  We study the welfare structure in two-sided large random matching markets. In
the model, each agent has a latent personal score for every agent on the other
side of the market and her preferences follow a logit model based on these
scores. Under a contiguity condition, we provide a tight description of stable
outcomes. First, we identify an intrinsic fitness for each agent that
represents her relative competitiveness in the market, independent of the
realized stable outcome. The intrinsic fitness values correspond to scaling
coefficients needed to make a latent mutual matrix bi-stochastic, where the
latent scores can be interpreted as a-priori probabilities of a pair being
matched. Second, in every stable (or even approximately stable) matching, the
welfare or the ranks of the agents on each side of the market, when scaled by
their intrinsic fitness, have an approximately exponential empirical
distribution. Moreover, the average welfare of agents on one side of the market
is sufficient to determine the average on the other side. Overall, each agent's
welfare is determined by a global parameter, her intrinsic fitness, and an
extrinsic factor with exponential distribution across the population.
