Introduction to a Weird Game

So here is a game that recently came to mind. The two-player case is almost trivial, but let's begin with it anyway.

There are players A and B. The game is turn based, at the beginning of each turn they bet on the result of the turn: A wins or B wins (so a person is allowed to bet on herself), where win is defined as scoring more points than the other person. The scoring rules are:

1. An "A wins" bet gives one point to A; the same goes for B.
2. Betting on the right winner gives a point (i.e. if B bets that A wins and A does win, then B scores a point)

At the end of the turn, calculate the points. A "draw" is called if either logical contradiction is resulted, or the two players tie in score. When there is a draw, a new turn is played. As soon as there is a loser, the game ends. There are four scenarios:

1. A bets A wins; B bets B wins
2. A bets B wins; B bets A wins
3. Both bets are A wins
4. Both bets are B wins

Let's investigate the outcomes.

For scenario 1, each player gains one point, so it's a draw.
For scenario 2 also, each player gains one point, so it's a draw.
For scenario 3, A scores 3 points versus B scores 1 point, A wins.
For scenario 4, A scores 1 point versus B scores 3 points, B wins.

So assuming everyone wants to be the winner, the best strategy is to bet on oneself. Now we introduce a new rule to make the game more fun: one can bet against a certain player, i.e. bet that she loses:

1. An "A loses" bet deducts one point from A; the same goes for B.
2. Betting on the right loser gives a point (i.e. if B bets that A loses and A does lose, then B scores a point)

It seems that for a 2-player game, the new rules do not change the nature of the game much. Thus the best strategy is to bet on oneself (or, equivalently, to bet that the other person loses). What if there are three players? Would it still be the case that the best course of action is to always bet on oneself?