Attitudes to risk and fairness: how they are related and how they evolved - theoretical and experimental studies

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May 27, 2003 Vítězslav Babický

Attitudes to risk and fairness: how they are

related and how they

evolved - theoretical and

experimental studies

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Proposed committee: Dirk Engelmann Axel Ockenfels Avner Shaked

Tentative chair: Andreas Ortmann Plan of the presentation:

• Questions - motivation for papers

• Models from recent literature

• Fairness under risk: Insights from dictator games

• Proposed experiments

• Modelling the “game of life”

• Conclusion, discussion

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1. Are attitudes to fairness and to risk related and if so, how?

• Fairness under risk: Insights from dictator games - 1st paper for my dissertation

(accepted for CERGE-EI working paper series)

Questions - motivation for papers

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2. Are there alternatives to fairness attitudes and how do they evolve?

• Mutual shareholding and inequity aversion -

theoretical and experimental analysis (joint work with Werner Gueth, Wiebke Kuklys and Andreas Ortmann) - theory completely worked out

- first experiments already done in Jena

• Evolutionarily stable positive inequity aversion in the mutual shareholding environment for the composition of the game of life from dictator, ultimatum and market games (joint work with Werner Gueth)

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3. How do risk attitudes evolve and how is the

result of evolution affected by various composition of the game of life?

• Evolution of risk attitudes on 2x2 games

- risk-loving players advantaged in some games, while risk-averse players advantaged in other

games

(joint work with Andreas Ortmann)

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Starting point: existing theories of other-regarding preferences

– decisions about ex-ante known pies, possibly with information constraint

People’s life circumstances are diverse and subject to chance events (sometimes there is need to decide

earlier)

So, introduce risk (sticking to use of expected utility theory, prospect theory may be a possibility)

Note: issue of procedural fairness studied in recent literature (Bolton, Brandts, Ockenfels, 2001)

Models from recent literature

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To recall the main contenders:

1) Theory of fairness, competition and cooperation [FCC] (Fehr, Schmidt, 1999)

- proposed model: aversion to differences in payoffs Utility function in the form

U_i(x) = x_i– (_i/(n–1))∑_kmax(x_i–x_k,0) – – (_i/(n–1))∑_kmax(x_k–x_i,0)

n=number of players (=2 for the dictator game)

Example: the graph of U_i(x_k|x_i) , e. g. _i=0.5, _i= 2 Empirical results for various classes of games can be explained in accordance with this fairness model

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What the FCC theory predicts for the dictator game:

Prediction with FCC is still s = 0 if ₁< 0.5

and s = 0.5 if ₁> 0.5 while in empirical studies both 0 and 0.5 are the modal outcomes

Problem: LINEAR inequity aversion

Another consequence of linearity: In case of decision to sacrifice share k on the pie, the expected utility of any lottery for the size of a pie is the same as for the sure pie size in (linear) FCC.

Need: Concavity in the amount of advantageous inequality

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2) Theory of Equity, Reciprocity and Competition [ERC] (Bolton, Ockenfels, 2000)

Agents maximize V_i = V_i(y_i, _i) (motivation function) where y_i is their payoff and

_i = _i(y_i, c, n) is the relative payoff (c = y_itotal payoff)

_i = y_i/c if c > 0 and 1/n otherwise

Under this utility function agents have narrow self-interest: V_i1  0, V_i11  0,

V_i22 < 0, V_i2 = 0 for _i = 1/n

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Examples: V_i(c_i, _i) = a_ic_i – b_i/2(_i– 0.5)² a_i, b_i>0 given constants

Additively separable V_i(y_i, _i) = u(y_i) – v(_i) u concave, v convex – given functions Works for Dictator game (and many others) Also works under incomplete information:

Response to proposal in the ultimatum game is done without knowing actual size of the pie

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The dictator game is the simplest one – the decision is affected only by preferences of the decision-maker

- we can study a single agent’s preferences separately Risk: Only a probability distribution of the pie

size is known to the dictator in the moment

of decision. The decision is to give a percentage share on the actual pie to the recipient.

Example: Prenuptial agreement

Fairness under risk: Insights from dictator games

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Pilot experiment:

Dictator game with ex-ante unknown size of pie in two groups. Both have the same expected size but different variance (note:

symmetric distributions were used).

Subjects asked to decide both absolute and relative part of the pie to transfer to the

second player. They are also asked about the preference of one of the decision.

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Intuition:

People would like to transfer part of the risk to the recipient

Hypothesis 1: More preference for relative over absolute offers.

Dictator wants to get a risk premium

Hypothesis 2: Lower offers in high variance treatment than in the low variance one.

Do the results conform to intuition?

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Results of the pilot experiment

(25 subjects, CERGE 1st year students):

For group A (low variance, 13 subjects):

4 decisions for absolute offer (risk captured by proposer)

Average offers: 2108 absolutely

28% relatively (2506 expected value) For group B (high variance, 12 subjects):

1 decisions for absolute offer

Average offers: 1425 absolutely

17% relatively (1533 expected value)

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Theory modifications:

Need - concave utility function

For the Fehr-Schmidt model, I can introduce exponent γ>1 and a concave function u into

(note: higher share is always preferred to lower share by dictator, i.e. x_dictator≥x_recipient always holds) U_i(x) = u(x_i) – _i(x_i-x_k)^ (i  k  {1,2})

Let C be random variable for the size of pie and p be the decision of the dictator – the share on the pie transferred to the recipient

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For simplicity, let u(x)=sgn(r) x^r , r<1, r

≠0

be an increasing concave CRRA function Measurement of risk: E(C^)

If EC₁=EC₂ then E(C_i^) determines variance of C_i (since Var C_i= E(C_i^) – [E(C_i)]^)

Obviously, Var C₁< Var C₂ iff E(C₁^) < E(C₂^).

Typically, also E(C₁^) < E(C₂^) for any γ>1;

E(C₁^r) > E(C₂^r) for any r  (0,1) and

E(C₁^r) < E(C₂^r) for all r<0 (namely for symmetric distributions)

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Similarly, for the ERC model, dictator maximizes : EV(p)= sgn(r) (1–p)^r E(C^r) – (1–2p)^

The solution:

(1–2p)^(1–p)^r = r sgn(r)E(C^r)/(2γ) The comparative statics:

the left-hand side is decreasing in p (since γ>1, r<1)

Dictator giving p is higher for lower variance treatments if r<0, it goes in the opposite direction if r  (0,1)

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Remark: if r=0 … the corresponding CRRA function is u(x)=log x

Then, the dictator maximizes expected utility EV(p)= log(1–p) + E(log C) – k(1–2p)^

i.e. the solution does not depend on changes

in the size of pie to be distributed in this situation.

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The solution for Fehr-Schmidt type model:

(1–2p)^(1–p)^r = r sgn(r)E(C^r)/(2γE(C^)) Left-hand side is decreasing in p (γ>1, r<1),

i.e. the dictator giving is higher if the right-hand side is lower (the decision depends on parameters γ, r)

THE PAPER CONCLUDES:

Attitude to fairness is related with attitude to risk

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In the laboratory controlled conditions,

computerized using z-Tree (Fischbacher, 1999)

• Experiments on risky dictator games to confirm the theory

• Possible refinements using prospect theory

(arbitrary division among absolute and relative offer also measurable by a new experiment and testable)

Proposed experiments

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Experiments about the distribution concern, working with efficiency gains

- important difference between Fehr-Schmidt and Bolton-Ockenfels models: if number of players n>2 then distribution matters only for inequity aversion in differences (Fehr-Schmidt) - experimental evidence about total efficiency concern (Engelmann, Strobel, 2002 with n=3), social welfare model by Charness and Rabin, AER 2002.

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My goal: to measure total efficiency concern against the distribution and own-payoff

concerns

Source of funding: grant proposal

Experiments in Jena - effects of mutual

shareholding on the level of inequity aversion Hypothesis: stronger degrees of mutual

shareholding weakens inequity aversion;

comparison for dictator, ultimatum and market games - the weakest effect in market games

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Evolutionary studies based on the evolution of

preferences - Gueth, Yaari, 1992 - indirect evolutionary approach

Principle: The game G (game of life) consist from

a composition of predefined different games; 2 players Preferences of agents depend on parameter 

R(, ’) is the payoff of player with -preferences when she is involved in the game G with ’-type player

* is evolutionarily stable type if R(*, *)> R(, *) for any *

Modelling the “game of life”

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- evolution of risk attitudes on a habitat of 2x2 games Baseline: Strobel, 2001 showed advantage of

risk-loving attitudes for a class of chicken games

Engelmann, 2003 proved advantage of risk-aversion for another class of 2x2 games (with mixed strategies equilibrium only)

Goal: for an arbitrary mix of 2x2 games prove the existence and uniqueness theorem for evolutionary stable

parameter of risk aversion for CRRA utilities;

comparative statics of the resulting parameter on the composition of the game of life

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Similar approach: evolution of fairness attitudes under mutual shareholding in the habitat composed from three types of interactions - dictator, ultimatum and market games.

The aim: dependence on the results of evolution depending on the composition of the habitat

(influence of different mixtures of games on resulting risk and fairness attitudes)

Preliminary results: border fairness attitude for dictator and ultimatum games; interior evolutionary stable

inequity aversion parameter resulting from the market game

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DISCUSSION

• Comments

• Suggestions

• Questions