Utility Curve Aggregation
lanquetuit.cyril@gmail.com, Université de Cergy Pontoise
Table des matières
1 Concepts 3
1.1 Nash equilibrium "Rock < Paper < Scissor" . . . 3 1.2 Pareto optimum : "Self carafon" . . . 3 1.3 Nash equilibrium vs Pareto optimum or "How to Cooperate ?" . 4 1.4 From cooperative household models to Hausedor mean . . . 4
2 Hausdor and Fréchet distances 4
2.1 Distances . . . 5 2.2 minuscule & MAJUSCULE . . . 6
Abstract
Thinking Collective Rationnalisation [2] with Chiappori, and according Sti- glitz that Economists should limit themselves to identifying Pareto ecient al- locations we will discuss How couples cooperate studying Nash [6] vs Pareto [8]
equilibrium in Prisoner's Dilemma. After a pur econometric approach of hou- sehold cooperation we cross scientic elds and try to introduce a computer vision tool to modelise preferences aggregation in couple : Norway's Hausdor dimension≃1,52[5]... Fractal in Econometry ? We are not so far ! Hausdor dis- tance which makes ∞ Koch snowake curve converged might give us a good theorical shape for household utility function.
Introduction
As shown by Thaler & Sunstein in "Nudge" Human perception is deformed (not perfect) :
"Nous ne sommes pas des écones !" [9]
Figure 1 Which is the longer table ? [7]
Allais Paradox [1], rst mentioned in 1953 by french economist and Nobel Prize
A : win 100¿ surely vs B : win 150¿ with probability p=90%
C : win 100¿ with probability p=10% vs D : win 150¿ with probability p=9%Between A and B, majority choose A option with a certain gain.
Between C and D, majority choose D option because prize appears higher and probability dierence appears negligeable.
We are not pure homo economicus : we deform probability. So only mathe- matics could not describe human behaviour and we need to understand par- tially psychology choice in comportemental economy. For example, Prelec For- mula [11] : w(p) =pγ/[pγ+ (1−p)γ]
1
γ with0,5<γ<1; 0<p<1in Kahneman
& Tversky try to modelise the human probability deformation perception.
Humans are risk averse : it's easier to discriminate a change between 3° and 6° than between 13° and 16° like it makes happier to increase gains from 100 to 200 than from 1100 to 1200 => concavity of utility function : we hypothesize that the value function for changes of wealth is normally concave above the reference point (v"(x) < 0, for x > 0) and often convex below it (v"(x) > 0, for x < 0)
First we will introduce concept of concave utility function for a typic ris- quophobe individual Second we will show through Prisonner's Dilemma that for a couple Pareto optimum is better than Nash equilibrium, in other words parteners win more being altruist and cooperative than egoïstic comportement does. Fird we will introduce Fréchet and Hausdor distance and nally show how Hausdor mean could give a good shape for household aggregation utility function.
Figure 2 Risk-lover vs Risk-averse
1 Concepts
Mazzocco [3] has identied several household models, all these models consi- der consumption, leisure and ressources, we are not trying to setup another better household model : we do not consider consumption or ressources we only focus on expected utility and how a couple combine their both utility curve to create a common couple utility curve. So we do not reject all literature about non-cooperative or cooperative household model but our approach is a bit dif- ferent, we'll try to resume briey what is a non-cooperative couple through Nash equilibrium example and what is a cooperative couple through a Pareto optimum example.
1.1 Nash equilibrium "Rock < Paper < Scissor"
This childy game do not admit "pure winning strategy" : play Rock, paper will win...
On the contrary, the "mixte strategy" consisting in playing at random one of each 3 choice with probability 1/3 constitute a Nash equilibrium.
1.2 Pareto optimum : "Self carafon"
Alice and Bob try to ll their carafon with a fountain which debit max is one carafon per minute
Precipitation Gallantry
Alice 2 min 1 min
Bob2 min 2 min
Total 4 min 3 min
Precipitation (try to ll two carafon simultaneously) does not lead to a Pareto optimum : we can minimise Alice waiting without increasing Bob one's.
1.3 Nash equilibrium vs Pareto optimum or "How to Co- operate ?"
Cooperative model of couple through Prisonner's Dilemma TalkSilence
Talk(-1, -1) (-2, 1)
Silence (1, -2) (0, 0)
Nash equilibrium : no one regret his choice
Pareto optimum : increasing one decrease the other We will considere that in cooperative couple model its members negotiate to reach Pareto optimum and therefore maximise common happiness
1.4 From cooperative household models to Hausedor mean
According to Priscille Touraille [10] if tall man genes are more common today on the earth it's for historical reason : in prehistory hunters male used to have a more easier access to proteines and women restricted themselves rst in starvation period to enable children to subsist. Today small women genes are more common but we are not still in prehistory and it seems wealthy to not do such a beastial dierence between men and women consumption : we can consider that today, the two members of a couple consume in the same way, those members could be either of the same sex and by the way the high heels price buy by women to be as tall as his man counterbalance beefsteak consumption dierence beetween two sex.
To study how to aggregate two utility function, we will need two asexual ctives characters : minuscule & MAJUSCULE which can either represent he- terosexual or same sex couple members. We will also suppose that we can let to prehistory models which try to distinguish minuscule's from MAJUSCULE's consumption pretexting than MAJUSCULE is bigger than minuscule and should therefore consume more. We will modelise our objectif in this following way : how from two curves (which could be therefore be utility curves) nd the curve resulting of the aggregation of these two last one ? We will in this way, in next session introduce a distance which can help us to solve the previous question as an optimisation problem. In the following next session we will consider to sim- plify explanation that minuscule and MAJUSCULE curve are trajectories and that more we are far away from our preferred trajectory more our personnal
"inconfort" is bigger.
2 Hausdor and Fréchet distances
Fréchet distance is the leach lenght allowinng to walk a dog in one way.
Hausdor distance is the leach length if backward move is allowed.
Figure 3 Fréchet - Hausdor
⇒For concave utility Fréchet or Hausdor distance are equivalent [4]
2.1 Distances
LetA andB be two given curves inS. Then, the Fréchet distance between AandBis dened as the inmum over all reparameterizationsαandβ of[0,1]
of the maximum over all t ∈ [0,1] of the distance in S between A(α(t)) and B(β(t)). Fréchet distance is F(A, B) =inf
α,β
t∈[0,1]max{d(A(α(t)), B(β(t)))}
LetX andY be two non-empty subsets of a metric space(M, d) Hausdor distance isH(X, Y) =max{sup
x∈Xinf
y∈Yd(x, y), sup
y∈Yinf
x∈Xd(x, y)}
Figure 4 Hausdor mean
The Fréchet distance is bounded from below by the Hausdor distance for any given pair of piecewise-linear curves (δF ≥δH) because for convex polygonal curves the Fréchet and Hausdor distances are equal while for other path geo- metries the Fréchet distance can become arbitrarily larger than the Hausdor distance.
2.2 minuscule & MAJUSCULE
minuscule & MAJUSCULE are walking in the space.
Figure 5 minuscule & MAJUSCULE Both of them have a prefered path.
If they travel together instead of walking alone they have to choose a common trajectory which minimise their "inconfort".
The "inconfort" of a point in space could be seen as the square of distance to "pleasant" (prefered) position : in each blue point, minuscule's "inconfort" is the square of distance to green curve.
NB : We could write more rigorous and formal equations but also heavier and boring if we want to demonstrate more formally that α=1/2, integrating
"inconfort" along curvilign abscisse which depends of time on each of the three curve, red, green, blue, the supposition α(t1) <α<α(t2) lead in integrate on neighbourhood of t1 and t2 to show that Pareto optimum is well reached in condition that blue curve should be "parallel" to red and green curve in the sense of parameterisation, it is equivalent to assume the existence ofαconstant such as "blue = "red" +α"green"
The last assumption : minimise α2+ (1−α)2 implies α = 1/2 correspond to the fact that we try to minimise the collective "inconfort" of minuscule and MAJUSCULE
Figure 6 The "inconfort :"
Let's demonstrate that α(t) = α
Assuming the existence of α(t1) < α < α(t2) we could therefore minimise minuscule's
"inconfort" without increasing MAJUSCULE one's
So in order to have a Pareto optimum in blue, it necessite α(t) =α
Minimisingα2+ (1−α)2implies α=1/2
Conclusion
Hausdor mean seems to be a good candidate to modelise couple preference aggregation through a common utility function.
We extend in this way reexion on collective model of household.
Paper limits :
We do not demonstrate such a utility function exists but simply show a plausible hypothetic theorical shape of this one.
We have just highlight a theorical shape for common utility of minuscule and MAJUSCULE who are two ctive characters. To validate the model, it should be necessary to led a complete study through questionnary like "Risk Tolerance" which would be asked to "single" individuals panel and then asked again when they get married... this issue lead to a huge scientic protocol dilemma, our result stay therefore purely theorical and to be experimentally validated.
Hausdor mean seems to be a better candidate to modelise preferences aggre- gation than a naïve arithmetic mean of individual curve constituting a couple.
Figure 7 Curve Aggregation 4D
Références
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[2] Pierre-André Chiappori. Rationnal household labor supply. Econometrica, 56 :6390, 1988.
[3] Pierre-André Chiappori and Maurizio Mazzocco. Static and intertemporal household decisions. Journal of Economic Literature, 55 :9851045, 2017.
[4] Carola Wenk Kevin Buchin, Maike Buchin. Computing fréchet distance between simple polygons. Elsevier, 41 :220, 2008.
[5] Benoit Mandelbrot. How long is the britain coast ? Science, 156 :636638, 1967.
[6] John Forbes Nash. Non-Cooperative Games. PhD thesis, Princeton, 1950.
[7] Roger N. Shepard. Mind Sights. Macmillan Learning, 1990.
[8] Joseph E. Stiglitz. Pareto ecient and the new new welfare economics.
Elsevier, 2 :9911042, 1987.
[9] Richard Thaler & Cass R. Sunstein. Nudge. Chicago & Harvard University, 2008.
[10] Priscille Touraille. Hommes grands, femmes petites : une évolution cou- teuse. Les régimes de genre comme force sélective de l'évolution biologique.
Editions de la Maison des Sciences de l'Homme, 2008.
[11] Daniel Kahneman & Amos Tversky. Advances in prospect theory. Econo- metrica, 5 :297323, 1992.
