1. Sequential Expected Utility Theory: Sequential Sampling in Economic Decision Making under Risk Andrea Isoni Andrea Isoni (Warwick) Graham Loomes Graham Loomes (Warwick) Daniel Navarro-Martinez Daniel Navarro-Martinez (LSE) University of Warwick, April 2012

2. Introduction Modern economics is largely silent about decision making processes (e.g., EUT, PT) Psychologists have dedicated substantial efforts to study decision processes Psychological process models: Decision Field Theory, Decision by Sampling, Query Theory, Elimination by Aspects, Priority Heuristic Some of the models/evidence suggest the idea of a deliberation process Sequential sampling models (e.g., Decision Field Theory, Decision by Sampling) Explain decision times (e.g., decision time decreases significantly as choice probability approaches certainty) Virtually all economic decision models are silent about deliberation processes and decision times

3. Introduction In this paper: We take EUT and introduce in it the idea of sequential sampling (deliberation). We show what such a model can do. We investigate experimentally some aspects of it. Similarity to Decision Field Theory Presentation:  Explain the Sequential EUT model  Illustrate its implications (simulation)  Show some experimental evidence

4. The Model: Sequential EUT Binary choice Based on a random preference EUT specification:

5. People sample repeatedly from the choice options to accumulate evidence, until it is judged to be enough to make a choice Use certainty equivalent (CE) differences: After each sample, individuals conduct a sort of internal test of the null hypothesis that D(L1, L2) is zero If the hypothesis is not rejected, sampling goes on; if it’s rejected, sampling stops and the individual chooses the favoured option Introducing sequential sampling

6. After each sample k, an evidence statistic E k is computed: The null hypothesis of zero difference is rejected if: Essentially a sequential two-tailed t-test of the null hypothesis that the difference in value between the options is zero Sampling is psychologically costly, so CONF decreases with sampling: We assume C = 1 Only one additional free parameter (d) Introducing sequential sampling

7. The model can address 4 main behavioural constructs: 1)Choice probabilities:, probability that the null hypothesis is rejected with E k > 0 instead of E k < 0 2)Response times (RTs): Increasing function of the samples taken to reach the threshold (n) and the number of outcomes: 3)Confidence (CONF): The desired level of confidence in the last test 4)Strength of preference (SoP): Absolute value of the average CE difference sampled: The model constructs

8. Simulation (50,000 runs per choice) Three main aspects:  Comparing a risky lottery to an increasing sequence of sure amounts  Effects of changes in the three free parameters ( α, β, d )  Behaviour in specific lottery structures (dominance, deviations from EUT) Illustration of the model’s implications

9. Choosing between a fixed lottery and a series of monotonically increasing amounts of money Increasing sure amount Lot. 1Lot. 2αβdPr(1, 2)CONFSoPRTPr Core (20, 1)(40, 0.8; 0, 0.2)0.351.000.100.000.857.917.600.04 (22, 1)(40, 0.8; 0, 0.2)0.351.000.100.000.826.008.310.08 (24, 1)(40, 0.8; 0, 0.2)0.351.000.100.020.784.119.590.15 (26, 1)(40, 0.8; 0, 0.2)0.351.000.100.120.722.4311.500.26 (28, 1)(40, 0.8; 0, 0.2)0.351.000.100.520.661.6913.140.45 (30, 1)(40, 0.8; 0, 0.2)0.351.000.100.940.722.7211.500.73 (32, 1)(40, 0.8; 0, 0.2)0.351.000.101.000.814.468.760.97 (34, 1)(40, 0.8; 0, 0.2)0.351.000.101.000.846.377.671.00

10. Lot. 1 αβdPr(1, 2)CONFSoPRTPr Core (30, 1)(40, 0.8; 0, 0.2)0.051.000.100.040.751.6210.480.20 (30, 1)(40, 0.8; 0, 0.2)0.101.000.100.120.711.3111.670.28 (30, 1)(40, 0.8; 0, 0.2)0.151.000.100.280.681.1112.700.36 (30, 1)(40, 0.8; 0, 0.2)0.201.000.100.510.661.1313.130.45 (30, 1)(40, 0.8; 0, 0.2)0.251.000.100.710.671.4312.990.55 (30, 1)(40, 0.8; 0, 0.2)0.301.000.100.860.691.9712.390.64 (30, 1)(40, 0.8; 0, 0.2)0.351.000.100.940.722.7311.530.73 (30, 1)(40, 0.8; 0, 0.2)0.401.000.100.980.743.6610.700.80 Changing the location of the distribution of risk aversion coefficients ( α ) Changing the free parameters (1)

11. Lot. 1Lot. 2αβdPr(1, 2)CONFSoPRTPr Core (30, 1)(40, 0.8; 0, 0.2)0.300.250.101.000.810.998.630.94 (30, 1)(40, 0.8; 0, 0.2)0.300.400.100.980.761.1110.240.82 (30, 1)(40, 0.8; 0, 0.2)0.300.550.100.950.721.2711.270.75 (30, 1)(40, 0.8; 0, 0.2)0.300.700.100.910.711.4611.840.70 (30, 1)(40, 0.8; 0, 0.2)0.300.850.100.880.691.7012.170.66 (30, 1)(40, 0.8; 0, 0.2)0.301.000.100.860.691.9712.380.64 (30, 1)(40, 0.8; 0, 0.2)0.301.150.100.840.682.3212.510.62 (30, 1)(40, 0.8; 0, 0.2)0.301.300.100.830.682.6912.540.61 Changing the range of the distribution of risk aversion coefficients ( β ) Changing the free parameters (2)

12. Lot. 1Lot. 2αβdPr(1, 2)CONFSoPRTPr Core (30, 1)(40, 0.8; 0, 0.2)0.301.000.050.930.711.9120.410.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.100.860.691.9812.330.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.150.820.662.019.830.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.200.800.632.018.570.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.250.790.602.007.750.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.300.780.562.047.410.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.350.770.532.087.010.64 (30, 1)(40, 0.8; 0, 0.2)0.301.000.400.770.522.076.610.64 Changing the confidence level decrease rate ( d ) Changing the free parameters (3)

13. Dominance (α = 0.24, β = 1.00, d = 0.05) Specific lottery structures Lot. 1Lot. 2Pr(1, 2)CONFSoPRTPr Core (50, 0.5; 0, 0.5)(60, 0.5; 0, 0.5)0.000.923.9410.120.00 (50, 0.5; 0, 0.5)(51, 0.5; 0, 0.5)0.000.920.3910.140.00

14. Common ratio (α = 0.24, β = 1.00, d = 0.05) Specific lottery structures Lot. 1Lot. 2Pr(1, 2)CONFSoPRTPr Core (30, 1)(40, 0.8; 0, 0.2)0.750.671.2023.020.53 (30, 0.25; 0, 0.75)(40, 0.2; 0, 0.8)0.370.680.1529.950.53 Deviations from EUT (common ratio and common consequence effects) Kahneman and Tversky (1979) Common consequence (α = 0.26, β = 1.00, d = 0.05) Lot. 1Lot. 2Pr(1, 2)CONFSoPRTPr Core (24, 1) (25, 0.33; 24, 0.66; 0, 0.01) 0.66 0.0531.100.48 (24, 0.34; 0, 0.66)(25, 0.33; 0, 0.67)0.310.670.0330.400.48

15. Distribution of CE differences

16. Experimental evidence Focus on one experiment: 44 students, University of Warwick Focus on subset of choice structures:  Common ratio  Dominance 4 different tasks:  Binary choice (with response times)  Confidence  Strength of preference  Monetary strength of preference

17. The choices Common ratio: Dominance: ChoicesLottery ALottery B 1(30, 1)(40,.80; 0,.20) 2(30,.95; 0,.05)(40,.76; 0,.24) 3(30,.25; 0,.75)(40,.20; 0,.80) 4(30,.05; 0,.95)(40,.04; 0,.96) ChoicesLottery ALottery B 1(35,.35; 0,.65)(36,.35; 0,.65) 2(35,.35; 0,.65)(45,.35; 0,.65) 3(35,.35; 0,.65)(35,.36; 0,.64) 4(35,.35; 0,.65)(35,.45; 0,.55)

18. The tasks (1)

19. The tasks (2)

20. The tasks (3)

21. The tasks (4)

22. Results Common ratio: Dominance: ChoicesLottery ALottery B Prop. A. 1(30, 1)(40,.80; 0,.20) 0.84 2(30,.95; 0,.05)(40,.76; 0,.24) 0.84 3(30,.25; 0,.75)(40,.20; 0,.80) 0.43 4(30,.05; 0,.95)(40,.04; 0,.96) 0.16 ChoicesLottery ALottery B Prop. A 1(35,.35; 0,.65)(36,.35; 0,.65) 0.00 2(35,.35; 0,.65)(45,.35; 0,.65) 0.00 3(35,.35; 0,.65)(35,.36; 0,.64) 0.00 4(35,.35; 0,.65)(35,.45; 0,.55) 0.00

24. Parameters α : 0.27 β : 1.18 d: 0.05

25. Conclusions We have introduced sequential sampling (deliberation) in a standard economic decision model (Sequential EUT) Just one additional parameter Can explain important deviations from EUT, by simply assuming that people sample sequentially from EUT Makes predictions about additional behavioural measures related to deliberation (response times, confidence) Experimental evidence shows that these measures follow quite systematic patterns Sequential EUT can explain most of the patterns obtained Potential to be extended to other economic decision models, and other types of tasks (e.g., CE valuation, multi-alternative choice)