Described in this way, the predictions of these two theories are difficult to distinguish. Although they make their predictions for different reasons, both theories seem to predict the same general result: punishment will cause a decrease in the punished behavior. Once they are translated into mathematical form, however, the different predictions of the two theories can be seen more easily. Deluty (1976) andde Villiers (1977, 1980) developed two different quantitative models of punishment, which can be viewed as mathematical versions of the avoidance theory of punishment and the negative law of effect, respectively. Both models begin with Herrnstein’s (1961) matching law, but then proceed in different directions.

In its simplest form, the matching law can be written as follows:

1where

B_{1}andB_{2}are the rates of response on reinforcement schedules 1 and 2, andR_{1}andR_{2}are the rates of reinforcement on these two schedules. This equation has often been applied to choice situations in which the two alternatives are variable-interval (VI) schedules of food reinforcement. Imagine that a pigeon responds on two keys, with Key 1 delivering 75 reinforcers per hour and Key 2 delivering 25 reinforcers per hour, so Equation 1 predicts that the pigeon will make 75% of its responses on Key 1. Now suppose that in addition to producing food, responses on both keys begin to deliver punishers (electric shocks) at a rate of 20 shocks per hour for each key. How can Equation 1 be expanded to deal with this situation?According to de Villiers (1977), if punishment is the opposite of reinforcement, as the negative law of effect states, then the punishers delivered by each alternative should be subtracted from the reinforcers delivered by that alternative:

2where

P_{1}andP_{2}are the rates of punishment on the two keys.In contrast, Deluty (1976) took the view that punishing one response increases the reinforcement for other responses, as proposed by the avoidance theory of punishment. Therefore, in his equation, the punishers for one alternative are added to the reinforcers for the other alternative:

3To keep this example simple, one shock is given the same weight as one food delivery, but both models could easily give food and shock different weights by multiplying

P_{1}andP_{2}by some constant other than 1. Using such a constant would not change the general conclusions presented here. In this example, withR_{1}= 75,R_{2}= 25, andP_{1}=P_{2}= 20, Equation 2 predicts that the percentage of responses on Key 1 should increase from 75% to 92% when the shocks are added to both keys. Conversely, Equation 3 predicts that the percentage of responses on Key 1 should decrease to 68% when the shocks are added. In an experiment with pigeons, de Villiers (1980) found that preference for the key that delivered more reinforcers increased when shocks were added to both keys with equal frequency. This result therefore favors the predictions of Equation 2 over those of Equation 3.It should be clear that the issue here is more fundamental than simply whether a plus sign or a minus sign should be used in an equation. These two models are based on two very different conceptions of how punishment exerts its effects on behavior. The experimental evidence suggests that punishment exerts its effect by weakening the target behavior, as the negative law of effect stipulates, not by strengthening alternative behaviors, as the avoidance theory proposes. This example illustrates how two psychological theories that seem to make similar predictions when stated verbally actually may make very different predictions when they are presented in mathematical form.

Mathematical Models and the Experimental Analysis of BehaviorCorrespondence should be addressed to James E. Mazur, Psychology Department, Southern Connecticut State University, New Haven, Connecticut 06515, e-mail: mazurj1@southernct.eduReceived July 21, 2005; Accepted October 3, 2005.

In a commentary about some competing mathematical models of timing, Killeen (1999) wrote: “If you think models are about the truth, or that there is a best timing model, then you are in trouble. There is no best model, any more than there is a best car model or a best swimsuit model, even though each of us may have our favorites. It all depends on what you want to do with the model” (p. 275). Those who do not enjoy studying mathematical models might take this statement (from a preeminent mathematical modeler) as an excuse to avoid them. Why bother putting in the time and effort to understand current mathematical models of behavior when there is no best model, and when they all have their weaknesses and limitations? Killeen addresses this issue by asserting that “all understanding involves models—reference to systems that exist in a different domain than the thing studied. Loose models make vague reference to ambiguous and ad hoc causes. Tighter models are more careful about definitions and avoid gratuitous entities. Models of phenomena are not causes of phenomena; they are descriptions of hypothetical structures or functions that aid explanation, prediction, and control” (p. 276).