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Jekyll & Hyde

Marie-Edith Bissey (Università Piemonte Orientale, Italy)

John Hey (LUISS, Italy and University of York, UK)Stefania Ottone (Econometica, Università Milano

Bicocca, Italy)

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Dynamic Inconsistencies

Non-EU people may be dynamically inconsistent.

Are such people aware of their dynamic inconsistency?

If not, they presumably behave naively. If they are aware, how do they behave? Do they behave resolutely or

sophisticatedly?

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Aims of the Experiment

Inducing dynamic inconsistencies in the lab, taking into account the fact that dynamically inconsistent people are really several different selves.

Check how people behave when facing dynamic inconsistencies.

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Jekyll & Hyde

We look at the way an individual makes his consumption/saving decision when the rate at which he discounts the future is not constant over time

Jekyll and Hyde were two different people inside the same body.

Each was aware of the other – but they clearly had different preferences.

As a whole “Jekyll & Hyde” was a dynamically inconsistent person.

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The Context

We analyse the behaviour of Jekyll & Hyde (two different subjects) within the context of a simple life-cycle consumption/savings model.

Each period Jekyll & Hyde receives an income M which he/they can consume or save.

They take it in turns to decide on consumption.

Accumulated savings earn interest.

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They both get utility out of consumption and both try to maximise expected discounted lifetime utility of consumption.

They have different discount factors – that of Hyde (1) lower than that of Jekyll (2).

Objective (payment) function (where ρi is the discount factor of player i):

u(c1) + ρiu(c2) + ρi2u(c3) + …

where u(c) = c.

The Model

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Naivety?

Assuming naivety – each ignores the other, or – equivalently - assumes that the other uses the same discount factor as himself.

The optimal solution is: 2[ /( 1)] where (1 )C W M r r

andρ = ρ1 in odd periods ρ = ρ2 in even periods

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Resoluteness?

Assuming resoluteness – the first player imposes the solution (how?)

The optimal solution is:

2[ /( 1)] where (1 )C W M r r

andρ = ρ1

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Assuming sophistication – players take into account the behaviour of the other.

We assume that Jekyll/Hyde :receives income M each period;can save at rate of return r. We assume that:

Sophistication?

1/ 2 1/ 2( ) [ /( 1)] ( ) [ /( 1)]i ii iY W W M r and Z W W M r

1/ 2

1/ 2 1/ 2

( ) max[ [( ) ]

max[ [( ) /( 1)]

i ii

C

i iC

Y W C Z W C r M

C W C r Mr r

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The solution (Sophistication)

We can show that the optimal consumption strategy is given by:

1 21

/( 1)W M rc

2 2 1/ 2 2 2 1/ 21 1 1 2 2 2

1 2 1 2 2 1 2 11 22 2 1/ 2 2 2 1/ 2

2 2 1 1

(1 ) (1 )

1 1

(1 ) (1 )

r r

r r

r r

2 22

/( 1)W M rc

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Experimental Design (I) 2n subjects and n projects. In each project there is an odd player - who plays in odd

periods and has the lower continuing probability... ...and an even player – who plays in even periods and has

the higher continuing probability. The session as a whole has an ‘earthquake’ which occurs

with probability e. When the earthquake happens the whole session finishes.

Player i has a ‘continuing probability’ equal to his or her ρi = pi - e. If does not continue, exits the project.

When a player exits from a project he or she is re-assigned to a new project in which he or she has not been before (if there is a space).

A subject may change project and role several times during the experiment...

... and may not be in a project.

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Experimental Design (II) Each project is endowed with a given stock

of wealth in tokens. Each period the subject chooses how much

of his accumulated wealth to convert into money - the conversion scale: pence = (conversion)

Payment for each period to both players is the converted value.

Subjects paid the total of all these payments over the duration of the session.  

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Experimental Design (Implications)

No participant will ever find him or herself in any particular project more than once. Thus the way a participant behaves in any one project cannot affect the earnings that he or she gets from any other project participants should consider each project as completely independent of any other project that he or she may be in.

Each period in each project contributes to the determination of the final payment.

Jekyll’s behavior influences Hyde’s payoff and vice-versa.

Unconverted tokens left when the experiment finishes become worthless.

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Experimental Design (Parameters)

T1. ρ1 = 0.85 and ρ2 = 0.85 T2. ρ1 = 0.82 and ρ2 = 0.85 T3. ρ1 = 0.85 and ρ2 = 0.88 T4. ρ1 = 0.82 and ρ2 = 0.88 Note that Treatment 1 is not a case of dynamic

inconsistency since both players have the same stopping probability.

M = 100 r = 1.25 e = 0.02 10 pounds show-up fee

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Experimental Procedure

Overall, 106 subjects. 26 players in T1, 28 subjects both in the T2 and in T3, 24 in T4

2 hours per session Average payoff: 13 pounds

Written instructions Power Point presentation with sample screens Table with conversion scale and calculator on the

screen Control Questions Questionnaire

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Odd Participant – Odd period

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Even participant – Odd period

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Example of control questions A) You are in project 2. You are the odd participant. Your partner is the

even participant. At the end of each period, what is the chance that you exit the project?

10% 13% 20%

At the moment, the stock of tokens associated with project 2 is 1600. It is your turn and you decide to convert 36 tokens.

How many pence do you earn? 6 36 72

How many pence does your partner earn? 0 6 36

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The Optimal apcsTt. ρ1 ρ2 apc1 apc2

1 0.85 0.85 0.097 (S)0.097 (N)0.097 (R)

0.097 (S)0.097 (N)0.097 (R)

2 0.82 0.85 0.163 (S)0.159 (N)0.159 (R)

0.101 (S)0.097 (N)0.159 (R)

3 0.85 0.88 0.104 (S)0.097 (N)0.097 (R)

0.040 (S) 0.032 (N)0.097 (R)

4 0.82 0.88 0.178 (S)0.159 (N)0.159 (R)

0.053 (S)0.032 (N)0.159 (R)

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Results (I)

Result 1. It is generally true that the subject with the lower continuing probability has a higher apc.

Actual apcs

Tt. ρ1 ρ2 apc1 apc2

1 0.85 0.85

0.184 0.178

2 0.82 0.85

0.177 0.157

3 0.85 0.88

0.157 0.173

4 0.82 0.88

0.182 0.146

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Results (II)

Result 2. Naïve fits better during the first periods while, over time, Sophisticated shows a better performance.

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Mean absolute deviations from optimal

Tr Naïve Sophisticated Resolute

S 1 S 2 S 1 S 2 S 1 S 2

1 157 (39) 162 (40) 157 (39) 162 (40) 157(39) 162 (40)

2 165 (48)

167 (48) 164 (49) 167 (50) 165 (56) 168 (60)

3 171 (38)

174 (38) 168 (39) 171 (39) 157 (40) 160 (41)

4 159 (38)

157 (38) 159 (40) 157 (41) 163 (58) 163 (61)

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Results (III) Result 3. The treatment effects are not in accordance with

our theory

Tt. ρ1 ρ2 apc1 apc2

1 0.85 0.85 0.097 (S)0.097 (N) *0.184*0.097 (R)

0.097 (S)0.097 (N) *0.178*0.097 (R)

2 0.82 0.85 0.163 (S)0.159 (N) *0.177*0.159 (R)

0.101 (S)0.097 (N) * 0.157*0.159 (R)

3 0.85 0.88 0.104 (S)0.097 (N) *0.157*0.097 (R)

0.040 (S) 0.032 (N) *0.173*0.097 (R)

4 0.82 0.88 0.178 (S)0.159 (N) *0.182*0.159 (R)

0.053 (S)0.032 (N) *0.146*0.159 (R)

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Results (IV)

Players play erratically – there is a big difference between different subjects.

Some understand the nature of the problem and try and save.

Others do not understand and consume all or most every period.

Some subjects realise that there is a trade-off: the more you consume now the higher the present utility, the less you consume now the higher the utility in the future. They (probably) cannot find the correct trade-off, so they consume in some periods and save in others.

Players are influenced by the time effect and their previous experience.

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Some examples of actual play

From treatments 1, 2 and 4. The person with the higher stopping probability plays in

the odd periods. The green graph is what they did. The black graph is the optimal strategy. The solid red line is the naive response of the player in

the odd periods. The solid blue line is the naive response of the player in

the even periods. The dashed red line is the average response of the

players in the odd periods. The dashed blue line is the average response of the

players in the even periods. The numbers at the top indicate the subject.

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Further analysis

Random effects tobit regression time and the fact of being in a new project

Correlation between subjects decisions and partners decisions 25 players over 106

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Main conclusions

We need a good analysis of data!

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Further research

Analyse the behavior of each player in each project and try and define a trend for each player over time.

Why is there so much noise in the data? Since subjects do not know with whom

they are playing it may be best to assume an irrational partner?

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Thank you