Can someone explain to me the proof on page 7/20 of the original paper about potential games ([https://www.cs.tau.ac.il/~mansour/sem-game-02-03/monderer-potential-96.pdf](https://www.cs.tau.ac.il/%7Emansour/sem-game-02-03/monderer-potential-96.pdf))?


It is about why two potentials of a game G can only differ by a constant.



Why do two potentials of a game only differ by a constant?
This is what it looks like to me: to get $H(y)$, we change strategy profile $z$ into $y$ one player at a time, summing the changes in utility. For any exact potential $P$ (which is actually equation 2.2 with weights $1$), each change in utility is equal to $P(a\_{i-1}) - P(a\_{i})$. This is a telescoping sum, so $H(y) = P(y) - P(z)$. This didn't depend on which exact potential $P$ we chose, so $P\_1(y) - P\_1(z) = P\_2(y) - P\_2(z)$ for all $y$. In other words $P\_1(y) = P\_2(y) + c$ where $c = P\_1(z) - P\_2(z)$.


