Is this game symmetric?

Question

Strategies $\vec{x} \in \mathbb{R}^3_+$ such that $\sum_{i=1}^3x_i = 1$ with a payoff for player 1 defined as $P_1(\vec{x}, \vec{y}) = \sum_{i=1}^3sgn(x_i - y_i)$. I understand that this is zero-sum - but I am struggling to verify symmetry.

Answer 1

In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the strategies employed, not on the identities of the players playing them.

In your case,

$$P_1(\vec{x}, \vec{y}) = \sum_{i=1}^3sgn(x_i - y_i)$$

$$P_2(\vec{y}, \vec{x}) = -\sum_{i=1}^3sgn(y_i - x_i)=\sum_{i=1}^3sgn(x_i - y_i)=P_1(\vec{x}, \vec{y}) $$

This means that the payoff matrix is symmetric, hence the game is symmetric.

https://en.wikipedia.org/wiki/Symmetric_game#:~:text=In%20game%20theory%2C%20a%20symmetric,then%20a%20game%20is%20symmetric.