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Maximizing the Score in "Ticket to Ride"

Elizabeth J. Schaefer, Andrew J. Schaefer

TL;DR

The paper tackles the problem of computing the maximum possible score in Ticket to Ride under a fixed train-car budget, ignoring player competition. It develops both undirected and directed graph models and a mixed-integer programming formulation to maximize edge points plus completed-ticket bonuses, constrained by the budget $\beta$. Computational results for the original board ($\beta=45$) yield an optimal score of $285$ (with $223$ from tickets) and extend analyses up to $\beta=50$, while revealing frequently chosen tickets and routes that could guide game balancing. By situating Ticket to Ride within fixed-charge network design and prize-collecting problems, the work demonstrates how optimization can elucidate the trade-offs between local route-building and global ticket rewards, informing both theoretical analysis and practical balance adjustments.

Abstract

We give two graph-theoretic models and a mixed-integer program to calculate the maximum achievable score in the popular board game "Ticket to Ride." In Ticket to Ride, players compete to claim railway routes on a map, with points awarded based on the length of each route and the successful completion of destination tickets connecting specific city pairs. Each player has 45 train cars available, and each route can be chosen by only one player. Using the mixed-integer programming model, we examine the optimal solution with the 45 allocatable train cars, leading to an optimal score of 285 points. We also calculate the optimal solutions for up to 50 train cars. We determine the most frequently chosen tickets and routes over these 50 instances, giving insight into how optimization might be used to balance games. In particular, we identify several instances in which the point values can be adjusted to better balance the game.

Maximizing the Score in "Ticket to Ride"

TL;DR

The paper tackles the problem of computing the maximum possible score in Ticket to Ride under a fixed train-car budget, ignoring player competition. It develops both undirected and directed graph models and a mixed-integer programming formulation to maximize edge points plus completed-ticket bonuses, constrained by the budget . Computational results for the original board () yield an optimal score of (with from tickets) and extend analyses up to , while revealing frequently chosen tickets and routes that could guide game balancing. By situating Ticket to Ride within fixed-charge network design and prize-collecting problems, the work demonstrates how optimization can elucidate the trade-offs between local route-building and global ticket rewards, informing both theoretical analysis and practical balance adjustments.

Abstract

We give two graph-theoretic models and a mixed-integer program to calculate the maximum achievable score in the popular board game "Ticket to Ride." In Ticket to Ride, players compete to claim railway routes on a map, with points awarded based on the length of each route and the successful completion of destination tickets connecting specific city pairs. Each player has 45 train cars available, and each route can be chosen by only one player. Using the mixed-integer programming model, we examine the optimal solution with the 45 allocatable train cars, leading to an optimal score of 285 points. We also calculate the optimal solutions for up to 50 train cars. We determine the most frequently chosen tickets and routes over these 50 instances, giving insight into how optimization might be used to balance games. In particular, we identify several instances in which the point values can be adjusted to better balance the game.

Paper Structure

This paper contains 10 sections, 4 theorems, 9 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Graph Representation and Optimization Model
  3. Undirected Graph Representation
  4. Relationship to Other Problems in the Literature
  5. Directed Graph Formulation
  6. Mixed-Integer Programming Model
  7. Computational Study
  8. The Optimal Solution for 45 Train Cars
  9. Sensitivity Analyses
  10. Conclusion

Key Result

Proposition 1

Given a subset of edges $E' \subseteq E$, for every ticket $k \in K$, there exists a directed cycle in $H'_k$ that contains the arc $(t_k,s_k)$ if and only if nodes $s_k$ and $t_k$ are connected in the undirected subgraph $G'$.

Figures (4)

  • Figure 1: Graph formulations of the Ticket to Ride problem. Each edge in Figure \ref{['UndirectedGraphExample']} is replaced by a forward and backward arc, and there is a "dummy" arc from 9 to 1 corresponding to the ticket defined on cities 1 to 9. Figure \ref{['DirectedGraphExample']} shows the transformation from the undirected graph given in Figure \ref{['UndirectedGraphExample']} to a directed graph.
  • Figure 2: Map of the optimal solution with 45 train cars.
  • Figure 3: Optimal scores as the number of train cars varies.
  • Figure 4: The ten most frequently chosen routes.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Corollary 2