Constrained transmission lines are known to be able to economically isolate submarkets from the competition of players located elsewhere on the network. This paper examines the type of oligopolistic competition that is likely to take place in these submarkets. It shows, using simple models, how static or intertemporal Nash equilibria can rise in a framework of price or supply function competitions, found to be more realistic than Cournot models in the particular case of short term competition in the electric power market. This paper shows also how transmission constraints can play a direct role in the outcome of the oligopolistic competition and encourage strategic behavior by the generators. Transmission lines that would not be constrained if the players did not know of their thermal limits may be strategically driven to operate at these limits in order to maximize the profits of the players who have market power, leaving the others to cope with the consequences of such behavior.