AMSS Natural Migration, Satiation and Lagged-Effect Mechanics

Summary

The non-media events in AMSS are: a market-size event that steers the in-market share to a target ${\rho }_t$ (seasonality times trend) with minimal migration; a de-satiation event at rate $\lambda$; and four natural-migration events that pull activity, favorability, loyalty and availability back to an equilibrium. The key result is about carryover. A natural-migration matrix produces no lag iff it is row-identical. $Q_{\alpha }=(1-\alpha )Q_0+\alpha I$ keeps a fraction $\alpha$ of any marketing-induced displacement each period. That is a geometric decay with rate $\alpha$, generated by the population dynamics and not imposed on spend.

Overview

Natural forces set the baseline that marketing builds on (§2.2.1, p.9). When many people are in the market, media has more people to work on. When satiation is high or the market is small, media-driven activity migrations are capped, because out-of-market or satiated people must stay inactive (§2.2.3, p.10). Natural migration also decides how long marketing effects last: the stronger the pull back to equilibrium, the shorter the effect (§2.2.4, p.10). This is the mechanism behind the paper’s lagged-effects experiment (AMSS MMM Evaluation - Simple Scenario and Lagged-Effects Study).

Sets used throughout (App. B, Eqs. 8-9; App. C.1):

$$\mathcal{A}=\{s:s_1=\text{in-market}\},\mathcal{B}=\{s:s_2=\text{unsatiated}\},\mathcal{D}=\{s:s_4=\text{favorable}\},\mathcal{E}=\{s:s_5=\text{loyal}\}.$$

Main Content

Event $k=1$: changing market size (§2.2.1; App. B.1)

Market rate and target (§2.2.1, p.9)

The market rate before event $k$ is $r_{t,k}=n_{t,k,\mathcal{A}}/n_{t,k,\mathcal{S}}$. The modeler specifies target rates $\rho =({\rho }_t{)}_{1:T}$, factored as ${\rho }_t={\rho }_t^{(\text{seas})}\cdot {\rho }_t^{(\text{trend})}$. AMSS reaches the target in expectation with minimal migration:

• If $r_{t,1}>{\rho }_t$: in-market people leave with probability $(r_{t,1}-{\rho }_t)/r_{t,1}$, so $E[r_{t,2}]=r_{t,1}(1-\frac{r_{t,1}-{\rho }_t}{r_{t,1}})={\rho }_t$.
• If $r_{t,1}<{\rho }_t$: out-of-market people enter with probability $({\rho }_t-r_{t,1})/(1-r_{t,1})$, so $E[r_{t,2}]=r_{t,1}+(1-r_{t,1})\frac{{\rho }_t-r_{t,1}}{1-r_{t,1}}={\rho }_t$.
• If $r_{t,1}={\rho }_t$: no migration.

Market entry and exit transition matrices (App. B.1, Eq. 10, pp.32-33)

Case 1, $r_{t,1}<{\rho }_t$ (entry). The affected population is $a_{t,1,s}=n_{t,1,s}{1}_{\mathcal{S}\setminus \mathcal{A}}(s)$. Entrants keep every dimension except market and, if unsatiated, activity. They land in activity state $j$ with a modeler-specified probability ${\gamma }_j$, where ${\sum }_{j\in S_3}{\gamma }_j=1$. Satiated entrants must stay inactive. Define

$$h_j^{(1)}(s)=\{\begin{array}{ll}(\text{in-market},s_2,j,s_4,s_5,s_6{)}^{\top }& s\in \mathcal{B}\\ (\text{in-market},s_2,\text{inactive},s_4,s_5,s_6{)}^{\top }& \text{otherwise}\end{array}\text{\hspace{1em}(10)}$$ $$q_{s,s^{\prime }}^{(t,1)}=\frac{1-{\rho }_t}{1-r_{t,1}}{1}_{\{s\}}(s^{\prime })+\sum _{j\in S_3}{\gamma }_j\frac{{\rho }_t-r_{t,1}}{1-r_{t,1}}{1}_{\{h_j^{(1)}(s)\}}(s^{\prime }).$$

Case 2, $r_{t,1}>{\rho }_t$ (exit). The affected population is $a_{t,1,s}=n_{t,1,s}{1}_{\mathcal{A}}(s)$. Leavers also become inactive: $h_{s_3}^{(1)-1}(s)=(\text{out-of-market},s_2,\text{inactive},s_4,s_5,s_6{)}^{\top }$, and

$$q_{s,s^{\prime }}^{(t,1)}=\frac{{\rho }_t}{r_{t,1}}{1}_{\{s\}}(s^{\prime })+\frac{r_{t,1}-{\rho }_t}{r_{t,1}}{1}_{\{h_{s_3}^{(1)-1}(s)\}}(s^{\prime }).$$

Migrations are then drawn by Eq. 1 and applied by Eq. 2.

Note that the paper reuses the symbol $\gamma$. The entry distribution over activity states ${\gamma }_j$ here is unrelated to competitor strength ${\gamma }_{t,s}$ in the sales event.

Event $k=2$: de-satiation (§2.2.2; App. B.2)

De-satiation (§2.2.2, p.10; App. B.2, Eq. 11, pp.33-34)

Purchasers become satiated in the sales event. At $k=2$ each satiated person de-satiates with probability $\lambda$ (the de-satiation rate). The affected population is $a_{t,2,s}=n_{t,2,s}{1}_{\mathcal{S}\setminus \mathcal{B}}(s)$. In-market de-satiators also redraw their activity state from ${\gamma }_j$, the same distribution as in B.1. With

$$h_j^{(2)}(s)=\{\begin{array}{ll}(s_1,\text{unsatiated},j,s_4,s_5,s_6{)}^{\top }& s\in \mathcal{A}\\ (s_1,\text{unsatiated},\text{inactive},s_4,s_5,s_6{)}^{\top }& \text{otherwise}\end{array}\text{\hspace{1em}(11)}$$

the transition probabilities are

$$q_{s,s^{\prime }}^{(t,2)}=(1-\lambda )1_{\{s\}}(s^{\prime })+\lambda \sum _{j\in S_3}{\gamma }_j{1}_{\{h_j^{(2)}(s)\}}(s^{\prime }).$$

A small $\lambda$ means slow recovery, as with durables and travel. A large $\lambda$ means fast recovery, as with perishables. Satiation lets AMSS reproduce post-promotion sales dips (p.10).

Event $k=3,\dots ,6$: natural migration towards equilibrium (§2.2.4; App. B.3)

Natural migration in dimension $l$ (App. B.3, Eq. 12, p.34)

For activity ($l=3$), only consumers in $\mathcal{A}\cap \mathcal{B}$ (in-market and unsatiated) migrate: $a_{t,3,s}=n_{t,3,s}{1}_{\mathcal{A}\cap \mathcal{B}}(s)$. A modeler-specified stochastic matrix ${\mathcal{Q}}^{(3)}=({\varrho }_{i,j}{)}_{S_3\times S_3}$ with ${\sum }_j{\varrho }_{i,j}=1$ gives the probability of moving from activity state $i$ to $j$. Migration changes only dimension $l$. With $h_j(s)=(s_1,s_2,j,s_4,s_5,s_6{)}^{\top }$ (Eq. 12),

$$q_{s,s^{\prime }}^{(t,3)}=(1-{\varrho }_{s_3,s_3^{\prime }})1_{\{s\}}(s^{\prime })+{\varrho }_{s_3,s_3^{\prime }}{1}_{\{h_{s_3^{\prime }}(s)\}}(s^{\prime }).$$

Favorability, loyalty and availability work the same way. Only the eligible origin sets differ:

• favorability: only switchers and competitor-loyal consumers ($\mathcal{S}\setminus \mathcal{E}$). “Ad exposure won’t change the favorability state of an individual who is ‘loyal’” (p.34);
• loyalty: only “favorable” consumers can become “loyal” ($\mathcal{D}$);
• availability: unrestricted.

Each ${\mathcal{Q}}^{(l)}$ does two jobs. It fixes the natural equilibrium of dimension $l$ (its stationary distribution) and the strength of the pull back towards it, which is the mixing speed.

Lagged effects as resistance to equilibrium (§2.2.4; App. B.3.1)

No-lag characterisation (App. B.3.1, Eq. 13, p.35)

Let $C=\mathcal{A}\cap \mathcal{B}$, split by activity into $C_1$ (inactive), $C_2$ (exploratory) and $C_3$ (purchase), and let $n=(n_{t,3,C_1},n_{t,3,C_2},n_{t,3,C_3}{)}^{\top }$. A natural-migration matrix brings any distribution back to equilibrium $x$ in one step, so marketing-induced changes have no lag, iff it is row-identical:

$${\mathcal{Q}}^{(3)\top }n=\Vert n\Vert x\forall n⟺{\mathcal{Q}}^{(3)}=(x,x,x{)}^{\top }.$$

Here $\Vert n\Vert$ is the total count (the ${\ell }_1$ norm). Example (Eq. 13): $Q_0$ with every row equal to $(0.50,0.30,0.20)$ gives $Q_0^{\top }n=\Vert n\Vert (0.50,0.30,0.20{)}^{\top }$.

Proof sketch. ($\Leftarrow$) If every row is $x^{\top }$, then $Q^{\top }n={\sum }_i{n}_{i}x=\Vert n\Vert x$. ($\Rightarrow$) Apply the identity to $n=e_i$, which puts everyone in state $i$. Then row $i$ of $\mathcal{Q}$ equals $x$ for every $i$. The paper states the equivalence without a proof; this argument fills it in.

Interpolated lag family (App. B.3.1, Eq. 14, p.35)

For $\alpha \in [0,1]$ let

$$Q_{\alpha }=(1-\alpha )Q_0+\alpha I_3.\text{\hspace{1em}(14)}$$

Then for any segmentation $n$,

$$Q_{\alpha }^{\top }n=(1-\alpha )\Vert n\Vert x+\alpha n.$$

So $\alpha =0$ gives no lag, and $\alpha =1$ ($Q_1=I_3$) gives permanent effects, since $Q_1^{\top }n=n$: the intervention’s impact “propagates into all subsequent time intervals, until new marketing interventions change the distribution once again” (p.35). For $\alpha \in (0,1)$, the deviation from equilibrium satisfies $Q_{\alpha }^{\top }n-\Vert n\Vert x=\alpha (n-\Vert n\Vert x)$. A fraction $\alpha$ of any displacement survives each period, so after $j$ periods without further marketing ${\alpha }^j$ of it remains. That is exactly the geometric-adstock decay ${\alpha }^j$.

General rule: any valid ${\mathcal{Q}}^{(3)}$ (entries in $[0,1]$, rows summing to 1) works. Larger diagonal entries ${\varrho }_{i,i}$ mean stronger persistence and longer lag. Larger off-diagonal entries mean faster mixing and weaker lag (p.35).

The step from $Q_{\alpha }^{\top }n$ to geometric decay uses ${\alpha }^j$ iteration. That is our derivation from Eq. 14; the paper does not spell it out. It holds in expectation and ignores multinomial noise and the media events that follow in the same week.

Why this matters for MMM validation. In a regression MMM, carryover is a transform of spend: geometric or delayed adstock (Carryover (Adstock) Functional Forms, Carryover Effects and Distributed Lags). In AMSS, carryover is a property of mindset persistence. It acts on people moved, and how many people are moved depends non-linearly on reach and frequency. The two coincide only roughly. An adstock MMM fitted to AMSS data with $\alpha >0$ is therefore misspecified in a realistic way, which is what makes the lag experiment informative.

Post-purchase migration (§2.2.5)

The sales event (the $K$-th) also migrates people. All purchasers become satiated, and favorability and loyalty can update with satisfaction. Satiation then decays at rate $\lambda$ in later weeks. Formal details are in ^def-amss-post-purchase.

Which dimensions marketing can move (§2.2.3)

Marketing-actionable dimensions (§2.2.3, p.10)

Marketing can move activity, favorability, loyalty and availability ($l=3,\dots ,6$). It cannot move market or satiation ($l=1,2$). People who are out of the market or satiated are stuck in “inactive”, so a small market or heavy satiation caps the number of activity migrations a channel can drive. How effective an intervention is depends on the pre-event segmentation and on the intervention itself, including audience size, frequency and creative efficacy.

Examples

Lag arithmetic with the Eq. 13 matrix

Setup. Equilibrium $x=(0.5,0.3,0.2)$ over (inactive, exploratory, purchase), with 1,000 eligible people. A TV burst moves the distribution to $n=(400,350,250)$, a displacement of $(-100,+50,+50)$ from $\Vert n\Vert x=(500,300,200)$.

$\alpha =0$. Next week $Q_0^{\top }n=(500,300,200)$. The 50 extra purchase-state people are gone, so there is no carryover.

$\alpha =0.4$. $Q_{0.4}^{\top }n=0.6(500,300,200)+0.4(400,350,250)=(460,320,220)$. The displacement is now $(-40,+20,+20)$, which is $0.4$ of the original. The week after, it is $(-16,+8,+8)$, and so on.

Interpretation. Raising the diagonal weight produces geometric carryover. The paper uses the same device on favorability ($Q_{\alpha }^{(4)}$) in §4.2.

Seasonal market rate in the simple scenario (§4.1.1, Fig. 7, pp.23-24)

${\rho }_t$ is a 52-week sinusoid centred at 0.42 with amplitude 0.175, multiplied by an AR(1) series with mean 1, SD 0.1 and autocorrelation 0.3. Fig. 7 shows the market rate oscillating between about 0.25 and 0.70 over the 208 weeks. The market-size event follows this target, so the pool that media can act on swings by a factor of about 2.5 within each year.

Connections

• Adstock as an emergent property. Compare the parametric adstock families in Carryover (Adstock) Functional Forms and Carryover Effects and Distributed Lags. Experimental-design issues caused by carryover are discussed in Q - How Adstock Breaks Switchback and Sequential Test Assumptions and Q - Carryover Dynamics and the Timing of Sequential Media Experiments.
• Seasonality as a confounder. Market-size seasonality drives both search spend (through queries) and sales. Its structural-time-series counterpart is Local Linear Trend and Seasonality.
• Markov-chain mixing. The lag parameter $\alpha$ is the second eigenvalue of $Q_{\alpha }$ when $Q_0$ is rank one. Stronger mixing means a shorter memory.

See Also

• AMSS Hidden-State Consumer Segmentation and Migration Formalism: Eqs. 1-2, which all these events instantiate
• AMSS MMM Evaluation - Simple Scenario and Lagged-Effects Study: uses $Q_{\alpha }^{(4)}$ to vary lag strength
• Zhang Vaver 2017 - Aggregate Marketing System Simulator - Overview