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Decision making in basketball

This analysis explores the performance of Kobe Bryant and Chris Bosh during the 2006-07 NBA season, focusing on their scoring averages, shot attempts, and success rates. Kobe averaged 31.6 PPG for the Los Angeles Lakers, showcasing his scoring prowess, while Chris Bosh averaged 26.3 PPG for the Toronto Raptors. The study utilizes the matching law to examine decision-making in basketball, emphasizing the influence of shot types (2-point vs. 3-point) on point maximization. We delve into the neural basis of the matching law and synaptic plasticity's role in reward-based learning.

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Decision making in basketball

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  1. Kobe Bryant LA Lakers 31.6 PPG (2006-7) Chris Bosh Toronto Raptors 26.3 PPG (2006-7) 3P attempts: 398 (23%) 2P attempts: 1,359 (77%) 35 (3%) 1,059 (97%) 3P success:34% 2P success : 50% 34% 50% Decision making in basketball • 2-point shot: easier, fewer points • 3-point shot: more difficult, more points

  2. The matching law NBA best 100 players (2006-2007) Bryant N2,3 = # of 2,3 points shots I2,3= # 2,3 points earned Bosh

  3. The reward schedule

  4. The matching law 1 1 Herrnstein, JEAB, 1961

  5. The matching law Sugrue, Corrado & Newsome, Science,2004

  6. The matching law Gallistel et al., unpublished

  7. The matching law Nj = # of attempts at alternative j investment inj Ij = # of points earned from alternative j  income fromj equal returns

  8. The matching law is very general. It is found in many animal types as well as humans, under very different experimental conditions.

  9. Example: addiction model E[R|A=drugs] freq [drugs] 1–freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991

  10. matching Example: addiction model E[R|A=drugs] E[R|A=work] freq [drugs] 1–freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991

  11. Example: addiction model E[R|A=drugs] E[R|A=work] E[R] maximizing matching freq [drugs] 1–freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991

  12. Question: What is the neural basis of the matching law?

  13. 0.4 μm It is generally believed that learning is due to changes in the efficacy of synapses Kennedy, Science, 2000

  14. Question: What is the neural basis of the matching law? Question: What microscopic plasticity rules underlie adaptation to matching behavior?

  15. Question: What is the neural basis of the matching law? Hypothesis: the matching law results from synaptic plasticity that is driven by the covariance of reward and neural activity

  16. Question: What is the neural basis of the matching law? Hypothesis: the matching law results from synaptic plasticity that is driven by the covariance of reward and neural activity

  17. Covariance is a measure of dependence • two random variables X, Y • covariance: • correlation coefficient:

  18. Covariance

  19. Hypothesis:the matching law results from synaptic plasticity that is driven by the covarianceof reward and neural activity

  20. Synaptic plasticity • Local signals affect synaptic efficacies. Popular theory: Hebbain plasticity • Global signals affect synaptic efficacies. Popular theory: dopamine gates Hebbian plasticity (Wickens)

  21. Schultz, Dayan & Montague, Science, 1997

  22. Synaptic plasticity • Local signals affect synaptic efficacies. Popular theory: Hebbain plasticity • Global signals affect synaptic efficacies. Popular theory: dopamine gates Hebbian plasticity (Wickens) • Popular theory: dopamine codes the mismatch between reward and expected reward (Schultz)

  23. Average trajectory approximation Synaptic plasticity

  24. Covariance-based plasticity rules N=Spre , N=Spost , N=SpreSpost Average trajectory approximation:

  25. Hypothesis: covariance-based synaptic plasticity The matching law outline: Stationary state of covariance-based plasticity The matching law

  26. Assumptions neurons N1 1. E[N|A=i] ≠E[N|A≠i] 2. The dependence of the reward R on neural activity N is through the action A. action reward N2 A R N3 N5 N4 hidden variables

  27. Theorem Suppose that Assumptions 1 and 2 are satisfied The matching law

  28. neuron action reward N A R Intuition • In general R depends on A • If, as a result of the policy used by the subject, R becomes independent of A then R also becomes independent of N

  29. Summary Hypothesis: Covariance based synaptic plasticity underlies the matching law Theorem: The matching law Loewenstein & Seung, PNAS, 2006 Loewenstein, PLoS Comp Biol, 2008 Disclaimer: There are learning rules that converge to Cov[R,N]=0 that are not driven by covariance

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