Story
I'm trying to write a baseball elimination problem in terms of integer programming. The basic problem looks like this:
Basically $x_{ij}$ is the future wins of team $i$ over team $j$, $g_{ij}$ is the overall games left between teams $i$ and $j$, $k$ is the team we're interested in, $p_j$ is the points a team has so far. Variable $z_j$ is just auxiliary, it's $1$ when team $j$ is ahead of $k$ (by total wins) and $0$ if behind.
The minimization function is basically the number of points team $k$ can possibly get. The first constraint simply says that in each game either team $i$ wins or team $j$ wins and no ties possible ($=$ sign, instead of inequality). $t_j$ is the total number of points in the end of the tournament, so $t_j-t_k\leq M z_j$ simply defines binary variable $z_j$, and we demand that no teams are ahead of us by the end of the tournament by the last constraint.
Problem
I wanted to add some tie-breaking rules, to adjust the constraints, so that if, e.g., $t_j = t_k$, it compares, lets say, $x_{jk}$ with $x_{kj}$ to decide whether team $j$ is ahead of $k$. The thing is, that python's PuLP that I use works only with linear integer problems, so I wanted to write this tie-breaking conditions using the same linear programming language (linear combination of $x_{ij},~z_{j}$ + maybe other variables).
Question
So basically I need a condition that kicks in only under particular circumstances. Maybe I can do this by introducing some other variable? What would you suggest?