Awareness as a Function
Awareness as a Function
Awareness as a Function: Complete Mathematical and Theoretical Foundations
Awareness operates as the gating function between stimulus and response across all human decision making. This comprehensive mapping reveals that marketing science and cognitive science converged on remarkably similar mathematical insights: awareness is scarce, capacity limited, probabilistically distributed, and subject to threshold effects. From E. St. Elmo Lewis’s 1898 sales funnel to Karl Friston’s 2010 free energy principle, the core problem remains unchanged — how do signals break through noise to enter conscious processing? The frameworks documented here represent the complete intellectual lineage informing modern attention measurement, from GRP formulas used in media buying to d prime equations in psychophysics laboratories.
Part 1: Marketing and Brand Awareness Frameworks
The hierarchy of effects tradition (1898–1980)
The foundational idea that awareness precedes action emerged from direct mail advertising in the late 19th century. E. St. Elmo Lewis (1872–1948) first articulated the principle in an 1898 Printers’ Ink article: “attract attention, maintain interest, create desire.” The fourth element — “get action” — appeared later. Lewis drew explicitly on William James’s psychology of attention, making this the first documented bridge between cognitive science and advertising theory. The AIDA acronym itself was coined by C.P. Russell in 1921, though Lewis’s conceptual framework preceded it by over two decades.
Robert Lavidge and Gary Steiner formalized the concept in their landmark 1961 paper “A Model for Predictive Measurements of Advertising Effectiveness” (Journal of Marketing). Their six stage hierarchy — Awareness → Knowledge → Liking → Preference → Conviction → Purchase — explicitly mapped onto three psychological domains: cognitive (thinking), affective (feeling), and conative (doing). This tripartite structure influenced all subsequent models. The same year, Russell Colley’s DAGMAR framework (Defining Advertising Goals for Measured Advertising Results, ANA, 1961) introduced the revolutionary principle that advertising should be measured by communication effects, not sales. His four stage model — Awareness → Comprehension → Conviction → Action — spawned modern brand tracking methodology.
Richard Vaughn’s FCB Grid (1980, Journal of Advertising Research) integrated these hierarchies with involvement theory, creating a 2×2 matrix crossing High/Low Involvement with Thinking/Feeling. The key insight: the Think Feel Do sequence reorders based on product category. High involvement/thinking products (cars, insurance) follow Learn→Feel→Do; low involvement/feeling products (beer, snacks) follow Do→Feel→Learn. This remains the standard framework in advertising planning.
Model Author(s) Year Sequence Key Innovation
AIDA E. St. Elmo Lewis 1898 Attention→Interest→Desire→Action First hierarchical model
Hierarchy of Effects Lavidge & Steiner 1961 6 stages mapped to cognition/affect/conation Psychological domains
DAGMAR Russell Colley 1961 Awareness→Comprehension→Conviction→Action Measurable objectives
FCB Grid Richard Vaughn 1980 Variable by quadrant Involvement × processing mode
Critiques: Vakratsas and Ambler’s 1999 meta analysis of 250+ papers found little empirical support for strict sequential processing. Modern neuroscience confirms that cognitive and affective processing occur simultaneously, not sequentially.
Media mathematics: Reach, frequency, and carryover effects
The quantitative infrastructure of media planning rests on equations developed between 1950 and 1980 that remain industry standard today.
Gross Rating Points (GRP) emerged organically in 1950s television advertising:
GRP = Reach (%) × Average Frequency
One GRP represents reaching 1% of the target population once. The formula’s elegance masks its limitation: it treats all exposures as equivalent regardless of attention quality.
Herbert Krugman’s Three Exposure Theory (1972, “Why Three Exposures May Be Enough,” Journal of Advertising Research) proposed that advertising has only three psychological states: (1) “What is it?” (curiosity), (2) “What of it?” (evaluation), and (3) reminder/decision. Krugman argued there is “no such thing as a fourth exposure psychologically — fours, fives, etc., are repeats of the third.” Michael Naples codified this as the “3+ frequency” standard in his 1979 ANA report, which dominated media planning for two decades.
Erwin Ephron’s Recency Theory (1995, “More Weeks, Less Weight,” Journal of Advertising Research) challenged effective frequency orthodoxy. Ephron argued that a single exposure within the purchase decision window matters more than accumulated frequency. His recommendation — maximize weekly reach rather than building frequency — transformed media planning for FMCG brands. John Philip Jones’s single source research confirmed that one exposure in the seven days before purchase has far greater impact than multiple earlier exposures.
Simon Broadbent’s Adstock Model (1979, Journal of the Market Research Society) provided the mathematical framework for modeling awareness decay:
Adstock_t = Advertising_t + λ × Adstock_(t 1)
Where λ (0 < λ < 1) is the decay parameter. The half life calculation:
Half life = ln(0.5) / ln(λ)
Industry benchmarks: FMCG brands average 2.5 weeks half life; academic studies suggest 7–12 weeks for brand awareness effects. This equation underlies all modern marketing mix models.
Advertising response functions follow three patterns:
Linear: Sales = α + β × Advertising (rarely observed)
Concave/Diminishing Returns: Sales = α × Advertising^β where 0 < β < 1 (most common empirically)
S Curve/Logistic: Threshold effect before response, then saturation
Response = Max × (Spend^α) / (K^α + Spend^α) [Hill Function]
The Vidale Wolfe Model (1957, Operations Research) introduced dynamics:
dS/dt = r × A × (M S)/M δ × S
Where S = sales rate, A = advertising spend, M = market saturation, r = response constant, δ = decay rate. This was the first differential equation model of advertising effects.
Share of Voice and the Ehrenberg Bass revolution
John Philip Jones (1990, Harvard Business Review) documented the empirical relationship between Share of Voice (SOV) and Share of Market (SOM):
Extra Share of Voice (ESOV) = SOV SOM
Challenger brands require positive ESOV to grow; market leaders can profit with negative ESOV. Les Binet and Peter Field’s IPA research suggests:
Expected Annual Market Share Growth ≈ 0.5% per 10% ESOV
The Ehrenberg Bass Institute (founded by Andrew Ehrenberg) transformed brand science through rigorous empirical analysis. Their NBD Dirichlet Model (Goodhardt, Ehrenberg & Chatfield, 1984, Journal of the Royal Statistical Society) describes purchase behavior with three parameters: M (mean category purchase rate), K (purchase frequency diversity), and S (brand propensity diversity). Key equations:
Brand Penetration:
b = 1 Σ(n=0→∞) P_n × p(0n)
Double Jeopardy Law: Smaller brands suffer twice — fewer buyers AND lower loyalty. Expression: w(1 b) = constant, where w = purchase frequency and b = brand penetration.
Duplication of Purchase Law:
b_XY = D × b_X
Brand Y’s buyers who also buy brand X equals the Duplication coefficient times brand X’s penetration. This implies no systematic brand segmentation in most categories.
Byron Sharp’s “How Brands Grow” (2010) synthesized Ehrenberg Bass findings into marketing’s most influential modern text. Key constructs:
Mental Availability: Brand’s propensity to be noticed or retrieved in buying situations
Physical Availability: Ease of finding and buying the brand
Category Entry Points (CEPs): The cues buyers use to access memory (developed by Jenni Romaniuk)
Mental Market Share = Brand’s CEP associations / Total category CEP associations
Kevin Lane Keller’s CBBE Pyramid (1993, 2001, Journal of Marketing) provides an alternative framework. Brand Salience forms the foundation, defined by depth (how easily recalled) and breadth (range of situations triggering recall). The pyramid ascends through Performance/Imagery → Judgments/Feelings → Resonance.
Attention economics: From Simon to digital metrics
Herbert Simon established the theoretical foundation in his 1971 lecture “Designing Organizations for an Information Rich World”:
> “A wealth of information creates a poverty of attention and a need to allocate that attention efficiently among the overabundance of information sources that might consume it.”
Simon connected attention scarcity to his earlier work on bounded rationality (1955, Quarterly Journal of Economics): humans “satisfice” rather than optimize because cognitive capacity is limited. This remains the master concept underlying all attention economics.
Michael Goldhaber (1997, Wired) declared attention “the currency of the New Economy,” arguing that unlike information, human attention is “truly individual and scarce.” Georg Franck (1998, Ökonomie der Aufmerksamkeit) developed the concept of “mental capitalism” — attention as literal capital that earns interest. Celebrity functions as a “stock exchange of attention capital.”
Thomas Davenport and John Beck (The Attention Economy, 2001) operationalized these ideas for business: “Understanding and managing attention is now the single most important determinant of business success.”
Modern digital attention measurement has evolved beyond simple impressions. The IAB/MRC Viewability Standards (2014) established minimum thresholds:
Display: ≥50% pixels visible for ≥1 second
Video: ≥50% pixels visible for ≥2 seconds
But viewability ≠ attention. Lumen Research (Mike Follett, founded 2013) pioneered eye tracking based metrics:
APM = (% of ads viewed × average viewing time) × 1000 impressions
aCPM = CPM / APM × 1,000
Their finding: only 35% of “viewable” ads are actually looked at.
Adelaide (Marc Guldimann, 2019) developed the AU (Attention Unit) metric — a 0–100 score combining eye tracking data, media quality signals, and outcome data to predict attention probability rather than duration. This addresses the “Attentive Audience Paradox” where older audiences dwell longer but don’t necessarily engage more.
Karen Nelson Field (Amplified Intelligence, The Attention Economy and How Media Works, 2020) established that 1.5 seconds of active attention is sufficient for memory encoding. Her research shows attention metrics are 7× more effective at predicting brand awareness than viewability.
Part 2: Cognitive Science and Attention Theory
Signal detection and information theory foundations
Signal Detection Theory (SDT) separates perceptual sensitivity from response bias. Green and Swets (Signal Detection Theory and Psychophysics, 1966) formalized:
d’ = z(Hit Rate) z(False Alarm Rate)
Where z() is the inverse normal CDF. The d’ (d prime) measures discriminability independent of criterion setting. Criterion measures:
c = 0.5 × (z(HR) + z(FAR)) [criterion location]
β = exp((z(FA)² z(H)²) / 2) [likelihood ratio]
ROC curves plot Hit Rate against False Alarm Rate across criterion levels; Area Under Curve (AUC) quantifies overall discrimination ability.
Claude Shannon’s Information Theory (1948, Bell System Technical Journal) established channel capacity:
C = B × log₂(1 + S/N)
Where C = capacity in bits/second, B = bandwidth, S/N = signal to noise ratio.
George Miller (1956, Psychological Review) applied information theory to cognition, discovering that immediate memory span is limited to 7 ± 2 chunks — independent of information bits per chunk. Miller emphasized chunking as the mechanism for expanding effective capacity. Nelson Cowan (2001, Behavioral and Brain Sciences) revised this to 4 ± 1 items when rehearsal is prevented:
K = (hit rate + correct rejection rate 1) × N
Selective attention: Filter models and their evolution
Donald Broadbent’s Filter Model (1958, Perception and Communication) proposed that a selective filter operates on physical characteristics (location, pitch, voice) early in processing, blocking unattended information before semantic analysis:
Stimuli → Sensory Buffer → FILTER → Limited capacity Processor → Response
Anne Treisman’s Attenuation Theory (1964, British Medical Bulletin) modified this: instead of blocking, the filter merely “turns down the volume.” Important stimuli (one’s name, danger words) have permanently low thresholds and break through:
If (Attenuated Signal) ≥ Threshold → Conscious Awareness
Threshold varies by: subjective importance, recent activation (priming), contextual expectancy, and biological significance.
Deutsch and Deutsch’s Late Selection Model (1963, Psychological Review) proposed all stimuli receive full semantic processing; selection occurs later based on importance/pertinence. Evidence: conditioned words produce galvanic skin responses even in unattended channels (Moray, 1969).
Daniel Kahneman’s Capacity Model (1973, Attention and Effort) shifted focus from filtering to resource allocation. Attention is a limited, undifferentiated pool of mental “fuel”:
Total_Capacity = g(Arousal) [inverted U function]
Performance = f(Resources_allocated, Task_demands). When demands exceed capacity, performance suffers.
Christopher Wickens’ Multiple Resource Theory (1980, 2002) identified four dimensions of resource separation:
Stages: Perceptual/Cognitive vs. Response
Modalities: Auditory vs. Visual
Codes: Spatial vs. Verbal
Visual Channels: Focal vs. Ambient
Prediction: interference increases with shared resources across dimensions. This explains why auditory verbal + visual spatial tasks combine well (different tanks), while visual verbal + visual spatial compete (same tank).
Computational models of visual attention
Anne Treisman’s Feature Integration Theory (Treisman & Gelade, 1980, Cognitive Psychology) distinguished:
Pre attentive processing: Features (color, orientation) processed in parallel; targets “pop out”
Focused attention: Required to bind features together (solving the “binding problem”)
Search slope equations:
Feature search (parallel): RT = a + b (flat, ~0 ms/item slope)
Conjunction search (serial): RT_positive = a + b × N/2; RT_negative = a + b × N
The Itti Koch Saliency Model (1998, IEEE PAMI) operationalized bottom up attention computationally:
1. Input decomposed into 9 spatial scales via Gaussian pyramids
2. Feature channels extracted: Intensity, Color (RG, BY opponency), Orientation (4 angles)
3. Center surround operations create feature maps (42 total)
4. Normalization operator N(·) promotes maps with few strong peaks
5. Conspicuity maps combined: S = (1/3)[N(Ī) + N(C̄) + N(Ō)]
6. Winner take all network selects most salient location
The model predicts first ~250ms of viewing (pre attentive processing). Software: SaliencyToolbox (MATLAB), pySaliencyMap (Python).
Jeremy Wolfe’s Guided Search (1989–2021) bridges bottom up saliency with top down guidance:
A_i = w_BU × BU_i + Σ_d(w_d × TD_d,i) + noise
Guided Search 6.0 (2021, Psychonomic Bulletin & Review) integrates five priority sources: bottom up salience, top down guidance, history/priming, reward/value, and scene semantics. Selection rate: ~20 Hz attention deployment.
Deep learning models now outperform classical saliency. DeepGaze IIE (Linardos et al., 2021) achieves ~93% of human inter observer consistency using pretrained vision transformer features without task specific training.
Working memory, consciousness, and predictive processing
Baddeley’s Working Memory Model (1974, updated 2000) comprises:
Central Executive: Attentional control, coordinates subsystems
Phonological Loop: Verbal/auditory information (~2 seconds rehearsal capacity)
Visuospatial Sketchpad: Visual/spatial information
Episodic Buffer (2000): Integrates information across subsystems and long term memory
Cognitive Load Theory (John Sweller, 1988, Cognitive Science) distinguishes:
Total Load = Intrinsic Load + Extraneous Load
Overload occurs when Total Load > Working Memory Capacity. Implications for advertising: minimize extraneous load through design optimization.
Global Workspace Theory (Bernard Baars, 1988) uses a theater metaphor: consciousness is the “spotlight” on a limited “stage,” broadcasting selected information to a vast “audience” of unconscious processors. Dehaene and Changeux (1998, 2003) provided neural implementation: long range workspace neurons in prefrontal parietal cortex show “ignition” — sudden, late (~300ms), sustained firing when threshold crossed. Signatures of consciousness: P3b wave, gamma synchrony, long distance coherence.
Predictive Processing (Rao & Ballard, 1999, Nature Neuroscience) models the brain as a hierarchical prediction machine:
Prediction: ŷ_i = W_i · r_(i+1)
Prediction Error: e_i = y_i ŷ_i
Feedforward connections carry errors; feedback carries predictions. Attention = precision weighting of prediction errors.
Karl Friston’s Free Energy Principle (2010, Nature Reviews Neuroscience) provides a unified framework:
F = E_q[ln q(s) ln p(s,o)]
Equivalently: F = KL divergence between beliefs and true posterior + surprisal. The brain minimizes free energy through: (1) perceptual inference (updating beliefs), (2) learning (updating model), (3) action (changing inputs). Attention modulates precision:
Precision (π) = 1/variance
High precision prediction errors gain more weight in updating beliefs.
Integrated Information Theory (Giulio Tononi, 2004) proposes consciousness = integrated information (Φ):
Φ = min_partition D(p(X_t+1X_t) Π_i p(X^i_t+1X^i_t))
The theory predicts the posterior cortical “hot zone” is more conscious than prefrontal cortex. Major critique: calculating Φ is computationally intractable (NP hard) for systems larger than ~15 elements.
Part 3: Cross Domain Synthesis
How cognitive theory shaped advertising science
The bridges between cognitive science and advertising are surprisingly well documented:
1. E. St. Elmo Lewis explicitly cited William James’s attention psychology in developing AIDA (1898)
2. Herbert Krugman based three exposure theory on the mere exposure effect and latent inhibition from cognitive psychology
3. Information theory (Shannon, Miller) informed message design: keep key points under 4–7 items
4. Signal Detection Theory provides framework for measuring ad noticeability independent of response bias
5. Feature Integration Theory explains why distinctive visual features (bright colors, motion) capture pre attentive attention
6. Kahneman’s capacity model explains why cognitively loaded consumers process ads superficially
7. Multiple Resource Theory informs multimodal ad design: combine audio verbal with visual spatial
The attention funnel: Unified mathematical framework
Across domains, awareness functions as the first filter in a multi stage process:
Marketing Funnel:
Exposure → Attention → Awareness → Processing → Attitude → Behavior
Cognitive Funnel:
Sensation → Feature Detection → Saliency Competition → Working Memory → Decision
Shared mathematical properties:
Capacity limits: ~4 items in working memory; ~7 brands in consideration set
Decay functions: Adstock λ ≈ 0.75 (awareness); trace decay ~1–2 seconds (sensory memory)
Threshold effects: Minimum exposure for awareness; ignition threshold for consciousness
Probabilistic retrieval: Brand salience = CEP based probability; memory retrieval = activation based probability
Integration of metrics across paradigms
Marketing Metric Cognitive Equivalent Mathematical Relationship
Brand Awareness (%) Recognition probability p(recall) = f(activation strength, threshold)
Effective Frequency (3+) Memory consolidation Repetition strengthens trace
Adstock decay (λ) Memory decay Exponential forgetting curve
Mental Availability Spreading activation Network accessibility
Attention time (seconds) Fixation duration Encoding time = f(frequency, eccentricity)
d’ (sensitivity) Ad discriminability Signal detection independent of bias
Master Reference: Key Thinkers and Equations
Marketing and Advertising
Framework Author(s) Year Key Equation/Model
AIDA E. St. Elmo Lewis 1898 Attention→Interest→Desire→Action
Hierarchy of Effects Lavidge & Steiner 1961 6 stages: Awareness→Knowledge→Liking→Preference→Conviction→Purchase
DAGMAR Russell Colley 1961 Awareness→Comprehension→Conviction→Action
FCB Grid Richard Vaughn 1980 2×2: Involvement × Thinking/Feeling
GRP Industry standard 1950s GRP = Reach × Frequency
Three Exposures Herbert Krugman 1972 Psychological threshold model
Effective Frequency Michael Naples 1979 3+ exposures optimal
Recency Theory Erwin Ephron 1995 One exposure in purchase window
Adstock Simon Broadbent 1979 A_t = T_t + λ × A_(t 1)
Vidale Wolfe Vidale & Wolfe 1957 dS/dt = r×A×(M S)/M δ×S
SOV SOM John Philip Jones 1990 ESOV = SOV SOM
NBD Dirichlet Ehrenberg et al. 1984 Brand choice probability model
Mental Availability Byron Sharp/Romaniuk 2004–2010 CEP based salience
CBBE Pyramid Kevin Lane Keller 1993 Salience→Meaning→Response→Resonance
AU Metric Adelaide/Guldimann 2019 0–100 attention probability score
Cognitive Science
Framework Author(s) Year Key Equation/Model
Channel Capacity Claude Shannon 1948 C = B × log₂(1 + S/N)
7 ± 2 Chunks George Miller 1956 Immediate memory span
Filter Model Donald Broadbent 1958 Early physical selection
Attenuation Anne Treisman 1964 Signal reduction + threshold
Late Selection Deutsch & Deutsch 1963 Post semantic selection
Signal Detection Green & Swets 1966 d’ = z(HR) z(FAR)
Capacity Model Daniel Kahneman 1973 Flexible resource pool
Feature Integration Treisman & Gelade 1980 Parallel features + serial binding
Multiple Resources Christopher Wickens 1980 4 D interference model
Global Workspace Bernard Baars 1988 Broadcast to unconscious processors
Cognitive Load John Sweller 1988 Total = Intrinsic + Extraneous
Itti Koch Saliency Itti & Koch 1998 S = (1/3)[N(Ī) + N(C̄) + N(Ō)]
Predictive Coding Rao & Ballard 1999 Prediction error minimization
4 ± 1 Chunks Nelson Cowan 2001 True capacity without rehearsal
Free Energy Karl Friston 2010 F = E_q[ln q(s) ln p(s,o)]
Guided Search 6.0 Jeremy Wolfe 2021 5 source priority map
— -
Conclusion: The unified logic of awareness
The intellectual archaeology of awareness reveals a remarkable convergence: marketing practitioners and cognitive scientists independently discovered the same fundamental constraints. Herbert Simon’s “poverty of attention” (1971) echoes through every subsequent framework — from Broadbent’s bottleneck to Adelaide’s AU metric. The mathematical formalizations differ in precision (d prime equations vs. GRP calculations) but share underlying logic: awareness is a competitive, capacity limited, threshold gated process where signals must overcome noise to achieve conscious representation.
Three principles emerge across all frameworks:
First, awareness is probabilistic, not deterministic. Whether measured as brand recall rate, fixation probability, or d’ sensitivity, awareness is always a probability distribution, not a binary state. The NBD Dirichlet model, signal detection theory, and predictive processing all treat awareness as stochastic.
Second, awareness has architectural constraints. Miller’s 7±2, Cowan’s 4±1, Wickens’ multiple resources, and working memory models all identify hard limits on simultaneous processing. These constraints explain both cognitive load effects in advertising and the effectiveness of chunking in brand design.
Third, awareness involves active prediction, not passive reception. From Treisman’s thresholds to Friston’s precision weighting, modern theory emphasizes that prior expectations shape what enters awareness. This validates marketing practices around brand consistency, mental availability building, and contextual targeting.
The field continues evolving toward attention based measurement (Adelaide, Lumen, Amplified Intelligence) and computational prediction (DeepGaze, saliency models). But the fundamental insight remains what Simon articulated fifty years ago: in an information rich world, awareness is the scarce resource that determines all downstream effects. Every equation documented here — from GRP = Reach × Frequency to F = E_q[ln q(s) ln p(s,o)] — represents an attempt to model that scarcity mathematically.
The Core Insight
f(Awareness) = P(signal → conscious_processing | noise, capacity, threshold)Awareness is not binary. It is a probability distribution gated by:
- Noise — Competing signals in the environment
- Capacity — Hard limits on simultaneous processing (4±1 items)
- Threshold — Minimum activation required for conscious access
Part 1: Marketing and Brand Awareness Frameworks
1.1 The Hierarchy of Effects Tradition (1898–1980)
The foundational idea that awareness precedes action emerged from direct-mail advertising in the late 19th century.
f(AIDA) — E. St. Elmo Lewis (1898)
E. St. Elmo Lewis (1872–1948) first articulated the principle in an 1898 Printers’ Ink article:
f(AIDA) = Attention → Interest → Desire → ActionLewis drew explicitly on William James’s psychology of attention, making this the first documented bridge between cognitive science and advertising theory. The AIDA acronym itself was coined by C.P. Russell in 1921, though Lewis’s conceptual framework preceded it by over two decades.
Key insight: The funnel begins with attention — without it, nothing downstream can occur.
f(Hierarchy_of_Effects) — Lavidge & Steiner (1961)
Robert Lavidge and Gary Steiner formalized the concept in their landmark 1961 paper “A Model for Predictive Measurements of Advertising Effectiveness” (Journal of Marketing).
f(Hierarchy) = Awareness → Knowledge → Liking → Preference → Conviction → PurchaseTheir six-stage hierarchy explicitly mapped onto three psychological domains:
DomainStagesFunctionCognitive (Thinking)Awareness, KnowledgeInformation processingAffective (Feeling)Liking, PreferenceAttitude formationConative (Doing)Conviction, PurchaseAction tendency
f(DAGMAR) — Russell Colley (1961)
Russell Colley’s DAGMAR framework (Defining Advertising Goals for Measured Advertising Results, ANA, 1961) introduced the revolutionary principle that advertising should be measured by communication effects, not sales.
f(DAGMAR) = Awareness → Comprehension → Conviction → ActionThis spawned modern brand tracking methodology — the idea that you can measure awareness independent of purchase.
f(FCB_Grid) — Richard Vaughn (1980)
Richard Vaughn’s FCB Grid (1980, Journal of Advertising Research) integrated these hierarchies with involvement theory:
| THINKING | FEELING
------------|------------------|------------------
HIGH | Learn→Feel→Do | Feel→Learn→Do
INVOLVEMENT | (cars, insurance)| (jewelry, fashion)
------------|------------------|------------------
LOW | Do→Learn→Feel | Do→Feel→Learn
INVOLVEMENT | (household items)| (beer, snacks)Key insight: The Think-Feel-Do sequence reorders based on product category. This remains the standard framework in advertising planning.
Summary: Hierarchy Models
ModelAuthor(s)YearSequenceKey InnovationAIDAE. St. Elmo Lewis1898Attention→Interest→Desire→ActionFirst hierarchical modelHierarchy of EffectsLavidge & Steiner19616 stages mapped to cognition/affect/conationPsychological domainsDAGMARRussell Colley1961Awareness→Comprehension→Conviction→ActionMeasurable objectivesFCB GridRichard Vaughn1980Variable by quadrantInvolvement × processing mode
Critiques: Vakratsas and Ambler’s 1999 meta-analysis of 250+ papers found little empirical support for strict sequential processing. Modern neuroscience confirms that cognitive and affective processing occur simultaneously, not sequentially.
1.2 Media Mathematics: Reach, Frequency, and Carryover
The quantitative infrastructure of media planning rests on equations developed between 1950 and 1980 that remain industry standard today.
f(GRP) — Gross Rating Points (1950s)
f(GRP) = Reach(%) × Average_FrequencyOne GRP represents reaching 1% of the target population once. The formula’s elegance masks its limitation: it treats all exposures as equivalent regardless of attention quality.
f(Three_Exposures) — Herbert Krugman (1972)
Herbert Krugman’s Three-Exposure Theory (1972, “Why Three Exposures May Be Enough,” Journal of Advertising Research) proposed that advertising has only three psychological states:
Exposure 1: "What is it?" (curiosity)
Exposure 2: "What of it?" (evaluation)
Exposure 3: Reminder/decision (action trigger)Krugman argued there is “no such thing as a fourth exposure psychologically” — fours, fives, etc., are repeats of the third.
Michael Naples codified this as the “3+ frequency” standard in his 1979 ANA report, which dominated media planning for two decades.
f(Recency) — Erwin Ephron (1995)
Erwin Ephron’s Recency Theory (1995, “More Weeks, Less Weight,” Journal of Advertising Research) challenged effective frequency orthodoxy.
f(Recency) = Maximize(weekly_reach) over Maximize(frequency)Ephron argued that a single exposure within the purchase decision window matters more than accumulated frequency. John Philip Jones’s single-source research confirmed that one exposure in the seven days before purchase has far greater impact than multiple earlier exposures.
Recommendation: Maximize weekly reach rather than building frequency.
f(Adstock) — Simon Broadbent (1979)
Simon Broadbent’s Adstock Model (1979, Journal of the Market Research Society) provided the mathematical framework for modeling awareness decay:
f(Adstock)_t = Advertising_t + λ × Adstock_(t-1)Where λ (0 < λ < 1) is the decay parameter.
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Half-life calculation:
Half_life = ln(0.5) / ln(λ)Industry benchmarks:
- FMCG brands: ~2.5 weeks half-life
- Brand awareness effects: 7–12 weeks (academic studies)
This equation underlies all modern marketing mix models.
f(Response_Curves) — Advertising Response Functions
Linear (rarely observed):
Sales = α + β × AdvertisingConcave/Diminishing Returns (most common empirically):
Sales = α × Advertising^β where 0 < β < 1S-Curve/Logistic (threshold + saturation):
Response = Max × (Spend^α) / (K^α + Spend^α) [Hill Function]f(Vidale_Wolfe) — Dynamic Response (1957)
The Vidale-Wolfe Model (1957, Operations Research) introduced dynamics:
dS/dt = r × A × (M - S)/M - δ × SWhere:
- S = sales rate
- A = advertising spend
- M = market saturation
- r = response constant
- δ = decay rate
This was the first differential equation model of advertising effects.
1.3 Share of Voice and the Ehrenberg-Bass Revolution
f(ESOV) — John Philip Jones (1990)
John Philip Jones (1990, Harvard Business Review) documented the empirical relationship between Share of Voice (SOV) and Share of Market (SOM):
f(ESOV) = SOV - SOM- Challenger brands require positive ESOV to grow
- Market leaders can profit with negative ESOV
Les Binet and Peter Field’s IPA research suggests:
Expected_Annual_Market_Share_Growth ≈ 0.5% per 10% ESOVf(NBD_Dirichlet) — Ehrenberg, Goodhardt & Chatfield (1984)
The Ehrenberg-Bass Institute (founded by Andrew Ehrenberg) transformed brand science through rigorous empirical analysis.
Their NBD-Dirichlet Model (Goodhardt, Ehrenberg & Chatfield, 1984, Journal of the Royal Statistical Society) describes purchase behavior with three parameters:
- M = Mean category purchase rate
- K = Purchase frequency diversity
- S = Brand propensity diversity
Brand Penetration:
b = 1 - Σ(n=0→∞) P_n × p(0|n)Double Jeopardy Law:
w(1-b) = constantSmaller brands suffer twice — fewer buyers AND lower loyalty.
Duplication of Purchase Law:
b_XY = D × b_XBrand Y’s buyers who also buy brand X equals the Duplication coefficient times brand X’s penetration.
f(Mental_Availability) — Byron Sharp & Jenni Romaniuk (2004–2010)
Byron Sharp’s “How Brands Grow” (2010) synthesized Ehrenberg-Bass findings into marketing’s most influential modern text.
Key constructs:
- Mental Availability: Brand’s propensity to be noticed or retrieved in buying situations
- Physical Availability: Ease of finding and buying the brand
- Category Entry Points (CEPs): The cues buyers use to access memory (developed by Jenni Romaniuk)
f(Mental_Market_Share) = Brand_CEP_associations / Total_category_CEP_associations1.4 Attention Economics: From Simon to Digital Metrics
f(Attention_Scarcity) — Herbert Simon (1971)
Herbert Simon established the theoretical foundation in his 1971 lecture “Designing Organizations for an Information-Rich World”:
“A wealth of information creates a poverty of attention and a need to allocate that attention efficiently among the overabundance of information sources that might consume it.”
Simon connected attention scarcity to his earlier work on bounded rationality (1955): humans “satisfice” rather than optimize because cognitive capacity is limited.
This remains the master concept underlying all attention economics.
f(Attention_Economy) — Michael Goldhaber (1997)
Michael Goldhaber (1997, Wired) declared attention “the currency of the New Economy”:
“Unlike information, human attention is truly individual and scarce.”
Georg Franck (1998, Ökonomie der Aufmerksamkeit) developed the concept of “mental capitalism” — attention as literal capital that earns interest. Celebrity functions as a “stock exchange of attention capital.”
f(Viewability) — IAB/MRC Standards (2014)
Digital attention measurement evolved beyond simple impressions. The IAB/MRC Viewability Standards (2014) established minimum thresholds:
FormatThresholdDisplay≥50% pixels visible for ≥1 secondVideo≥50% pixels visible for ≥2 seconds
But viewability ≠ attention.
f(APM) — Lumen Research (Mike Follett, 2013)
Lumen Research pioneered eye-tracking based metrics:
f(APM) = (% of ads viewed × average viewing time) × 1000 impressionsf(aCPM) = CPM / APM × 1,000Key finding: Only 35% of “viewable” ads are actually looked at.
f(AU) — Adelaide (Marc Guldimann, 2019)
Adelaide developed the AU (Attention Unit) metric — a 0–100 score combining:
- Eye-tracking data
- Media quality signals
- Outcome data
This predicts attention probability rather than duration.
f(Active_Attention) — Karen Nelson-Field (2020)
Karen Nelson-Field (Amplified Intelligence, The Attention Economy and How Media Works, 2020) established:
f(Memory_Encoding) requires ≥ 1.5 seconds active attentionHer research shows attention metrics are 7× more effective at predicting brand awareness than viewability.
Part 2: Cognitive Science and Attention Theory
2.1 Signal Detection and Information Theory
f(Signal_Detection) — Green & Swets (1966)
Signal Detection Theory (SDT) separates perceptual sensitivity from response bias:
f(d') = z(Hit_Rate) - z(False_Alarm_RateWhere z() is the inverse normal CDF.
Criterion measures:
c = -0.5 × (z(HR) + z(FAR)) [criterion location]
β = exp((z(FA)² - z(H)²) / 2) [likelihood ratio]ROC curves plot Hit Rate against False Alarm Rate across criterion levels; Area Under Curve (AUC) quantifies overall discrimination ability.
f(Channel_Capacity) — Claude Shannon (1948)
Claude Shannon’s Information Theory (1948, Bell System Technical Journal) established channel capacity:
f(C) = B × log₂(1 + S/N)Where:
- C = capacity in bits/second
- B = bandwidth
- S/N = signal-to-noise ratio
f(7±2) — George Miller (1956)
George Miller (1956, Psychological Review) applied information theory to cognition, discovering that immediate memory span is limited to:
f(Chunk_Capacity) = 7 ± 2 itemsIndependent of information bits per chunk. Miller emphasized chunking as the mechanism for expanding effective capacity.
f(4±1) — Nelson Cowan (2001)
Nelson Cowan (2001, Behavioral and Brain Sciences) revised Miller’s estimate to:
f(True_Capacity) = 4 ± 1 items (when rehearsal is prevented)K = (hit_rate + correct_rejection_rate - 1) × N2.2 Selective Attention: Filter Models
f(Filter) — Donald Broadbent (1958)
Donald Broadbent’s Filter Model (1958, Perception and Communication) proposed early selection based on physical characteristics:
Stimuli → Sensory Buffer → FILTER → Limited-capacity Processor → ResponseThe filter blocks unattended information before semantic analysis.
f(Attenuation) — Anne Treisman (1964)
Anne Treisman’s Attenuation Theory (1964, British Medical Bulletin) modified this: instead of blocking, the filter merely “turns down the volume.”
If (Attenuated_Signal) ≥ Threshold → Conscious_AwarenessThreshold varies by:
- Subjective importance
- Recent activation (priming)
- Contextual expectancy
- Biological significance
Important stimuli (one’s name, danger words) have permanently low thresholds and break through.
f(Late_Selection) — Deutsch & Deutsch (1963)
Deutsch and Deutsch’s Late Selection Model (1963, Psychological Review) proposed all stimuli receive full semantic processing; selection occurs later based on importance/pertinence.
f(Capacity) — Daniel Kahneman (1973)
Daniel Kahneman’s Capacity Model (1973, Attention and Effort) shifted focus from filtering to resource allocation:
f(Total_Capacity) = g(Arousal) [inverted-U function]Attention is a limited, undifferentiated pool of mental “fuel”:
Performance = f(Resources_allocated, Task_demands)When demands exceed capacity, performance suffers.
f(Multiple_Resources) — Christopher Wickens (1980, 2002)
Christopher Wickens’ Multiple Resource Theory identified four dimensions of resource separation:
DimensionOptionsStagesPerceptual/Cognitive vs. ResponseModalitiesAuditory vs. VisualCodesSpatial vs. VerbalVisual ChannelsFocal vs. Ambient
Prediction: Interference increases with shared resources across dimensions.
This explains why auditory-verbal + visual-spatial tasks combine well (different tanks), while visual-verbal + visual-spatial compete (same tank).
2.3 Computational Models of Visual Attention
f(Feature_Integration) — Treisman & Gelade (1980)
Anne Treisman’s Feature Integration Theory distinguished:
Pre-attentive processing: Features (color, orientation) processed in parallel; targets “pop out”
Focused attention: Required to bind features together (solving the “binding problem”)
Feature search (parallel): RT = a + b (flat, ~0 ms/item slope)
Conjunction search (serial): RT_positive = a + b × N/2
RT_negative = a + b × Nf(Saliency) — Itti & Koch (1998)
The Itti-Koch Saliency Model (1998, IEEE PAMI) operationalized bottom-up attention computationally:
- Input decomposed into 9 spatial scales via Gaussian pyramids
- Feature channels extracted: Intensity, Color (RG, BY opponency), Orientation (4 angles)
- Center-surround operations create feature maps (42 total)
- Normalization operator N(·) promotes maps with few strong peaks
- Conspicuity maps combined:
f(Saliency) = (1/3)[N(Ī) + N(C̄) + N(Ō)]- Winner-take-all network selects most salient location
The model predicts first ~250ms of viewing (pre-attentive processing).
f(Guided_Search) — Jeremy Wolfe (1989–2021)
Jeremy Wolfe’s Guided Search bridges bottom-up saliency with top-down guidance:
f(Activation)_i = w_BU × BU_i + Σ_d(w_d × TD_d,i) + noiseGuided Search 6.0 (2021, Psychonomic Bulletin & Review) integrates five priority sources:
- Bottom-up salience
- Top-down guidance
- History/priming
- Reward/value
- Scene semantics
Selection rate: ~20 Hz attention deployment
2.4 Working Memory, Consciousness, and Predictive Processing
f(Working_Memory) — Baddeley & Hitch (1974, 2000)
Baddeley’s Working Memory Model comprises:
ComponentFunctionCentral ExecutiveAttentional control, coordinates subsystemsPhonological LoopVerbal/auditory information (~2 seconds rehearsal)Visuospatial SketchpadVisual/spatial informationEpisodic Buffer (2000)Integrates across subsystems and LTM
f(Cognitive_Load) — John Sweller (1988)
Cognitive Load Theory (John Sweller, 1988, Cognitive Science) distinguishes:
f(Total_Load) = Intrinsic_Load + Extraneous_LoadOverload occurs when Total Load > Working Memory Capacity.
Implications for advertising: Minimize extraneous load through design optimization.
f(Global_Workspace) — Bernard Baars (1988)
Global Workspace Theory uses a theater metaphor:
- Spotlight = Consciousness
- Stage = Limited working memory
- Audience = Vast unconscious processors
Selected information is broadcast to all processors simultaneously.
Dehaene and Changeux (1998, 2003) provided neural implementation: long-range workspace neurons in prefrontal-parietal cortex show “ignition” — sudden, late (~300ms), sustained firing when threshold crossed.
Signatures of consciousness: P3b wave, gamma synchrony, long-distance coherence
f(Predictive_Processing) — Rao & Ballard (1999)
Predictive Processing models the brain as a hierarchical prediction machine:
Prediction: ŷ_i = W_i · r_(i+1)
Prediction Error: e_i = y_i - ŷ_i- Feedforward connections carry errors
- Feedback connections carry predictions
- Attention = precision-weighting of prediction errors
f(Free_Energy) — Karl Friston (2010)
Karl Friston’s Free Energy Principle (2010, Nature Reviews Neuroscience) provides a unified framework:
f(F) = E_q[ln q(s) - ln p(s,o)]Equivalently: F = KL-divergence between beliefs and true posterior + surprisal.
The brain minimizes free energy through:
- Perceptual inference (updating beliefs)
- Learning (updating model)
- Action (changing inputs)
Attention modulates precision:
Precision (π) = 1 / varianceHigh-precision prediction errors gain more weight in updating beliefs.
Part 3: Cross-Domain Synthesis
3.1 How Cognitive Theory Shaped Advertising Science
The bridges between cognitive science and advertising are well-documented:
YearCognitive SourceAdvertising Application1898William James (attention)Lewis’s AIDA model1956Miller (7±2 chunks)Message design limits1966Signal Detection TheoryAd noticeability measurement1972Mere-exposure effectKrugman’s three exposures1973Kahneman (capacity)Cognitive load in ads1980Feature IntegrationDistinctive brand assets1980Multiple ResourcesMultimodal ad design
3.2 The Attention Funnel: Unified Framework
Across domains, awareness functions as the first filter in a multi-stage process:
Marketing Funnel:
Exposure → Attention → Awareness → Processing → Attitude → BehaviorCognitive Funnel:
Sensation → Feature Detection → Saliency Competition → Working Memory → DecisionShared mathematical properties:
PropertyMarketingCognitiveCapacity limits~7 brands in consideration set~4 items in working memoryDecay functionsAdstock λ ≈ 0.75Trace decay ~1–2 secondsThreshold effectsMinimum exposure for awarenessIgnition threshold for consciousnessProbabilistic retrievalCEP-based probabilityActivation-based probability
3.3 Metric Integration Across Paradigms
Marketing MetricCognitive EquivalentMathematical RelationshipBrand Awareness (%)Recognition probabilityp(recall) = f(activation, threshold)Effective Frequency (3+)Memory consolidationRepetition strengthens traceAdstock decay (λ)Memory decayExponential forgetting curveMental AvailabilitySpreading activationNetwork accessibilityAttention time (seconds)Fixation durationEncoding timed’ (sensitivity)Ad discriminabilitySignal detection
Master Reference Tables
Marketing and Advertising Frameworks
FrameworkAuthor(s)YearKey Equation/ModelAIDAE. St. Elmo Lewis1898Attention→Interest→Desire→ActionHierarchy of EffectsLavidge & Steiner19616 stages: Awareness→Knowledge→Liking→Preference→Conviction→PurchaseDAGMARRussell Colley1961Awareness→Comprehension→Conviction→ActionFCB GridRichard Vaughn19802×2: Involvement × Thinking/FeelingGRPIndustry standard1950sGRP = Reach × FrequencyThree ExposuresHerbert Krugman1972Psychological threshold modelEffective FrequencyMichael Naples19793+ exposures optimalRecency TheoryErwin Ephron1995One exposure in purchase windowAdstockSimon Broadbent1979A_t = T_t + λ × A_(t-1)Vidale-WolfeVidale & Wolfe1957dS/dt = r×A×(M-S)/M — δ×SSOV-SOMJohn Philip Jones1990ESOV = SOV — SOMNBD-DirichletEhrenberg et al.1984Brand choice probability modelMental AvailabilityByron Sharp/Romaniuk2004–2010CEP-based salienceCBBE PyramidKevin Lane Keller1993Salience→Meaning→Response→ResonanceAU MetricAdelaide/Guldimann20190–100 attention probability scoreActive AttentionKaren Nelson-Field20201.5s threshold for memory encoding
Cognitive Science Frameworks
FrameworkAuthor(s)YearKey Equation/ModelChannel CapacityClaude Shannon1948C = B × log₂(1 + S/N)7 ± 2 ChunksGeorge Miller1956Immediate memory spanFilter ModelDonald Broadbent1958Early physical selectionAttenuationAnne Treisman1964Signal reduction + thresholdLate SelectionDeutsch & Deutsch1963Post-semantic selectionSignal DetectionGreen & Swets1966d’ = z(HR) — z(FAR)Capacity ModelDaniel Kahneman1973Flexible resource poolFeature IntegrationTreisman & Gelade1980Parallel features + serial bindingMultiple ResourcesChristopher Wickens19804-D interference modelGlobal WorkspaceBernard Baars1988Broadcast to unconscious processorsCognitive LoadJohn Sweller1988Total = Intrinsic + ExtraneousItti-Koch SaliencyItti & Koch1998S = (1/3)[N(Ī) + N(C̄) + N(Ō)]Predictive CodingRao & Ballard1999Prediction error minimization4 ± 1 ChunksNelson Cowan2001True capacity without rehearsalFree EnergyKarl Friston2010F = E_q[ln q(s) — ln p(s,o)]Guided Search 6.0Jeremy Wolfe20215-source priority map
Conclusion: The Unified Logic of Awareness
The intellectual archaeology of awareness reveals a remarkable convergence: marketing practitioners and cognitive scientists independently discovered the same fundamental constraints.
Herbert Simon’s “poverty of attention” (1971) echoes through every subsequent framework — from Broadbent’s bottleneck to Adelaide’s AU metric. The mathematical formalizations differ in precision (d-prime equations vs. GRP calculations) but share underlying logic:
Awareness is a competitive, capacity-limited, threshold-gated process where signals must overcome noise to achieve conscious representation.
Three Unifying Principles
1. Awareness is probabilistic, not deterministic.
Whether measured as brand recall rate, fixation probability, or d’ sensitivity, awareness is always a probability distribution, not a binary state. The NBD-Dirichlet model, signal detection theory, and predictive processing all treat awareness as stochastic.
2. Awareness has architectural constraints.
Miller’s 7±2, Cowan’s 4±1, Wickens’ multiple resources, and working memory models all identify hard limits on simultaneous processing. These constraints explain both cognitive load effects in advertising and the effectiveness of chunking in brand design.
3. Awareness involves active prediction, not passive reception.
From Treisman’s thresholds to Friston’s precision-weighting, modern theory emphasizes that prior expectations shape what enters awareness. This validates marketing practices around brand consistency, mental availability building, and contextual targeting.
The Opportunity: New Math for a New Era
The core mathematical frameworks documented here were established between 1950 and 1980. We are operating under pre-internet, pre-digital, pre-AI assumptions.
The field continues evolving toward:
- Attention-based measurement (Adelaide, Lumen, Amplified Intelligence)
- Computational prediction (DeepGaze, saliency models)
- Real-time optimization (ML-driven media buying)
But the fundamental insight remains what Simon articulated fifty years ago:
In an information-rich world, awareness is the scarce resource that determines all downstream effects.
Every equation documented here — from GRP = Reach × Frequency to F = E_q[ln q(s) - ln p(s,o)]—represents an attempt to model that scarcity mathematically.
The next breakthrough will come from synthesizing these traditions with modern computational methods.
References
Primary Sources — Marketing
- Binet, L. & Field, P. (2013). The Long and the Short of It. IPA.
- Broadbent, S. (1979). One Way TV Advertisements Work. Journal of the Market Research Society.
- Colley, R. (1961). Defining Advertising Goals for Measured Advertising Results. ANA.
- Ehrenberg, A., Goodhardt, G., & Barwise, P. (1990). Double Jeopardy Revisited. Journal of Marketing.
- Ephron, E. (1995). More Weeks, Less Weight. Journal of Advertising Research.
- Jones, J.P. (1990). Ad Spending: Maintaining Market Share. Harvard Business Review.
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- Nelson-Field, K. (2020). The Attention Economy and How Media Works. Palgrave Macmillan.
- Sharp, B. (2010). How Brands Grow. Oxford University Press.
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Primary Sources — Cognitive Science
- Baars, B. (1988). A Cognitive Theory of Consciousness. Cambridge University Press.
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- Broadbent, D. (1958). Perception and Communication. Pergamon Press.
- Cowan, N. (2001). The Magical Number 4 in Short-Term Memory. Behavioral and Brain Sciences.
- Friston, K. (2010). The Free-Energy Principle: A Unified Brain Theory? Nature Reviews Neuroscience.
- Green, D.M. & Swets, J.A. (1966). Signal Detection Theory and Psychophysics. Wiley.
- Itti, L. & Koch, C. (1998). A Model of Saliency-Based Visual Attention. IEEE PAMI.
- Kahneman, D. (1973). Attention and Effort. Prentice-Hall.
- Miller, G.A. (1956). The Magical Number Seven, Plus or Minus Two. Psychological Review.
- Shannon, C.E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal.
- Simon, H.A. (1971). Designing Organizations for an Information-Rich World. In Computers, Communication, and the Public Interest.
- Treisman, A. (1964). Selective Attention in Man. British Medical Bulletin.
- Treisman, A. & Gelade, G. (1980). A Feature-Integration Theory of Attention. Cognitive Psychology.
- Wolfe, J.M. (2021). Guided Search 6.0. Psychonomic Bulletin & Review.
f(Awarenes) function by FunnelFunction.com. All rights reserved. Armstrong Knight, Abullah Khan, Claude.ai
