Is the definition of fairness subject to rational choice?

Niall Flynn

i) Motivation

ii) Outline of experiment

iii) Inequity averse utility function and test of self-serving point of equity

iv) Results

MotivationInterdependent preferences:

i) Non-experimental literature → -ve effects, e.g. Duesenberry (1949), Frank (1985), Rayo and Becker (2007).

ii) Experimental literature → +ve and -ve effects, e.g. Inequity Aversion

MotivationInterdependent preferences:

i) Non-experimental literature → -ve effects, e.g. Duesenberry (1949), Frank (1985), Rayo and Becker (2007).

ii) Experimental literature → +ve and -ve effects, e.g. Inequity Aversion

In practice: inequity = inequality:

“For most economic experiments it seems natural to assume that an equitable allocation is an equal monetary payoff for all players. Thus inequity aversion reduces to inequality aversion” : Fehr and Schmidt (2005).

MotivationInterdependent preferences:

i) Non-experimental literature → -ve effects, e.g. Duesenberry (1949), Frank (1985), Rayo and Becker (2007).

ii) Experimental literature → +ve and -ve effects, e.g. Inequity Aversion

In practice: inequity = inequality:

“For most economic experiments it seems natural to assume that an equitable allocation is an equal monetary payoff for all players. Thus inequity aversion reduces to inequality aversion” : Fehr and Schmidt (2005).

Inequity Aversion:

If individual is above/below point of equity the payoffs of those below/above point of equity have +ve/-ve effect.

Choice of self-serving definition → lowers potential to be above the “fair” allocation → lowers potential for positive externalities and increases potential for negative externalities.

Outline of experimentStage 1: Subjects undertake a 20 minute test of the same 7 “spot-the-difference” questions. Each additional correct answer increases size of endowment used in dictator game. Test is scored via increasing returns to performance production function in order to create different possible definitions of fairness.

Y= yi + yj = f(qi) + f(qj); i,j = 1,2,i j ; f(qi)>0, f(qi)>0

Where: Y = dictator game endowmentqi = number of questions answered correctly by player iyi = f(qi) = income generated by player i

Scoring system as follows:1 question correct: generates £0.502 question correct: generates £1.003 question correct: generates £2.004 question correct: generates £4.005 question correct: generates £8.006 question correct: generates £16.007 question correct: generates £32.00

Outline of experimentStage 2: Each subject receives results from stage 1, then one playerfrom each pair is randomly chosen to allocate their endowment.

Results are presented to subjects in the following way:

By YouBy Your Partner

Total

Number of questions answered correctly 6 7 13

Proportion of questions answered correctly 46% 54% 100%

Money generated £16 £32 £48

Proportion of money generated 33% 67% 100%

Allocation decisions are made in percentiles.

Outline of experiment

Possible to split subjects into 3 sets: qd>qr, then d is high productivity (H) qd<qr, then d is low productivity (L) qd=qr, then d is equal productivity (E)

Three obvious conceptions of fairness:

1. Equal split – i.e. ½

2. Output ratio – i.e. yi/Y

3. Input ratio – i.e. qi/Q

Increasing returns to performance production function means:

H: yd/Y > qd/Q > ½

L: ½ > qd/Q > yd/Y

E: ½ = qd/Q = yd/Y

Outline of experimentCreating pairs:

Pairs of subjects are chosen after stage 1 so as to achieve:

i) For every dictator in set H there is a unique dictator in set L making an allocation decision for the same endowment – i.e. the same question pairing.

ii) Set E contains dictators uniquely matched to those in H and L, but playing for higher endowments.

Inducing inequity aversion:

• No show up fee.

• Non-neutral language of instructions.

• Dictator decision made in same room as recipient → very limited anonymity.

Inequity-averse utility function

i

i

iiii Y

xgYhxfU .

xi = income – i.e. absolute amount kept in dictator game

Yi = endowment – i.e. amount dictator game is played for

Φi = proportional point of equity – 0 ≤ Φi ≥ 1

f(xi)>0

gx(.)>0, for xi/Yi > Φi

gxx(.)>0, for xi/Yi > Φi

h(Yi)>0, for Yi>0

h(Yi) ≤ 0

(1)

Inequity-averse utility function

i

i

ii Y

xgYh .

i

i

iiii Y

xgYhxfU .

xiΦi.Yi

Yi=Y1 Yi=Y2

Where: Y1< Y2

(1)

Test of self-serving point of equity

ii

i

i

i

Yh

Ybg

Y

x

)(.1 (2)

For a given endowment and point of equity, and assuming f(.) is linear such that f(xi) = a + bxi, the FOC of (1) gives an optimal proportional self allocation of:

Test of self-serving point of equity

1 1 1 1

2i i

ii H i H i EH H i E i

x x

n n Y n Y

ii

i

i

i

Yh

Ybg

Y

x

)(.1

Summing (2) within each set and averaging, and assuming that Φi = ½ for all i in E, gives a lower bound estimates for the average point of equity in set H of:

(2)

(3)

For a given endowment and point of equity, and assuming f(.) is linear such that f(xi) = a + bxi, the FOC of (1) gives an optimal proportional self allocation of:

Average point of equity in Set H

Average proportion kept in Set H

Average deviation in set E

Test of self-serving point of equity

1 1 1 1

2i i

ii H i H i EH H i E i

x x

n n Y n Y

ii

i

i

i

Yh

Ybg

Y

x

)(.1

Summing (2) within each set and averaging, and assuming that Φi = ½ for all i in E, gives a lower bound estimates for the average point of equity in set H of:

As sets L and H are perfectly matched their points of equity should sum to one if there is no self serving bias, which gives:

(2)

(3)

(4)

For a given endowment and point of equity, and assuming f(.) is linear such that f(xi) = a + bxi, the FOC of (1) gives an optimal proportional self allocation of:

Average point of equity

Average proportion kept

Average deviation in set E

0211

Ei i

i

ELi i

i

LHi i

i

H Y

x

nY

x

nY

x

n

ResultsH L E

Matched Set No.

Q (d,r) Y x/Y Q (d,r) Y x/Y Q (d,r) Y x/Y

1 7,6 48 0.90 6,7 48 1.00 7,7 64 1.00

2 7,6 48 0.75 6,7 48 0.80 7,7 64 0.95

3 7,6 48 0.73 6,7 48 0.80 7,7 64 0.8

4 7,6 48 0.67 6,7 48 0.50 7,7 64 0.65

5 7,6 48 0.67 6,7 48 0.50 7,7 64 0.50

6 7,6 48 0.67 6,7 48 0.50 7,7 64 0.50

7 7,6 48 0.67 6,7 48 0.44 7,7 64 0.50

8 6,5 24 1.00 5,6 24 0.45 6,6 32 1.00

9 5,4 12 1.00 4,5 12 1.00 5,5 16 0.75

10 5,4 12 1.00 4,5 12 0.90 5,5 16 0.65

11 5,4 12 0.67 4,5 12 0.75 5,5 16 0.50

12 5,4 12 0.67 4,5 12 0.50 5,5 16 0.50

13 5,4 12 0.67 4,5 12 0.50 5,5 16 0.50

14 5,4 12 0.67 4,5 12 0.50 5,5 16 0.50

15 5,4 12 0.67 4,5 12 0.33 5,5 16 0.50

16 4,3 6 1.00 3,4 6 0.5 4,4 8 0.80

Mean 6 28.13 0.775 5 28.13 0.623 6 37.50 0.662

SD 0.146 0.215 0.194

E Dictators Allocation to Self

E: ½ = qd/Q = yd/Y

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

n

E Dictators Allocation to Self

E: ½ = qd/Q = yd/Y

L Dictators Allocation to Self

L: ½ > qd/Q > yd/Y

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

n

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

n

E Dictators Allocation to Self

E: ½ = qd/Q = yd/Y

L Dictators Allocation to Self

L: ½ > qd/Q > yd/Y

H Dictators Allocation to Self

H: yd/Y > qd/Q > ½

0

1

2

3

4

5

6

7

8

9

10

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

% self allocation

ResultsRemoving the four highest allocations from each of the three sets gives a data set of solely interior solutions, which can be interpreted as tangency points. This process leaves comparable data sets of: nE=13; nL=10; nH=12

The test derived for self-serving points of equity is tested against a null of:

0 : 2. 0H L EH

1 : 2. 0H L EH

t = 1.17; p=0.121

ResultsAssume utility function of:

ii i i

i

xbU x

Y Y

11

111

..

ii

i

xY

Y b

0 1 2 3 4. . . .iL H i i

i

xY g

Y

Which gives an optimal allocation of:

Which gives a regression equation of:

Where: δL = dummy variable equal to one if dictator in set L, 0 otherwiseδH = dummy variable equal to one if dictator in set H, 0 otherwiseg = gender, f=0; m=1

ResultsDependent Variable: xi/Yi – i.e. proportional allocation to self

1 2 3 4

Constant (β0)0.5255 (0.000)

0.5426 (0.000)

Constrained to equal ½

Constrained to equal ½

Low Prod Dummy (β1)-0.0356 (0.440)

-0.0371(0.416)

-0.0251 (0.534)

-0.0139(0.720)

High Prod Dummy (β2)0.1282***

(0.006)0.1291***(0.005)

0.1369*** (0.001)

0.1503***(0.000)

Endowment0.0010 (0.285)

0.0009(0.332)

0.0013** (0.045)

0.0015**(0.019)

Gender0.0234 (0.539)

-0.0334 (0.295)

N 35 35 35 35

R2 0.3554 0.3471 0.6210 0.6072

P values in brackets

2. H0: (β0 + β1)+(β0 + β2)= 1 ; H1: (β0 + β1)+(β0 + β2) > 1

F(1,30) = 2.48; p=0.063

4. H0: β1 + β2= 0 ; H1: β1 + β2 > 0

F(1,31) = 3.02; p=0.046