Logit Purchase Decision Model

Summary

Karakaya et al. (2011) use a two-stage purchase decision model combining a deterministic utility threshold with a stochastic logit function. A consumer purchases if their utility exceeds a buying threshold $\alpha$ AND a logit-transformed probability exceeds a random draw. This captures bounded rationality — consumers with high utility are likely but not certain to buy.

Overview

The purchase decision model addresses the empirical observation that consumers do not always act rationally. Even when a product satisfies expectations, a consumer may not purchase — and conversely, a consumer may purchase impulsively despite low expected utility. The model uses a threshold from Granovetter (1978) combined with a logit function (Anderson, de Palma & Thisse 1992) to introduce calibrated randomness.

Main Content

The Logit Function

Definition: Logit Function for Purchase Decisions (Karakaya et al. 2011, Eq. 7)

$$\text{Logit}(u)=\frac{1}{1+e^{k(\alpha -u)}}$$

where:

• $u$: utility of the consumer
• $k$: smoothing constant ($k=5$ in experiments)
• $\alpha$: buying threshold ($\alpha =0.7$ in experiments)

The logit function maps utility to a purchase probability in $[0,1]$.

Purchase Decision Rule

Definition: Purchase Decision Rule (Karakaya et al. 2011, Eq. 8)

Consumer $i$ purchases the product if and only if:

$$U_i>\alpha \text{and}\text{Logit}(U_i)\ge {\tau }_i$$

where:

• $\alpha =0.7$: buying threshold (minimum utility for consideration)
• ${\tau }_i\sim U(0,1)$: random number generated for consumer $i$

Two-Stage Mechanism

Stage 1 — Threshold gate: The consumer’s utility must exceed $\alpha$ to even be considered. This represents a minimum acceptable level of satisfaction — consumers below this threshold are completely uninterested regardless of randomness.

Stage 2 — Stochastic decision: Among consumers who pass the threshold, the logit function converts their utility into a purchase probability. Higher utility leads to higher probability, but the outcome is still stochastic. A consumer with $U_i$ slightly above $\alpha$ has approximately 50% chance of purchasing, while a consumer with $U_i$ well above $\alpha$ has near-certain purchase probability.

Properties of the Logit

The logit function has useful properties for this application:

• Sigmoid shape: Smooth transition from low to high purchase probability
• Centered at threshold: $\text{Logit}(\alpha )=0.5$ — exactly 50% purchase probability at the threshold
• Steepness controlled by $k$: Higher $k$ makes the transition sharper; $k\to \infty$ recovers a deterministic step function
• Bounded: Always in $[0,1]$, interpretable as a probability

Post-Purchase Behavior

Once a consumer purchases:

• They use the product until the last time step ($T=20$)
• They do not make another purchasing decision in consecutive time steps
• Their utility does not decay due to external factors after purchasing
• They begin disseminating WOM (positive or negative) based on their satisfaction

Examples

Example: Logit at Different Utility Levels

Setup: With $k=5$ and $\alpha =0.7$:

Utility $U_i$	$\text{Logit}(U_i)$	Interpretation
0.5	0.27	Low utility — 27% chance of purchase
0.7	0.50	At threshold — coin flip
0.9	0.73	Above threshold — 73% chance
1.2	0.92	Well above — 92% chance
1.5	0.98	Very high — near certain

Interpretation: The logit creates a gradual transition rather than a sharp cutoff, with the steepness determined by $k=5$.

Connections

• This model operationalizes Agent Decision Rules and Bounded Rationality in the Karakaya framework
• The utility input comes from Consumer Utility Function Components
• The threshold concept relates to Granovetter’s (1978) threshold models of collective behavior, also used in Product Adoption and Diffusion Models
• The sensitivity analysis of threshold and smoothing parameters is covered in Population Initialization and Parameter Sensitivity

See Also

• Consumer Utility Function Components — the utility that feeds into this decision
• Agent Decision Rules and Bounded Rationality — comparison with other decision architectures
• Karakaya et al 2011 - Overview — paper context