, the outcome of the game can be considered as random as any observer will not be able to predict the outcome of the game with certainty. It is this unpredictability the keeps the game interesting. In fact, theoretical investigations of this game have been shown to have important applications, for example, in biology (Sinervo and Lively 1996; Kerr et al. 2002; Reichenbach et al. 2007; Kirkup and Riley 2004; Menezes 2021; Lewin-Epstein and Hadany 2020; Schreiber and Killingback 2013). This same type of unpredictability is present in most games of strategy, including chess, checkers, and backgammon.

Just as randomness is present in some form or another in the games we play, randomness flows throughout our lives. There are few days where we can exactly predict what will happen to us. We may arrive late to work or school because of traffic or an accident on the roadways. We might accidentally step wrong on a flight of stairs and walk with a limp for the rest of the day. The person in front of you in the drive-through line at a fast-food restaurant may decide to pay for your meal as a sign of humanity. Humans thrive on and live with randomness, but we often have a difficult time describing exactly what it is. What makes something random?