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

Win Rate in Simulated Games Wins by Strategy in 20,000 Games 0.35 0.30 0.25 Win Rate 0.20 0.15 0.10 0.05 0.00 Hungry Path One Step Long Route Strategy

How should routes be valued? Idea: value = expected time to collect 3 3 measured 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 N k := number of cards until k blue cards are found C := all cards B := blue cards x ∈ C \ B if x appears before the k th blue card � 1 I x , k := 0 otherwise Then � N k = k + I x , k x ∈ C \ B

Expected number to find k blue cards � N k = k + I x , k x ∈ C \ B Taking expectation, E [ N k ] = k + ( | C \ B | ) × E [ I x , k ] Recall E [ I x , k ] = 1 × P ( I x , k = 1) + 0 = P ( I x , k = 1). Then E [ N k ] = k + ( | C \ B | ) × P ( I x , 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 ( I x , k = 1) is k / ( | B | + 1)

Expected number to find k blue cards k E [ N k ] = k + ( | C \ B | ) × | B | + 1 � 1 + 110 − 12 � = k 12 + 1 = 111 13 k Thus our scoring should be linear!!

Choosing a scalar Proportion of Wins by Str ategy and Points per Train 0.45 0.40 0.35 0.30 Proportion of Wins Hungry 0.25 Path One Step 0.20 Long Route 0.15 0.10 0.05 0.00 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) Perhaps somewhere between 3.5 and 5?

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

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 � (flow on route) 2 resistance = min flows routes More, shorter paths → lower resistance

Effective Resistance! Destination Tickets by Difficulty and Reward Colored by Difference from Expected Proportion of Wins 0.09 1 Denver /El Paso 2 Houston/Kansas City 3 Atlanta /New York 22 30 4 Calgary/Salt Lake City 29 5 Chicago/New Orleans 20 28 27 6 Duluth/Houston 0.06 7 Helena /Los Angeles 8 Nashville/Sault St. Marie Reward (Length of Minimum Path ) 18 9 Chicago/Santa Fe 1 0 Atlanta /Montreal 26 25 11 Oklahoma City/Sault St . Marie 16 24 1 2 Los Angeles /Seattle 0.02 13 Miami/Toronto 1 4 Duluth/El Paso 14 1 5 Denver /Pittsburgh 23 22 21 16 Phoenix/Portland 12 20 19 17 Dallas/New York 1 8 Little Rock/Winnipeg -0.01 15 18 16 17 19 Boston /Miami 10 14 13 20 Houston/Winnipeg 2 1 Santa Fe /Vancouver 11 9 12 10 22 Montreal /New Orleans 8 7 8 6 23 Calgary/Phoenix 4 5 24 Chicago/Los Angeles -0.04 25 Atlanta/San Francisco 6 3 26 Nashville/Portland 2 27 Montreal/Vancouver 2 8 Los Angeles /Miami 4 1 29 Los Angeles/New York 30 New York/Seattle -0.08 0.2 0.4 0.6 0.8 1.0 Difficulty (Effective Resistance)

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 Destination Tickets by Path Length and Wins Destination Tickets by Resistance and Wins Pearson coefficient: 0.64 Pearson coefficient: -0.204 p -value: 0.00014 p-value: 0.281 0.10 0.10 Proportion of Wins 0.05 Proportion of Wins 0.05 0.00 0.00 −0.05 −0.05 −0.10 −0.10 4 9 14 19 0.2 0.4 0.6 0.8 Path Length Resistance Destination Tickets by Residual and Wins Pearson coefficient: 0.766 p-value: 8.25e-07 0.10 Proportion of Wins 0.05 0.00 −0.05 −0.10 −5 0 5 10 Residual

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