Statistical Tools VI: Pinball Machines

"Because chaos is deterministic, or nearly so, games of chance should not be expected to provide us with simple examples, but games that appear to involve chance ought to be able to take their place. Among the devices that can produce chaos, the one that is nearest of kin to the coin or the deck of cards may well be the pinball machine. It should be an old-fashioned one, with no flippers or flashing lights, and with nothing but simple pins to disturb the free roll of the ball until it scores or becomes dead." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Dynamical systems that vary continuously, like the pendulum and the rolling rock, and evidently the pinball machine when a ball’s complete motion is considered, are technically known as flows. The mathematical tool for handling a flow is the differential equation. A system of differential equations amounts to a set of formulas that together express the rates at which all of the variables are currently changing, in terms of the current values of the variables." (Edward N Lorenz, "The Essence of Chaos", 1993)

"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)

"In a complex society, individuals, organizations, and states require a high degree of confidence - even if it is misplaced - in the short-term future and a reasonable degree of confidence about the longer term. In its absence they could not commit themselves to decisions, investments, and policies. Like nudging the frame of a pinball machine to influence the path of the ball, we cope with the dilemma of uncertainty by doing what we can to make our expectations of the future self-fulfilling. We seek to control the social and physical worlds not only to make them more predictable but to reduce the likelihood of disruptive and damaging shocks (e.g., floods, epidemics, stock market crashes, foreign attacks). Our fallback strategy is denial." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)