AMSS Paid Search Module (Queries, Auction, Organic vs Paid)

Summary

Paid search in AMSS is driven by demand. The budget sets campaign settings (spend cap, bid, keyword-list length). Segment-specific query propensities produce a query volume that depends on the hidden mindset distribution. A simplified auction maps the bid to CPC and maximum share of voice, and these give available impressions, clicks (segment CTRs) and spend, truncated by the cap. Effects differ across three exposure types: organic only, paid impression without a click, and paid click, weighted by $\psi =({\psi }_w,{\psi }_u,{\psi }_v)$. Because spend follows queries and queries follow latent demand, search spend is endogenous. Because organic search is modelled, the incremental value of paid over organic is well defined: zero the search budget.

Overview

Search differs from TV in that inventory is limited and priced by auction (§2.3.2, p.14). Its volume and effectiveness also depend on what earlier events did to the population. Activity state raises the probability of generic queries. Activity and brand states raise the probability of branded queries. Favorability and loyalty raise CTR. Organic results exist when no paid ad is shown, which “allows modelers to identify the incremental effect of paid search over organic search” (p.14).

One module, several channels (§2.3, p.11). A channel is a parameterisation of a module. Different query rates and CTRs by segment let the same paid search module produce separate branded and generic search events, each sequenced as its own event.

Figure 4 (p.15) is a dependency diagram. Inputs (ovals): expected reach by segment, query rate, competitor bid range, budget-to-campaign-settings mapping functions, maximum effect, relative effectiveness by exposure type, CTR by segment, budget parameters. Observed (unshaded): query volume; impressions, clicks and spend; budget. Hidden (shaded): audience, share of voice and CPC, campaign settings, effect scaling, migration, and the population segmentations before and after.

Main Content

Paid search module, event $k$ in interval $t$ (§2.3.2, pp.15-17; App. C.2, pp.39-41)

1. Campaign settings from budget. The modeler supplies three functions of the budget: (a) the weekly spend cap $x_t^{(M)}$, which can be the budget split evenly or uncapped ($\infty$); (b) the bid $c_t^{\ast }$, a single campaign-level number rather than per-keyword bids (footnote 5), e.g. rising linearly with the yearly budget; (c) the keyword-list length: a longer list matches more queries. App. C.2 (p.39) adds two points. The bid is expressed as a maximum cost per click, which the paper says “is equivalent to a bid”. A budget cut can be realised in two ways. (i) Lower bids, which lower both the CPC and the SOV through the auction in step 4. (ii) Throttling paid search volume through a lower spend cap, a shorter keyword list, or any combination of these.
2. Audience. People in each segment who make queries matching the keyword list, $a_{t,k,s}$. The draw is analogous to TV, but the query propensity depends on activity and brand state, which is needed for branded campaigns, and is scaled by keyword coverage. Generic ads mainly reach exploratory and purchase consumers. Branded ads reach high favorability and loyalty. “Modeling multiple channels with different levels of targeting allows modelers to explore challenges related to selection bias” (p.16).
3. Query volume. Each audience member queries as a Poisson process with rate ${\lambda }^{(k)}$: $w_{t,\mathcal{S}}\mid a_{t,k,\cdot }\sim \text{Poisson}({\lambda }^{(k)}{a}_{t,k,\mathcal{S}})$.
4. Auction: CPC and maximum share of voice. The competitive landscape is summarised by a bid range $[{\eta }_t^{(m)},{\eta }_t^{(M)}]$. Below ${\eta }_t^{(m)}$ the advertiser loses every auction. At or above ${\eta }_t^{(M)}$ it wins every auction. $$c_t={\eta }_t^{(m)}\vee c_t^{\ast }\wedge {\eta }_t^{(M)},p_t^{(u)}=0\vee \frac{c_t^{\ast }-{\eta }_t^{(m)}}{{\eta }_t^{(M)}-{\eta }_t^{(m)}}\wedge 1.$$ $\vee$ is max and $\wedge$ is min. Maximum SOV is “the fraction of search queries matching the keyword list whose auctions the advertiser can win at the current bid” (footnote 6). The realised SOV can be lower when the cap binds. For branded campaigns, which win cheaply thanks to quality-score relevance, set a low $[{\eta }^{(m)},{\eta }^{(M)}]$.
5. Available impressions. $u_{t,\mathcal{S}}^{\ast }\mid w_{t,\mathcal{S}}\sim \text{Binomial}(w_{t,\mathcal{S}},p_t^{(u)})$.
6. Spend cap → click cap. $v_t^{(M)}=x_t^{(M)}/c_t$. The overall CTR is the audience-weighted $p_t^{(v)}={\sum }_s{a}_{t,k,s}{\pi }_s^{(v)}/a_{t,k,\mathcal{S}}$, using segment CTRs ${\pi }_s^{(v)}$ that are higher for favourable and loyal states. The impressions needed to hit the click cap are drawn as $u_t^{(M)}-v_t^{(M)}\sim \text{NegBin}(v_t^{(M)},p_t^{(v)})$ (failures before the $r$-th success).
7. Paid impressions and clicks. $u_{t,\mathcal{S}}=\min \{u_{t,\mathcal{S}}^{\ast },u_t^{(M)}\}$. If $u_t^{(M)}\le u_{t,\mathcal{S}}^{\ast }$ (the cap binds), clicks are $v_t^{(M)}$. Otherwise $v_{t,\mathcal{S}}\sim \text{Hyper}(v_t^{(M)},u_t^{(M)}-v_t^{(M)},u_{t,\mathcal{S}}^{\ast })$, the successes in $k$ draws from $m$ successes and $n$ failures. This tends to a binomial as $m,n\to \infty$.
8. Allocate volumes to segments (multinomials): clicks $v_{t,\cdot }\mid v_{t,\mathcal{S}}\sim \text{Mult}(v_{t,\mathcal{S}},(a_{t,k,s}{\pi }_s^{(v)}{)}_s/{\sum }_s{a}_{t,k,s}{\pi }_s^{(v)})$; non-click impressions $u_{t,\cdot }-v_{t,\cdot }\sim \text{Mult}(u_{t,\mathcal{S}}-v_{t,\mathcal{S}},(a_{t,k,s}(1-{\pi }_s^{(v)}){)}_s/{\sum }_s{a}_{t,k,s}(1-{\pi }_s^{(v)}))$; queries without an ad $w_{t,\cdot }-u_{t,\cdot }\sim \text{Mult}(w_{t,\mathcal{S}}-u_{t,\mathcal{S}},a_{t,k,\cdot }/a_{t,k,\mathcal{S}})$.
9. Spend. $x_{t,s}=v_{t,k,s}{c}_t$, i.e. paid clicks × CPC.
10. Effect. See below. Migrate by Eqs. 1-2 with the effect-scaled matrix, as in the traditional module.

Search effect by exposure type (§2.3.2 step 4, p.17; App. C.2, p.41)

In each segment, count the people (not the queries) who: made a matching query, $n_{t,s}^{(w)}$; saw a paid impression, $n_{t,s}^{(u)}$; made a paid click, $n_{t,s}^{(v)}$. The counts are drawn in a chain (App. C.2, p.41):

1. Query-makers. Given the segment’s query total $w_{t,s}$ spread over its query-making audience $a_{t,s}$, the queries per individual follow $\text{Multinomial}(w_{t,s},(1/a_{t,s},\dots ,1/a_{t,s}{)}^{\top })$. The first and second moments of $n_{t,s}^{(w)}$, the number with at least one query, are computed from this, and $n_{t,s}^{(w)}$ is drawn from a normal approximation.
2. Impression-viewers. The same calculation, conditioned on the impression total $u_{t,s}$ and the $n_{t,s}^{(w)}$ query-makers, gives $n_{t,s}^{(u)}$.
3. Clickers. The same calculation, conditioned on the paid-click total $v_{t,k,s}$ and the $n_{t,s}^{(u)}$ impression-viewers, gives $n_{t,s}^{(v)}$.

Each step is the occupancy calculation used for TV reach in AMSS Traditional Media Module (Reach, Frequency, Hill Effect): the moments of the number not reached, with mean ${\mu }_1$ and variance ${\mu }_2-{\mu }_1^2$. Exposure types are disjoint: (a) organic only, $n^{(w)}-n^{(u)}$; (b) paid impression without a click, $n^{(u)}-n^{(v)}$; (c) paid click, $n^{(v)}$. The effect size is

$$e_{t,s}={\psi }_w^{(k)}\frac{n_{t,s}^{(w)}-n_{t,s}^{(u)}}{a_{t,k,s}}+{\psi }_u^{(k)}\frac{n_{t,s}^{(u)}-n_{t,s}^{(v)}}{a_{t,k,s}}+{\psi }_v^{(k)}\frac{n_{t,s}^{(v)}}{a_{t,k,s}}.$$

Search uses no frequency curve: a person “will feel the full effect of any search volume at a frequency of 1.” Usually ${\psi }_v=1$. The maximal matrix ${\mathcal{Q}}^{(k)}$ is built from per-dimension ${\mathcal{Q}}^{(k,l)}$ exactly as in the traditional module.

Incrementality of paid over organic (§2.3.2, p.17; App. C.2, p.41)

${\psi }_w$ is the effect of organic results seen by searchers who were not shown the paid ad (footnote 9). Setting the search budget to zero puts every searcher in group (a). The paid-vs-zero-budget counterfactual therefore isolates the incremental effect of paid search. A high ${\psi }_w$ simulates branded search where the paid ad mostly replaces the organic click and adds little.

Simplified competition (App. C.2, p.39). In reality competitors make all the same decisions. AMSS summarises them by $[{\eta }_t^{(m)},{\eta }_t^{(M)}]$, so competitors do not react to the advertiser’s bids. The paper flags this as a simplification.

Uncapped spend is not settable (§3.1, p.22). When the cap does not bind, spend = clicks × CPC moves with query volume, SOV and CPC, and query volume moves with every upstream event. Several combinations of campaign settings can produce the same spend with different effects on consumers. So “fixing search spend” is not a well-defined intervention. Counterfactuals must be set on the budget or the campaign settings. See ^def-amss-counterfactual-types.

Inconsistency: weekly vs daily spend cap

§2.3.2 step 1 and footnote 4 (p.15) describe a weekly spend cap. App. C.2 (p.39) says advertisers choose “a daily spend cap”. The simulator works in weekly intervals, and the algorithm above uses the weekly cap $x_t^{(M)}$. Read “daily” in the appendix as the real-world setting and “weekly” as what the simulator models.

Examples

Search in the simple scenario (§4.1.1, pp.25-26)

Targeting. Ads are shown on queries from 1% of inactive, 30% of exploratory and 40% of purchasing people. CTR. 0.005 for inactive, 0.08 for exploratory, 0.10 for purchase. Neither query rates nor CTRs depend on favorability, so that TV cannot raise search effectiveness and there is no TV × search interaction. Settings. Spend cap and keyword list are fixed and spend is uncapped. The bid rises linearly with budget from 0.8 at budget 0. The search budget is larger each year to create variation. Effect on activity only:

$${\mathcal{Q}}^{(search,3)}=\left(\begin{array}{ccc}0.05& 0.95& 0\\ 0& 0.85& 0.15\\ 0& 0& 1\end{array}\right),{\psi }^{search}=(0.0,0.1,1.0).$$

Solution. A click moves an inactive person to exploratory with probability $1.0\times 0.95=95\%$, and an exploratory person to purchase with probability $1.0\times 0.15=15\%$. An impression without a click gives 9.5% and 1.5%. Organic-only exposure gives 0%. Interpretation. The 95% looks huge, but footnote 7 notes it applies only to the small group of inactive people who see and click a paid ad. The campaign is targeted and reaches few inactive people.

What the simulated search data look like (Figs. 10-11, p.27)

Fig. 10. Weekly TV spend (black) is spiky, up to about $11M a week with many zero weeks. Search spend (blue) is smooth, about $0.5-2M, and tracks the seasonal market rate. Fig. 11. Query volume (solid, about 1-2.5×10⁷ a week), paid impressions (dotted) and paid clicks (dashed, about 10⁶) all follow seasonality and are highly correlated within each year. As the bid rises each year, SOV rises and the paid-impressions line approaches the query line. Interpretation. Search spend is a near-proxy for seasonal demand. A model that does not control for seasonality would credit search with seasonal sales. That is why the paper hands the model observed seasonality.

Connections

• Endogenous search spend. The aggregate analogue of the selection problem measured in the eBay paid-search experiment (Blake Nosko Tadelis 2015 - Paid Search Effectiveness - Overview), where brand-keyword ads were close to zero incremental. In AMSS, a high ${\psi }_w$ for a branded campaign reproduces that finding.
• Selection bias in observational MMM. Chan Perry 2017 - Challenges in Media Mix Modeling - Overview discusses search as the canonical selection-bias channel. Activity Bias in Advertising and Experimental Benchmarks for Observational Ad Measurement cover the user-level version.
• Predictive importance vs causal effect. A predictive model will give search spend large attribution because it co-moves with demand. See Dillon et al - SHAP and Causal Insights - Overview and Dillon et al - Observed Confounding and Double ML for Ad Spend.

See Also

• AMSS Traditional Media Module (Reach, Frequency, Hill Effect)
• AMSS Ground Truth - Counterfactual Scenarios and Simulated ROAS: why spend cannot be the intervention
• Zhang Vaver 2017 - Aggregate Marketing System Simulator - Overview
• Blake 2015 - Brand Keyword Experiments and Paid-Organic Substitution — empirical paid/organic substitution rates