Stochastic Choice with Advertising

Does an Ad Change Your Mind or Just Your Focus?

When you browse an e-commerce site and see a "Top Pick" or "Recommended" badge, does that product sell because you actually like it more, or simply because it is the only thing you noticed? Distinguishing between these two drivers is a fundamental challenge in digital marketing. Most current models struggle to untangle whether advertising shifts a consumer's actual preference or merely directs their limited attention toward a specific subset of options.

A new study by Henrik Petri and Kai Wang proposes a mathematical framework to solve this ambiguity. The authors report that by extending traditional choice models, they can create a system that separates the "attention channel" from the "preference channel" using only observed choice data.

The Dual Role of Advertising

At its core, the paper investigates how digital interfaces—like Amazon’s "Sponsored Products"—influence decision-making. The authors argue that advertising acts through two distinct psychological mechanisms.

The first is the attention channel. This is an informative role where advertising makes certain items more salient (noticeable). It acts like a spotlight. Much like a person entering a crowded room who only notices people wearing bright red shirts, a consumer might restrict their consideration to the advertised items.

The second is the preference channel. This is a persuasive role where advertising changes how much a consumer values a product. For example, seeing a "Premium" label might change a consumer's utility (a numerical representation of an item's attractiveness). The central problem in existing research is that when an advertised product sees a spike in sales, it is difficult to tell if the spike came from the spotlight or the persuasion.

The Mechanics of LMA and A-LMA

To address this, the authors build upon the Luce model. This is also known as the multinomial logit model. The Luce model is a classic framework in discrete choice modeling. It predicts the probability of choosing an item based on its utility relative to the total utility of all available options.

The authors first introduce the Luce Model with Advertisement (LMA). In this version, they posit that a consumer follows a set-directed attention rule. With a certain probability $\lambda$, the consumer ignores the full menu and focuses only on the advertised items. If they do not follow the advertisement, they consider everything. The authors demonstrate that this model is uniquely identifiable. Specifically, they show that the model is characterized by two axioms: Block-IIA (where relative choice frequency is independent of the menu if items are both advertised or both not) and Advertising Boost. These properties allow researchers to recover the original utility values and the attention parameter from choice data.

However, the LMA only accounts for attention. To capture the full complexity of human behavior, the authors develop the A-LMA (Advertisement-dependent Luce Model). In this richer framework, the utility $u$ of an item is no longer a fixed value. Instead, it becomes a function of the advertised set $B$. This allows the model to account for "spillover effects," where advertising one product makes a similar, unadvertised product more attractive. It also accounts for "negative own effects," where an intrusive ad actually makes the product feel less desirable.

Optimizing the Spotlight

Beyond theoretical modeling, the paper tackles a practical management problem. How should a retailer design their "Recommended" list to maximize profit? The authors derive specific design rules based on how consumer attention behaves.

They first consider "Attention Scarcity." This is the idea that as you add more items to an advertisement, the consumer's ability to focus on any single one diminishes. This is akin to trying to listen to one speaker in a room that is getting increasingly noisy. Under this assumption, the authors report a striking rule: it is often optimal to advertise only a single item. Specifically, they suggest featuring the one item with the highest profit margin.

The paper also explores "Inverted-U-Shape Attention." Here, increasing the number of advertised items initially helps capture interest. However, it eventually leads to "choice overload." This happens when the consumer becomes overwhelmed and stops following the recommendations. For this scenario, the authors provide a method to efficiently calculate the optimal number of items to feature. This reduces a massive computational problem to a manageable, ranked list of candidates.

Separating Attention from Preference

The most significant breakthrough presented in the paper is the ability to perform "point identification" in a specialized version of the model called the Menu-Invariant A-LMA (MI-A-LMA).

By assuming that the utility of a "default option" (the choice to buy nothing) remains constant regardless of the advertisement, the authors show that the two channels can be decoupled. They demonstrate that by comparing the odds of choosing an unadvertised item against the "buy nothing" option, they can isolate the preference shift. Once the preference shift is known, the remaining boost in sales can be attributed to the attention parameter.

The authors illustrate this with a hypothetical example. They show that a product might appear to be a huge success after an ad campaign. However, the math reveals a hidden story. The ad might have actually decreased the product's inherent appeal (a negative own effect). Simultaneously, it might have boosted the appeal of a competitor (a positive spillover). The increase in sales was driven entirely by the attention spotlight. This masked the underlying drop in preference.

Limits of the Framework

While the A-LMA provides a powerful toolkit, the authors note its boundaries. The general version of the A-LMA is only "partially identified." Without the specific structural assumptions used in the MI-A-LMA, you can only establish bounds on the effects. You cannot pinpoint exact numbers for attention versus preference.

Furthermore, the model assumes that the advertisements themselves are clearly observable to the researcher. It does not currently account for different mediums, such as the difference between a TV spot and a digital banner. It also does not explore advertisement formats other than a specific set of items. Finally, the model is built on the assumption that the consumer's decision-making process follows the Luce rule. This may not hold in all complex psychological environments.