Kingmaker Scenario

In game theory, a kingmaker scenario in a game of three or more players is an endgame situation where a player who is unable to win has the capacity to determine which player among others will win. This player is referred to as the kingmaker or spoiler. No longer playing for themselves, they may make game decisions to favor a player who played more favorably (to them) earlier in the game. Except in games where interpersonal politics, by design, play a decisive role, this is undesirable.

Gladiator example

Consider this simple game: Three gladiators are playing, with strengths 3, 4, 5. In each turn, each gladiator must engage another, and they begin combat. The result of combat is that the weaker player is eliminated, and the stronger player loses strength equal to that of the weaker player. (For example, if "5" attacks "3", "3" will die and "5" will have strength 2.) The winning gladiator is the last one standing.

Each round of combat eliminates one gladiator, so there will be two rounds of combat. The first round of combat will eliminate one participant and weaken the other to a strength no greater than 2. The nonparticipant's strength is at least 3, so they are guaranteed to win the second round of combat, and the entire contest. Therefore, the game collapses: The winning gladiator is the one not involved in the first battle.

Hence, the gladiator whose turn comes first is the kingmaker. They must be involved in the first battle, hence cannot win, but with the liberty of choosing their opponent in that battle, can elect either of the other two players to be the winner of the contest.