$N$ people are at a party and decide to play a cooperative game. They begin by standing in a circle. The game proceeds in turns. In each turn, one person is chosen to perform one of the following actions:
- Shake hands with someone adjacent to them
- Swap positions with someone adjacent to them
The game ends when every pair of players has shaken hands, and the aim of the game is to minimize the number of swaps required. What is the optimal strategy?
I've tried working through some small examples (e.g. $N = 3,4,5,6$ require $0,1,3,5$ swaps minimum but I can't really find a pattern in the strategy).