Aspiration-Weighted Influence

TL;DR

The paper develops the Aspiration-Weighted Luce Model (AWLM) to explain directed social influence when an influencer operates over a richer menu than the follower. It formalizes influence as a mix of idiosyncratic Luce propensities and the influencer’s distribution, followed by normalization onto the follower’s feasible set, introducing aspirational dampening via the feasible-share parameter $q_S$. The main contributions are a rigorous axiomatic characterization, a microfoundation, and a constructive identification strategy that recovers the influence strength $\alpha$ and Luce weights $u$ from as few as two exposure regimes, with overidentification tests when more data are available. The model yields testable predictions about how infeasible exposure shapes choices and offers practical implications for aspirational marketing and consumer analysis in the presence of unequal opportunity sets.

Abstract

We study directed social influence when an influencer chooses from a richer menu than a constrained follower (decision maker, the DM). The DM selects from a feasible set, while the influencer displays a distribution over a superset that includes infeasible alternatives. We propose the Aspiration-Weighted Luce Model (AWLM): the DM forms a convex combination of her idiosyncratic Luce preferences within the feasible set and the influencer's distribution, then renormalizes this attempt target onto the feasible set. This renormalization generates an aspirational dampening effect: holding the influencer's within-feasible composition fixed and shifting exposure toward infeasible alternatives attenuates influence on feasible choices. We provide an axiomatic characterization based on proportional responses to shifts in feasible exposure and a unit-slope leverage restriction across different levels of feasible share. The model allows for point identification of influence strength and idiosyncratic preferences from two exposure regimes, yielding testable overidentifying restrictions for empirical application.

Figures (12)

• Figure 1: AWLM as "mix then normalize." The influencer's exposure $q$ is mixed with the idiosyncratic feasible distribution $p_0(\cdot\mid S)$ to form the attempt-level target $M=(1-\alpha)p_0+\alpha q$ on the chord $[p_0,q]$. The realized choice $p(\cdot\mid S;q,\alpha)$ is the normalization of $M$ onto the feasible set $S$, represented by the dotted conditioning ray through $M$. The point $q(\cdot\mid S)$ is the analogous projection of $q$ onto $S$.
• Figure 2: Benchmark: influence within the feasible set ($S=I$). When the influencer's menu coincides with the DM's feasible set, $q_S=1$ and no normalization is needed. The DM's influenced choice $p$ lies on the chord joining $p_0$ and $q$ at position $p = (1-\alpha)p_0 + \alpha q$. This is the standard convex-mixture form of influence Chambers_Cuhadaroglu_Masatlioglu_2022.
• Figure 3: Changing $\alpha$ at fixed exposure. Holding $(S,p_0,q)$ fixed, increasing $\alpha$ shifts the attempt target $M$ toward $q$ and moves the realized feasible choice $p(\cdot\mid S;q,\alpha)$ along the segment joining $p_0(\cdot\mid S)$ and $q(\cdot\mid S)$. The displacement is governed by the effective weight $\beta(q_S)$.
• Figure 4: Aspirational dampening (Proposition \ref{['prop:dampening']}). The within-$S$ composition $q(\cdot\mid S)$ is held fixed (orange), while the feasible share $q_S$ changes, moving $q$ (red) along the ray toward the infeasible vertex. Since $p(\cdot\mid S;q,\alpha)=(1-\beta(q_S))p_0+\beta(q_S)q(\cdot\mid S)$ with $\beta(\cdot)$ increasing in $q_S$, a lower $q_S$ reduces $\beta(q_S)$ and pulls the realized choice $p$ (green) toward $p_0$ (blue).
• Figure 5: Intuition for Controlled Collinearity (Axiom \ref{['ax:A1']}). $S=\{x,y,z\}$; $I=\{x,y,z,w\}$ The influencer's exposures $q_1,q_2$ (red) lie in the interior of $\Delta(I)$. Each exposure mixes with the baseline $p_0$ (blue) to form attempt targets $M_1,M_2$ (gray), which are projected onto $S$ to yield realized choices $p_1,p_2$ (green). Axiom \ref{['ax:A1']} requires $p_2-p_1 \parallel (q_2-q_1)|_S$.

Theorems & Definitions (31)

• Definition 1: Aspiration-Weighted Luce Model (AWLM)
• Lemma 1: Intra-aspiration irrelevance
• proof
• Proposition 1: Sampling-until-feasible implies AWLM
• proof
• Proposition 2: Aspirational dampening
• Lemma 2: Controlled multiplier and level-affine form
• proof
• Theorem 1: Per-menu AWLM Characterization
• proof