One colleague and me were discussing this problem during lunch today, and I did a little bit digging for several hours after returning to my office.
Fact: For an $(m,n,k)$-game, there does not exist a strategy that assures that the second player will win. For example, in Five-in-a-row, first player wins with perfect play.
Now our question is: How to modify the rules to make the game fair for both players?
I googled a bit more, found an MSE question that addresses the fairness of this kind of games.
I think the fairness in that question can be strenghthen by the following definition: A game is fair if and only if that two players play with perfect strategy, the game will always be a draw. This is fair in that this game favors player with less mistakes. For example, Tic-tac-toe is fair.
Several proposals to modify the rules to enforce the fairness:
Impose more restrictions on the player who plays first. For example, Five-in-a-row bans black to play "three and three", "four and four", and "overlines".
Change the $(m,n,k)$-game to an $(m,n,k,p,q)$-game: $k$-in-a-row on an $(m\times n)$-board, first player put $p$ stones on board, in subsequent moves, players put $q$ stones on board. For example, [Connect 6].
Go to higher dimension. For example, Five-in-a-row played in a $19\times 19\times 19$ "board".
I am no expert in combinatorial game theory or computational complexity in board games, but it is always nice to learn some new things in summer time. My question is: Any papers or treatises dealing the fairness of this kind of games? or more specifically, any proof of the fairness using above three modifications of rules? especially, for Five-in-a-row, do those additional rules make the game fair?
Any analysis of fairness on simpler cases like Connect 4 is welcome too.
Lastly just out of curiousity, as an avid Go player myself, I wonder is there any mathematical analysis on addressing the fairness using komi in the Go game? Or Go is just too complex to analyze...