# Pure Pairwise Nash Equilibriums in multiplayer games

• Xian
In summary, the conversation discusses a game with k players where only two players are allowed to make changes to their strategies. The question posed is whether the existence of pairwise pure nash equilibriums implies the existence of a nash equilibrium for all k players, and if additional assumptions are needed to determine this. The response suggests searching for papers on "multiplayer Nash equilibrium" for more information.
Xian
Hey everyone,

Not sure if this is the right section to post this in, but makes the most sense to me so here goes.

Consider a game with k players. Now suppose that if only two of the players are allowed to make changes to their strategies (i.e. k-2 players are frozen) then those two players have a pure strategy nash equilibrium. Phrased another way; every pair of players has a pure nash equilibrium.

My question is does this imply anything about a global nash equilibrium, specifically; if pairwise pure nash equilibriums exist, must there also exist a nash equilibrium for all k players? Do we need strengthen our assumptions to determine anything? What about in the case of 3 players?

Thanks.

I'm afraid it is not as easy. Search for "multiplayer Nash equilibrium", there are plenty of seminar or lecture papers.

## 1. What is a Pure Pairwise Nash Equilibrium in multiplayer games?

A Pure Pairwise Nash Equilibrium (PPNE) is a solution concept in game theory that represents a state in which no player in a multiplayer game can improve their payoff by unilaterally changing their strategy, given the strategies chosen by all other players.

## 2. How is a Pure Pairwise Nash Equilibrium different from a Nash Equilibrium?

A PPNE differs from a Nash Equilibrium in that it applies specifically to multiplayer games, whereas a Nash Equilibrium can apply to both single and multiplayer games. Additionally, a PPNE considers only pairwise comparisons between players, while a Nash Equilibrium considers the strategies of all players simultaneously.

## 3. How is a Pure Pairwise Nash Equilibrium determined in a multiplayer game?

To determine a PPNE, all possible combinations of strategies for each pair of players are evaluated. The combination that results in the highest payoff for each player is considered a PPNE.

## 4. Can there be more than one Pure Pairwise Nash Equilibrium in a multiplayer game?

Yes, a multiplayer game can have multiple PPNEs. In fact, it is common for games to have multiple PPNEs, and it is up to the players to choose which one to play.

## 5. What are the implications of Pure Pairwise Nash Equilibriums in multiplayer games?

PPNEs can help players make strategic decisions in multiplayer games by identifying the most optimal strategies for each player. It also helps to predict the possible outcomes of the game and understand the potential actions of other players.

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