April 1, 2004 Helen Belogolova Amy Daitch

1. M☺deling of User Behavior In Matching Task Based on Previous Reward History and Personal Risk Factor April 1, 2004 Helen Belogolova Amy Daitch

3. Project Summary • Experiment: • Subjects given matching task in which they choose between button A and B • Received reward based on predetermined reward functions • Our Model: • Subject’s memory decay: leaky integration • Personal Risk Factor • Cumulative Risk Factor

4. Method for Modeling the Behavior • General Method • Part I. Exploratory Phase • P(A) = 0.5, P(B) = 0.5 • First 10 trials have an equal probability of choosing A or B • Part II. Choices Based on Past Reward History • Reward function took into account 40 trial buffer updated after each trial • Vector of rewards weighted based on leaky integrator model with decay parameter d: weighted_rewards_vector = [exp(1*d) exp(2*d) … exp(240*d) ]’ * reward_vector • Most recent reward carries most influence on subject’s next move

5. Method for Modeling the Behavior • To choose between A and B we sum up the weighted rewards after A button presses (rewA) and B button presses (rewB) P(A) = rewA/(rewA+rewB) P(B) = 1-P(A) • Based on these total rewards the next choice is generated like this: if rand(1) < p(A)  choice A else  choice B

6. Method for Modeling the Behavior • Model Accounting for Risk • Risk = subject’s willingness to deviate from optimal choice based on past trials • Personality Risk • Constant in experiment, Range from 0 to 1 • Function of personality = willingness to take risks in general • Cumulative Risk, Range from 0 to 1 • Increases as the Cumulative Reward increases cumulative_risk(trial) = cumulative_reward(trial)/max_cumulative_reward • Maximum Cumulative Reward in our case was 6

7. Method for Modeling the Behavior • Weights of Personal Risk Factor and Cumulative Reward Risk Factor make up Total Risk Factor: total_risk = personal_risk*personal_risk_weight + cumulative_risk*cumulative_risk_weight • With the total risk parameter as above, the decisions are made like this and the choice of A or B is generated the same way as in the general model: p(A) = rewA/(rewA + rewB) – (rewA/(rewA + rewB) – 0.5)*2*total_risk p(B) = 1 – p(A)

8. Results • Ran experiment on model, varying one parameter at a time • Since stochastic decisions, ran experiment several times for each set of parameters to diminish the effects of randomness • A subject could produce somewhat different results if experiment done more than once = we ran the experiment on the model many times to see how a subject with certain characteristics would behave.

9. Results • We then plotted the ratio of the subject’s button press within the buffer vs. the trial number and observed that: • Varying only personal risk factor = most successful when risk factor very high or very low (same in this experiment) • Below: personal risk, cumulative risk = 0

10. Personal risk = 0.5, Cumulative Risk = 0

11. Personal Risk = 1, Cumulative Risk = 0

12. Results • Varying only cumulative reward risk factor = more successful as cumulative reward risk increases • Below(cumulative risk = 0.25, personal risk = 0)

13. Cumulative risk = 1, Personal risk = 0

14. Results • Decay rates 0.5, 1.0, and 2.0 while keeping risk factor zero • At decay rate of 2.0 succeeded the most • At the decay rate of 0.5 had the least success. • This suggests that the most important rewards to remember are the ones in the immediate past

15. Decay rate = 0.5

16. Decay Rate = 2.0

17. Comparison of Results With Real Data • Compared choices of our model with those of the tested subjects • Cross correlated the choice vector of the subject (real data) with the choice vectors we generated by our model for all of the variations

18. Comparison of Results With Real Data • Observed strong correlation between our subjects and the models with very high personal risk factors and very low personal risk factors (below: p-risk, c-risk = 0)

21. Comparison of Results With Real Data • For the cumulative reward risk parameter we found that as it increased, with personal risk constant at zero, the correlation improved (below: cumulative risk = 0.25)

22. Cumulative risk = 1, Personal risk = 0

23. Comparison of Results With Real Data -Changing the decay rate in the model didn’t appear to affect correlation between model and subject generated data (decay rate = 1)

24. Decay rate = 2