Cooperation in Team Games
Brian Macdonald1
1 Network
Chris Arney1
Elisha Peterson2
Science Center, and Department of Mathematical Sciences
United States Military Academy, West Point, NY
2 Johns
Hopkins University Applied Physics Lab, Laurel, MD
Sunbelt XXXII - 2012 Sunbelt Social Networks Conference
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Outline
1
Introduction to Subset Team Games
2
Definitions with Examples
3
A Hockey Example
4
Future Work
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Cooperation in Subset Team Games
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A game where players are trying to accomplish some
common goal.
I
Each player has some contribution toward the goal.
I
Contributions of a player or subset of players are classified
as competitive or altruistic (selfish/unselfish, greedy/ not
greedy).
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Cooperation in Pursuit and Evasion Games.
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Cooperation in Pursuit and Evasion Games.
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Goals
I
Try to understand cooperation within organizations and
teams
I
I
I
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What pairs of players (or groups of players) yield best
results?
All altruistic/unselfish players?
All competitive/selfish players?
Some combination?
I
Mathematically rigorous
I
Applicable to a wide variety of situations
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Some Definitions.
Notation and definitions [AP08]:
T A team of players
A, B A subset of players
Ac Complement of A, T \A. All players not in A.
u Value function. Assigns value to each outcome.
uA (B) Represents the value to A when B participates.
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Example of Value
Pursuit and Evasion Games.
Value: # of wildebeasts caught by A when B plays.
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Value, Marginal Contribution
uA (B) Represents the value to A when B participates.
Two special cases:
uT (T ) Value to the team when the whole team
participates
uAc (Ac ) Value to everyone but A when everyone but A
participates.
In other words
uT (T ) How the team does when everyone plays
uAc (Ac ) How the team does when A doesn’t play.
The marginal contribution of A is
m(A) = uT (T ) − uAc (Ac )
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Example of Value
Pursuit and Evasion Games.
Value - # of wildebeasts caught.
A - a single lion.
uT (T ) - measures # of WB caught by T when whole team plays.
uAc (Ac ) - measures # of WB caught by the rest of the team
when A doesn’t play.
m(A) - difference in how the team does with and without A
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The marginal contribution of A is
m(A) = uT (T ) − uAc (Ac )
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The marginal contribution of A is
m(A) = uT (T ) − uAc (Ac )
We decompose marginal contribution:
m(A) = c(A) + a(A)
c(A) Competitive Contribution - how much A contributes
a(A) Altruistic Contribution - everything else. Or, the
difference in how much A’s teammates contribute
when A plays versus when A does not play.
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Example of Value
Pursuit and Evasion Games.
Value: # of wildebeasts caught. Suppose A is a single lion.
c(A) measures # of WB caught by A.
a(A) measures how well do the other lions do when A is
playing vs not playing.
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Hockey Example
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A is a single player
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uT (T ) = goals per 60 minutes scored by the team
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uAc (Ac ) = goals per 60 minutes scored by A’s teammates
when A doesn’t play.
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m(A) = uT (T ) − uAc (Ac )
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Hockey Example
I
A is a single player
I
uT (T ) = goals per 60 minutes scored by the team
I
uAc (Ac ) = goals per 60 minutes scored by A’s teammates
when A doesn’t play.
I
m(A) = uT (T ) − uAc (Ac )
Decompose marginal contribution, m(A) = c(A) + a(A)
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c(A) = goals per 60 minutes scored by A
I
a(A) = everything else. Or, the difference in goals per 60
minutes scored by A’s teammates when A does and does
not play.
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Cooperation Space
0.1
0.2
0.3
0.4
Forwards
Defensemen
Average F
Average D
0.0
Competitive Contribution
0.5
c(A) vs a(A) for forwards and defensemen
−0.4
−0.2
0.0
0.2
0.4
Altruistic Contribution
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Correlations
c(A) and a(A) are not correlated.
0.2
0.0
0.1
Competitive
0.3
0.4
Competitive vs Altruistic
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
Altruistic
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Correlations
a(A) and Assists per 60 minutes are correlated.
0.0
−0.1
−0.2
−0.3
−0.5
−0.4
Altruistic Contribution
0.1
0.2
Altruistic Contribution vs Assists
0.5
1.0
1.5
2.0
Assists
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Some things to consider
I
c(A), a(A) depend on teammates that player A never plays
with.
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You can’t just remove A. Must replace A with someone
else.
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Correlations after applying remedies
After applying remedies:
Competitive vs Altruistic
1.5
S. Crosby
1.0
0.5
Competitive
D. Briere
0.0
M. Koivu
−0.2
0.0
0.2
0.4
0.6
0.8
Altruistic
Correlation: 0.10
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Future Work
Can we use c(A) and a(A) to predict future m(A)?
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Preliminary results: yes.
What kinds of pairs of players have high m(A1 ∩ A2 )?
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Is m(A1 ∩ A2 ) ≥ m(A1 ) + m(A2 )?
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Form networks based on A1 “likes playing with” A2 .
Predict which players will “like playing with” each other.
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Should player A1 and A2 play on the same line or on
different lines? [Wil12].
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Future Work
We’ve used only goals. They are other ways to contribute.
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Use shots.
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Estimate how different box score stats contribute to team
success. Get Expected Goals. Use those instead of goals.
[Mac12]
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References I
David C. Arney and Elisha Peterson.
Cooperation in Social Networks: Communication, Trust, and
Selflessness, 2008.
http://elishapeterson.wikidot.com/papers:
cooperation-in-social-networks:
communication-trust-an.
Brian Macdonald.
An Expected Goals Model for Evaluating NHL Teams and Players.
Proceedings of the 2012 MIT Sloan Sports Analytics Conference, 2012.
http://www.sloansportsconference.com/?p=6157, Accessed
2-20-2012.
Murphy Pfomann and Jessica Sexauer.
Modeling Cooperation Through the Use of Pursuit and Evasion Teams,
2012.
In progress. Senior reserach project.
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References II
Peter Williams.
Effects of Chemistry and Cooperation on Team Success, 2012.
In progress. Senior reserach project.
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Questions?
?
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