The article examines the problem of ensuring the political stability of a democratic social system with a shortage of a vital commodity (like vaccine against COVID-19). In such a system, members of society - citizens assess the authorities. Thus, actions by the authorities to increase the supply of this commodity can contribute to citizens' approval and hence political stability. But this supply is influenced by random factors, the actions of competitors, etc. Therefore, citizens do not have sufficient information about all the possibilities of supplying, and it is difficult for them to make the right decisions. Such citizen unawareness can be exploited by unscrupulous politicians to achieve personal targets. Therefore, it is necessary to organize public control in order to motivate politicians to use all available opportunities in supplying. The goal of the paper is to build such a digital mechanism of public control of the politicians by citizens, which would best assess and stimulate the activities of the authorities to improve the supply of a vital commodity. In the age of artificial intelligence, such digital public control in the face of uncertainty can be based on digital machine learning. In addition, it is necessary to take into account and model the activities of politicians associated with the presence of their own targets that do not coincide with public ones. Such politicians can use the learning of citizens for their own targets. The objective of the article is to build an optimal digital mechanism of public control in a two-level model of a democratic social system – a digital society. At its top level there is the Citizen, who gives an assessment for the Politico located at the lower level. In turn, the Politico can influence the supplying of a vital commodity. Political stability is guaranteed if the Citizen regularly approves of the Politico's actions to increase this supply. But the Politico may not use the opportunities available to him to offer a commodity to achieve a personal target. To avoid this, the Politico’s control mechanism is proposed. It includes the procedure for digital learning of the Citizen, as well as a procedure for assessing the Politico activity. Sufficient conditions have been found for the synthesis of such the Politico’s control mechanism, at which stochastic possibilities of increasing the supply of a vital commodity are used. The example of such the Politico’s control mechanism is considered on the case of supply of the COVID-19 vaccine in England.