drafting territories in the board game...
TRANSCRIPT
![Page 1: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/1.jpg)
Drafting Territories in the Board Game Risk
Presenter: Richard Gibson
Joint Work With: Neesha Desai and Richard Zhao
AIIDE 2010October 12, 2010
![Page 2: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/2.jpg)
Outline
RiskDrafting territories
How to draft territories in Risk?UCT + machine-learned evaluation function
Empirical results
Conclusions + Future Work
![Page 3: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/3.jpg)
Risk
http://sillysoft.net/lux
Classic multi-player board game
A number of computer implementations, including Lux Delux by Sillysoft Games
Popular!
![Page 4: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/4.jpg)
Risk
Researchers are also interested: Using multi-agent system technology in risk bots, Johansson and Olsson, 2006.
Mixing search strategies for multi-player games, Zuckerman, Felner, and Kraus, 2009.
Both papers use non-standard variant where territories assigned randomly to begin the game.
![Page 5: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/5.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 6: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/6.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 7: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/7.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 8: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/8.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 9: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/9.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 10: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/10.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 11: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/11.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 12: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/12.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
![Page 13: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/13.jpg)
Drafting Territories in Risk
http://sillysoft.net/lux
Players take turns selecting territories until all 42 territories are owned.
Problem: How should we draft territories?
![Page 14: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/14.jpg)
Drafting Territories in Risk Does territory drafting even matter?
http://sillysoft.net/lux
![Page 15: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/15.jpg)
Drafting Territories in Risk Does territory drafting even matter?
http://sillysoft.net/lux
Still, does territory drafting really matter?
![Page 16: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/16.jpg)
Drafting Territories in Risk
What about the rest of the game after the draft?Lux Delux provides several Risk bots. We will use the “Quo” bot for all post-draft playand replace its drafting algorithm with our own.
Others have worked on how to play the rest of the game, but all ignore the drafting phase.
Territory drafting is all we care about here.
We are only going to play 3-player Risk.
![Page 17: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/17.jpg)
How to Draft Territories in Risk?
Rule-based:
All bots supplied with Lux Delux are rule-based drafters.
Go for Australia, no matter what!
http://sillysoft.net/lux
![Page 18: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/18.jpg)
How to Draft Territories in Risk?
Minimax search?
Really only applies to 2-player games...
Artificial Intelligence: A Modern Approach, Russell and Norvig, 2003.
![Page 19: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/19.jpg)
How to Draft Territories in Risk?
maxn search?
Large branching factor (42, then 41, then 40, etc.)
Would require good evaluation function of all draft states
A
B C
D E F G
P1
P2
P3
4,1,-2 3,5,0 -4,2,9 6,7,7 3,1,0 -5,1,3 0,0,-5 1,-1,2
3,5,0 -4,2,9 -5,1,3 1,-1,2
3,5,0 -5,1,3
3,5,0
a1
a2
b1
b2
c2
c1
d1
d2
e2
e1
f2
f1
g2
g1
An algorithmic solution of n-person games, Luckhart and Irani, 1986.
![Page 20: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/20.jpg)
How to Draft Territories in Risk?
UCT? (Upper Confidence Bounds applied to Trees)A
B C
D
P1
P2
P3
1,4,3
E
F
0,4,6
2,4,4 0,4,12
...
0,1,0 1,7,0
Update averages along path
Simulate actions randomly
Simulate action from state s to stateargmax s ' V i s ' c logn s
ns ' After many simulations, go to state argmaxs 'V i s '
Bandit based Monte-Carlo planning, Kocsis and Szepesvari, 2006.
![Page 21: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/21.jpg)
How to Draft Territories in Risk?
UCT? (Upper Confidence Bounds applied to Trees)A
B C
D
P1
P2
P3
1,4,3
E
F
0,4,6
2,4,4 0,4,12
...
0,1,0 1,7,0
Update averages along path
Simulate actions randomly
Simulate action from state s to stateargmax s ' V i s ' c logn s
n s ' After many simulations, go to state argmaxs 'V i s '
Better at handling large branching factor Typically requires no evaluation function
Bandit based Monte-Carlo planning, Kocsis and Szepesvari, 2006.
![Page 22: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/22.jpg)
Applying UCT to Risk Drafting
Typically with UCT, the more simulations that are run to completion, the more informative the decision.
Big Problem: Risk can be a very long gameGame may never end through random play,and so we may not even complete one simulation.
![Page 23: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/23.jpg)
Applying UCT to Risk Drafting Solution: Terminate simulations at draft end.
A
B C
D
P1
P2
P3 E
F
0,4,6
2,4,4 0,4,12
...
0,1,0 1,7,0
Update averages along path
1,4,3
All terminal states are “simple” easier to evaluate
Fixed simulation length
![Page 24: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/24.jpg)
Evaluating Draft Outcomes
S2 = (Aus-0, SA-2, Afr-6, NA-0, Eur-2, Asia-4, Pos-2, 13, 15)
Continent counts
http://sillysoft.net/lux
For any draft outcome, define feature set Si for
player i by just 4 types of features:
Turn order Friendly Neighbours
Enemy Neighbours
![Page 25: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/25.jpg)
Evaluating Draft Outcomes
For any draft outcome, define feature set Si for
player i by just 4 types of features:
The number of territories owned in each continentThe player's position in the turn orderThe number of distinct enemy neighboursThe number of friendly neighbours
![Page 26: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/26.jpg)
Evaluating Draft Outcomes
S1,S
2,S
3
S1,S
2,S
3
S1,S
2,S
3
Random Drafts(7,394)
![Page 27: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/27.jpg)
Evaluating Draft Outcomes
S1,S
2,S
3
S1,S
2,S
3
S1,S
2,S
3
Random Drafts(7,394)
Play Riskx100
Play Riskx100
Play Riskx100
Quo vs Quo vs Quo
(S1,47)
(S2,23)
(S3,30)
(S1,0)
(S2,0)
(S3,100)
(S1,92)
(S2,7)
(S3,1)
![Page 28: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/28.jpg)
Evaluating Draft Outcomes
S1,S
2,S
3
S1,S
2,S
3
S1,S
2,S
3
Random Drafts(7,394)
Play Riskx100
Play Riskx100
Play Riskx100
Quo vs Quo vs Quo Training Set
SupervisedMachineLearning
f (Si)
ϵ [0,100]
~
Adapted from Automated action set selection in Markov decision processes, Lee, 2004.
(S1,47)
(S2,23)
(S3,30)
(S1,0)
(S2,0)
(S3,100)
(S1,92)
(S2,7)
(S3,1)
![Page 29: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/29.jpg)
Evaluating Draft Outcomes
Used linear regression to obtain f
Final evaluation function:
Vi( ) = f +(S
1) + f +(S
2) + f +(S
3)
f +(Si )
where f +(Si ) = max{ 0, f (S
i ) }
![Page 30: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/30.jpg)
A
B C
D
P1
P2
P3 E
F
...
Update averages along path
Evaluating Draft Outcomes
Vi( ) = f +(S
1) + f +(S
2) + f +(S
3)
f +(Si )
V1( ), V
2( ), V
3( )
![Page 31: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/31.jpg)
Evaluating Draft Outcomes Weights of features from linear regression:
0 1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
Number of Territories
Wei
ght
Europe
North America
Asia
Africa
South America
Australia
![Page 32: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/32.jpg)
Evaluating Draft Outcomes Weights of features from linear regression:
Feature Weight
First to play 13.38
Second to play 5.35
Third to play 0.00
Enemy neighbours (multiplier) -0.07
Friendly neighbours (multiplier) 0.48
![Page 33: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/33.jpg)
Empirical Evaluation
The good guy:UCT-Quo: UCT + ML evaluation function Quo
The bad guys (most difficult bots in Lux Delux):Killbot: Directs attacks/defence at viable continentsQuo: Tries to slowly expand a cluster of territoriesEvilPixie: Similar to Killbot, different parametersBoscoe: Similar to Quo, plus targets runaway leaders
Some other guys:Greedy-Quo: 1-ply maxn + ML evaluation function QuoRandom-Quo: Drafts randomly Quo
![Page 34: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/34.jpg)
Empirical Evaluation
50 rounds played, 6 games per round (all 3! orderings)
UCT runs 3000 simulations with exploration constant c = 0.01 in less than 1 second on personal laptop
![Page 35: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/35.jpg)
Empirical Evaluation
Round robin tournament (all 10 3-player match-ups), 50 rounds per match-up, 6 games per round (all 3! orderings)
UCT runs 3000 simulations with exploration constant c = 0.01 in less than 1 second on personal laptop
![Page 36: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/36.jpg)
Empirical Evaluation
50 rounds played, 6 games per round (all 3! orderings)
UCT runs 3000 simulations with exploration constant c = 0.01 in less than 1 second on personal laptop
![Page 37: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/37.jpg)
Conclusions Simple machine-learned evaluation function can generalize fairly well
Combining UCT with a machine-learned evaluation function works well for drafting territories in Risk
Our UCT-Quo bot outperforms all of the strongestbots supplied with Lux Delux
Territory drafting is an important stage in Risk
Our approach could be appealing to commercial Risk AI programmers
Makes good decisions very quickly
![Page 38: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/38.jpg)
Future Work
Generalize the evaluation function to more players
Adapt to other types of games, perhaps those that involve drafting-type scenarios
In particular, apply to drafting in sports leaguesReal-life rookie / waiver / expansion draftsVideo gamesFantasy sports
![Page 39: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/39.jpg)
Real-Life Sports League Drafts
Teams take turns selecting players from a pool Create an automated draft assistant? Mock drafts against automated opponents?
Wikimedia Commons – Alexander Laney
![Page 40: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/40.jpg)
Drafting in Video Games
Create more intelligent computer opponents to draft against?
EA Sports “NHL 10”
![Page 41: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/41.jpg)
Fantasy Sports Drafts
Fantasy sports are a multi-billion dollar business Implement a drafting coach?
Yahoo! Sports Fantasy Hockey
![Page 42: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/42.jpg)
References
Johansson, S., and Olsson, F. 2006. Using multi-agent system technology in risk bots. In Laird, J., and Schaeffer, J., eds., AIIDE, 42-47. AAAI Press.
Kocsis, L., and Szepesvari, C. 2006. Bandit based Monte-Carlo planning. In 15th European Conference on Machine Learning, 282-293.
Lee, G. 2004. Automated action set selection in Markov decision processes. Master's thesis, University of Alberta.
Luckhart, C., and Irani, K. 1986. An algorithmic solution of n-person games. In AAAI-86, 158-162.
![Page 43: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/43.jpg)
References Russell, S., and Norvig, P. 2003. Artificial Intelligence: A Modern Approach. Upper Saddle River, New Jersey: Prentice Hall, second edition.
Sillysoft. Lux Delux – The best Risk game there is. http://sillysoft.net/lux/ . Accessed 28-Sept-2010.
Zuckerman, I.; Felner, A.; and Kraus, S. 2009. Mixing search strategies for multi-player games. In IJCAI, 646-651.
Acknowledgements: We would like to thank Vadim Bulitko for his helpful pointers throughout this project.Funding provided by NSERC and iCore, now part of AlbertaInnovates – Technology Futures.
![Page 44: Drafting Territories in the Board Game Riskrichardggibson.appspot.com/static/work/10aiide-risk/10aiide-risk... · Drafting Territories in Risk What about the rest of the game after](https://reader033.vdocuments.mx/reader033/viewer/2022060523/605348b74e4b0e14d4162df7/html5/thumbnails/44.jpg)
Thanks for Listening!
Richard Gibson PhD Student Department of Computing Science, University of Alberta Email: [email protected] Website: http://sites.google.com/site/richardggibson/
Go for North America!
http://sillysoft.net/lux