My question: is the set of potential games closed under convex combinations?


An n player game with action set $A = A\_1 \times \ldots \times A\_n$ and payoff functions $u\_i$ is called an exact potential game if there exists a potential function $\Phi$ such that:
$$\forall\_{a\in A} \forall\_{a\_{i},b\_{i}\in A\_{i}} \Phi(b\_{i},a\_{-i})-\Phi(a\_{i},a\_{-i}) = u\_{i}(b\_{i},a\_{-i})-u\_{i}(a\_{i},a\_{-i})$$


A game is a general (ordinal) potential game if there exists a potential function $\Phi$ such that: 
$$\forall\_{a\in A} \forall\_{a\_{i},b\_{i}\in A\_{i}} sgn(\Phi(b\_{i},a\_{-i})-\Phi(a\_{i},a\_{-i})) = sgn(u\_{i}(b\_{i},a\_{-i})-u\_{i}(a\_{i},a\_{-i}))$$


Potential games are interesting because they always have pure strategy Nash equilibria: in particular, a sequence of best-responses must eventually converge to one.


Say that we have two games on the same action set, with utility functions $u\_i$ and $u'\_i$ respectively, for each player $i$. For any $0 \leq p \leq 1$, there is a convex combination of these two games, again on the same action set, where the utility function for each player $i$ is now $u^p\_i(\cdot) = (1-p)u\_i(\cdot) + pu'\_i(\cdot)$. 


Clearly, the convex combination of two exact potential games is also an exact potential game: just take the same convex combination of the two potential functions. 


But is it possible to have two (general) potential games such that their convex combination is not a potential game?



Is the convex combination of two potential games a potential game?
No, the set of ordinal potential games is not convex. I will think about an example, but it follows from a little fact that I've yet to get around to publishing:


Theorem: Any convex set of games strictly containing the exact potential games contains a game with no pure Nash equilibrium.


In particular, since every ordinal potential game has a pure Nash equilibrium, convexity of the set thereof would contradict this theorem.


EDIT: Bimatrix games A and B below are ordinal potential games, but the average of these two games is the zero-sum game "matching pennies" which obviously has no pure Nash equilibrium and so is not an ordinal potential game.



```
Game A:                     Potential A:
 (4,-1)    (0,1)              3    4
(-2,1)    (-2,-1)             2    1

Game B:                     Potential B:
(-2,-1)   (-2,1)              1    2
 (0,1)     (4,-1)             4    3

Matching pennies:           (No potential)
 (1,-1)   (-1,1)
(-1,1)     (1,-1)

```

