Perhaps the most relevant critical theory to examine the story "Games at Twilight" would be Psychoanalytic. Based upon the fact that the work itself is deemed psychological, examining the text from a Psychoanalytic perspective would seem natural.
Psychoanalytical criticism examines a text from the theologies applied by Freud and others involved in the field of psychology. This criticism is used to examine the unconscious desires and anxieties of the author. At the same time, the text is examined in regards to links made between the characters own mentalities and those of the author. Basically, how do the characters reflect the author himself/herself.
Another criticism one could use would be Reader-Response Criticism. This criticism examines the reader's reaction to the text and determines how the reader's response is critical to understanding the text itself. In the Reader-Response method, reader's are still responsible for using a "lens" of another theory by which they read the text through. This method, in essence, use two criticisms to examine a text.