The theme of loyalty is shown throughout The Hunger Games. First of all, when Katniss Everdeen, the protagonist of the story, hears her twelve year old sister Prim's name called for the lottery, she chooses to replace her in the Hunger Games. This is clearly an act of loyalty. Katniss knows that her sister's gentle nature would be her demise in the games and chooses to sacrifice herself to protect her.
Peeta shows loyalty throughout the book to Katniss. First, in chapter two, Peeta literally saves Katniss's life several times in the novel. First, he saves her life when he intentionally burns bread in the family bakery and risks his mother's wrath. This is done so that he can throw it to her, and it literally saves her life. Peeta's elaborate plans to protect Katniss in the arena clearly demonstrate his loyalty to her, although she doesn't always realize. By teaming up with the Careers, Peeta actually saves her life in chapter 14. Later, Katniss repays him by nursing him back to health.
Katniss also shows loyalty to Rue, the girl in the games who reminds her of Prim, her sister. Rue watches over Katniss after she has been attacked by tracker jackers and is hallucinating. After, Katniss shares her body heat to keep Rue warm after. Later, when Rue is captured in a net and speared by a Career, she risks her own death by singing to Rue until she dies. Rue's favorite thing in the world is music, so this eases her passing.
Loyalty is shown throughout all three books. The government counts on the Hunger Games to keep the populations loyalty. That is why they hold the Games in the first place. Peeta also shows loyalty to Katniss throughout all three books. In the first book he shows loyalty when he pretends to side with the other players' alience, just to be ready to save Katniss when the time arises. Katniss shows loyalty to Peeta when she risks her life to leave Peeta in the cave and retrieve the medicine he needs. This continues thoughout the entire series.