This phenomenon is explained with the help of a model. First of all Assume an economy composed of n citizens, with one of them serving the role of a self interested public official. This official is considering whether to implement a project. Here we take a situation where there is only one citizen i who derives positive utility from the project Ui > 0. All other citizens in the economy including the public official derive utility U-i ≤ 0.
The public official may corrupt citizen i by proposing a bribe, which the citizen may accept or reject. For each unit of bribe b, we also assume citizen i suffers a cost (1+ά), with ά ≥ 0. We assume Ui ÷ 1+ ά > U-i and complete information.
Sub-game Nash Equilibrium is used in dynamic games. More informally, we can say that it means if the players used to play any smaller game that consisted of one part of the larger game then their behavior symbolizes Nash equilibrium of the smaller game. In any finite game like one mentioned above, we implement a common method of backward induction in which one considers the last actions and outcomes of the game first and determines which actions would be required to maximize utility in possible circumstances. For example the public official is proposing bribe and corrupt the citizen i keeping in mind his own benefits. However, the citizen may accept or reject the bribe. In case if the citizen accepts the bribe then it will include in corruption on the part of the public official as well as the citizen to maximize their own benefits. Furthermore, this corruption results in social inefficiency and harming the overall economic growth of the country.
Now, suppose the project is a driving license for citizen i and citizen i is a dangerous driver. In such a case there would be negative externalities of the project for all other citizens. However, these negative externalities are small.