Brackets for tournament in a three player game
It's virtually identical to 2-player games, except there are two "loser" brackets instead of one, and an extra game could be needed because you can get into a situation where everyone lost twice.

________  \
________  -&gt;01  ________  \
________  /                |
                           |
________  \                |
________  -&gt;02  ________  -&gt;16  ________  \
________  /                |               |
                           |               |
________  \                |               |
________  -&gt;03  ________  /                |
________  /                                |
                                           |
________  \                                |
________  -&gt;04  ________  \                |
________  /                |               |
                           |               |
________  \                |               |
________  -&gt;05  ________  -&gt;17  ________  -&gt;23  ________  -----------------\
________  /                |               |                               |
                           |               |                               |
________  \                |               |                               |
________  -&gt;06  ________  /                |                               |
________  /                                |                               |
                                           |                               |
________  \                                |                               |
________  -&gt;07  ________  \                |                               |
________  /                |               |                               |
                           |               |                               |
________  \                |               |                               |
________  -&gt;08  ________  -&gt;18  ________  /                                |
________  /                |                                               |
                           |                                               |
________  \                |                                               |
________  -&gt;09  ________  /                                                |
________  /                                                                |
                                                                           |
                                                                           |
                                                __12P2__  \                |
                                                           |               |
                                __16P2__  \                |               |
                                __17P2__  -&gt;19  ________  -&gt;24  ________  -&gt;26  ________
                                __18P2__  /                |               |
                                                           |               |
                __01P2__  \                                |               |
                __02P2__  -&gt;10  ________  \                |               |
                __03P2__  /                |               |               |
                                           |               |               |
                                           |               |               |
                __04P2__  \                |               |               |
                __05P2__  -&gt;11  ________  -&gt;20  ________  /                |
                __06P2__  /                |                               |
                                           |                               |
                                           |                               |
                __07P2__  \                |                               |
                __08P2__  -&gt;12  ________  /                                |
                __09P2__  /                                                |
                                                                           |
                                                                           |
                                                __12P3__  \                |
                                                           |               |
                                __16P3__  \                |               |
                                __17P3__  -&gt;21  ________  -&gt;25  ________  /
                                __18P3__  /                |
                                                           |
                __01P3__  \                                |
                __02P3__  -&gt;13  ________  \                |
                __03P3__  /                |               |
                                           |               |
                                           |               |
                __04P3__  \                |               |
                __05P3__  -&gt;14  ________  -&gt;22  ________  /
                __06P3__  /                |
                                           |
                                           |
                __07P3__  \                |
                __08P3__  -&gt;15  ________  /
                __09P3__  /


[xxPy refers to the person who placed in position y of in game x.]


If 23P1 doesn't win 26, this is their first loss. Replay 26. [27]
If 24P1 won 26 and 25P1 wins 27, everyone lost twice. Replay 27. [28]
If 25P1 won 26 and 24P1 wins 27, everyone lost twice. Replay 27. [28]


An in-depth view of games 26..28:

                 Game Winner      Total Loss Count  Result
                 --------------   ----------------  ------------
                 26   27   28     23P1  24P1  25P1

Before game 26:                   0     1     1     =&gt; Game 26

After game 26:   23P1             0     2     2     =&gt; 23P1 wins
                 24P1             1     1     2     =&gt; Game 27
                 25P1             1     2     1     =&gt; Game 27

After game 27:   24P1 23P1        1     2     3     =&gt; 23P1 wins
                 24P1 24P1        2     1     3     =&gt; 24P1 wins
                 24P1 25P1        2     2     2     =&gt; Game 28

                 25P1 23P1        1     3     2     =&gt; 23P1 wins
                 25P1 24P1        2     2     2     =&gt; Game 28
                 25P1 25P1        2     3     1     =&gt; 25P1 wins

After game 28:   24P1 25P1 23P1   2     3     3     =&gt; 23P1 wins
                 24P1 25P1 24P1   3     2     3     =&gt; 24P1 wins
                 24P1 25P1 25P1   3     3     2     =&gt; 25P1 wins

                 25P1 24P1 23P1   2     3     3     =&gt; 23P1 wins
                 25P1 24P1 24P1   3     2     3     =&gt; 24P1 wins
                 25P1 24P1 25P1   3     3     2     =&gt; 25P1 wins


Yes, that means the best players may have a chance of winning after being eliminated twice, but I don't think that's a problem, and I'm not sure there's a solution if it is.