Behavioral contingency analysis

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Behavioral contingency analysis

Francis Mechner*
The Mechner Foundation, 200 Central Park South, 18E, New York, NY 10019, United States

Received 4 September 2007; accepted 17 January 2008

Abstract

This paper presents a formal symbolic language, with its own specialized vocabulary and grammar, for codifying any behavioral contingency, including the complex multiparty contingencies encountered in law, economics, business, public affairs, sociology, education, and psychotherapy. This language specifies the “if, then” and temporal relationships between acts and their consequences for the parties involved. It provides for the notation of the probabilities, magnitudes, positive or negative valences, or time delays of the consequences for the parties, and for the parties that would perceive, misperceive, not perceive, predict, mispredict, or not predict events. The language’s fractal-like hierarchical and recursive grammar provides for the flexible combination and permutation of the modifiers of the language’s four nouns: acts, consequences, time intervals, and agents of acts; and its four verbs: consequate, prevent, perceive, and predict-thereby giving the language the ability to describe and codify various nuances of such complex contingencies as fraud, betting, blackmail, various types of games, theft, crime and punishment, contracts, family dynamics, racing, competition, mutual deterrence, feuding, bargaining, deception, borrowing, insurance, elections, global warming, tipping for service, vigilance, sexual overtures, decision making, and mistaken identity. Applications to the management of practical situations and techniques for doing so, as well as applications in current behavior analysis research and neuroscience, are discussed.

© 2008 Elsevier B.V. All rights reserved.

Keywords: Formal contingency language; Behavioral economics; Competition and cooperation; Cognitive-behavior therapy; Neuroscience; Behavior notation and codification; Social interaction codification; Operations research; Simulation

1. Behavioral contingencies

1.1. What are they?

Behavioral contingencies ${ }^{1}$ are the ubiquitous “if, then” situations that influence what people do and do not do. The “if” part specifies some behavior; the “then” part some consequence for the involved party(ies): if you drop a glass (the behavior), it may break (the consequence), and a price label on a product in a store states that if you pay that price, you get the product. The price label contingency might have effects on customer behavior, sales of that product, and competitive pricing by other stores. Further contingencies may be implicit, e.g., if you wait too long, the product may no longer be available.

^[1]Behavioral contingencies exist independently of any behavior or dynamics that may occur as a result of their existence. The dropped glass would break if dropped even if no one ever dropped it or picked it up, and the price label contingency is in effect even if no one ever sees the price label or buys anything. In general, however, a behavioral contingency is of practical interest because of its possible behavioral effects.

1.2. Behavioral contingencies in human affairs

Behavioral contingencies, often rather complex ones, are a common element of all the behavioral sciences and are at the root of the behavioral phenomena that are of concern in such diverse realms as law, business, education, economics, management, public affairs, therapy, the social sciences, child management and everyday interactions between people.

1.2.1. Prominent examples

• Laws consist, in general, of “if, then” statements of the form, "If a person does or doesn’t perform certain

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1. • Tel.: +1212765 7899/718 5442001.

   E-mail address: fmechner@panix.com.
   ${ }^{1}$ B.F. Skinner (1969) applied the term behavioral contingency to the “if, then” conditional relationship, “If a response, then a consequence” originally to describe the basic paradigm of operant conditioning, “If a response, then a consequence.” The more traditional usage - temporal contiguity regardless of conditionality - is not the sense in which the term is used here. ↩︎

acts, certain consequences for that person shall follow". Laws are, in essence, behavioral contingency statements intended to regulate, modify, or influence behavior in a society.

• Education systems involve the behavioral contingencies that govern the interactions of students, teachers, parents, administrators, unions, textbook publishers, and members of the community.
• When business managers seek to improve operations by means of incentive compensation systems, work flow systems, and safety practices, they operate on the behavioral contingencies involved.
• The rules of games, ranging from tic-tac-toe to baseball, bridge, poker, or chess, are behavioral contingencies that determine how the games are played.
• Many everyday interactions between people involve behavioral contingency statements of the general type “If you do A, I will do B,” including promises, requests, enticements, and threats, sometimes with reference to other parties, time periods, probabilities, or other qualifications.

Unlike the other major determiners of behavior - the personal histories of the involved parties and the immutable realities of physics and biology - behavioral contingencies can be modified and designed.

1.3. The roles of signals and experience

The effects of a prevailing behavioral contingency depend on the individual’s history or previous experience with that contingency or with related ones.

Example. The effect of a price tag depends on ability to read and on prior exposure to the product and to price tags. For verbal individuals, a contingency can be communicated by verbal descriptions, including “rules”. For non-verbal individuals, a contingency can be communicated by other kinds of signals ${ }^{2}$ and the information conveyed about a contingency can be accurate or inaccurate, complete or incomplete, well understood or only partially understood.

1.4. The value of a formal behavioral contingency language

The present paper presents a formal language, with its own specialized vocabulary, grammar, and syntax, for the systematic codification and analysis of any behavioral contingency. The practical application of this language falls into the province of behavioral technology, ${ }^{3}$ but given the central place of behavioral contingencies in behavioral science and its applications,

^[1]behavioral contingencies are also worthy of study in their own right. ${ }^{4}$

It is true that formal codifications of behavioral contingencies can often also be expressed, sometimes with less effort, by means of natural language, as can the codifications of other formal symbolic languages like those of mathematics or symbolic logic. But in the case of complex behavioral contingencies, formal codification, with the attendant identification of key variables and features, is often a necessary precondition for their systematic study and analysis, whether by means of experiments with living subjects, computer simulation, operations research, or other methods.

Formal symbolic languages, including the present one, also provide the advantage of cutting across all natural languages (universality), with codifications that are succinct and avoid ambiguity. They can reveal relationships among elements that would not be revealed as clearly by natural language descriptions, and in science they can aid in the development of classification systems and theories.

While the antecedent of the present language (Mechner, 1959) was developed as a tool for the specification of independent variables and experimental procedures (Mechner and Latranyi, 1963; Mechner et al., 1963), and has found pedagogic uses like those cited in Section 1.5, the present expanded language is designed for the codification of any behavioral contingency, including complex multiparty contingencies. The language is also well suited for the specification of the situations and contingencies that are the subject matter of behavioral economics (Camerer and Lowenstein, 2003; Becker, 1976, 1995, 1997), including some (described in Section 5) that have become the subject of current behavior analysis research.

Clearly, this type of language cannot generate new knowledge about behavior or make empirical assertions of any kind. Behavioral contingencies do not describe or predict what behavior will actually occur-they only specify consequences for the parties involved if some specified behavior occurs. From the standpoint of experimental science, behavioral contingencies have the status of independent variables, and if used as such in experiments, their empirically observed behavioral effects would have the status of dependent variables.

1.5. Related work

While this paper may be the first effort to develop a general language for codifying any behavioral contingency, specialpurpose notation systems for behavioral contingencies have been proposed in the past. ${ }^{5}$ The original language and notation sys-

^[2]

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1. ${ }^{2}$ The technical term for “signal” is “discriminative stimulus.” The term “stimulus” is avoided here because of its connotation of stimulus-response and reflex phenomena.
   ${ }^{3}$ The application of behavioral contingencies is key in such applied areas as educational technology, behavior management, clinical interventions, and business applications. Sidman (2004) pointed out that the application of behavior analysis to human affairs has still fallen far short of the potential. ↩︎

2. ${ }^{4}$ The distinction between independent and dependent contingencies proposed in Weingarten and Mechner (1966), and hierarchical contingency structures, may be examples of types of issues that a science of contingencies might address.
   ${ }^{5}$ Some behavioral notation systems or paradigms (e.g., Chisholm and Cook, 1995; Findley, 1962; Goldwater and Acker, 1995; Keller and Schoenfeld, 1950; Malott et al., 1993; Mattaini, 1995) have been used to codify behavior analysis principles for instructional purposes and for describing behavioral phenomena that actually occur, as opposed to behavioral contingencies as defined here. These behavioral notation systems have no direct relevance to the theme of the present paper. ↩︎

tem for codifying behavioral contingencies (Mechner, 1959) dealt with examples drawn mainly from the experimental psychology literature on schedules of reinforcement, escape and avoidance behavior, discrimination learning, and maze learning. Weingarten and Mechner (1966) took an initial step in extending the language to contingencies involving multiple parties.

Skinner (1958) proposed a notation for describing certain specific schedules of reinforcement graphically. Snapper adapted “state notation” (Snapper et al., 1982; Stephens and van Haaren, 1977; Tanaka and Nomura, 1993) for the computer control of certain types of behavioral experiments. In a detailed comparison of the Mechner (1959) notation system and state notation, Michael reported his use of the Mechner language in the undergraduate and graduate courses he taught at Arizona State University (Michael and Shafer, 1995). Michael also used the Mechner (1959) language for his laboratory manual (Michael, 1963), and it was used in the textbook Behavior Analysis and Learning (Pierce and Epling, 1995). A strippeddown version of it was used in a laboratory instruction manual (Millenson and Leslie, 1979).

2. The language’s vocabulary, grammar, and syntax

2.1. The elements of the language

This paper attempts to show how the proposed language’s sparse vocabulary and simple grammar can describe a wide range of complex behavioral contingencies in diverse domains.

The units of any formal symbolic language reflect the conceptualizations of its field: the basic units of chemical notation are atoms; those of musical notation are notes; and those of the present language are acts $A,{ }^{6}$ time intervals $T$, consequences $C,{ }^{7}$ and the involved parties (designated here by arbitrarily chosen lower case letters).

These are the language’s six basic elements:

1. $A \rightarrow$, read as, “If act $A$ occurs then …”. Thus, every $A$ is understood to be preceded by an implied “If”.
2. Agent of the $A$ : The agent(s) of an act $A$ are designated by arbitrarily chosen letters placed in front of the $A$ (e.g., $a A$, $b c A) .^{8}$
3. $T \rightarrow$, read as “upon termination of time $T \ldots$…”. The time period $T$ is part of the situation, as opposed to a property of the behavior.
4. A consequence $C$ is any situation or event resulting from an $A \rightarrow$ or a $T \rightarrow$. The term refers broadly to all relevant aspects of the prevailing environment.

^[1]5. Prevention of a consequence: A consequence is prevented when a horizontal arrow leading to it is cut by a vertical arrow originating from another $A$ or $T$, like this: $\square^{9}$.
6. Simultaneity of onset: A bracket enclosing vertically listed $A s, T s$, or $C$ s indicates simultaneity of onset of the listed conditions. For instance, $\mathbf{C} \quad$ would be read as “When $C$ is present then if $A$, then …”. The order in which the conditions are listed has no significance.

Only $A$ s or $T$ s can consequate ${ }^{9} C$ s or further $A \rightarrow$ or $T \rightarrow$ contingencies. Thus, arrows can never emanate from a $C$, only from an $A$ or a $T$.

2.2. Architecture of the grammar: the four quadrants

Every element of the language can have up to four modifier categories, as shown in the four quadrants of each symbol in Fig. 1. Each quadrant is allocated to one of the four modifier categories in a consistent way. The modifiers can, in turn, have the same modifiers in the corresponding quadrants, in a theoretically infinite hierarchical and recursive regress, fractal mode. ${ }^{10}$

The subscript (lower right quadrant, e.g., the numeral 5 in $a A_{5}$ ) is an arbitrary numeral indexed to a legend where the referenced entity is described (see Section 2.3).

The superscript (upper right quadrant) is allocated to the entity’s attributes, which can be viewed as the language’s adjectives and adverbs-probability, magnitude, valence, ${ }^{11}$ duration, or variability, as explained in more detail in Section 2.4.

The pre-subscript (lower left quadrant) of an entity indicates the party(ies) that would perceive ${ }^{12}$ the entity (see Sections 2.5-2.8), and the pre-superscript (upper left quadrant) indicates the party(ies) that would predict ${ }^{13}$ the entity (see Section 2.9). One of the theses of this paper is that the verbs perceive and predict are a unique (necessary and sufficient) and parsimonious key to the generality of the language’s applicability.

2.3. Subscript: indexing to a legend

Arbitrary subscripts (usually numerals) can index the diagram’s symbols to descriptions in a legend. They can also provide differentiation between different $A s, C$ s, or other enti-

^[2]

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1. ${ }^{6}$ The term “act” is used here, rather than “response,” which has the undesired connotation of a stimulus-response reflex, and usually of a brief response performed by a single individual.
   ${ }^{7}$ The term “consequence” is used in favor of the traditional term “stimulus,” again to avoid the “stimulus-response” connotation. Consideration was given to using 5 for “situation,” but the term consequence conforms better to behavioranalytic thinking.
   ${ }^{8}$ The term “agent” or “party” refers to an individual, group, team, company, populace, or country that would perform the designated act. ↩︎

2. ${ }^{9}$ The verb conseguate has been used in behavior analysis and is useful in the present context. The alternative verbs “produce,” “generate,” “result in,” or “cause” tend to carry undesired epistemological baggage.
   ${ }^{10}$ These are also characteristics of such formal languages as symbolic logic and mathematics, as well as of most natural languages.
   ${ }^{11}$ A $C$ 's valence indicates whether the $C$ is rewarding or punishing, desirable or undesirable, beneficial or harmful, positively or negatively reinforcing.
   ${ }^{12}$ The verb “perceive” is understood to encompass all of the learned or genetically encoded discriminative responses to the stimuli or situation comprised by the perceived entity.
   ${ }^{13}$ The verb “predict,” as used here, refers only to the behavioral effects of the party’s prior contact and experience with the contingency involved, or with the signals (verbal or non-verbal discriminative stimuli) that were previously associated with the contingency. No other connotations are implied. ↩︎

Fig. 1. Diagrammatic representation of the hierarchical/recursive quadrant-based grammar of the behavioral contingency language. The center box containing the language’s four nouns has four quadrants, shown as shapes that represent modifier entities (e.g., perceive, predict, attributes, index numbers to the legend), each of which in turn has its own analogous quadrants.
ties. For example, to indicate that the duration of $C_{1}$ is $T$, a new consequence $C_{2}$ is shown replacing $C_{1} \cdot \mathrm{C}_{1}$

When descriptions consist of one or two words, it may be convenient to show those words in the subscript positions within the diagram, while lengthier descriptions can be provided in the legend. In $a_{7} A_{5}$, the legend might describe the nature of party $a$ as a particular individual, team, population, or government under the arbitrary numeral 7, and the nature of act $A$ under the numeral 5. Without subscripts and legends, every diagram would represent the infinite number of contingencies that have the same structure. Specificity is achieved only by the use of descriptive subscripts.

2.4. Attributes of entities

2.4.1. Valence

The notation $C^{a+}$ or $C^{a-}$ means that $C$ has a positive or negative valence for party $a . C^{a+, b-}$ means that the valence is positive for $a$ and negative for $b$, and $C^{a b-}$ means that the negative valence applies to both. $C^{a t, t, 1}$ means that the valence for $a$ can be either positive or negative.

2.4.2. Magnitude $M$

The magnitude attribute for act $A$, as in $A^{M}$, can refer to such intensive dimensions as effort, force, loudness, duration, or an amount of money. When $M$ refers to the act’s duration, a lower case $t$ replaces the $M$. The duration $t$ of an act is measured from its beginning to its termination. In $A_{\text {start }} \rightarrow A_{\text {end }}^{t}, A^{t}$ is read as “end of act A’s duration.” The term $A_{\text {start }}$ can be omitted, since every act that ends must also have started.

The duration attribute is particularly significant when an act is repetitive and can continue or end at any time. Examples of such acts, often called activities, are walking, running, waiting, foraging, working, or practicing a skill. In behavior research, a latency or inter-response time would be codified as an act whose terminating event is $A^{t}$.

The magnitude of a $C$ would usually refer to the magnitude of its positive or negative valence, but can also refer to any scalable dimension like value or amount of money, or to some qualitative dimension.

For time periods, $T^{M}$ refers to the duration of $T$.
A magnitude of a consequence or of a valence that changes as a function of some variable $x$ can be represented

by $M=f(x)$ ) shown in the attribute position of the consequence.

2.4.3. Probability

A lower case $p$, or an actual value of a probability shown in an entity’s upper right quadrant, indicates the probability, in the analyst’s view, that the entity would occur.

2.4.4. Variability

$T^{c}$ or $M^{c}$ indicate that the modified entity is variable.

2.4.5. Negation

The tilde sign $\sim$, representing the negation attribute “not”, is placed above the entity’s symbol. ${ }^{14}$

Although these attributes are fully sufficient for codifying the wide and diverse range of situations analyzed in this paper, additional attributes may, in the future, be required to meet challenges presented by specialized applications in other fields like law, business, economics, sociology, public affairs, or psychotherapy. ${ }^{15}$

2.5. The notation of perception

In $A \rightarrow{ }_{a b} C$, the lower left quadrant of $C$ indicates that both parties $a$ and $b$ would ${ }^{16}$ perceive the consequence $C$ of act $A$. To show that party $a$ would not perceive $C$ but that $b$ would, the notation is $A \rightarrow{ }_{a b} C$, with the $a$ having a tilde sign above it meaning "not $a$ ".
$a A \rightarrow{ }_{a b} T \rightarrow{ }_{a b} C$ is read as, “If party $a$ does $A$, then $C$ occurs after time lag $T$. The $C$ would be perceived by both $a$ and $b$, and the time lag would be perceived by $b$ but not by $a$.” The $T$ would be shown without any pre-subscripts if the analyst considers perception of the time lag not relevant to his focus.

The contingency $a A_{1} \rightarrow b A_{2} \rightarrow$ is read as “If $a A_{1} \rightarrow$ then if $b A_{2} \rightarrow$ then …”. In the case where the desired meaning is “If $a A_{1} \rightarrow$ then $b A_{2}$ would occur,” the $b A_{2}$ term would have the status of a consequence $C$.

^[1]

2.6. The notation of misperception

${ }_{a} C$ implies that party $a$ 's perception of $C$ would be “correct” in the sense that it corresponds to the analyst’s belief, that being the sense in which the terms “correct” or “actual” will be used throughout. If the analyst wishes to show that the perception would be a “misperception” or idiosyncratic, ${ }^{17}$ the misperceiving party’s designator, still shown in the lower left quadrant, would have the letter $x$ in its own upper right (attribute) quadrant. For example, $a^{x} b^{\mathrm{C}}$ means that $a$ would misperceive $C$ and that $b$ would perceive it correctly. To elaborate on the misperception, the analyst could explain under, say, the legend’s numeral $5, \boldsymbol{a}_{5}^{\boldsymbol{b}} \boldsymbol{b}^{\boldsymbol{C}}$ that $a$ would, for example, misperceive a neutral comment as hostile. The term “misperceives” is used broadly here for any perception that is idiosyncratic or “subjective,” in the sense that it deviates from the analyst’s belief.

When time is an attribute of an act (i.e., its duration) rather than a feature of the situation, the party’s perception, misperception, or non-perception of the act’s duration would be shown as the pre-subscript of the $t$, like this: $\boldsymbol{a} \boldsymbol{A}^{\boldsymbol{a}^{t}} \boldsymbol{t}_{\text {or }} \boldsymbol{a} \boldsymbol{A}^{\boldsymbol{a}^{t}}$. The latter diagram could mean " $a$ not keeping track of how long its act $A$ was taking."

2.7. Perception, misperception, and non-perception of valences

If a party would perceive (or misperceive or not perceive) a consequence $C$, would that party then also perceive (or misperceive or not perceive) that same $C$ 's valence? The answer is that it would, based on the distributivity rule which states that any perception, non-perception, or misperception of the $C$ applies also to that $C$ 's valence and any of its other attributes.

Example. $a A \rightarrow{ }_{a b} C^{b \cdots a+}$ could mean that $a$ would harm $b$ and benefit himself, with both $a$ and $b$ perceiving this consequence and its valences. If either $a$ or $b$ would not perceive either or both of these valences, that party’s designator would be shown with a tilde over it, in the lower left quadrant of the non-perceived valence term, to override the distributivity rule. For example, to show that $a$ would not perceive the $C$ 's valence $b-$ in spite of perceiving the $C$ itself, the $b-$ valence symbol would need the pre-subscript $\tilde{a}, \boldsymbol{a} \boldsymbol{b}^{\mathrm{C}}{ }_{\tilde{a} b}, \boldsymbol{a}+{ }_{a b} C^{b-}$ would mean that $a$ would perceive neither $C$ nor its valence.

To show that a party would misperceive a $C$ 's valence but not the $C$ itself, that party’s designator, with the letter $x$ in its upper right quadrant, is shown in the valence’s lower left quadrant. Familiar examples are Adam and Eve, or Snow White, who might have perceived the apple correctly but misperceived its valence.

^[2]

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1. ${ }^{14}$ It is more convenient to place it above the entity than in its upper right quadrant, which is where it would be placed based on the placement rule for attributes. In symbolic logic it is usually placed in front of the character, but here this placement would have the drawback of the multiplicity of meanings (including “approximately”) that such a placement has in various other contexts.
   ${ }^{15}$ It may be possible, for example, to extend the language to the description of the environmental contingencies that result in speciation and evolution-the adaptation of species to contingencies imposed by the environment. The consequences would then be characteristics like camouflage or immunity, and the valences might pertain to competitive advantages or disadvantages related to reproduction and survival. The language may also be applicable to the contingencies involved in the evolution of cultural practices (Glenn and Madden, 1995; Glenn, 2003, 2004), and perhaps to any system involving conseqation with feedback.
   ${ }^{16}$ The term “would” or “will” is used throughout, to maintain consistency with the conditional “if, then” logic of contingency statements, e.g., “If act $A$, then party(ies) $a, b, c$ would or will perceive or predict a designated entity.” A statement of the form that $a, b$, or $c$ actually do perceive or predict an entity, without the qualification of conditionality, would be inconsistent with the definition of a behavioral contingency statement. ↩︎

2. ${ }^{17}$ In behavior analysis terminology, misperception or idiosyncratic perception means responding to a stimulus (here termed consequence) as if it were a different stimulus. ↩︎

The valence shown is the one the analyst considers the real one. The precise nature of the misperception he has in mind can be described in a legend. Here are some combinations (remember that the $C$ 's pre-subscript applies not only to the $C$ but also to its attributes):
$a^{\mathrm{C}^{\mathrm{a}-}}$ a perceives both the $C$ and its negative valence for itself.
$a^{\mathrm{C}^{\mathrm{a} a}-}$ a perceives the $C$ and misperceives its negative valence.
$a^{\mathrm{a}^{\mathrm{a}}} \mathrm{C}^{\mathrm{a}-}$ a misperceives both the $C$ and its negative valence.
$a^{\mathrm{C}^{\mathrm{a}}} \mathrm{C}^{\mathrm{a}-}$ a perceives the $C$ but not its negative valence.
$\hat{a}^{\mathrm{C}^{\mathrm{a}}}-a$ does not perceive either the $C$ or its negative valence.

2.8. Misperception of a party: who done it?

$b^{a} A_{\text {means that } b}$ perceives that $a$ is the agent. $b^{a} a^{A}$ means that $b$ misperceives the fact that $a$ is the agent. This distinction could come up in false accusations or in criminal law where there can be divergent assertions regarding the agent of an act. Again, the nature of the misperception could be described in a legend. Analogously, the notation ${ }_{a} T$ means that $a$ would misperceive $T$, and the notation ${ }_{a} T$ means that $a$ would not perceive $T$ at all.

2.9. Prediction

The term “predict,” as used here, encompasses all of the behavioral effects of the party’s prior experience with similar contingencies, or with signals (called verbal or non-verbal discriminative stimuli by behavior analysts) that were previously associated with similar contingencies. Signals consisting of verbal statements of contingencies are sometimes called rules. When the analyst indicates that a party would predict a consequence, he therefore reflects his beliefs regarding the party’s relevant history.

A signal that might occasion a party’s prediction of a consequence has the status of a $C$.

2.10. The notation of “predict”

The party(ies) that would predict a $C$ are designated in the $C$ 's upper left quadrant, like this: ${ }^{a} C . \quad a A \rightarrow{ }_{a}^{a} C$ means that $a$ would perceive $C$ when it occurs and would also predict it. $a A \rightarrow{ }_{a}^{a} C_{\text {means that }} a$ would predict $C$ but would not perceive it (note the $\hat{a}$ in $C$ 's lower left quadrant).

Examples. Committing suicide, predicting the consequence of sending a letter without perceiving that consequence when the letter reaches its destination, predicting the consequences of issuing an order without perceiving the order being carried out, dropping a bomb on a target. The reverse, $a A \rightarrow{ }_{a}^{a} C$, means
that $a$ would perceive $C$ when it occurs but would not have predicted it. Here one might say that $C$ would surprise $a$.

Depending on the desired emphasis and focus, the analyst may or may not indicate the party(ies) that do or do not perceive or predict a particular consequence.

2.11. Perceiving or predicting another party’s perception or prediction

$a A \rightarrow{ }^{b} C^{a-}$ means that party $b$ would predict that $a$ would hurt himself. The notation ${ }^{a} \mathrm{C}^{a-}$ means that $a$ would mispredict it. To show that $b$ would also perceive that $a$ was mispredicting $C^{a-}$, the $b$ would be shown in the lower left quadrant of the $a^{x}$, like this ${ }^{a^{a}} \mathrm{C}^{a-}$. The diagram ${ }^{a^{a}}, b^{a}$ would show that $b$ would, in addition, predict correctly that $a$ 's act would be hurtful to $a$.

When the verb predict is applied to a probability, the meaning is similar to that of the verb “estimate.” The expression ${ }^{a} p_{1}$, for instance, would be read as “party a’s prediction/estimation of $p_{1}$.”

2.12. Nuances of meaning involving probabilities and predictions

This diagram could mean that if $a$ asks $\left(a A_{1}\right) b$ for a favor, then if $b$ complies $\left(b A_{2}\right)$ with the request ${ }_{b} C_{4}$, the consequence $a A_{1} \rightarrow{ }_{b}^{b} C_{4} \rightarrow C_{3}$
would be $C_{3}$. The analyst can express the following nuances regarding $b A_{2}$ :
$\left(b A_{2}\right)^{p}:$ The probability of $b A_{2}$ occurring is less than one.
$b^{a} A_{2}: p$ is the probability that $b$ (rather than another party) would do $A_{2}$.
${ }^{a}\left(b A_{2}\right): a$ would predict $b A_{2}$. Replacing the $a$ with $a^{p}$ would mean that the analyst considers the probability to be $p$ that $a$ would predict $b A_{2}$.

2.13. Why the terms “aware,” “intend,” and “expect” are not used

The term “aware,” when applied to a contingency, is imprecise in that it could apply equally to the act only, to the consequence only, or to the party’s prediction of the consequence only, and when applied to the act or to the consequence it does not distinguish between its perception and prediction. The terms perceive and predict, on the other hand, are precise in that they can be applied with pinpoint precision to acts, consequences, and attributes, and unlike the term “aware,” they can be modified by a probability or as a misperception or misprediction.

The terms “intend” or “expect” ${ }^{18}$ are also imprecise in that they do not tell us if an intent or expectation is still present if the consequence is modified by a probability or delayed by a time interval. The contingency language avoids such ambigu-

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1. ${ }^{18}$ For other discussions of related terminological issues see Chisholm (1957), Baum and Heath (1992), Foxall (2007) and Hineline (2003). ↩︎

ity. It is able to express these concepts fully, along with their many possible nuances of meaning, by specifying that the act’s agent predicts, mispredicts, or does not predict the act’s consequence, with further nuances provided by recursively assigned modifiers.

2.14. Vertical arrows that terminate and change contingencies

A vertical arrow (initiated by an $A$ or a $T$ ) cutting a horizontal arrow, like this, $\square$ terminates the contingency represented by that horizontal arrow. This diagram could describe the contingency involved in catching a plane or train or in meeting a deadline.

To show that a vertical arrow not only prevents a consequence but also creates a new one, the vertical arrow is shown pointing to the new consequence $C_{2}$, as in the diagram on the left.

Note that at the intersection of the arrows, only the contingency represented by the horizontal arrow is terminated or changed. The horizontal arrow does not cut or affect the contingency represented by the vertical arrow.

2.15. Recycling contingencies

To show that a contingency remains in effect and can repeat, a recycling arrow is used. For example, when a CD has been played, it can be played again. To indicate that an act can recycle $n$ times, the $n$ is written above the recycling arrow: $\overbrace{}^{n} \mathrm{~A}_{\rightarrow}^{n} \mathrm{C}$.

3. Other features of the contingency language

3.1. Distributivity relations among the quadrants

The default rule is that an entity’s subscript (the numeral that indexes a legend item) is distributive across all of the entity’s modifiers, but the analyst may sometimes want to use separate subscripts for individual modifiers.

The verbs perceive and predict also extend to the modified entity’s attributes (shown in the upper right quadrant)-i.e., any entity that is perceived or predicted includes all of that entity’s attributes, these being similarly perceived or predicted.
For example, $b^{a} \hat{b} \mathrm{C}^{a+}$
means that $b$ would perceive that $p$ is the probability that $a$ would perceive $C$ and $C$ 's valence.

All other relations are non-distributive. For example, in $\stackrel{0}{a} \mathrm{C}$ $a$ would not also perceive the fact that $b$ would predict $C$, that is, the perception by $a$ would not be distributive to the upper left quadrant, and vice versa. To show that $a$ would also perceive that $b$ would predict $C$, the $a$ would have to be shown again in the $b$ 's lower left quadrant.

3.2. The grammar of consequences

A general default feature is that only one consequence $C$ is present at one time, because every $C$ is presumed to include all of the relevant features of the prevailing situation. Thus, any change of $C$ is a new, again all-inclusive, $C$ produced by a further $A$ or $T$.

Suppose, for instance, that $A_{1}$ consequates $C_{\text {tone }}$, and $C_{2}$ $\mathrm{A}_{1} \rightarrow \mathrm{~T}_{\rightarrow} \mathrm{C}_{2}$
$\mathrm{C}_{\text {tone }}$ is a light coming on subsequently. $C_{2}$ then replaces $C_{\text {tone }}$ (tone alone), and may then consist either of both the light and the tone together or of the light alone. The legend would state which, if the analyst considers the distinction relevant. In either case, $C_{\text {tone }}$ (tone alone), having been replaced by $C_{2}$, is no longer present.

3.3. When all consequences are represented by one $C$

The analyst would always narrow his analysis to those consequences that are relevant to his focus and would represent these with a single $C$, notwithstanding the fact that all real-life acts have multiple and innumerable consequences, in the sense of the proverbial wing flap of the butterfly in China ultimately affecting the weather on the other side of the earth. A party performing an act would never perceive or predict all of the act’s consequences.

A mundane example. If I open the refrigerator and pour myself some juice, I would probably predict that I would be drinking juice in a few seconds and then perceive it. I would probably not perceive or predict all of the physical, chemical, and thermal consequences of opening the refrigerator or of the juice on my stomach chemistry.

A weightier example. If a company’s directors close down a factory, they would predict and/or perceive certain of the consequences and their time lags, but not numerous others.

3.4. Diverse consequences: when more than one consequence is shown

On the other hand, when the modifiers of the consequences are heterogeneous and yet relevant to the analyst’s focus, a single $C$ cannot always represent all of them. The analyst would then represent them with more than one $C$ :
$\underset{\leftarrow}{a A_{1}} \rightarrow \underset{\leftarrow}{b} \mathrm{C}_{2}$
Examples. (1) Party $a$ introduces parties $b$ and $c$ to each other. The result would be two consequences $-{ }_{b} C_{2}$ ( $b$ 's perception of the situation that includes party $c$ ), and ${ }_{c} C_{3}$ ( $c$ 's perception of the situation that includes party $b$ ). Also, $C_{2}$ and $C_{3}$ may have different valences for the three parties $a, b$, and $c$, which may also have different predictions or perceptions of those valences; (2) a business executive $a$ assigns a task to $b$ and $c$. When $b$ and $c$ divide the work and each takes a different part of the job, the consequence for each of them would necessarily be different.

In some cases, the analyst may want to distinguish between diverse consequences of a single act.

Example. Shooting at a target and hitting it. In this case, $A_{1}$ is pulling the trigger, $C_{1}$ is the gun firing, and $C_{2}$, is the bullet’s point of impact. A party could perceive and/or predict either of these consequences and not the other.

There are also cases where the analyst may want to $a A_{1}^{D_{2}} \cdot b C_{2}^{a_{4}}$
$b A_{2} \rightarrow C_{4}$ assign different probabilities to the performance of the act and to its consequence occurring (e.g., to the act of shooting at a target and to the probability of hitting it).

In the real world, the flow of acts and their consequences is continuous and seamless. In codifying a situation, exactly where in that flow the analyst chooses to draw the demarcation line between an act and its consequence depends on the desired focus. When the consequence of $a A_{1}$ serves as the cue for party $b$ 's act, this cue can be just the sight or some other consequence of $a$ performing the act.

3.5. Consequation by external agencies

A situation $C$ may be consequated by an external agency $e$ as well as by one of the parties involved in the contingency proper.

Examples of externally consequated $C$ s. The hand that a bridge player is dealt, a test item presented to a test taker, or any situation presented by the physical environment. In these cases, the external agency would be the card dealer, the presenter of the test item, or the physical environment. Acts by such external agencies $e A$ would be shown only if the analyst considers this aspect of the contingency relevant to his intended focus.

The diagram at the left shows that $a$ would perceive that he has an e-mail, ${ }_{e} C_{3}$, sent by an unknown external agency $e A_{1}$.
$e A_{1} \cdot a A_{2} \cdot b C_{4}$
$a C_{3} a A_{5} \rightarrow b C_{6}^{a}$ If $a$ opens the e-mail $\left(a A_{2}\right), a$ will predict and perceive its display $C_{4}$ on the screen, including an attachment. The $a$ shown in $C_{4}$ 's upper left quadrant means that $a$ would predict that opening the e-mail would consequate the screen image ${ }_{e} C_{4}$. This $a$ pertains only to $C_{4}$ and has no bearing on the $a A_{5} \rightarrow C_{6}^{a_{-}}$contingency, namely on whether $a$ would also perceive or predict that if $a A_{5}$ (clicking on the attachment), the computer would be infected with a virus $C_{6}^{a-}$. The $\breve{a}$ designator in the upper left quadrant of ${ }^{a} C_{6}^{a-}$ indicates that $a$ would not predict that opening the attachment would incur a virus. To show that $a$ would predict it, the $a$ would need to be shown in the upper left quadrant, like this, ${ }^{a} C_{6}^{a-}$, without the tilde.

3.6. The notation of uncertainty

Uncertainty is indicated by a question mark after the entity to which the uncertainty applies. The specific nature of the uncertainty, or the reason for it, can be elaborated in the legend. The numeral indexing the legend can be shown in the question mark’s lower right quadrant: $b ? 5$.
${ }^{a} \mathrm{C}$ means that the analyst is uncertain that $a$ would predict $C .\left({ }^{a} ? b\right.$ means that the analyst is uncertain as to whether $a$ would perceive the $C$. When a party, rather than the analyst, is uncertain, the party’s designator is shown in the question mark’s attribute quadrant: $\left(a \rightarrow^{a}\right) b \mathrm{C}$. (The logic underlying this rule is that the uncertain party is like an attribute of the uncertainty.) To show that the analyst is uncertain regarding the correctness of a perception or prediction, the question mark is shown in the party’s attribute quadrant: ${ }^{a} \mathrm{C}$. To show that it is $a$ who is uncertain about the correctness of $a$ 's own prediction, an additional $a$ is shown in the question mark’s attribute quadrant: $a^{p^{a}} \mathrm{C}$. To indicate that the analyst is uncertain as to the valence for $b$ but not for $a$, the notation is $\mathbf{C}^{a_{+}, b(+,-)}$ ?

3.7. Uncertainty and probability

When to use the question mark and when to use the probability notation: a question mark indicates the lack of some existing information where the uncertainty would be eliminated if that information were provided. An indication of a probability, on the other hand, suggests that the event depends on an external process like the tumbling of dice, the order of the cards in the deck, or the results of decision processes of others. In such cases, a second branch of the consequating arrow can show the consequence for the $1-p$ complement of the probability, if doing so is relevant to the focus. A probability can have as a modifier a party’s prediction/estimation (shown as the $p$ 's pre-superscript), but an uncertainty cannot.

Notations of uncertainty or probability are used only when they are relevant to the analyst’s focus. Although all consequences would necessarily be probabilistic to some degree, relatively small uncertainties are usually unimportant in relation to the focus, and the analyst may therefore not indicate them.

3.8. Simplifying assumptions and abbreviations

Formal symbolic statements are inherently stripped down to presumed essentials through the use of simplifying assumptions intended to eliminate features that might mask the intended focus. With that intent, the analyst simplifies the contingencies being represented by selecting some features and omitting others. The result bears the same type of relationship to reallife contingencies that a drawing of an object bears to the real object.

A common simplifying assumption is the omission of the $T$ s representing the time lags between all acts and their consequences. When the time lag is relevant, the contingency might be shown as $A \rightarrow T \rightarrow C$.

Abbreviations provide another way to simplify diagrams. For example, the symbol $C^{o+}$ is an abbreviation. If the analyst considers the specific reasons for the valence being positive for $a$ important, he can elaborate them in the diagram proper or in a separate diagram indexed to the point of insertion.

Examples of reasons. Upon the occurrence of $C, a$ might be able to avert an impending negative consequence or procure a positive consequence by certain further acts.

3.9. “And” relationships: cooperation, contracts, agreements

When a consequence depends on the occurrence of all of two or more events, the symbolic logic symbol $\cap$ for the intersection of sets (the “and” relationship) is used. ${ }^{19}$
Example. Mother says to child, "I will read you a story $\left(C^{+}\right)$ in $5 \min \left(T_{3}\right)$ if you have brushed your teeth $\left(A_{1}\right)$ and gotten into bed $\left(A_{2}\right)$ in those $5 \mathrm{~min} ":\left(A_{1} \cap A_{2} \cap T_{3}\right) \rightarrow C^{+}$. The $\cap$ s mean that all three conditions must be met.

The $\cap$ symbol is also used in codifying cooperation, as when $a$ and $b$ perform separate acts $a A_{1}$ and $b A_{2}$ when they cooperate.
Example. If $a$ and $b$ make an agreement $\left(a A_{1} \cap b A_{2}\right)$ by exchanging promises, undertakings, signatures, or money, and agree to perform further acts $\left(a A_{3} \cap b A_{4}\right)$, namely the undertakings stated in the agreement, presumably for the parties’ mutual benefit $C^{a b+}$, this would be represented as $\left(a A_{1} \cap b A_{2}\right) \rightarrow\left(a A_{3} \cap b A_{4}\right) \rightarrow C^{a b+}$. The specific terms of the agreement can be elaborated in the legend. If $a A_{1}$ and $b A_{2}$ are thought of as a single act rather than as separate acts, the notation $a b A$ would be used instead of the expression with the $\cap$ symbol.

3.10. Durations of acts

In many contingencies the consequence is a function of the amount of time that has elapsed or that is consumed by an act. A Chinese proverb relates the contingency of being afraid to dismount the tiger that keeps getting hungrier. Here $t$ is the duration of $a A_{\text {dismounts }}$ (time since mounting), and $M$ and/or $p$ are direct functions of $t$, where $M$ is the magnitude of the negative consequence and $p$ is its probability.

Examples. Procrastinating having a dental or medical problem taken care of, fear of reporting a worsening business situation to one’s shareholders, fear of confessing a crime, delaying paying a tax penalty, or not fixing a leaky roof.

In the analogous contingency where the valence is positive, patience is rewarded, or the magnitude or probability of the reward increases the longer one waits.

^[1]Foraging, too, can be described by this general contingency, where finding food may depend on the time $t$ spent foraging.

3.11. Repetitive recycling of an act: learning and practicing

Familiar examples of possible behavioral effects of such recycling contingencies are the progressive decrease in $t$ (i.e. an increase in the speed of the act) $\mathrm{L}_{\mathrm{d}} \mathrm{T} \mathrm{A}_{\mathrm{d}}^{\mathrm{T}} \mathrm{C}$ seen in learning and practicing, or an increase in $t$ seen in fatigue or drug effects. Such observed effects can then be described by the function $t=f\left(n_{i}\right)$.

3.12. Behavioral history factors

Some aspects of a behavioral contingency description are independent of the histories of the involved parties, and some are not. The specification of the acts, the time periods, the consequences, the parties, and certain of the entities’ attributes like probabilities and magnitudes, reflect the analyst’s beliefs about the situation’s objective or physical features, without consideration of the parties’ histories. But such modifiers as perceive, predict, and the valences of consequences do depend on the parties’ histories and reflect the analyst’s beliefs about the effects of those histories. For example, the glass will break if dropped regardless of any party’s history, but a party’s prediction of that consequence, and its valence for that party, would depend on the party’s history. Thus, parties’ histories can be part of the contingency. The representation of any given situation would therefore depend not only on the analyst’s focus and analytic intent but also on his beliefs regarding the parties’ histories and motivations, and would, accordingly, be different for different parties, and for the same party(ies) at different times.

4. Examples of applications

4.1. Wrong predictions: getting swindled, wrong number, “friendly fire”

A party might mispredict the consequence of its act $A$.

Examples. Dialing an incorrect phone number in error, paying a swindler for nonexistent real estate, and a soldier shooting one of his own men. These situations would all be represented as $a A \rightarrow a^{c} C^{a}$. The $a^{c}$ in the $C$ 's upper left quadrant shows that $a$ would mispredict $C^{a-}$. These examples make the point that only consequences that would actually occur (i.e., in the analyst’s view) - calling the wrong party, obtaining an invalid deed, and shooting a comrade - can be codified, in this case represented by the $a-$. Consequences that would not occur (e.g., “expectations”), cannot. Only the legend would state what the parties may have intended or expected-calling the desired party, obtaining a valid deed, and shooting an enemy. The diagram can show only

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1. ${ }^{19}$ As explained in Mechner (1959), the “and” symbol is an abbreviation for a diagram that would show the possible occurrence of any of the permutations of the sequence of the events in question, with reciprocal vertical arrows showing that the occurrence of each sequence prevents the subsequent occurrence of the others. ↩︎

that the party would have predicted a consequence that differs from the actual one. ${ }^{20}$

To show that $a$ would also perceive the actual consequence if and when it occurred in each of the above examples, the $a$ would also be shown in the lower left quadrant of the $C$, as in $a A \rightarrow{ }_{a}^{a} C^{a^{-}}$.

4.2. Unpredicted effect on another party’s behavior

When a party $a$ misuses a word in an unfamiliar language or gives someone faulty driving instructions, the consequence would be an unpredicted effect on the behavior of another party. When a word is misused, the consequence might be puzzlement by the other party. When faulty driving instructions are given, it might be the other party getting lost and complaining. In both cases, the actual consequence is negative for $a$.

If $a A$ is uttering the wrong word and
$a A_{\text {utterm }} \stackrel{a}{\left\lvert\, b^{a} A_{\text {wuth }}{ }^{a}\right.}+{ }_{a} C_{\text {wound of "huh" }}^{a-}$
$C_{\text {wrong word }}$ is the utterance of that word perceived (heard) by $b$, it is $b$ 's puzzled reaction “huh?” to the word that would constitute the negative consequence for $a$, not $C_{\text {wrong word }}$.

If the listener $b$ nods in spite of puzzlement (see diagram)
$a A_{\text {utterm }} \stackrel{a}{\left\lvert\, b^{a} A_{\text {wuth }}{ }^{a}\right.} \underset{\text { right of nod }}{ } C_{\text {nigh }}^{a+}$
as indicating that $b$ understood. The consequence of the nod is positive for $a$ because of $a$ 's misperception of it. If $b$ predicted $a$ 's misperception, as indicated by the $b$ in the upper left of the $a^{a}$, the nod would constitute a form of deception by $b$.

4.3. When nuances determine the meaning: setting a trap

In this diagram, $C_{3}$ 's valence is negative for $b$.
$a A_{1} \rightarrow b A_{2} \rightarrow a b C_{3}^{b-}$
The two negation symbols $\bar{b}$ show that $b$ would not predict $C_{3}$ or perceive the warning signal $C_{4}$. This could be described as $a$ setting of a trap for $b$. The $a$ in the upper left of $C_{3}$ shows that $a$ would predict that $b A_{2}$ would consequate $\mathbf{C}_{3}^{b-}$ and would therefore have set the trap “intentionally,” as one might say. Additional nuances or emphases could be indicated by showing $a$ 's in the upper left quadrants of either or both $\bar{b}$ s (in $C_{4}$ and $C_{3}$ ), to show that $a$ would predict $b$ 's non-perception of $C_{4}$ and non-prediction of $\mathbf{C}_{3}^{b-}$. Nuances like these can assume great significance in legal matters.

^[1]

4.4. Probability of a consequence: creating a hazard

The assignment of probability $p$ to $C_{3}^{b-}$, as an additional attribute, defines the contingency as a hazard.
$a A_{1} \rightarrow b^{a} A_{2} \rightarrow b^{b} C_{3}^{b-, p}$
The $a$ in the upper left quadrant of $b$ means that $a$ would predict that $b$ would be the agent. The $a$ in the upper left quadrant of $A_{2}$ means that $a$ would predict $A_{2}$ regardless of who does it. The $b$ in the upper left quadrant of $C_{3}$, in addition to the $a$, means that both $a$ and $b$ would predict that $A_{2}$, if it occurs, will consequate $C_{3}$ and its attributes, which include $p$. Would $a$ and $b$ then also be predicting that $C_{3}$ 's probability is $p$ ? The distributivity rule (that the modifiers perceive and predict apply also to the entity’s attributes) says that they would. An ${ }^{a} p$ in $C_{3}$ 's attribute quadrant would override the distributivity rule, meaning that $a$ would then not predict $p .{ }^{a b} p$ would mean that $b$ still would.

4.5. Theft

If $b$ takes the cookie out of $a$ 's lunch box $\left(b A_{4}\right)$ before $a$ does $\left(a A_{3}\right), a$ would be prevented (vertical cutting arrow) from doing so.

(The fact that the cookie is in $a$ 's lunch box in the first place $\left(C_{1}\right)$ would have been consequated by the act of an external agent, $e A$, not shown.) Since only $b$ would predict that $b$ will end up with the cookie $\left(C_{2}\right)$, and since $\bar{a}$ shows that $a$ would not perceive that the cookie is in $b$ 's possession, $b A_{4}$ would represent a theft. This example makes the additional point that a rigorous definition of the concept of theft, especially from a legal standpoint, requires specification of the perceptions and predictions of both parties, and the valences of the consequences.

4.6. Applications in therapy

To achieve desired results, therapists (psychotherapists, psychoanalysts, behavior therapists, cognitive-behavior therapists, coaches, or counselors) often seek to understand the contingencies involved in the patient’s interactions with others (Erickson and Achilles, 2004; Wachtel, 1997). Such contingencies can be generated in part by the interactions themselves, including their reverberations at higher levels of the recursion hierarchy.

Family therapy in particular is an area where modifiers of modifiers can be important. In dealing with family dynamics, the therapist may analyze the contingencies that parents, siblings, and related parties create for each other. These often involve multiple levels of regress of each other’s perceptions, mis-/nonperceptions, predictions and mis-/non-predictions of acts and their consequences, as well as of their valences and magnitudes. Since all of these are the result of the parties’ histories, one of the therapist’s challenges is to modify the effects of such histories,

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1. ${ }^{20}$ When the act’s agent would also predict the act’s consequence, one might say that the consequation is intentional. However, as was explained in Section 2.13, terms relating to intentionality, as well as expectations, hopes, visualizations, goals, etc., are not needed. ↩︎

an endeavor in which a detailed behavioral contingency analysis can be useful.

4.7. Consequence of omitting an act: failure to pay a bill

There are situations where failure to perform an act has a significant consequence. In such cases $C$ would be consequated by a $T \rightarrow$ or an $A \rightarrow$ or by another (sometimes external) party. For example, if a phone bill is not paid by the end of time $T$,
the phone company will shut off service.

Here, $b$ would be the agent of the preventive act. The vertical arrow shows that $b A_{3}$ would avert $C_{2}$ by terminating the $T \rightarrow C_{2}$ contingency. Non-performance of $b A_{3}$ would permit $C_{2}$ to occur.

The legend indexed to the subscripts might read:

1. If phone company a threatens to cut off service after time $T$.
2. Service would be cut off.
3. (If) customer $b$ pays the phone bill within time $T$.
4. Service would continue.

The consequence $C$ of the termination, with all of its modifiers, can be shown at the end of the terminating arrow.

4.8. Reciprocal vertical arrows: decision making and competition

When each of two consequences prevents the occurrence of the other, as in decision making and certain types of competition, reciprocal vertical arrows are required (see diagram).

taking a multiple choice test,

the student may be confronted with a question $C$ together with simultaneously available choices (acts), as in the above diagram. The external agency $e$ that presents (consequates) the $C$ may be a teacher, a computer, or the student himself turning a page. The reciprocal vertical arrows show that each of the three acts terminates the availability of the others.

4.10. Two interrelated time intervals: the roast in the oven

The diagram shows a situation where $a$ put a roast in the oven and left the house without turning the oven off $\left(a A_{1}\right)$. The roast will burn $\left(C_{5}{ }^{a-}\right)$, unless the oven is turned off $\left(A_{8}\right)$ within time $T_{4}$.

If the oven is turned off $\left(A_{8}\right)$ after $T_{7}$ and before $T_{4}$, the roast will be done. But the oven will get turned off only if $a$ asks $\left(a A_{6}\right)$ her neighbor $b$ to do so before $T_{4} . T_{7}$ and $a A_{6}$ therefore have an “and” relationship.

Legend:

$$\begin{aligned} & a A_{1}: \text { If } a \text { leaves the roast in the oven on leaving home } \\ & \text { ... } \\ & { }_{a} C_{5}: \text { The roast would be in the oven with the oven on. } \\ & T_{4}: \text { Time after which the roast would burn. } \\ & T_{7}: \text { Time after which the roast would be done. } \\ & C_{5}^{a-}: \text { Burnt roast. } \\ & a A_{6}: \text { If } a \text { calls her neighbor } b \text { and leaves her a message } \\ & \text { ... } \\ & { }_{a b} C_{9}: \text { Message to turn off the oven after time } T_{7} \text {. } \\ & b A_{6} \rightarrow: \text { If } b \text { turns off the oven after } T_{7} \text { and before } T_{4} \ldots \\ & C_{5}^{a+}: \text { The roast is done and } C_{5}{ }^{a-} \text { is averted. } \end{aligned}$$

The $b$ in ${ }_{a b} C_{9}$ 's lower left quadrant implies that $b$ got the message. The ${ }^{a} p$ attributes of $A_{8}$ and $C_{3}$ represent $a$ 's

prediction/estimation of the probability that $b$ will actually do $b A_{8}$ and that the roast will therefore be taken out in time.

4.11. Types of “or” relationships: achieving priority, putting out a fire

One kind of “or” relationship is where either one of two or more acts can result in a given consequence. Another is where a single act can result in either one of two or more consequences. Both can in turn be divided into exclusive (either, or, but not both) and inclusive (either, or, or both) “or” relationships.

$$\left[\begin{array}{l} a A_{1} \\ b A_{2} \end{array}\right] \rightarrow C_{\text {priority }} \quad a A_{1} \cup b A_{2} \rightarrow C_{\text {fire out }}$$

Diagram on the left: an exclusive “or” relationship shown by a merging of the horizontal arrows. The consequating entities in such diagrams can be $A$ s and/or $T$ s. This diagram could represent two parties competing to consequate $C$, where whoever gets there first obtains the only available $C$ (as when parties compete for priority in applying for a patent or in reaching the South Pole).

Diagram on the right: either one of two parties or both together can put out the fire-the inclusive “or,” represented by the union symbol $\cup$.
$\left(a A_{1} \cup b A_{2}\right) \rightarrow\left(a A_{3} \cap b A_{4}\right) \rightarrow C^{a b+}$ would be read as, “If either $a$ or $b$ or both (note the inclusive “or” symbol $\cup$ ) create a situation in which cooperation between $a$ and $b$ (note the $\cap$ symbol for “and”) can occur, the consequence would be $C^{a b+}$.”

4.12. Probabilities of alternative consequences: Russian roulette

Another type of “or” situation is where a single act $A$ (e.g., the tossing of dice) consequates two or more possible outcomes with different probabilities. In “Russian roulette” there is a $1 / 6$ probability of the consequence “bullet in chamber” occurring. If the focus is on the two alternative scenarios, the arrow can be shown as branching, with the other branch consequat-

ing the complementary probability $5 / 6$.

In general, contingencies involving alternative consequences are represented by a multi-pronged fork with two or more arrows pointing to the different possible Cs.

The analyst may sometimes wish to show that a modifier like perceive and predict, or a valence, has an “if” in front of it. For example, he may want ${ }_{a} C$ to be read as “If $a$ would perceive $C$ " rather than the standard " $a$ would perceive
C.” He would then show the two possibilities
as the branches of an “or” fork. For example, when considering whether or not to commit a crime (Becker, 1976, 1995, 1997) a party might weigh the consequences of the crime being perceived $(p)$ or not perceived $(1-p)$ by society. (For a fuller analysis, see Mechner, 2008).

4.13. Sequential multiple discriminations: traffic lights, naming people

In a sequential multiple discrimination, signals appear one at a time.

Example. We stop when the traffic light is red and proceed when it is green, and call different people by their respective names.

To codify this type of contingency we first ask how the circumstances for the acts are consequated. Traffic lights are controlled by timers, and telephone equipment transmits the voices of people we may want to name when they call us. In both cases external agencies $e$ consequate the situations in an “or” manner-the traffic light is either red or green, and we receive phone calls from one person or another.

Again, the “or” relationship is represented by a fork. When the multiple discrimination involves more than two signals, the fork would have correspondingly more branches.

4.14. Misperception of $C$ : mistaken identity

If policeman $a$ sees a suspicious character $b$, he may try to arrest him. If $b$, claiming mistaken identity, reached into his pocket $\left(b A_{\text {reaches }}\right)$ for his identification $\left({ }_{a b} C_{\text {ID }}\right)$, then in $T$ seconds this might take, the policeman could misperceive the visual consequence ( $\boldsymbol{a}^{\mathrm{a}} \mathrm{C}_{2}$ ) of $\left(b A_{\text {reaches }}\right)$ as pulling out a gun, and shoot $b$ dead, mispredicted by $b$. The $b$ term means that $b$ would not predict $a$ 's misperception.

4.15. Misperception of a missile test

A similar diagram could describe a misperception by country $a$ in response to a missile test by a hostile power $b$, and the associated mispredictions by $b$.

Here, $\frac{\hat{a}_{a}}{a^{a}} \mathrm{C}_{\text {test }}$ shows $a$ 's non-prediction and misperception of the test. If $a$ responds with $a A_{\text {attacks }}$, the consequence would be negative for both parties. The $b^{x}$ in the upper left quadrant of the $a^{x}$ shows that $b$ would mispredict $a$ 's misperception of the test. The $b^{x}$ in the upper left of $A_{\text {attacks }}$ shows that $b$ would mispredict the attack by $a$. If it were desired to show that $b$ would not predict rather than mispredict $a$ 's misperception, $\hat{b}$ would replace $b^{x}$ in $a^{x}$ 's upper left quadrant.

Many types of situations that involve misperception, and misprediction of misperceptions, are of great concern to societies and nations, as well as in interpersonal relationships. Behavioral contingency analysis helps identify the critical variables whose effects can then be studied by various methods.

4.16. Concatenated probabilities in systems analysis: the Trojan horse

If Homer had been the analyst, he might have shown Odysseus - the general of the besieging Greeks - as party $a$ scheming the following contingency:
“If we build a giant hollow wooden horse and leave it for the Trojans to find ( $a A_{\text {offers }}$ ), the Trojans (b) may misperceive the horse (as being empty rather than filled with soldiers). If the Trojans thereupon take the horse into their city ( $b A_{\text {takes in }}$ ), our soldiers hidden inside the horse may be able to emerge from the horse during the night ( $a A_{\text {emerge }}$ ) and open the gates for the rest of us to enter and sack the city, thereby winning the war.”

The $\hat{b}$ in the upper left quadrant of $C_{\text {horse }}$ shows Odysseus’s bet that the Trojans would not predict this “gift.” The $b^{x}$ in the lower left quadrant of $C_{\text {horse }}$ shows $b$ 's possible misperception, and the ${ }^{a} p_{1}$ in $b^{x}$ ‘s attribute quadrant shows Odysseus’ prediction/estimation of the probability of this misperception. Odysseus might also have predicted/estimated the probabilities of various subsequent events- ${ }^{a} p_{2}$ for the act $b A_{\text {takes in }}$, and ${ }^{a} p_{3}$ for $a A_{\text {emerge }}$ (e.g., the Trojans might discover the ruse, or the horse’s door might jam). The $\hat{b}$ symbols in the two verb quadrants of $a A_{\text {emerge }}$ show that $b$ would not perceive or predict $a A_{\text {emerge }}$. The attributes quadrant of $C_{\text {sucked }}$ shows that this consequence would be negative for $b$, positive for $a$, and that $a$ would have predicted/estimated its probability as $p_{4}$.

If Odysseus had indeed performed this analysis and considered the product of the four probabilities, he might have concluded that his scheme was very (too?) risky.

4.17. Vigilance ${ }^{22}$ and the analysis of interrelated probabilities

If $\left(p_{1}\right) e$ plants a bomb ( $e A_{\text {plants }}$ ) that has probability $p_{7}$ of going off after unknown time $T^{M 7}$, and if ( $a A_{\text {vigilance }}$ ) shows that

^[1]$a$ is probably $\left(p_{2}\right)$ engaged in vigilance/detection activity, then $a$ will detect (perceive) ${ }_{a} C_{\text {bomb }}$ with probability $p_{3}$. If $a$ thereupon alerts ( $a A_{\text {alerts }}$ ) the bomb squad $b$ (probability $p_{4}$ ) with a signal $\left({ }_{a b} C_{\text {alert }}\right)$, the bomb squad will probably $\left(p_{5}\right)$ attempt to disarm the bomb ( $a A_{\text {disarm }}$ ), with $p_{6}$ being the probability of doing so successfully.

4.18. Modification of contingencies: mitigating a danger

This diagram shows a situation in which two or more parties cooperate to avert a threat. Again, $C^{a b-}$ can come after an unpre-
dictable time $T^{M 7}$ with probability $p$. $\left[\begin{array}{c}\mathrm{T}^{\mathrm{at}} \\ (\mathrm{aA} \cap \mathrm{bA})^{\mathrm{T}}\end{array}\right] \rightarrow \mathrm{C}^{\mathrm{at}-\mathrm{p}}$
By acting cooperatively $(a A \cap b A)$ (e.g., by exercising vigilance, building levees, or storing provisions), $a$ and $b$ can prevent $C^{a b-}$.

To show that $(a A \cap b A)$ merely reduces the probability of $C^{a b-}$ from $p_{1}$ to $p_{2}$, rather than to zero, the changed contingency would be shown at the end of the vertical arrow with the new probability $p_{2}$, as in the diagram.

Here $(a A \cap b A)$ would change both the timing and the probability of the terminated contingency. If the new contingency requires a bracket, the arrow would point to the bracket.

4.19. Risky choices: thinking ahead in a game

The risks that a chess, checker, or go player takes when choosing between moves are the uncertainties regarding (1) how accurately or completely he identified the opponent’s possible responses, (2) which of these the opponent will actually choose, and (3) the valence of the outcomes of these possibilities.

If $a$ considers two possible moves $a A_{1}$ and $a A_{9}$, and thinks two moves ahead in considering $b$ 's possible responses, $a$ would be uncertain as to whether, in response to $a A_{1}, b$ would choose $b A_{3}$ (say, a particular identified move) or $b A_{5}$ (say, a possible

unidentified move),

$\mathrm{C}_{12}$ or whether in response to $a A_{9}, b$ would choose $b A_{7}$ (say, another particular identified

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1. ${ }^{22}$ Experimental studies of vigilance and observing behavior and the associated contin gencies include Dinsmoor (1983), Dinsmoor et al. (1972), Case and Fantino (1989); and Nevin et al. (2005). ↩︎

move) or $b A_{10}$ (say, another possible unidentified move). Furthermore, $a$ would be uncertain regarding the valences of the resulting situations (note the $a \mathrm{~s}$ in the attribute quadrants of the question marks).

4.20. Divergent contingencies for two parties: a sexual overture

If $a$ makes a sexual overture $a A_{1}$ to $b$, the two possible outcomes, from $a$ 's point of view, are that $b$ would accept (probability $p$ ) and that $b$ would decline $(1-p)$. The reciprocal vertical arrow is needed to show that $b$ 's accepting and declining
are mutually exclusive.

From $b$ 's point of view the issue is whether to accept or decline, rather than a probability issue. This type of examination of two divergent perspectives can be useful in fields like law, business (see Section 6.7), economics, or public affairs, where the differences in the contingencies for each party can be important.

4.21. Game theory situations and bluffing: blackmail and kidnapping

If blackmailer $a$ threatens to harm $b$ unless $b$ pays, the probability of the harm actually occurring, even if $a A_{\text {executes }}$, might be $p_{2}$. If $a A_{\text {demand }}$ stated that $b$ can avert harm by $b A_{\text {pays }}$, then the probability that $a$ would do no harm if $b$ pays is $p_{4}$, and

the probability that $a$ would actually execute the threat if $a$ does not pay is $p_{1}$, that is, $a$ is bluffing. The probability that $b A_{\text {pays }}$ would consequate a recycling of the entire contingency is $p_{3}$. When a contingency recycles, all components of that contingency are restored for the next cycle, including ones terminated by vertical arrows in the preceding cycle.

The same diagram could also describe hostage taking or kidnapping. This example and many of those that follow illustrates how behavioral contingency analysis can serve as a tool for identifying the variables and their relationships, including utility, that would be considered in the types of economic analyses proposed by G.S. Becker (1976, 1992, 1995).

4.22. Predictions/estimations of $p$ : the basis for a wager

Here $a A^{M 1}$ and $b A^{M 2}$ represent bets that $a$ and $b$ can place in situation ${ }_{a b} C_{5}$ on, say, the outcome of a ball game. The external agency $e$ could be two ball teams producing one of the two possible consequences.

The attributes $M_{1}$ and $M_{2}$ of the bets can be the amounts of money $a$ and $b$ would be betting. The symbol $\cap$ indicates the “and” relationship of the bets, as in a contract. The notations ${ }^{(a, b)} p_{1}$ and ${ }^{(a, b)} p_{2}$, with commas between the $a$ and $b$, show that $a$ and $b$ have different predictions/estimations of $p_{1}$ and $p_{2}$ (they would not be betting unless they did). If there were no comma, just $a b$, it would mean that $a$ and $b$ have a joint prediction/ estimation.

4.23. Perceptions of $M$ : tipping a service provider

If customer $b$ has received a service $\left(a A_{1}\right)$ from $a$ (say, a waiter), the magnitude $M$ of the tip $b$ gives $\left(b A^{M}\right)$ may be $M_{1}$, $M_{2}$, or $M_{3}$. The three tips are $C_{3}, C_{7}$, and $C_{11}$, and $a$ may have subjective/idiosyncratic perceptions of their magnitudes.

The magnitudes are attributes of the three $C$ s and the idiosyncratic perceptions of the magnitudes (indicated by $a^{x}$ s in the $M$ s’ lower left quadrants), are shown in the three references, a technique used because if these attributes were scaled to the size of the main diagram and inserted, they would be hard to read.

The fact that the two sets of three $M$ s have the same (lower right) subscripts means that they represent the same amounts of money. (The three-way reciprocal vertical arrow shows that these three possible consequences are mutually exclusive.) The legend might elaborate that $a$ would idiosyncratically perceive $M_{1}$ as too small, $M_{2}$ as acceptable, and $M_{3}$ as generous. These three idiosyncratic perceptions have probabilities $p_{1}, p_{2}$, and $p_{3}$, respectively, which $b$ might predict/estimate, as shown by the $b$ s in the upper left quadrants of the three probabilities.
$a$ 's possible reactions $a A_{4}$ and $a A_{9}$ would consequate $C_{5}$ or $C_{10}$, which $b$ would perceive and whose respective probabilities $b$ would predict/estimate as $p_{4}$ and $p_{5}$. $C_{5}$ might be an insult and $C_{10}$ might be an expression of gratitude.

These and many other intricate social and cultural variables at play in many tipping situations have been discussed by Green et al. (2003), and a brief review of the relevant literature is presented in Lin (2007).

4.24. Misperception of the $M$ of a C: perpetration of a fraud

If seller $a$ offers $\left(a A_{1}\right)$ to sell $b$ a counterfeit painting, $C_{3}, a$ would perceive the value of the painting to be $M_{4}$ (as indicated by the $a$ in the lower left quadrant of $C_{3}$, which is distributive to $C_{3}$ 's attribute $M_{4}$ ),