Word of the Week 08/04/19: Kingmaker

(in the context of game theory):

From Wikipedia:

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. Said 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.

Consider this simple game: Three gladiators play, with strengths 3, 4, 5. In 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.

Because they allow the outcome of the game to be determined by a player of (presumably) inferior strategy, kingmaker scenarios are usually considered undesirable, though to some extent they may be unavoidable in strategy games. Of course the argument can be made that this means the winner, chosen by the kingmaker, played with the additional restriction of not annoying the other players as much, presumably a more difficult task. In these games, the game mechanics, players' outcomes and strategies are often so interconnected that to eliminate all possibilities of this situation is almost impossible.

In tournament situations where for instance the first few teams proceed to the next round, a player that is already guaranteed to proceed can experience a situation similar to a kingmaker. They can sometimes influence who of the remaining players comes in second (when 2 players proceed). For such a player it can be profitable to make sure the weakest player proceeds, because this reduces his competition in subsequent rounds. This is often seen as undesirable because it conflicts with the concept that the strongest few are allowed to proceed to the next round.