# Decision making in basketball 2-point shot: easier, fewer Decision making in basketball 2-point shot: easier, fewer points 3-point shot: more difficult, more points Kobe Bryant LA Lakers 31.6 PPG (2006-7) 1 Chris Bosh Toronto Raptors 26.3 PPG (2006-7) 3P attempts:

398 2P attempts: 1,359 (77%) 1,059 (97%) 3P success: 34% 34%

2P success : 50% 50% (23%) 35 (3%) The matching law NBA best 100 players (2006-2007) N3

N 2 N3 1 0.8 0.6 0.4 Bryant 0.2 0

0 2 N2,3 = # of 2,3 points shots Bosh 0.2 0.4 0.6 0.8 I3

I 2 I3 1 I2,3 = # 2,3 points earned The reward schedule R t r A t , A t 1 , A t 2 , 3 The matching law 1

N1 N1 N 2 1 Herrnstein, JEAB, 1961 4 I1 I1 I 2 The matching law Sugrue, Corrado & Newsome, Science, 2004

5 The matching law Gallistel et al., unpublished 6 The matching law Nj = # of attempts at alternative j investment in j Ij = # of points earned from alternative j income from j N1 I1 N1 N 2 I1 I 2

I1 I2 N1 N 2 equal returns E R A 1 E R A 2 7 The matching law is very general. It is found in many animal types as well as humans, under very different experimental conditions. MATCHING MAXIMIZING

8 Example: addiction model E[R|A=drugs] 0 9 0.2 0.4 0.6

freq [drugs] 1freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991 0.8 1 Example: addiction model E[R|A=drugs] E[R|A=work] matching

0 10 0.2 0.4 0.6 freq [drugs] 1freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991

0.8 1 Example: addiction model E[R|A=drugs] E[R|A=work] E[R] maximizing 0 11 0.2

matching 0.4 0.6 freq [drugs] 1freq [work] after Herrnstein and Prelec, J Econ Perspect, 1991 0.8 1

Question: What is the neural basis of the matching law? 12 It is generally believed that learning is due to changes in the efficacy of synapses 0.4 m 13 Kennedy, Science, 2000 Question:

What is the neural basis of the matching law? Question: What microscopic plasticity rules underlie adaptation to matching behavior? 14 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 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 Covariance is a measure of dependence two random variables X, Y X X E[ X ] Y Y E[Y ] covariance:

Cov X , Y E[ X Y ] E[ X Y ] E[ X Y ] correlation coefficient: r 17 Cov[ X , Y ] Var[ X ]Var[Y ] Covariance Cov[X,Y ] 0 18

Cov[X,Y ] 0 Cov[X,Y ] 0 Hypothesis: the matching law results from synaptic plasticity that is driven by the covariance of reward and neural activity

19 Synaptic plasticity Local signals affect synaptic efficacies. Popular theory: Hebbain plasticity W S pre S post Global signals affect synaptic efficacies. Popular theory: dopamine gates Hebbian plasticity (Wickens) W DS pre S post 20

21 Schultz, Dayan & Montague, Science, 1997 Synaptic plasticity Local signals affect synaptic efficacies. Popular theory: Hebbain plasticity W S pre S post Global signals affect synaptic efficacies. Popular theory: dopamine gates Hebbian plasticity (Wickens) W DS pre S post

Popular theory: dopamine codes the mismatch between reward and expected reward (Schultz) 22 D R E R R Synaptic plasticity W DS pre S post D R W R S pre S post Average trajectory approximation E W E[ R S pre S post ] Cov[ R, S pre S post ]

23 Covariance-based plasticity rules W R E R N W R N E N N=Spre , N=Spost , N=SpreSpost Average trajectory approximation: E W Cov[ R, N ] 24 Hypothesis: covariance-based synaptic plasticity

The matching law outline: Stationary state of covariance-based plasticity Cov R, N 0 25 The matching law Assumptions neurons

N1 N2 N3 N5 action reward A R

N4 hidden variables 1. E[N|A=i] E[N|Ai] 2. The dependence of the reward R on neural activity N is through the action A. 26 Theorem Suppose that Assumptions 1 and 2 are satisfied j Cov R, N 0 E R | A 1 E R | A 2

The matching law 27 Intuition Cov R, N 0 E R | A i E R neuron action reward

N A R 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 28 W R S E S

29 W R S E S W RS 30 W R S E S W RS W ij R E R S i pre M j post 31 Summary

Hypothesis: Covariance based synaptic plasticity underlies the matching law Theorem: Cov R, N 0 The matching law Loewenstein & Seung, PNAS, 2006 Loewenstein, PLoS Comp Biol, 2008 Disclaimer: There are learning rules that converge to 32

Cov[R,N]=0 that are not driven by covariance