# What's the name of the theory that the player with the most options has the advantage?

A long time ago, I remember reading about a theory that sounds like this:

Players A and B have access to options X and Y. Player A also has access to option Z. Player A is, at worst, equal to and, at best, at an advantage to player B.

Ok, so that was poorly worded. Let me put it another way:

Having more options available to you in a two player game can't possibly hurt you.**

Example

Player A can attack from both sides. Player B must attack from the left. If attacking to the left is the best choice, then both players are equal. If attacking from the right is the best choice, then Player A has a distinct advantage. Player A is either tied or winning. Player B is either tied or losing.

Can anyone tell me the name of the theory I described?

** "... can't possibly hurt you" assumes optimal play from both players. It's not always the case the player with the most options will use them wisely. Intuition tells us that sometimes having too many options can make it difficult for the player to find the best one. Just assume that both players are making the best choices when this theory applies.

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Indeed, if we don't assume optimal play, then the result could easily be the exact opposite! Player B only has access to option X, and Player A has option X and another, worse option. Player B is in the better position. – GendoIkari Jun 20 '14 at 4:17
Are you sure this is actually part of formal game theory, or just a theory about games? – doppelgreener Jun 20 '14 at 7:45
@JonathanHobbs I'm not well-versed in formal game theory. When I wrote the question, I felt like a theory that relates "options available" to "advantage" should fall into that realm. Since it was the identify-this-game tag that led me to even ask this question here, I thought it would be appropriate to retag with a brand new tag called identify-this-theory. – Rainbolt Jun 20 '14 at 13:26
This question appears to be off-topic because it is about a trivial set theory result with the word "player" substituted in. It is not related to any particular board or card game. – murgatroid99 Jun 20 '14 at 18:28
OK then, let's ignore the triviality aspect. Your question is still of the class "Here is a description of <something vaguely related to games>. Is there a word for it?" I don't think that belongs here. – murgatroid99 Jun 20 '14 at 20:06

This sounds a lot like a strategy-stealing argument. It's often used to prove that one side of a game has an advantage, even if optimal strategy is not known. As a broad example, consider a game with the following properties:

• It is a turn-based game
• The starting positions are identical except Player A goes first
• Player A has the option of skipping the first turn

Assuming optimal play, the first player cannot be at a disadvantage. The proof is a strategy-stealing argument: Any hypothetical strategy that Player B could use to get an advantage could be countered by Player A just skipping the first turn and then using that exact strategy.

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Is this principle even true? Consider the following game:

A, B, and C, are dividing \$60. All players left standing at the end of the game receive an equal split.

Each player may shoot one other player. If a player gets shot twice, that player can no longer participate in the split.

Alternatively, C has the option to shoot another player, and then (after the first shots resolve) shoot both other players twice.

Obviously, the optimal strategy for C is to take the second option, but then the optimal strategy for A and B is to both shoot C (if they don't, they will get nothing). This means that if all players act rationally, A and B will get \$30 each and C will get \$0.

In other words, C having more options leads to a worse result for C. So the theory isn't even true, which I think is decent evidence that it doesn't have a name, especially given that nobody else who's come here so far has heard of it. :-P

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First, this isn't even an answer. Second, your counterexample involves more than two players. The theory still holds for two players. I edited the question to clarify. I know that it invalidates half of your answer, but it should have been a comment to begin with. – Rainbolt Jun 25 '14 at 2:06
I think the game must also have no hidden information and no randomness. I can elaborate if you're interested. – Neil Fitzgerald Jun 25 '14 at 2:37
This place doesn't work like a forum - check our tour for more information. Answers are expected to answer the question (in this case, what's the theory called?), whereas this just seems to just be discussing the theory itself. You're welcome to discuss that as part of an answer that otherwise answers the question, though. – doppelgreener Jun 25 '14 at 10:39
This doesn't pass the bar. At what point is this actually answering the question of what the theory is called? Even if it's an inaccurate or incorrect theory, it still may have a name. If the theory doesn't have a name, there either is no answer, or the answer is "it doesn't have one, and I can say that because {insert authoritative citation here}" - so, yes, the answer @Kevin describes is roughly what a correct answer would look like. Not having a name doesn't turn this into an opportunity for this to be a discussion forum. – doppelgreener Jun 26 '14 at 8:42
I'm not deleting this. Attempting to prove the theory invalid as an argument for why it doesn't exist is a reasonable answer. Whether the proof is any good is a different question. Feel free to vote appropriately. – ire_and_curses Jun 26 '14 at 18:33