AMSS Sales Event - Demand Curves, Competition and Post-Purchase Migration

Summary

The last event each week turns hidden segment counts into sales. Only purchase-state segments buy. Each has a linear demand curve in price $({\alpha }_s-{\beta }_s{p}_t)$ clipped to $[0,1]$. Competition is summarised by strength ${\gamma }_{t,s}$ and substitutability ${\omega }_{t,s}$. A multinomial splits each segment into no purchase, advertiser purchase and competitor purchase. Units per buyer are one plus a Poisson draw. Afterwards, all buyers become satiated, and brand favorability and loyalty can update. Media increases sales by moving people to segments with better demand curves. Price promotions move people along a demand curve.

Overview

The sales event (event $K$; footnote 3: “sales” from the advertiser’s view, “purchase” from the consumer’s) computes advertiser sales per segment $y_{t,\cdot }=(y_{t,s}{)}_{s\in \mathcal{S}}$ and competitor sales $z_{t,\cdot }$ (§2.4, p.18). It also produces the post-purchase segmentation $n_{t+1,1,\cdot }$. To avoid simulating competitors in as much detail as the advertiser, AMSS summarises them with two time-varying parameters (p.18).

Main Content

Competition-free purchase probability (§2.4.1, p.18)

For segments with $s_3=\text{purchase}$, define per-segment parameters:

• ${\alpha }_s\in [0,1]$: probability of buying the advertiser’s brand at price 0, with no competition;
• ${\beta }_s\in \mathbb{R}$: drop in purchase probability per unit price increase. Usually ${\beta }_s>0$.

At unit price $p_t>0$,

$$r_{t,s}^{(y^{\ast })}=({\alpha }_s-{\beta }_s{p}_t)\vee 0\wedge 1.$$

Segments with high favorability, loyalty or availability should have higher ${\alpha }_s$ and smaller ${\beta }_s$.

Figure 5 (p.19) shows competition-free demand curves for purchase-state, switcher, average-availability segments: favorable (solid, highest intercept), somewhat favorable (dash), neutral (dot), unaware (long dash) and negative (the caption’s word for Table 1’s unfavorable level) (dot-dash, lowest). The lines fall linearly in price to zero.

Competitor strength and substitutability (§2.4.1, pp.18-19)

• $\gamma =({\gamma }_{t,s})\in [0,1]$: probability a segment-$s$ consumer buys a competitor brand at time $t$ when the advertiser’s price is so high that it sells nothing.
• $\omega =({\omega }_{t,s})\in [0,1]$: how far brand and competitor sales replace each other.
  • $\omega =1$: competitor sales are unaffected by the advertiser’s price, and competitors take as much as possible. This is the default for competitor-loyal consumers (Fig. 6 top).
  • $\omega =0$: advertiser sales are unaffected by competition, and the advertiser takes as much as possible. This is the default for loyal consumers (Fig. 6 bottom).
  • $\omega =0.5$: the default for switchers (Fig. 6 middle).

Purchase probabilities are

$$r_{t,s}^{(z)}=\{{\gamma }_{t,s}-(1-{\omega }_{t,s})r_{t,s}^{(y^{\ast })}\}\vee 0,r_{t,s}^{(y)}=\max (r_{t,s}^{(y^{\ast })},{\gamma }_{t,s})-r_{t,s}^{(z)}.$$

Figure 6 (p.20) plots the probabilities against price. The light region is the competitor’s share $r^{(z)}$ and the dark region is the advertiser’s share $r^{(y)}$, shown for $\omega =1$, $1/2$ and $0$. Total category purchase is $\max (r^{(y^{\ast })},\gamma )$. $\omega$ decides how that total is split where the two curves overlap.

Sales draws (§2.4.1, p.19)

Let $a_{t,K,s}^{(w)}$, $a_{t,K,s}^{(y)}$ and $a_{t,K,s}^{(z)}$ be the numbers who buy nothing, buy the advertiser’s brand, and buy a competitor’s:

$$(a^{(w)},a^{(y)},a^{(z)}{)}_{t,K,s}\sim \text{Multinomial}(n_{t,K,s},(1-r_{t,s}^{(y)}-r_{t,s}^{(z)},r_{t,s}^{(y)},r_{t,s}^{(z)})).$$

With ${\lambda }^{(y)}$ the mean units per category purchaser, brand units are

$$y_{t,s}\sim a_{t,K,s}^{(y)}+\text{Poisson}(a_{t,K,s}^{(y)}({\lambda }^{(y)}-1)),$$

and revenue $=y_{t,s}{p}_t$. The paper describes competitor units $z_{t,s}$ but does not write them out; they presumably follow the same form.

Profit. §2.4.2 (p.20) opens with “In addition to calculating sales, revenue, and profit, the sales module also models post-purchase changes”. So profit is listed as a sales-module output alongside sales and revenue, but the paper never defines it. It gives no cost-of-goods or margin parameter, and the ROAS ground truth in §3 is stated in revenue terms. A natural reading is revenue minus unit costs, possibly net of media spend, but that is not in the source.

Post-purchase migration (§2.4.2, pp.20-21; §2.2.5, p.11)

Two migrations take $n_{t,K,\cdot }$ to $n_{t+1,1,\cdot }$:

1. Satiation. For unsatiated $s\in \mathcal{B}$, the affected population is every category purchaser, $a_{t,K,s}=a_{t,K,s}^{(y)}+a_{t,K,s}^{(z)}$. Each moves deterministically to the matching satiated segment $g(s)=(s_1,\text{satiated},s_3,\dots ,s_6{)}^{\top }$, i.e. $q_{s,s^{\prime }}^{(t,K)}=1_{\{g(s)\}}(s^{\prime })$. Apply Eqs. 1-2.
2. Brand-state update. Purchase experience can create habit or preference. Matrices ${\mathcal{Q}}^{(K,l)}$ for $l=4,5,6$ drive the migration as in a media event (C.1), “except the frequency is ignored.”

Satiated purchasers are then subject to de-satiation at rate $\lambda$ in later weeks (^def-amss-desatiation). That is the mechanism behind post-promotion dips.

Examples

Sales parameters in the simple scenario (§4.1.1, p.26)

Setup. Price is a constant $40. Everyone is a switcher with average availability, so purchase probability varies only with favorability. At $40 the advertiser purchase probabilities are $(0.014,0,0.2,0.3,0.9)$ for (unaware, negative [= Table 1’s unfavorable], neutral, somewhat favorable, favorable). $\gamma \equiv 0.8$ and $\omega \equiv 0$, which removes competitor effects on advertiser sales. Satiation is switched off: buyers stay in-market and active.

Implication. With $\omega =0$, $r^{(z)}=(0.8-r^{(y^{\ast })})\vee 0$ and $r^{(y)}=\max (r^{(y^{\ast })},0.8)-r^{(z)}=r^{(y^{\ast })}$. Advertiser sales therefore depend only on favorability. TV’s value comes entirely from moving people up the favorability ladder: from neutral (0.2) to somewhat favorable (0.3) is +0.1 per purchase-state person, and from somewhat favorable to favorable is +0.6.

Worked competition arithmetic (illustrative)

Setup. A switcher segment with ${\alpha }_s=0.6$, ${\beta }_s=0.01$, $p=$$20, so $r^{(y^{\ast })}=0.4$. $\gamma =0.5$, $\omega =0.5$.

Solution. $r^{(z)}=0.5-0.5\cdot 0.4=0.3$. $r^{(y)}=\max (0.4,0.5)-0.3=0.2$. No purchase: $1-0.2-0.3=0.5$.

Interpretation. Category purchase is 0.5. The advertiser keeps half of its competition-free 0.4. A price cut to $p=$$0 gives $r^{(y^{\ast })}=0.6$, $r^{(z)}=0.2$ and $r^{(y)}=0.4$: some sales are stolen from the competitor and the category grows to 0.6.

Connections

• Price effects in MMM. Price enters as a demand-curve movement, which is distinct from media’s segment-shifting effect. See Price and Distribution Effects. Competitive reactions, which AMSS simplifies away, are covered in Reaction Functions and Competitive Dynamics.
• Base vs incremental sales. Base sales are the sales under an all-budgets-zero counterfactual. They come out of this event fed by the natural-migration equilibrium. In the simple scenario they are about 95% of total sales (AMSS MMM Evaluation - Simple Scenario and Lagged-Effects Study).
• Availability as a sales lever. Availability affects sales only through ${\alpha }_s,{\beta }_s$. Distribution changes are modelled as availability migrations.

See Also

• AMSS Hidden-State Consumer Segmentation and Migration Formalism
• AMSS Ground Truth - Counterfactual Scenarios and Simulated ROAS: revenue $y\cdot p$ is the ROAS numerator
• Zhang Vaver 2017 - Aggregate Marketing System Simulator - Overview