How is Yu-Gi-Oh played with more than two players?
Play with the rules the show gave for free-for-all duels

The closest we have to official rules for Yu-Gi-Oh! with more than three players is Episode 122 of the original Yu-Gi-Oh! anime. It featured a four way free-for-all duel with the following rules (plus a rule involving the results of this duel determining seeds for the final four):


  Each Duelist must play with no more than 40 cards in his Deck.
  Each Duelist can attack or target any of the other three, but may not attack until one full turn has been played.
  
  The turn order is determined as follows: each player selects 1 monster from his Deck and removes it from play; the owner of the monster with the highest ATK goes first and the lowest ATK goes last. [All duelists had  different ATK, so it was not explained what happens in a tie]


Throughout the duel, there were temporary alliances that were made and broken. The duel ended when the first player got to 0 Life Points and everybody was ranked for tournament seeds. In other words, the highest remaining Life Points was ranked first, with others being ranked in relation. Yugi and Joey initially teamed up, then then turned against each other when they realized it was advantageous to do so in order to get favorable tournament seeds. Again, there were no ties, so it was unclear what would have happened in that case.



I would suggest adding the following rules:


Each player selects 1 monster from his Deck and removes it from play; the owner of the monster with the highest ATK goes first and the lowest ATK goes last. You cannot choose a monster with a ? on either ATK or DEF. 
In the case of a tie with the highest ATK, all tied players use the following tie-breakers with the cards they chose until there is no tie, with the winner going first.


Highest DEF
Highest Level (monsters without a printed Level, such as Xyz monsters, count as 0)
Highest Xyz Rank (monsters without a rank, which would be any monster aside from Xyz monsters, count as 0)
Highest Link Number (monsters without a Link Number, which would be any monster aside from Link Monsters, count as 0)
Each player rolls a die with the highest winning the roll, repeat as needed for ties.



Example: Yugi chooses Kuriboh (300/200, Level 1). Bandit Keith chooses Machine King Prototype (1600/1500, Level 3). Odion chooses Dragon Zombie (1600/0, Level 3). Mako chooses Fire Kraken (1600/1500, Level 4). Yugi goes last since his monster had the lowest attack, but the three remaining duelists' monsters tie with 1600 ATK, so we need to go into tiebreakers, starting with comparing DEF. Mako and Bandit Keith both have DEF 1500 monsters, while Odion has a DEF 0 monster. Odion will go after the other two. Since there is still a tie, Level is compared. Mako's monster is a Level 4 monster, but Bandit Keith's is a Level 3, so Mako goes before Bandit Keith (since Odion is already out of the running, his monster's Level is not compared). Thus the order is Mako, Bandit Keith, Odion, Yugi.

If you don't want to have the strategic layer of selecting monsters, you could just have it where all players roll a die with the highest roll going first. The disadvantage to this is that players do not have the ability to strategically determine whether they want to go first, last, or in the middle.

Also, you can play to the last man standing if desired.