Mathematics of motivation

When Ivan Pavlov and his dogs led to the discovery of learned behaviour through repeated exposure, and Edward Thorndike discovered the Law of Effect that stated that rewarded behaviours tended to increase, many psychologists were impelled to separate psychology from armchair introspection and formulated their theories as mathematical formulas.

• The Drive x Habit Theory. Clark Hull's formula was sEr = D x sHr, which states that excitatory tendency (E) is the result of the drive (D) combined with the habit (H). The drive is nonspecific, such as hunger or thirst. The habit, however, depends on the stimulus (s) and response (r). But the theory turned out to be wrong and even opposite in many cases. 
• Expectation x Value Theory. Drawing on ideas in economics and game theory, Edward Tolman and Kurt Lewis formulated an alternative account by evaluating motivation based on expectations. Tolman expressed the ideas as the mathematical formula: Subjective Expected Utility = Probability1 * Utility1 + P2U2 + P3U3 + … where subjective expected utility of an action equalled the motivation to act. But, if you expect a reward, why act and not simply passively wait for the expected reward?