Consider the following game on a directed weighted graph $G$ with a chip at some node.
All nodes of $G$ are marked by A or B.
There are two players Alice and Bob. The goal of Alice (Bob) is to shift the chip to a node that marked by A (B).
Initially Alice and Bob have $m_A$ and $m_B$ dollars respectively.
If a player is in a losing position (i.e., current position of the chip is marked by opposite letter) he or she can move the chip to a neighboring node. Such move costs some dollars (the weight of the corresponding edge).
The player loses if he or she is in a losing position and has not money to fix it.
Now consider the language GAME that consists of all directed weighted graphs $G$ (all weights are positive integers), initial position of the chip, and capitals of Alice and Bob that are given in the unary representation
such that Alice has a wining strategy at this game.
The language GAME belongs to P. Indeed, the current position of the game is defined by the position of the chip and the current capitals of Alice and Bob, so dynamic programming works (here it is important that initial capitals are given in the unary representation).
Now consider the following generalization of this game. Consider several directed weighted graphs $G_1, \ldots G_n$ with a chip at each graph. All nodes of all graphs are marked by A and B. Now Bob wins if all chips are marked by B and Alice wins if at least one chip marked by A.
Consider the language MULTI-GAME that consists of all graphs $G_1, \ldots, G_n$, initial positions and capitals $m_A$ and $m_B$ (in the unary representation) such that Alice wins at the corresponding game. Here it is important that capitals are common for all graphs so, it is not just several independent GAMEs.
Question What is the complexity of the language MULTI-GAMES? (Is it also belongs to P or there are some reasons to thing that this problem is hard?)
UPD1 Neal Young suggested to use Conway's theory. However I do not know is it possible to use this theory for several games with common capital.
UPD2 I want to show an example that shows that MULTI-GAME is not very simple. Let Alice split her capital $m_A$ to some $n$ terms $m_A = a_1 + a_2 + \ldots a_n$ (She is going to use $a_i$ dollars for $i$-th graph). Define $b_i$ as the minimal number such that in $i$-th game Bob wins if Alice and Bob have $a_i$ and $b_i$ dollars respectively. If $b_1 + \ldots b_n > m_B$ (for some spliting $m_A = a_1 + a_2 + \ldots a_n$) then Alice wins. However, the opposite is not true. Consider two copies of the following graph (initially the chip is at the left up A):
For one graph Bob wins if $m_A=0$ and $m_B=2$ or if $m_A=1$ and $m_B=3$. However for the game with two copies of this graph Bob loses if $m_A=1$ and $m_B=5$. Indeed, Bob have to spend $4$ or $5$ dollars to shift both chips to a node marked by $B$. Then Alice can shift at least one chip to a node marked by A. After that Bob have not money to save his position.
UPD3 Since the question for arbitrary graphs seems difficult consider specific graphs. Denote the nodes of some graph $G_i$ as $1, \ldots k$. My restriction is the following: for every pair $i<j$ there exists edge from $i$ to $j$ and there is no the reverse edge. Also there exists a restriction for the costs of edges: for $i<j<k$ the edge $j$ to $k$ is not greater than from $i$ to $k$.