The Dunbar theory suggests that individuals’ local social networks have a multilayer concentric structure, with outward layers featuring greater sizes and lower tie weights, and an empirically consistent scaling ratio of 3 between these layers. In the current paper, we propose a formal mechanistic model of network evolution, in which an individual (Ego) spreads their limited social resources among adjacent nodes (Alters) in a discrete-time stochastic process. The Ego faces a sequence of challenges modeled by random variables that measure the issues’ difficulties. To handle these challenges, the Ego requests help – activates their ties with Alters – and more difficult issues require stronger ties to be activated. Alters may refuse to help with a fixed probability, which is the key parameter of our model. If the assistance is successfully provided by an Alter, the corresponding tie undergoes a marginal weight increase. Using a combination of analytical approaches and numerical simulations, we found stationary states of the model and the model hyperparameter values that lead to multilayer ego-networks that follow the Dunbar theory. An interesting implication of our analysis is that tie weight heterogeneity is hardly achieved if the probability of refusal is zero. Conversely, when all Alters are perfectly reliable, our model has the following long-term behavior: it consistently converges to weight profiles, in which all weights except a few are zero, with the nonzero weights being equal to each other. One of these stationary states is universal across all settings. In this state, the Ego locates all their resources in only one tie—they repeatedly ask the same Alter for help and always receive it. Our findings indicate that uncooperative, non-reciprocal behavior is sufficient for tie weight heterogeneity in social networks.