Reputation through Purchase Decisions

We study a reputation game in which future customers observe earlier purchase decisions but neither product quality nor the private information behind those decisions.

A commitment type always supplies high quality, whereas a strategic seller chooses whether to do so before a short-lived customer receives a private signal and an idiosyncratic payoff shock.

The public state is reputation for the seller's fixed type, not expected current quality.

In the relevant demand region, more strategic quality makes purchase behavior less diagnostic of type: the reputations following a purchase and a non-purchase move toward one another, and the sales advantage of quality falls.

This posterior-compression mechanism gives a unique local seller response.

We identify the exact cost threshold at or above which the strategic seller never supplies high quality.

Just below that threshold, a locally unique stationary equilibrium supplies high quality only at intermediate reputations.

As cost approaches the threshold, the active region contracts on a square-root scale and quality provision vanishes on a linear scale.

After rescaling, every limiting quality profile has one peak and one positive interval.

Exact symbolic calculations and outward-rounded interval arithmetic verify the assumptions on a nonempty open set within a compact-support Beta family.