Strategy optimization in beachvolleyball applying a two scale - - PowerPoint PPT Presentation
Strategy optimization in beachvolleyball applying a two scale approach to the olympic games Susanne Hoffmeister Jrg Rambau Chair of Buisness Mathematics University of Bayreuth MathSport, Padua 2017 Hoffmeister (Bayreuth) Sport Strategy
Susanne Hoffmeister Jörg Rambau
Chair of Buisness Mathematics University of Bayreuth
MathSport, Padua 2017
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 1 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
◮ should capture the strategic question Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
◮ should capture the strategic question ◮ can be evaluated for future matches Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
◮ should capture the strategic question ◮ can be evaluated for future matches
give recommendations
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
◮ should capture the strategic question ◮ can be evaluated for future matches
give recommendations
◮ for past and future matches Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Sport Strategy Optimization
Introduction
identify a sport strategic decision of your team
◮ may be opponent dependend ◮ may change from match to match
model sport game as Markov Decision Process (MDP)
◮ should capture the strategic question ◮ can be evaluated for future matches
give recommendations
◮ for past and future matches ◮ for player performances varying with the day Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 2 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution model data detailled level g-MDP
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution model data detailled level g-MDP
skills
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution model data detailled level g-MDP
skills simulation
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution model data detailled level g-MDP
skills simulation validation
Sport Strategy Optimization MathSport 2017 3 / 17
Two Scale Approach
Sketch of General Procedure
strategic Question strategic recommendation model data rough level s-MDP analytical opt solution mathematical analysis
? numerical opt solution model data detailled level g-MDP
skills simulation validation
numerical opt solution
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 3 / 17
Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Strategic Question – Risky or Safe?
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Strategic Question – Risky or Safe?
Risky or save serve? (technique and target field)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Strategic Question – Risky or Safe?
Risky or save serve? (technique and target field) Risky or field attack? (technique and target field)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Strategic Question – Risky or Safe?
Risky or save serve? (technique and target field) Risky or field attack? (technique and target field) Blocking player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application
Beach Volleyball
Olympia 2012: Brink/Reckermann
Result: 2:1 (23:32, 16:21, 16:14)
Strategic Question – Risky or Safe?
Risky or save serve? (technique and target field) Risky or field attack? (technique and target field) Blocking player Serving on which opponent player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 4 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application – Beach Volleyball
s-MDP and g-MDP
actions are complete attacking sequences small number of states
pserve
a
ˆ pserve
a
pserve
a
s-MDP transitions
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 5 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Example Application – Beach Volleyball
s-MDP and g-MDP
actions are complete attacking sequences small number of states
pserve
a
ˆ pserve
a
pserve
a
s-MDP transitions positional information hitting techniques and moves billions of states
p
s u c c , ρ
(pos(ρ), S
∗
) pdev,ρ (pos(ρ), S∗) p
f a u l t , ρ
(pos(ρ), S
∗
) 1 1 − ω(target) ω(target) p
s u c c , σ
(pos(σ), r
∗
) pdev,σ (pos(σ), r∗) p
f a u l t , σ
(pos(σ), r
∗
)
g-MDP transitions
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 5 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Data Collection
Beach Volleyball Tracker
Sources: Videos of Olympic Games (2012)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 6 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Data Collection
Beach Volleyball Tracker
Sources: Videos of Olympic Games (2012) Software: Beach Volleyball Tracker
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 6 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
E W S N W E S N Q05 Q04 Q03 Q02 Q01 Q00 Q15 Q14 Q13 Q12 Q11 Q10 Q25 Q24 Q23 Q22 Q21 Q20 Q35 Q34 Q33 Q32 Q31 Q30 P00 P01 P02 P03 P04 P05 P10 P11 P12 P13 P14 P15 P20 P21 P22 P23 P24 P25 P30 P31 P32 P33 P34 P35 1m 3m 3m 1m 3.5m 4m 0.5m strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) number of observations Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) successful hit Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) execution fault or net hit Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – )
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Olympic games 2012 – Skills of Brink
target fields Q11-Q14 Q21-Q24 Q31-Q34 performance # ; succ fault # succ fault # succ fault Serve SF P01 - P04 33 0.91 (0.91) 0.00 (0.00) 43 0.88 (0.88) 0.12 (0.12)
32 0.94 (0.94) 0.00 (0.00) 18 0.78 (0.78) 0.17 (0.17)
FSM
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
0.89 ( – ) 0.03 ( – ) 0.89 ( – ) 0.03 ( – )
57 0.89 (0.89) 0.04 (0.04) 17 0.94 (0.94) 0.00 (0.00)
5 0.76 (0.60) 0.02 (0.00) 2 0.91 (1.00) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 0.77 ( – ) 0.06 ( – )
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
8 0.67 (0.63) 0.11 (0.13) 7 0.83 (0.86) 0.02 (0.00)
0.77 ( – ) 0.06 ( – ) 1 0.79 (1.00) 0.06 (0.00)
0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P11-P14 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) 0.96 ( – ) 0.04 ( – ) P21-P24 8 0.99 (1.00) 0.01 (0.00) 31 0.94 (0.94) 0.06 (0.06) 0.96 ( – ) 0.04 ( – ) P31-P34 2 0.96 (1.00) 0.04 (0.00) 2 0.96 (1.00) 0.04 (0.00) 0.96 ( – ) 0.04 ( – ) aggregated frequency Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Player Skills
All matches except final
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 7 / 17
strategic Question strategic recommendation s-MDP mathematical analysis g-MDP Strategy
Definition
Basic strategy guarantees play flow
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Risky or Safe?
Risky or save serve? Risky or field attack? Blocking player Serving on which opponent player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Risky or Safe?
Risky or save serve? ← πserve
h,tech, πserve h,target
Risky or field attack? Blocking player Serving on which opponent player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Risky or Safe?
Risky or save serve? ← πserve
h,tech, πserve h,target
Risky or field attack? ← πfield
h,tech, πfield h,target
Blocking player Serving on which opponent player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Risky or Safe?
Risky or save serve? ← πserve
h,tech, πserve h,target
Risky or field attack? ← πfield
h,tech, πfield h,target
Blocking player ← πb Serving on which opponent player
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
g-MDP Strategy
Definition
Basic strategy guarantees play flow specify parameter π ∈ [0, 1] for each decision
Risky or Safe?
Risky or save serve? ← πserve
h,tech, πserve h,target
Risky or field attack? ← πfield
h,tech, πfield h,target
Blocking player ← πb Serving on which opponent player ← πs
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 8 / 17
Estimated Strategy
Estimation from final matches
final GER BRA Brink Reckermann Alison Emanuel πserve
h,tech
19% 24% 68% 39% πserve
h,target
8% 17% 16% 23% πfield
h,tech
77% 69% 93% 82% πfield
h,target
27% 33% 41% 52% πb 2% 96% πs 23% 33% Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 9 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Strategy
Estimation from final matches
final GER BRA Brink Reckermann Alison Emanuel πserve
h,tech
19% 24% 68% 39% πserve
h,target
8% 17% 16% 23% πfield
h,tech
77% 69% 93% 82% πfield
h,target
27% 33% 41% 52% πb 2% 96% πs 23% 33% pre final GER BRA Brink Reckermann Alison Emanuel πserve
h,tech
40% 50% 36% 35% πserve
h,target
18% 36% 8% 18% πfield
h,tech
65% 71% 82% 86% πfield
h,target
36% 43% 36% 32% πb 2% 96% πs 50% 50% Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 9 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Estimated Strategy
Estimation from final matches
final GER BRA Brink Reckermann Alison Emanuel πserve
h,tech
19% 24% 68% 39% πserve
h,target
8% 17% 16% 23% πfield
h,tech
77% 69% 93% 82% πfield
h,target
27% 33% 41% 52% πb 2% 96% πs 23% 33% pre final GER BRA Brink Reckermann Alison Emanuel πserve
h,tech
40% 50% 36% 35% πserve
h,target
18% 36% 8% 18% πfield
h,tech
65% 71% 82% 86% πfield
h,target
36% 43% 36% 32% πb 2% 96% πs 50% 50% Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 9 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Simulation
calculate transition probabilities
pa, ¯ pa, ˆ pa
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 10 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 11 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% simulating the g-MDP with
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4% skills of final only 2% 2% 6% 10% 2%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4% skills of final only 2% 2% 6% 10% 2%
Bounds on estimated probabilities in terms of g-MDP input skills
pserve
estimatedStrat ∈ [0, 0.2291]
pserve
estimatedStrat ∈ [0, 0.1936]
ˆ pserve
estimatedStrat ∈ [0.5913, 1]
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4% skills of final only 2% 2% 6% 10% 2%
Bounds on estimated probabilities in terms of g-MDP input skills
pserve
estimatedStrat ∈ [0, 0.2291]
pserve
estimatedStrat ∈ [0, 0.1936]
ˆ pserve
estimatedStrat ∈ [0.5913, 1]
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4% skills of final only 2% 2% 6% 10% 2%
Bounds on estimated probabilities in terms of g-MDP input skills
pserve
estimatedStrat ∈ [0, 0.2291]
pserve
estimatedStrat ∈ [0, 0.1936]
ˆ pserve
estimatedStrat ∈ [0.5913, 1]
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – serving situation
pserve pserve qserve qserve
2% 4% 4% 14% 0% internet statistics
9% 15% 3% 12% 5% simulating the g-MDP with skills of all matches except final 8% 14% 6% 15% 5% skills of all matches 7% 11% 6% 14% 4% skills of final only 2% 2% 6% 10% 2%
Bounds on estimated probabilities in terms of g-MDP input skills
pserve
estimatedStrat ∈ [0, 0.2291]
pserve
estimatedStrat ∈ [0, 0.1936]
ˆ pserve
estimatedStrat ∈ [0.5913, 1]
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 12 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% simulating the g-MDP with
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% internet statistics 55%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% internet statistics 55%
51% 18% 50% 20% 3% simulating the g-MDP with
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% internet statistics 55%
51% 18% 50% 20% 3% simulating the g-MDP with skills of all matches except final 51% 21% 47% 21% 3%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% internet statistics 55%
51% 18% 50% 20% 3% simulating the g-MDP with skills of all matches except final 51% 21% 47% 21% 3% skills of all matches 51% 20% 49% 25% 3%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Validate simulation
The truth – field situation
pfield pfield qfield qfield
47% 24% 52% 21% 0% internet statistics 55%
51% 18% 50% 20% 3% simulating the g-MDP with skills of all matches except final 51% 21% 47% 21% 3% skills of all matches 51% 20% 49% 25% 3% skills of final only 50% 15% 48% 20% 4%
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 13 / 17
strategic Question strategic recommendation s-MDP mathematical analysis s-MDP
mathematical analysis
s-MDP can be solved analytically
Maximum total expected reward is monotone increasing in x and monotone decreasing in y.
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 14 / 17
strategic Question strategic recommendation s-MDP mathematical analysis s-MDP
mathematical analysis
s-MDP can be solved analytically
Maximum total expected reward is monotone increasing in x and monotone decreasing in y. Play the attacking plan that maximizes d(s) = arg max
a∈As
pfield
a
+ ˆ pfield
a
qfield 1 − ˆ pfield
a
ˆ qfield
Sport Strategy Optimization MathSport 2017 14 / 17
strategic Question strategic recommendation s-MDP mathematical analysis s-MDP
mathematical analysis
s-MDP can be solved analytically
Maximum total expected reward is monotone increasing in x and monotone decreasing in y. Play the attacking plan that maximizes d(s) = arg max
a∈As
pfield
a
+ ˆ pfield
a
qfield 1 − ˆ pfield
a
ˆ qfield
s-MDP
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 14 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Computational Results
prior to final match
Based on input parameters (skills) estimated from pre-final matches Strategy Germany serve risky risky safe safe pre-final estimation field risky safe risky safe pre-final estimation Strategy Brazil risky-risky 89% 94% 95% 98% 94% risky-safe 66% 76% 67% 80% 70% safe-risky 52% 80% 68% 90% 75% safe-safe 23% 45% 24% 48% 33% pre-final estimation 58% 78% 66% 86% 73%
recommend safe − safe strategy for Germany
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 15 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Computational Results
prior to final match
Based on input parameters (skills) estimated from pre-final matches Strategy Germany serve risky risky safe safe pre-final estimation field risky safe risky safe pre-final estimation Strategy Brazil risky-risky 89% 94% 95% 98% 94% risky-safe 66% 76% 67% 80% 70% safe-risky 52% 80% 68% 90% 75% safe-safe 23% 45% 24% 48% 33% pre-final estimation 58% 78% 66% 86% 73%
recommend safe − safe strategy for Germany recommend safe − safe strategy for Brazil
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 15 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Computational Results
prior to final match
Based on input parameters (skills) estimated from pre-final matches Strategy Germany serve risky risky safe safe pre-final estimation field risky safe risky safe pre-final estimation Strategy Brazil risky-risky 89% 94% 95% 98% 94% risky-safe 66% 76% 67% 80% 70% safe-risky 52% 80% 68% 90% 75% safe-safe 23% 45% 24% 48% 33% pre-final estimation 58% 78% 66% 86% 73%
recommend safe − safe strategy for Germany recommend safe − safe strategy for Brazil Nash equilibrium for both teams playing safe − safe
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 15 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Computational Results
after final match
Based on input parameters (skills) estimated from final match Strategy Germany serve risky risky safe safe actual strategy of final field risky safe risky safe actual strategy of final Strategy Brazil risky-risky 68% 57% 75% 65% 67% risky-safe 40% 28% 40% 29% 33% safe-risky 68% 68% 75% 75% 74% safe-safe 39% 38% 38% 38% 39% actual strategy of final 63% 62% 65% 63% 65%
actual mixed strategy of Germany achieves the highest possible winning probability
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 16 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Computational Results
after final match
Based on input parameters (skills) estimated from final match Strategy Germany serve risky risky safe safe actual strategy of final field risky safe risky safe actual strategy of final Strategy Brazil risky-risky 68% 57% 75% 65% 67% risky-safe 40% 28% 40% 29% 33% safe-risky 68% 68% 75% 75% 74% safe-safe 39% 38% 38% 38% 39% actual strategy of final 63% 62% 65% 63% 65%
actual mixed strategy of Germany achieves the highest possible winning probability would have recommended risky − safe strategy for Brazil
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 16 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
sensitivity analysis day performance for coaches
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
colour shows best strategy
1 pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
skill of player FSM=Smash probability of fault
5 1 pfault,ρ (pos(ρ), FSM)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
skill of player FSM=Smash probability of success
0.5 1
real skill Tea
psucc,ρ (pos(ρ), FSM)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
play safe!
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
play risky!
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
best strategy depends on skills
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM psucc,ρ (pos(ρ), FSM)
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
direct point probability
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
fault probability
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
perfect opponent doesn’t matter what we play!
Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis Bonus
Skill Strategy Score Cards
Example: SSSC for Smash (FSM)
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
0.5 1 0.5 1
real skill Team P
pfault,ρ (pos(ρ), FSM) psucc,ρ (pos(ρ), FSM) −0.4 −0.2 0.2 0.4 winProb(safe)-winProb(risky) Color key
qfield qfield 0.0 0.0
average opponent best strategy depends
Skill Strategy Score Cards – Beginning Hoffmeister (Bayreuth) Sport Strategy Optimization MathSport 2017 17 / 17
strategic Question strategic recommendation s-MDP mathematical analysis