Symbolism, Imagery, Allegory
No, we're not talking about some alternative to everyone's favorite moralizing kiddie board game, Shoots And Ladders… although the butlering strategy of Wheels and Ladders is kind of a game.
In a key passage, Stevens marks the distinction between his generation and his father's through the metaphors of a wheel and a ladder:
Butlers of my father's generation, I would say, tended to see the world in terms of a ladder […]. Our generation, I believe it is accurate to say, viewed the world not as a ladder, but more as a wheel. (4.4)
His father's generation viewed the world as a strict hierarchy, with royalty at the top and ordinary people at the bottom. The aim of butlers way back when was to try to serve at the highest possible rung of the ladder. Serving a king? Good on you—you've just won the butler sweepstakes.
Our main man Stevens believes his generation views the world more as a wheel, with the movers and shakers (who are not necessarily the highest in terms of social class) at the hub, and everyone else moving around them. For Stevens's generation, a butler tried to get as close to the power hub as possible.
This signals a shift away from the crazy-stratified social order of Stevens Sr.'s day. It shows that Stevens Jr. has more of an awareness of current events and global power than other people might think. Not only has Stevens Jr. seen his own employer, Mr. Darlington, butter up diplomats and other powerful non-nobility, he also ends up working for (gasp! our pearls need clutching!) an American. There ain't no nobility in the US of A: just powerful people and less powerful people.