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It's well known that in a 2 player, zero-sum game, a mistake by an opponent can only help you. It's also well known that this is not necessarily true in a 3 player game, even a zero-sum one: a mistake by an opponent can also hurt you, helping only the other player.

Are there any 3 player games where this property holds regardless?

I'm thinking there might not be.

  • Firstly, if interaction doesn't exist, then trivially there is no kingmaking. However, such a game barely counts as a multiplayer game: it's multiplayer solitaire. (And even then, multiplayer solitaire isn't immune to kingmaking! Say that you took your last round and scored 60 points, and opponent scored 50 points. The other opponent can either take a safe line that scores 55, or a risky line that scores 65 10% of the time and 45 90% of the time.)
  • Secondly if interaction does exist, and affects all opponents equally at all times, it's not very interesting. I'm not aware of any game where this property holds.
  • And then, if interaction exists that affects opponents unequally, you get kingmaking, although the severity of course depends on the game.

That said, I am not certain this argument is airtight. Some counter-mechanisms:

  • Interaction which affects opponents equally, but not at all times, perhaps with unilateral mechanisms. Example: Alice buys a punishable asset. Bob and Carol have the punishing cards, and use them to score Alice -2, Bob +1, Carol +1. This looks like two players working together to gang up on the third, but if either Bob or Carol can unilaterally force the punish in this case, then Alice cannot credibly accuse either one of kingmaking.
  • Challenge mechanisms: Alice and Bob are in a tight battle for first. Carol is clearly getting third. Carol makes a move, which Alice believes unfairly affects them. Alice challenges Carol, claiming Carol is getting 3rd no matter what. Carol has a choice:
    • Concede the challenge, in which case Carol is eliminated and the kingmaking problem is averted.
    • Fight the challenge, in which case Alice and Carol play a subgame to determine if the claim is correct. The penalty to the loser of the subgame is severe (worth more than 1 loss, so to speak) to encourage Carol to concede valid challenges and deter Alice from making invalid challenges.

I can imagine these mechanisms might be able to create a game where you are never punished for another player's mistakes.

So which is true?

  • In any multiplayer (3+p), zero-sum game, with interaction (with effect dependent on the board state) necessarily comes with some degree of kingmaking, in the sense that you may be worse off due to an opponent's mistake.
  • Using mechanisms (such as the challenge mechanism above), games exist where the entire cost of any mistake can be shifted to the person responsible for it.

EDIT: I don't believe the answers so far have answered the question as is. I'm asking about 3-player games where a "mistake" (defined as a move that lowers the EV of the player making it) can also "kingmake against" someone (defined as a mistake that also lowers the EV of someone not making that move.) Equivalently, I'm asking regarding zero-sum 3-player games, where the amount of EV players can guarantee for themselves sum to zero.

Under this definition, if the final score of the game were (1, 0, -1) based on final ranking alone, and someone were trying to move up the rankings, it would not count as a "mistake", and therefore not a "kingmaker move".

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  • Is it kingmaking if the interaction is random? Imagine a hypothetical game where a player can take an action "you and a randomly selected opponent both lose the game". Dec 15, 2023 at 15:00
  • I would say that it's not considered kingmaking if all opponents aren't worse off, EV-wise by taking the action. For example, it's the last round of the game, Alice is about to get first, Bob is about to get last, and then some joker takes your hypothetical action. I would say that Alice got kingmade against and Bob did not get kingmade against, and therefore kingmaking has occurred. I would say that regardless of who the randomly selected opponent actually was.
    – Han Guo
    Dec 15, 2023 at 15:46
  • As another example of "random kingmaking", imagine the following scenario in a hypothetical poker variant. Alice has a set of Aces, and checks the flop. Then, all hole cards are turned face up. Bob has a flush draw and checks. Carol with 72o shoves, denying Bob's equity, Alice calls, Bob folds. I'd say that Bob was kingmade against. As the cards lie, the turn completes the flush, but the S/P ratio is good enough for Alice to call Bob's shove, and the river competes the boat. In this case, Carol's shove saved Bob money, but I would still say Bob was kingmade against.
    – Han Guo
    Dec 15, 2023 at 15:56
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    There's a lot of current games that fight kingmaking by obscuring who is actually winning, which seems outside of your scope entirely.
    – Erik
    Dec 15, 2023 at 22:45
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    @PhilipKendall "you and a randomly selected opponent both lose the game" - in my mind this is certainly kingmaking. You acted to help a player win. This is kingmaking. The fact that you did not know in advance who you are making the king IMO does not matter, you were sure you were making someone the king. Dec 15, 2023 at 23:48

4 Answers 4

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Your title question is "Do all multiplayer board games necessarily contain a kingmaking component?" while the body of your question asks "Are there any 3 player games where this property holds regardless?" where "this property" presumably refers to "a mistake by an opponent can only help you."

So, first of all, I think that these are rather different questions. To me, "kingmaking" means that one player has no realistic chance of winning the game, and their actions are decided based on personal feelings as to who should win, rather than in-game incentives. That's different from simply engaging in non-optimal play.

A further issue is just what a "mistake" is. I suppose one could say that there is some Nash equilibrium, and this is "correct" play, but it's not feasible for a supercomputer, let alone a human, to calculate the Nash equilibrium of even a moderately complicated game. And if it were, then games would be rather pointless, since everyone would just mechanically engage in "optimal" play each turn, and there would be no real judgment calls.

Playing according to the Nash equilibrium requires not only that you can calculate what it is, but you have to be confident that it's so obvious that everyone else will also play according to it (and you have to be confident that everyone else is confident that you know it, and confident that everyone else is confident that you are confident that they know it, etc.)

Part of the strategy of multi-player games is trying to guess what the other players will consider "optimal" strategy. Alice may feel that it's unfair that Carol is giving the win to Bob, but apparently Bob is better at guessing what Carol will do than Alice, so Bob is in that sense "better" at playing the game.

As for the question of whether there's any way to have this property that opponents' "mistakes" (however that's defined) will always help you, it is theoretically possible, but very limiting.

For the issue of avoiding king-making, there are a few strategies. One is to have the results of the game be more finely grained that "winning" and "losing". For instance, in a cash game of poker, players don't just "win" or "lose", they win or lose different amounts of money. So a player still has the incentive of losing as little money as possible. Another strategy is making it so that good play doesn't assure victory. Poker is again an example; the probability of winning a knock-out round of poker is roughly proportional to the size of your chip stack, so even someone way behind isn't completely out of it. A third strategy is to keep victory points secret, so even if someone has no chance of winning, they don't know that.

There is another issue that is similar to king-making, but not quite the same, and that is collusion. Collusion is when two or more players work together to increase their chances at the expense of the others. This is more of an ethics issue than a game design one, although different game designs can make collusion easier and/or more tempting. King-making because one of the players is your partner is bad sportsmanship, but doing it because you think one of the players played a better game is not.

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    A kingmaker situation (a situation where you can't win, but can decide who can) is distinct from a kingmaker move (one that causes a player other than yourself to win). These do not always occur together. Example, Alice, Bob, and Carol are playing a Eurogame, and the game is close. Bob plays "You lose 30 VPs. An opponent of your choices loses 20 VPs." This is a kingmaker move, but there is no kingmaker situation before (since the game is even) or after (since the king is already made) the move. Also, this is an counterexample to your claim that you need to know the NE to determine a mistake.
    – Han Guo
    Dec 17, 2023 at 3:58
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    @HanGuo That is how games work, that is not a kingmaker move. Those cards would not be in the game in the first place if they are not intended to be played like that.
    – Joe W
    Dec 17, 2023 at 16:44
  • @HanGuo That type of action is not inherently a kingmaker. It depends on how the player uses it, and when. Also, that example does not represent a zero-sum game, as there is no player(s) gaining a sum of 50 points. Dec 21, 2023 at 7:26
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Before I give the answer I will start with a frame challenge, just because a player feels that they are getting ganged up on and kingmaking is happening that isn't always the case. Players are free to make moves that might not be the best choice but they feel could prevent them from coming in last and/or help them move up the rankings that hurt and help others.

Yes, any multiplayer game will contain elements that allow for kingmaking in them by the very nature of interaction between players. Lets take a look at your examples as to how they can still be used.

Random mechanics: A player chooses to not use something that will randomly punish other players if they feel it would impact the person they are trying to help win more then others. This is the same if it will impact all players equally.

Challenge mechanics: How do you tell the difference between being a kingmaker and trying to take down 2nd place so you can move from 3rd place to 2nd place?

As long as players are interacting and able to make choices they will be able to pick some to help and others to hurt.

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Most multiplayer games contain tiny kingmaking component that isn't a problem in most plays.

  • In most board games that people play for fun the issue of kingmaking do not rise. Indeed, someone could make a mistake that benefit another player, but if all players play in good sportsmanship then its a non-issue.
  • If you are looking for some mathematical theorem, then... I have no idea.
  • Some games have big kingmaking component, mostly in multiplayers wars, to name a few: Risk, Diplomacy and A Game of Thrones.
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There is inherently potential for kingmaking, but it can be mitigated and doesn't have to be utilized. The mechanism you described is unlikely to exist in a game as stated.

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So which is true?

1.) In any multiplayer (3+p), zero-sum game, with interaction (with effect dependent on the board state) necessarily comes with some degree of kingmaking, in the sense that you may be worse off due to an opponent's mistake.

2.) Using mechanisms (such as the challenge mechanism above), games exist where the entire cost of any mistake can be shifted to the person responsible for it.

Perhaps part of the confusion is how you are using established game theory terms, and substituting your own definitions for them. To clarify, in a zero-sum game, one player's gain is equivalent to the associated loss of all opponents. Kingmaking is where a player who is unable to win has the capacity to determine which player among others will win, and they intentionally make decisions which favor a specific player. You are defining kingmaking as an action which lowers the scores of the action taker and an opponent.

You want to know if there can be a game where it is impossible for someone to lower their score and the score of an opponent. Well, the simple answer is to not allow players to take actions which lower their own score. If the game does not have a method for the active player to lower their score, then according to your definitions of mistake and kingmaking, they cannot make a "mistake" to "kingmake" a player.

Let's consider the case where the actie player makes a decision which does not change their own score. This is now outside your definition of "mistake", so I don't know if you will be interested. Since this is a zero-sum game, some player(s) will lose points while the other player(s) gain points. This is inherently open to kingmaking if players have perfect information. It does not guarantee that kingmaking will happen, but it does allow for players to do so if they chose. However, this can be mitigated with mechanisms such as:

  • Hidden information regarding player points.
  • Addition of a non-scoring resource and a high cost for performing this type of action (skip a turn, pay $500, opponent who lost points gains some benefit).
  • Adding an element of randomness (random or target opponent gains or loses points).
  • Ability of players to counter undesirable actions to prevent being attacked.

Hopefully this satisfies your first question. Zero-sum games with interaction will have inherent potential for kingmaking, but the introduction of mechanisms and rules can mitigate/prevent kingmaking. Also, picking your playgroup can prevent kingmaking.

As for your second question, there are games where players can challenge an action and the loser of the challenge will be penalized. This is common in word games, and is typically a challenge about the accepted spelling or existence of a word. The loser of the challenge (be it the active player or the challenger) will normally lose points equivalent to the value they would have gained for the word. However, I have not heard of a mechanism such as the one you have described, nor do I believe it would be a good design choice:

Alice and Bob are in a tight battle for first. Carol is clearly getting third. Carol makes a move, which Alice believes unfairly affects them. Alice challenges Carol, claiming Carol is getting 3rd no matter what. Carol has a choice:

This design allows players to contest actions that they don't like on the grounds that they would be negatively affected by them. Playing a subgame would not determine whether or not the action was made with the intent of kingmaking, and does not resolve the issue. Depending on the likelihood of Alice to win over Carol, or the difference between suffering Carol's action or the subgame penalty, Alice may start contesting every action that Carol takes, regardless of Bob's standing.

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