It is a game with the following rules. It is loosely based off a game called goofspiel but I want to see if the rules can apply to a continuous variant.
- There are 2 players $A$ and $B$.
- Each defines a "play function" $\mathcal{P}$. $A$ defines $\mathcal{P}_A$ and B defines $\mathcal{P}_B$.
- A play function $\mathcal{P}$ must satisfy some constraints:
- The domain and codomain of $\mathcal{P}$ are $(0,1)$.
- $\int_0^1 \mathcal{P}(x)\;dx=0.5$
- $\int_0^1 \ln(\mathcal{P}(x)/x)\;dx=0$
- Given the "contest function" $\mathcal{C}$ as $$\mathcal{C}(x)=\begin{cases} \;1 & \text{if}\quad\mathcal{P}_A > \mathcal{P}_B \\ \;0 & \text{if}\quad\mathcal{P}_A = \mathcal{P}_B \\ -1 & \text{if}\quad\mathcal{P}_A < \mathcal{P}_B \\ \end{cases}$$ Player $A$ wins if $\int_0^1 x\;\mathcal{C}(x)\;dx > 0$, otherwise $B$ wins. (We can suppose a tie if it works out to exactly $0$, if that makes a difference.)
The game is zero sum and symmetric. In Goofspiel, there is no pure Nash equilibrium.
Does my variant of this game also have no pure Nash equilibrium or is there an exploit that I didn't account for?