Your assignment is to write a sociological criticism of some aspect of the story The Lady or the Tiger. Sociology is "a discipline that strives to comprehend social activity." Therefore, your assignment is to consider some moment of human interaction within the story. There are many possibilities.
You could examine the relationship of the king and the young man, analyzing why the king chooses this system of justice, how his subjects react to such proceedings, and what options the young man has. You could discuss the interaction between the young man and the princess, which is quite different before they are found out and after the king learns of their affair.
This royal maiden was well satisfied with her lover, for he was handsome and brave to a degree unsurpassed in all this kingdom, and she loved him with an ardor that had enough of barbarism in it to make it exceedingly warm and strong.
How could the king, who loved his daughter "above all humanity," subject one loved by her to such a hideous judgement? What motivations did the princess have to deal with in deciding how to shape her response to the young man's "quick and anxious glance"?