Effect of Multiple Unskilled Competitors on Skilled Player Success
Investigating the impact of unskilled competitors on a skilled player's performance in a spelling bee-like game. The study explores the role of ability versus randomness using matrix formulation and Monte Carlo simulations.
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Presentation Transcript
The Effect of Multiple Unskilled Competitors on the Success of a Skilled Player in a Model Spelling Bee-Like Game Robert Monahan Dr. Jeff Groff
Outline I) Purpose II) The Model A. 2 - Players B. 3 - Players III) The Method A. Matrix Formulation B. Monte Carlo Simulation C. Direct vs Simulation IV) Results V) Questions?
Purpose How much does ability matter? Will the most qualified succeed? What role does randomness play?
The Model 2 Players (?1 ,?2)with a skill level of ( 1 , 2) Where ? is a value between 0 and 1 (?1 ,?2) Binary Representation 1 = active player 0 = inactive player 1 2+ (1 1)(1 2) (1 ,1) 1(1 2) 1 1 2 (1 ,0) (0 ,1)
The Model 3 Players (?1 ,?2 ,?3)with a skill level of ( 1 , 2 , 3) Where ? is a value between 0 and 1 (?1 ,?2 ,?3) Binary Representation 1 = active player 0 = inactive player (1,1,1) (1,1,0) (0,1,1) (1,0,1) (0,1,0) (1,0,0) (0,0,1)
Matrix Formulation 0 0 1(1 2) 0 (1 1) 2 0 ? = For N players, the Markov Chain takes the form of a matrix with 2? 1 rows and 2? 1 columns. Example: For 12 players, the size of the Markov Chain Matrix is 4095 rows by 4095 columns called State Space Explosion!
Monte Carlo Simulation Simulates a game at a time and tracks the results Required to ensure that the Direct Matrix Calculation is accurate Faster than Direct Calculation for games with lots of players - still only an approximation
Monte Carlo vs Direct Calculation = 1 2 3 4 = [0.4 0.5 0.6 0.7]
Results A Diamond in the Rough Two types of players: Ace (high skill) Joker (low skill) Consists of 1 Ace and many Jokers ?1,?2,?3,?4, ?1,?1,?2,?3, How skilled must the Ace be to win at least 50% of the competitions?
Results A Diamond in the Rough Winning 50% of the Competitions 0.5 0.45 0.4 Skill Required of Ace 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 12 Number of Jokers
Results A Diamond in the Rough Diamond in the Rough Summary Diamond in the Rough Summary 1 1 0.5 0.9 0.9 y = 0.1529ln(x) + 0.5402 R = 0.9642 0.8 0.8 Skill Required y = 0.1944ln(x) + 0.3224 R = 0.9913 0.6 0.7 0.7 Skill Required y = 0.1935ln(x) + 0.2062 R = 0.9963 0.6 0.6 y = 0.1787ln(x) + 0.132 R = 0.9971 0.7 0.5 0.5 y = 0.1588ln(x) + 0.0794 R = 0.9928 0.4 0.4 0.3 0.8 0.3 0.2 0.5 0.6 0.7 0.2 0.1 0.9 0 0.1 0 2 4 6 8 10 12 0 Number of Jokers 0.8 0.9 0 2 4 Number of Jokers 6 8 10 12
Results A Diamond in the Rough How many Jokers does it take to overwhelm the Ace? 1) 21 Jokers 2) 33 Jokers 3) 61 Jokers 4) 127 Jokers 5) 330 Jokers
Conclusion How much does ability matter? Will the most qualified succeed? What role does randomness play?
