Applications of Graph Theory and Probability in the Board Game Ticket - - PowerPoint PPT Presentation

Applications of Graph Theory and Probability in the Board Game Ticket to Ride

• R. Teal Witter & Alex Lyford

Middlebury College

January 16, 2020

Ticket to Ride (USA)

Overview

Routes ◮ Long routes are overvalued ... and can be used to easily win. ◮ We can find a better route scoring scheme with indicator random variables. Destination Tickets ◮ Players with some Destination Tickets perform better than players with others ... why? ◮ We use regression to identify the best Destination Tickets.

Current Route Values

Route Length 1 2 3 4 5 6 Points Scored 1 2 4 7 10 15 Points per Train 1.00 1.00 1.33 1.75 2.00 2.50

Current Route Values

Route Length 1 2 3 4 5 6 Points Scored 1 2 4 7 10 15 Points per Train 1.00 1.00 1.33 1.75 2.00 2.50 Arguments for ◮ Collecting many trains of the same color is hard Arguments against ◮ Is it really that hard? ◮ Only one route can be claimed per turn ◮ Collecting multiple colors simultaneously helps

Games with routes of length at most k

For all games, all 45 trains will be collected over 23 turns.2 k Composition Points Turns Points per Turn 1 1 x 45 45 23 + 45 0.66 2 2 x 22, 1 x 1 45 23 + 23 0.98 3 3 x 15 60 23 + 15 1.58 4 4 x 11, 1 x 1 78 23 + 12 2.23 5 5 x 9 90 23 + 9 2.81 6 6 x 7, 3 x 1 109 23 + 8 3.52

2We ignore locomotives collected from the five face up cards.

Win Rate in Simulated Games

Hungry Path One Step Long Route Strategy 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Win Rate

Wins by Strategy in 20,000 Games

How should routes be valued?

Idea: value = expected time to collect 3

3measured in number of cards rather than turns

Expected number to find k blue cards

Without loss of generality, our goal is to calculate the expected numbers to find k blue cards.

Expected number to find k blue cards

Nk := number of cards until k blue cards are found C := all cards B := blue cards x ∈ C \ B Ix,k :=

• 1

if x appears before the kth blue card

• therwise

Then Nk = k +

• x∈C\B

Ix,k

Expected number to find k blue cards

Nk = k +

• x∈C\B

Ix,k Taking expectation, E[Nk] = k + (|C \ B|) × E[Ix,k] Recall E[Ix,k] = 1 × P(Ix,k = 1) + 0 = P(Ix,k = 1). Then E[Nk] = k + (|C \ B|) × P(Ix,k = 1)

Expected number to find k blue cards

Think of the deck as non-blue cards separated by blue cards into |B| + 1 piles (possibly of size 0): xxxbxbbxxxbxbb . . . xbxxxb Then P(Ix,k = 1) is k/(|B| + 1)

Expected number to find k blue cards

E[Nk] = k + (|C \ B|) × k |B| + 1 =

• 1 + 110 − 12

12 + 1

• k

= 111 13 k Thus our scoring should be linear!!

Choosing a scalar

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Points per Train (α) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Proportion of Wins

Proportion of Wins by Strategy and Points per Train

Hungry Path One Step Long Route

Perhaps somewhere between 3.5 and 5?

Destination Tickets and Wins

0.0 0.1 0.2 0.3 0.4 0.5 0.6 Proportion of Wins Montreal/Vancouver New York/Seattle Los Angeles/Miami Chicago/Los Angeles Denver/Pittsburgh Atlanta/San Francisco Los Angeles/New York Duluth/El Paso Calgary/Phoenix Phoenix/Portland Calgary/Salt Lake City Nashville/Portland Helena/Los Angeles Santa Fe/Vancouver Los Angeles/Seattle Oklahoma City/Sault St. Marie Chicago/Santa Fe Denver/El Paso Nashville/Sault St. Marie Dallas/New York Montreal/New Orleans Houston/Kansas City Houston/Winnipeg Little Rock/Winnipeg Atlanta/Montreal Chicago/New Orleans Miami/Toronto Atlanta/New York Duluth/Houston Boston/Miami

Destination Tickets

Players Two Four

Best and Worst

Best: Montreal/Vancouver, New York/Seattle Worst: Boston/Miami

Effective Resistance?

A measure of connectivity between two nodes on a graph: electric flow from one node to another resistance = min

flows

• routes

(flow on route)2 More, shorter paths → lower resistance

Effective Resistance!

0.2 0.4 0.6 0.8 1.0 Difficulty (Effective Resistance) 4 6 8 10 12 14 16 18 20 22 Reward (Length of Minimum Path)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Destination Tickets by Difficulty and Reward Colored by Difference from Expected Proportion of Wins

1 Denver/El Paso 2 Houston/Kansas City 3 Atlanta/New York 4 Calgary/Salt Lake City 5 Chicago/New Orleans 6 Duluth/Houston 7 Helena/Los Angeles 8 Nashville/Sault St. Marie 9 Chicago/Santa Fe 10 Atlanta/Montreal 11 Oklahoma City/Sault St. Marie 12 Los Angeles/Seattle 13 Miami/Toronto 14 Duluth/El Paso 15 Denver/Pittsburgh 16 Phoenix/Portland 17 Dallas/New York 18 Little Rock/Winnipeg 19 Boston/Miami 20 Houston/Winnipeg 21 Santa Fe/Vancouver 22 Montreal/New Orleans 23 Calgary/Phoenix 24 Chicago/Los Angeles 25 Atlanta/San Francisco 26 Nashville/Portland 27 Montreal/Vancouver 28 Los Angeles/Miami 29 Los Angeles/New York 30 New York/Seattle

• 0.08
• 0.04
• 0.01

0.02 0.06 0.09

Rankings by Minimum Path Length and Residual

Summary

Routes ◮ Long routes are overvalued ... and can be used to easily win. ◮ We can find a better route scoring scheme with indicator random variables. Destination Tickets ◮ Players with some Destination Tickets perform better than players with others ... why? ◮ We use regression to identify the best Destination Tickets.

Thank you!

Questions?

Correlations with overall wins

4 9 14 19 Path Length −0.10 −0.05 0.00 0.05 0.10 Proportion of Wins

Destination Tickets by Path Length and Wins Pearson coefficient: 0.64 p-value: 0.00014

0.2 0.4 0.6 0.8 Resistance −0.10 −0.05 0.00 0.05 0.10 Proportion of Wins

Destination Tickets by Resistance and Wins Pearson coefficient: -0.204 p-value: 0.281

−5 5 10 Residual −0.10 −0.05 0.00 0.05 0.10 Proportion of Wins

Destination Tickets by Residual and Wins Pearson coefficient: 0.766 p-value: 8.25e-07

References

• W. Ellens, F. Spieksma, P. Van Mieghem, A. Jamakovic, and
• R. Kooij. Effective Graph Resistance. Linear Algebra and its

Applications,435(10):2491–2506, 2011.

• F. de Mesentier Silva, S. Lee, J. Togelius, and A. Nealen.

Ai-based Playtesting of Contemporary Board Games. In Proceedings ofthe 12th International Conference on the Foundations of Digital Games, page 13. ACM, 2017

• A. Moon. Ticket to Ride. [Board Game]. Days of Wonder:

Los Altos, CA, 2004