a laboratory experiment on the heuristic switching model

A Laboratory Experiment on the Heuristic Switching Model

A Laboratory Experiment on theHeuristic Switching Model

Mikhail Anufrieva Aleksei Chernulicha Jan Tuinstrab

a University of Technology Sydneyb University of Amsterdam

Symposium for Experimental EconomicsDongbei University of Finance and Economics (DUFE)

29 October 2017

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SummaryExperiment focusing on one (of the two) keymechanisms of Heuristic Switching Models and ...

... testing implications of Brock-Hommes (ECMA 1997,JEDC 1998) model

Consistently with the model: high information cost ofrational rule cause instability

Evidence of endogenous change in switching

... consistent with Intensity of Choice parameter reactingon predictability of past returns ...

... leading to “moderately complex” dynamics

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Outline

1 Introduction

2 Experiment

3 Dynamics of the Stylized HSM and Hypotheses

4 Results of the Experiment

5 High (Large and Long) Treatment

6 Conclusion

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Introduction

Plan

1 Introduction

2 Experiment




6 Conclusion

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Introduction

Expectations in Economic Theoryeconomy is an expectation feedback system

expectations a�ect our decisions and realizationsexpectations may be a�ected by past experience

expectations play the key role in most economic models30s-60s naive and adaptive expectations70s-90s rational expectations

90s models of learning and bounded rationalityadaptive learning (OLS-learning)bayesian and belief-based learningreinforcement learning

2000s- heterogeneous expectations (HeterogeneousAgent Models)

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Introduction

Example: Model of Financial Marketdemand for the long-lived asset of a myopic MV trader

Dh(pt) =Eh,t[pt+1 + yt+1] − (1 + r)pt

a Vh,t[pt+1 + yt+1]

solve market clearing at time t, find equilibrium∑hDh(pt) = 0 ; pt =

11 + r

∑hEh,t[pt+1 + yt+1]

rational (homogeneous) expectations

pt =1

1 + rEt[pt+1+yt+1] ; (for i.i.d. dividends) pf =

yr

heterogeneous expectations

pt =1

1 + r

∑h

Eh,t

[1

1 + r

∑h′

Eh′,t+1[pt+2 + yt+2] + yt+1

]6 / 53


Introduction

Example (ctd): Heterogeneous Agent Modelthere are two types of investors

fundamentalists, Ef ,t [pt+1] = pf + v(pf − pt−1)chartists, Ec,t [pt+1] = pt−1 + g(pt−1 − pt−2) with g > 0

evolution of price

pt =1

1 + r

(nf ,t Ef ,t[pt+1] + nc,t Ec,t[pt+1]

)+

y1 + r

evolution of fractions

nf ,t+1 =exp [βπf ,t]

exp [βπf ,t] + exp [βπc,t]

profits πf ,t and πc,t are computed as their holdings timesreturn pt + yt − (1 + r)pt−1 and known to everybody

fundamentalists pay fixed cost C > 07 / 53


Introduction

Example (ctd): Simulation

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Introduction

HAMs and their Empirical ValidationHAMs assume several expectational rules (a�ecting tradingbehavior); these rules get reinforced from their past profit.Do the data support this theory?Empirical Studies

Branch (2004), Boswijk, Hommes and Manzan (2007),Goldbaum and Mizrach (2008), De Jong, Verschoor, andZwinkels (2009), Kouwenberg and Zwinkels (2010),Franke and Westerho� (2011), Chiarella, He andZwinkels (2014)

Experimental StudiesHommes et al (2005, 2008), Heemeijer et al (2009),Anufriev and Hommes (2012), Bao et al (2012), Pfajfarand Žakelj (2014), Assenza et al (2015), Anufriev, Bao,Tuinstra (2016, JEBO) 9 / 53


Introduction

Heuristic Switching ModelsRational Expectation Hypothesis: Restrictivetheoretical assumptions and Limited empirical validity.

Heterogeneous Agent Models (HAMs):1 agents use behavioral decision rules (“forecasting

heuristics”)2 agents switch between rules based on their past

performances (Brock and Hommes, 1997).

Applications: Financial markets (endogenous bubblesand crashes and a lot of “stylized facts”),Macroeconomics (persistence of inflation, di�erent policyimplications).

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Introduction

Brock-Hommes model (ECMA, 1997; JEDC, 1998)Setup

Supply/Demand-driven market where participants mustform expectations about future priceThe equilibrium is stable under costly Rationalexpectations and unstable under free Naive expectationsDiscrete choice is based on past profits

PredictionIf cost of RE is high, prices exhibit bubble/crash pa�erns

MechanismNear equilibrium two heuristics give similar forecastsand, due to fix cost of RE, majority uses naive ruleDynamics diverge and naive expectations get less preciseEventually majority switches to Rational expectationsand price returns towards equilibrium 11 / 53


Introduction

Role of Lab ExperimentsHAMs are empirically successful, tractable, intuitive......but the dynamics depends on the chosen heuristics andtheir cost and also parameters of switching.

Experiments with paid human subjects allow to1 test assumptions and implications2 pin down relevant heuristics (LtF)3 estimate parameters of switching

in a controlled environment.

Switching ExperimentsAnufriev, Bao, and Tuinstra (JEBO, 2016) tested switchingbetween 2, 3 or 4 heuristics on exogenous dataThis paper: binary choice on endogenous data

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Introduction

General setup of switching modelsagents’ choices are distributed over H di�erent heuristics.past payo�s of heuristics are known:

πht−1, π

ht−2, . . . , π

ht−`, . . .

fraction of agents using heuristic h at time t, is given bydiscrete choice model (Manski and McFadden, 1981)

nh,t =exp [αh + βπh,t−1]∑Hk=1 exp [αk + βπk,t−1]

,

where β > 0 is the Intensity of Choice and αh ≡ 0

Anufriev, Bao, and Tuinstra (JEBO, 2016) found that:(i) intensity of choice is not the same across treatments,but depends on past predictability of profits;(ii) model with predisposition (α1 > 0) provides be�er fit

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Introduction

General setup of switching modelsagents’ choices are distributed over H di�erent heuristics.past payo�s of heuristics are known:

πht−1, π

ht−2, . . . , π

ht−`, . . .

fraction of agents using heuristic h at time t, is given bydiscrete choice model (Manski and McFadden, 1981)

nh,t =exp [αh + βπh,t−1]∑Hk=1 exp [αk + βπk,t−1]

,

where β > 0 is the Intensity of Choice and αh ≡ 0Anufriev, Bao, and Tuinstra (JEBO, 2016) found that:(i) intensity of choice is not the same across treatments,but depends on past predictability of profits;(ii) model with predisposition (α1 > 0) provides be�er fit

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Introduction

Other Studies for HSMs Estimation andCalibration

on financial data: Boswijk, Hommes and Manzan (JEDC,2007), Goldbaum and Mizrach (JEDC, 2008), Chiarella,He and Zwinkels (JEBO, 2014), Cornea-Madeira,Hommes, and Massaro (JBES, 2017)

on survey data: Branch (EJ, 2004), Goldbaum andZwinkels (JEBO, 2014)

on experimental data: Hommes (JEDC, 2011), Anufrievand Hommes (AEJ-Micro, 2012), Anufriev, Hommes andPhilipse (JEE, 2013)

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Introduction

Objectives of the experiment1 Verify if aggregate switching behavior is well described

by the discrete choice model;

2 Provide estimations of the Intensity of Choice parameterfor calibration purposes;

3 Study possible e�ects of endogeneity in profits onswitching (cf., Anufriev, Bao, Tuinstra, JEBO, 2016)

4 Test a prediction of Brock-Hommes (E, 1997; JEDC, 1998)model about the e�ects of information cost di�erencebetween heuristics (e.g., rational expectations vs. naive)

Low: stable dynamicsHigh: locally unstable but bounded (bubbles and

crashes) 15 / 53


Experiment

Plan

1 Introduction

2 Experiment




6 Conclusion

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Experiment

Screen

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Experiment

ExperimentIndividual discrete choice experiment with group e�ect

10 Participants:choose between alternatives A and B during 40 periods inone Block and then during 40 periods in another Block

are informed that the profits of alternatives depend ontheir and other participants’ choices

are not informed about the functional forms of profitgenerating processes

are shown the history of past profits (graph and table)

Additional Sessions: 35 participants, 60 periods18 / 53


Experiment

DGP: Stylized version of the BH modelEndogenous Variables:

πA,t profit of “rational” (stabilizing but costly) heuristicπB,t profit of “naive” (cheap but destabilizing) heuristicxt “deviation of price from REE steady state”nB,t share of “naive” heuristic’s users, 1− nB,t = nA,t

State Variable Dynamics and Profits of Heuristics

xt = λnB,txt−1 + εt

πA,t = w + γAx2t , πB,t = W − γBx2twith λ = 2.1, γA + γB = 1, w < W , εt ∼ N (0, 0.1)

Exogenous Variable:W − w = C cost di�erenceC > 0 means that “rational” heuristic is more costly

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Experiment

DGP: Stylized version of the BH modelEndogenous Variables:

πA,t profit of “rational” (stabilizing but costly) heuristicπB,t profit of “naive” (cheap but destabilizing) heuristicxt “deviation of price from REE steady state”nB,t share of “naive” heuristic’s users, 1− nB,t = nA,t

State Variable Dynamics and Profits of Heuristics

xt = λnB,txt−1 + εt

πA,t = w + γAx2t , πB,t = W − γBx2twith λ = 2.1, γA + γB = 1, w < W , εt ∼ N (0, 0.1)Exogenous Variable:

W − w = C cost di�erenceC > 0 means that “rational” heuristic is more costly

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Experiment

Experiment Design and ParametrizationState variable dynamics

xt = 2.1nB,txt−1 + εt, εt ∼ N (0, 0.1), x0 = 0Two Blocks of 40 decision periods in each

High: C = 8πA,t = 1 + 0.6x2t , πB,t = 9− 0.4x2t

Low: C = 1πA,t = 4.95 + 0.6x2t , πB,t = 5.05− 0.4x2t

Two treatments with 4 sessions each and rematchingparticipants by the groups of 10

High block to Low blockLow block to High block

2 extra sessions for High Treatment (N = 35, T = 60)20 / 53


Dynamics of the Stylized HSM and Hypotheses

Plan

1 Introduction

2 Experiment




6 Conclusion

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Dynamics of the Stylized HSMAssuming discrete choice model with predisposition

Probability to choose B =1

1 + eα+β(πA,t−1−πB,t−1)

and large population (nB,t = Probability to choose B)

the state variable evolves as

xt =λxt−1

1 + eα+β(x2t−1−(W−w))

+ εt =2.1xt−1

1 + eα+β(x2t−1−C)

+ εt

In the experiment we vary the cost di�erence:High Block : C = 8Low Block : C = 0.1

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Properties of the Model Dynamics

xt = λxt−1

1 + exp[α + β(x2t−1 − C)]

x∗ = 0 is a steady state for all parameter values. Thissteady state is unique and globally stable for λ < 1.For λ > 0 the system undergoes a pitchfork bifurcationwhen λ = 1 + exp(α− βC).At the moment of the bifurcation two non-zero steadystates, x+ > 0 and x− < 0, are created, withcorresponding steady state fraction 1/λ.For λ < 0 the system undergoes a period doublingbifurcation when −λ = 1 + exp(α− βC).At the moment of the bifurcation a 2-cycle is created,with corresponding steady state fraction −1/λ.

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Bifurcation Diagrams: C = 8 vs C = 0.1Let λ = 2.1:

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

β

α

High Information Cost

0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4

β

α

Low Information Cost

Red: the zero steady state is stableBlue: the non-zero steady states are stable

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Bifurcation Diagrams: C = 8 vs C = 0.1Let λ = 2.1:

α = 0.4 α = 0 α = −0.425 / 53



Simulation: C = 8 vs C = 0.1Let λ = 2.1, α = 0, β = 5:

0 5 10 15 20 25 30 35 40time period

-5

-4

-3

-2

-1

0

1

2

3

4

5

stat

e va

riabl

e, x

parametrization of High blocks

0 5 10 15 20 25 30 35 40time period

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

stat

e va

riabl

e, x

parametrization of Low blocks

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

parametrization of High blocks

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

parametrization of Low blocks

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Hypotheses

H1. There is a di�erence in volatility of both nB,t andxt between the High blocks and the Low blocks.

H2. The endogenous variable nB,t can be described bya discrete choice model with one lag and apredisposition e�ect.

H3. There is no di�erence between the discretechoice models estimated for High and for Lowblocks. In particular, the Intensity of Choiceparameter is the same.

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Results of the Experiment

Plan

1 Introduction

2 Experiment




6 Conclusion

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Fraction of B-choices: High-Low Treatment I

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

High: Session 1, group 1

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

Low: Session 1, group 1

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


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Fraction of B-choices: High-Low Treatment II

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


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Fraction of B-choices: Low-High Treatment I

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


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Fraction of B-choices: Low-High Treatment II

0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


0 5 10 15 20 25 30 35 40time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice


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Histograms of B-choices

High: All groups

0 0.2 0.4 0.6 0.8 1

fraction of B choice

0

10

20

30

40

50

freq

uenc

y

Low: All groups

0 0.2 0.4 0.6 0.8 1

fraction of B choice

0

10

20

30

40

50

freq

uenc

y

33 / 53



State Variable: High - Low Treatment I

0 5 10 15 20 25 30 35 40time period

0

1

2

3

4

5

6

7

8

9

stat

e va

riabl

e, x

Session 1

High: group 1

High: group 2

Low: group 1

Low: group 2

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State Variable: High - Low Treatment II

0 5 10 15 20 25 30 35 40time period

0

1

2

3

4

5

6

7

8

9

stat

e va

riabl

e, x

Session 3

High: group 1

High: group 2

Low: group 1

Low: group 2

35 / 53



State Variable: Low - High Treatment I

0 5 10 15 20 25 30 35 40time period

0

1

2

3

4

5

6

7

8

9

stat

e va

riabl

e, x

Session 2

High: group 1

High: group 2

Low: group 1

Low: group 2

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State Variable: Low - High Treatment II

0 5 10 15 20 25 30 35 40time period

0

1

2

3

4

5

6

7

8

9

stat

e va

riabl

e, x

Session 4

High: group 1

High: group 2

Low: group 1

Low: group 2

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Descriptive statistics

Fraction of B-choices, nB State Variable, xData Mean Std. Dev. Mean Std. Dev.

Session 1. Group 1 0.58 0.27 2.43 1.52Session 1. Group 2 0.61 0.25 2.26 1.52

High Session 2. Group 1 0.60 0.27 2.51 1.47Session 2. Group 2 0.58 0.28 2.39 1.27...

......

......

All groups 0.59 0.26 2.43 1.52

Session 1. Group 1 0.56 0.23 0.26 0.22Session 1. Group 2 0.51 0.23 0.26 0.24

Low Session 2. Group 1 0.55 0.24 0.17 0.21Session 2. Group 2 0.56 0.24 0.13 0.28...

......

......

All groups 0.54 0.24 0.20 0.2438 / 53



Discrete Choice Model

IoC Predisposition Zero SS Non-Zero SS

Data Beta S.E. Alpha S.E. (x∗, n∗) f ′(x∗) (x+, n+) f ′(x+)

Session 1. Group 1 0.08 0.01 0.31 0.11 (0,0.58) 1.22 (2.31,0.48) 0.55Session 1. Group 2 0.12 0.02 0.38 0.11 (0,0.64) 1.35 (2.37,0.48) 0.29

High Session 2. Group 1 0.12 0.02 0.35 0.11 (0,0.65) 1.36 (2.42,0.48) 0.26Session 2. Group 2 0.17 0.02 0.28 0.11 (0,0.75) 1.57 (2.63,0.48) -0.23...

......

......

......

......

All groups 0.12 0.01 0.25 0.04

Session 1. Group 1 3.35 0.84 0.26 0.11 (0,0.52) 1.09 (0.23,0.48) 0.82Session 1. Group 2 5.24 0.93 0.09 0.11 (0,0.61) 1.27 (0.32,0.48) 0.45

Low Session 2. Group 1 11.35 1.66 -0.17 0.13 (0,0.79) 1.65 (0.35,0.48) -0.47Session 2. Group 2 8.67 1.47 0.10 0.11 (0,0.68) 1.43 (0.32,0.48) 0.10...

......

......

......

......

All groups 5.71 0.42 0.11 0.04

Table: Estimations of the DCM with one lag and predisposition.

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Estimations on the Bifurcation diagram

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

β

α


0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4

β

α

Low Information Cost

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High (Large and Long) Treatment

Plan

1 Introduction

2 Experiment




6 Conclusion

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Fraction of B-choices: Large/Long High T

0 10 20 30 40 50 60time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

High: Large - Long, group 1

0 10 20 30 40 50 60time period

0

0.2

0.4

0.6

0.8

1

frac

tion

of B

cho

ice

High: Large - Long, group 2

42 / 53



State Variable: Large and Long High Treatment

0 10 20 30 40 50 60time period

0

1

2

3

4

5

6

7

8

9

stat

e va

riabl

e, x

Large - Long Sessions

Session 1

Session 2

43 / 53



Estimations on the Bifurcation diagram

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

-0.2

0.0

0.2

0.4

β

α


44 / 53


Conclusion

Plan

1 Introduction

2 Experiment




6 Conclusion

45 / 53


Conclusion

Hypotheses and ResultsH1. There is a di�erence in volatility of both nB,t and

xt between the High blocks and the Low blocks.

Confirmed, especially for xt .

H2. The endogenous variable nB,t can be described bya discrete choice model with one lag and apredisposition e�ect.

Confirmed.

H3. There is no di�erence between the discrete choicemodels estimated for High and for Low blocks.

There is a di�erence in values of parameters.

46 / 53


Conclusion

Conclusion1 E�ect of information cost in Brock-Hommes model is

confirmed.

2 At the same time, participants adapt their choice to theenvironment they are in.

They become less sensitive to past profit di�erences inless stable environment.

As a result aggregate dynamics become only moderatelycomplex: e.g., asymmetric equilibrium or 2-cycle.

3 Theoretical literature of HSM may need to take it onboard and endogenize the Intensity of Choice.

47 / 53


Conclusion

THANK YOU!

48 / 53


Experimental results

Plan

7 Experimental results

49 / 53



Profits in High blocks

Figure: Profits dynamics 50 / 53



Profits in Low blocks

Figure: Profits dynamics 51 / 53



Table: Estimations of the discrete choice model with two lags andpredisposition.

Predisposition IoC IoCData Alpha S.E. Beta S.E. Beta2 S.E.

Session 1. Group 1 0.33 0.11 0.08 0.02 0.02 0.01Session 1. Group 2 0.40 0.11 0.12 0.02 0.02 0.02

High Session 2. Group 1 0.35 0.11 0.11 0.02 -0.02 0.02Session 2. Group 2 0.28 0.12 0.17 0.02 0.01 0.02

all groups and sessions 0.33 0.06 0.12 0.01 0.00 0.01

Session 1. Group 1 0.25 0.11 3.19 0.88 0.57 0.57Session 1. Group 2 0.04 0.11 5.20 0.97 -0.76 0.80

Low Session 2. Group 1 -0.17 0.13 11.35 1.73 1.13 1.31Session 2. Group 2 0.12 0.12 8.91 1.51 1.74 0.88

all groups and sessions 0.12 0.06 6.21 0.62 0.56 0.39

52 / 53



Table: Estimations of the discrete choice model with three lags andpredisposition.

Predisposition IoC IoC IoCData Alpha S.E. Beta S.E. Beta2 S.E. Beta3 S.E.

Session 1. Group 1 0.31 0.11 0.07 0.02 0.02 0.02 -0.01 0.01Session 1. Group 2 0.40 0.12 0.11 0.02 0.01 0.02 0.01 0.02

High Session 2. Group 1 0.32 0.11 0.11 0.02 -0.03 0.02 0.00 0.01Session 2. Group 2 0.26 0.12 0.16 0.02 0.01 0.02 -0.02 0.02

all groups and sessions 0.31 0.06 0.11 0.01 0.00 0.01 0.00 0.01

Session 1. Group 1 0.28 0.11 3.19 0.84 0.09 0.67 0.92 0.60Session 1. Group 2 -0.01 0.11 5.21 0.96 -0.85 0.83 -0.77 0.82

Low Session 2. Group 1 -0.12 0.13 11.45 1.73 1.64 1.36 -1.33 1.50Session 2. Group 2 0.17 0.12 9.14 1.53 2.11 0.89 -0.11 0.98

all groups and sessions 0.13 0.06 6.27 0.62 0.62 0.42 0.01 0.38

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a laboratory experiment on the heuristic switching model

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