Often when groups of the same gender live in close quarters with one another, there is a "pecking order" established; that is, the strongest--whether it be physical or mental dominance--is at the top, and the weakest at the bottom. Carlson is a brutish man with a gun; Candy is an old man with a hand missing. In this naturalistic environment, Candy turns desperately to Slim, with his "God-like eyes" and hierarchical position, hoping that the mule skinner will empower him:
Candy looked a long time at Slim to try to find some reversal. And Slim gave him none. At last Candy said softly and hopelessly, "Awright--take'im."
Candy virtually has no choice but to let the dog go because he would be shunned by the men and Carlson would continue to badger him, and then probably act on his own, anyway. When Carlson has previously asked Candy to let him shoot the dog he has said,
Got no teeth,” he said. “He’s all stiff with rheumatism. He ain’t no good to you, Candy. An’ he ain’t no good to himself. Why’n’t you shoot him, Candy?”
These words imply Candy's weakness for not putting the dog down; in addition, they foreshadow the final chapter of Steinbeck's novella as Lennie, who no longer can be of use and who has made one too many mistakes must go or be imprisoned.
Here is a video that describes Candy's character in-depth:
There is a hidden purpose behind Carlson's shooting of Candy's old dog. It is complicated. When Steinbeck was plotting his story he wanted to end it with George killing Lennie. The reason Steinbeck wanted George to kill Lennie was that he wanted to end the story quickly and with finality. He wanted to do this because he planned to adapt the book into a play immediately. But the play could only run for about an hour and a half. Therefore he could not write a big panoramic novel about the lives of itinerant farm workers, traveling on freight trains, working out in the fields, and so on. Both the book and the play came out in the same year, 1937. In order to kill Lennie, George would need a gun. It would have to be a handgun because a rifle would be too conspicuous, both when he stole it and when he used it on Lennie. The episode in which Carlson shoots Candy's dog was intended to establish that a handgun existed and was readily available. It was an unusual-looking pistol because it was a German Luger. No doubt Carlson was in World War I and brought it back as a souvenir. When George pulls out the gun to kill his friend, the reader will automatically realize that George stole it from under Carlson's bunk. The audience will also recognize it as Carlson's gun because a Luger is a very distinctive-looking automatic pistol. George could not logically kill Lennie any other way. George is a little guy and Lennie is a powerful giant. If George tried, for example, to hit him with a club, he might fail to knock him out, and Lennie might kill George instead. If George tried to cut Lennie's throat it would be too gruesome, and he might fail in the attempt. When Carlson kills the old dog he explains where he will place the shot to make death painless. After shooting the dog, Carlson cleans the gun carefully. George sees and hears all of this, so he knows how to use the gun on Lennie and how the complicated mechanism of the automatic pistol works. Candy had to agree to let Carlson shoot his old dog because the author mandated it for plot purposes. Everything that happens happens because the author decided to make it happen. Everybody thinks it was Lennie who stole the Luger after he killed Curley's wife and fled from the ranch. This is useful to the plot because it appears to the other men that George must have taken the gun away from Lennie and shot him in self-defense. That allows the story, and the play, to end cleanly. There is no investigation or interrogation. It ends right on the spot where George pulled the trigger. Ostensibly, Candy lets Carlson kill his dog because he succumbs to pressure from all the men in the bunkhouse, including Slim; Candy is also afraid of losing his job if all the men were to complain about the smelly old dog. But the real reason is that Steinbeck needed to show the gun to George and to let him see how it worked and hear where to point it. Steinbeck could have had Carlson just show George the gun and explain how the automatic worked, but he was too conscientious a writer for such shortcuts. He made a naturalistic dramatic scene out of it.