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Optimizing for Strategy Diversity in the Design of Video Games

Oussama Hanguir, Will Ma, Christopher Thomas Ryan, Jiangze Han

Abstract

We consider the problem of designing a linear program that has diverse solutions as the right-hand side varies. This problem arises in video game settings where designers aim to have players use different "weapons" or "tactics" as they progress. We model this design question as a choice over the constraint matrix $A$ and cost vector $c$ to maximize the number of possible \emph{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.

Optimizing for Strategy Diversity in the Design of Video Games

Abstract

We consider the problem of designing a linear program that has diverse solutions as the right-hand side varies. This problem arises in video game settings where designers aim to have players use different "weapons" or "tactics" as they progress. We model this design question as a choice over the constraint matrix and cost vector to maximize the number of possible \emph{supports} of unique optimal solutions (what we call "loadouts") of Linear Programs with nonnegative data considered over all resource vectors . 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.

Paper Structure

This paper contains 29 sections, 27 theorems, 94 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Video game motivation
  3. Some illustrative examples
  4. Our contributions
  5. Related work
  6. Organization of the paper
  7. Statement of the main results
  8. The Cyclic Polytope
  9. Statements of Main Results
  10. Roadmap for proving \ref{['thm:upperbound', 'thm:lowerbound']}
  11. Preliminaries
  12. Upper Bound (Proof of \ref{['thm:upperbound']})
  13. From equality loadouts to triangulations
  14. From cells of a triangulation to faces of a polytope
  15. Proof of Theorem 1
  16. ...and 14 more sections

Key Result

Theorem 1

Fix positive integers $n,m,k$ with $n>m\ge k\ge2$. Then the number of $k$-loadouts for any design $(A,c)$ with $A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R}^n$ satisfies

Figures (4)

  • Figure 1: An illustration of upper bound in \ref{['thm:upperbound']} and lower bound in \ref{['thm:lowerbound']}.
  • Figure 2: An illustration of the triangulations $\Delta_{c_1}(A)$ (left side of the figure) and $\Delta_{c_2}(A)$ (right side of the figure). In the figure, the third dimension (corresponding to the row of $1$'s in the matrix $A$ in \ref{['eqn:example-data']}) is suppressed since all objects are at the same height of $1$.
  • Figure 3: Representation of the cyclic polytope $\mathcal{C}(7,3)$.
  • Figure 4: The array associated with $n = 9$ and $L = \{1, 3, 4, 7, 8, 9\}$.

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Proposition 1
  • Example 1
  • Definition 2
  • Definition 3: Cyclic Polytope
  • Definition 4: $f$-vector
  • ...and 40 more