Optimizing for Strategy Diversity in the Design of Video Games

We consider the problem of designing video games (modeled here by choosing the structure of a linear program solved by players) so that players with different resources play diverse strategies. In particular, game designers hope to avoid scenarios where players use the same ``weapons'' or ``tactics'' even as they progress through the game. We model this design question as a choice over the constraint matrix $A$ and cost vector $c$ that seeks to maximize the number of possible supports of unique optimal solutions (what we call loadouts) of Linear Programs $\max\{c^\top x \mid Ax \le b, x \ge 0\}$ with nonnegative data considered over all resource vectors $b$. We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. The upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope. Our construction provides practical guidance to game designers seeking to offer a diversity of play for their plays.