Ok assume you're not the game starter but someone on the table, in order for you to lose the coin must pass over all the other players but not you, particularly it must reach the person on your left then go all the way around and reach the person on your right (or the opposite) the game will keep going until it reaches one of the two people beside you (let's say it reaches the one on your left first) my claim is that all the events up until that moment don't matter, why? take a look at this picture:

Does it matter (for you) if you were in situation A or B? no, you'd win if the coin comes to you and you'd lose if the coin goes around you and reaches the person on your right, it does not matter what happens to the people in the middle since the coin gets passed around in a continuous way it will pass over everyone if it reaches the person to your right.

So that shows if you were somewhere on the table the probability of you losing is independent of your location, let's call that p_{k}.

This is true for everyone except the two people directly to the left and right of the person who starts the game, what is the probability of one of them losing? well at the start of the game he has 0.5 chance of getting the first coin, and 0.5 chance of not getting it in which case his chance of losing is p_{k}, so for the two people next to the game starter the probability of losing is 0.5*p_{k}.

Since exactly one person has to lose the probabilities must sum up to one, so you'll find that:

(k-2)*p_{k} + 0.5*p_{k} + 0.5*p_{k} = 1.

And from that you'll get 0.5*p_{k} = 1/(k-1)