propagation mechanisms in markets the case of the altruistic meme1 to share or not to share: and gur...
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Propagation Mechanisms in Markets
The case of the altruistic meme 1
To share or not to share:
and Gur Yaari
”Cast thy bread upon the waters: for thou shalt find it after many days. Give a portion to seven, and also to eight; for thou knowest not what evil
shall be upon the earth.”,Solomon , Ecclesiastes 11,1-2
Sorin Solomon
Racah Institute of Physics, Hebrew University , Jerusalem, Israel.
Lagrange Complexity Lab , ISI Foundation, Turin, Italy.
that is the question!
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Propagation Mechanisms in Markets
The case of the altruistic meme 2
Altruism:
What is it??
Selfless concern for the welfare of others...
1) it is directed towards helping another,
2) it involves a high risk or sacrifice to the actor,
3) it is accomplished by no external reward,
4) it is voluntary.
How come it is evolutionary stable strategy?
Group selection,
Kin selection,
Reciprocity
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Game theory is against it
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22
0
0LOYAL
LOYAL
DEFECT
DEFECT
-1
-1
1
-2
1
-2
1 sentence reduced by 1 year
(or gain 1 M $)
0 nothing changes
-1 add one more year in prison
(or lose 1M $)
-2 add 2 more years in prison
(or lose 2 M $)
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Propagation Mechanisms in Markets
The case of the altruistic meme 5
Reality has larger gains and worse risks:
EXTREME EXAMPLE
each agent is undergoing, at random times the following dynamics:
Xi(t+1)=
0 with probability 1/2
3·Xi(t) with probability 1/2
i = 1..N
On average, the agents are suppose to grow like (3/2)T
However, each agent (with probability 1) will get to 0 with average life time of 2 steps...
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Propagation Mechanisms in Markets
The case of the altruistic meme 6
In A-synchronous updating mechanism even a group of two sharing individuals
will NEVER reach 0!And will grow exponentially faster then (3/2)T
Caricature of the paradox:
What happens if the agents can share their wealth after each iteration ?
In a synchronous updating mechanism, the average life time of each of the agents
in a group of N sharing individuals will multiply by 2N-1
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Propagation Mechanisms in Markets
The case of the altruistic meme 7
Let us consider the more general stochastic process:
Xi(t+1)=
a·Xi(t) with probability p
b·Xi(t) with probability q=1-p
i = 1..N
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Propagation Mechanisms in Markets
The case of the altruistic meme 8
<Xi(t+1)>
i=(p·a+(1-p)·b)·<X
i(t)>
i
naïve calculation:
Which leads to:
<Xi(T)>
i=(FT)·<X
i(0)>
i
{F≡Arithmetic mean
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Propagation Mechanisms in Markets
The case of the altruistic meme 9
<Xi(t+1)>
i=(p·a+(1-p)·b)·<X
i(t)>
i
naïve calculation:
Which leads to:
<Xi(T)>
i=(FT)·<X
i(0)>
i
{F≡Arithmetic mean
IS THAT SO ???
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Propagation Mechanisms in Markets
The case of the altruistic meme 10
...Only for exponentially large number of agents:
i.e. : M ~ eAT
For example: p=q=0.5, a=2, b=1/3, T=100
time evolution
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Propagation Mechanisms in Markets
The case of the altruistic meme 12
What is going on??
As some of you may know:
for long times: the typical value dominants
(the expectation of the log of the wealth)
Xi(T)=aW·bL=T-W·X
i(0)=
Xi(0)·eW·log(a)+L·log(b) =
Xi(0)·e{w·log(a)+l·log(b)}·T =
→Xi(0)·e{p·log(a)+q·log(b)}·T ≡X
i(0)·(ap·bq)T
w≡W/T
l≡L/T
T→∞ {r≡Geometric mean
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Propagation Mechanisms in Markets
The case of the altruistic meme 13
What is going on??
As some of you may know:
for long times: the typical value dominants
(the expectation of the log of the wealth)
U= p·log(a)+q·log(b)
Utility function; Morgenstern and Von Neumann
Bernoulli, Saint Petersburg
→Xi(0)·e{p·log(a)+q·log(b)}·T
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Propagation Mechanisms in Markets
The case of the altruistic meme 14
Partial summary:
If one takes the limit N→∞ then the arithmetic mean
determine the mean behaviour of the system.
If the limit T→∞ is taken first, then the geometric mean sets the mean dynamics of the system.
What we may remember from hi-school is that
arithmetic mean ≥ geometric mean
In this work we focused on the case where:
arithmetic mean >1> geometric mean
which brings the problem to a matter of life and death !!
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Propagation Mechanisms in Markets
The case of the altruistic meme 15
How could the arithmetic mean be restored without having exponential number of realizations??? (could it?)
The answer is ....YES – by sharing:
The average (over time) of the growth rate of N sharing individuals could be calculated to be:
It can be shown to behave like rN- log(F)=o(1/N)
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Propagation Mechanisms in Markets
The case of the altruistic meme 16
Partial summary:
In cases discussed before
(arithmetic mean >1> geometric mean)
There exist Ncrit
Which distinguishes between LIFE and DEATH
This make this altruistic meme a stable evolutionary strategy!!
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Propagation Mechanisms in Markets
The case of the altruistic meme 17
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Propagation Mechanisms in Markets
The case of the altruistic meme 18
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Propagation Mechanisms in Markets
The case of the altruistic meme 19
What would have J.Kelly though about all this?
The Gambler problem: invest 1$ and in case of winning one gets (d+1)$. In case of loosing- null.
Xi(t+1)=
(1+f·d)·Xi(t) with probability p
(1-f)·Xi(t) with probability q=1-p
When f=1 , the geometrical mean is 0
Kelly introduced myopic (static) strategy:
Do not risk all you have....(f≠1)
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Propagation Mechanisms in Markets
The case of the altruistic meme 20
P=0.55,
d=1
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Propagation Mechanisms in Markets
The case of the altruistic meme 21
A-synchronous: how should we approach it?
keeping a fraction (N-1)/N in a safe place:
i.e. In Kelly's terminology:
f→f/N
t→t∙N
Calculating the growth rate for N>>1 A-synchronous sharing individuals gives us result that is actually
BETTER than the arithmetical mean!!
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Propagation Mechanisms in Markets
The case of the altruistic meme 22
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Propagation Mechanisms in Markets
The case of the altruistic meme 23
Limited Generosity:
If one decides to share only a fraction D out of the difference between it's wealth and the average wealth,
what happens?
In formula:
After each “reaction step” we make diffusion step:
Xi(t+1)-X
i(t)=D·{<X
i(t)>
i-X
i(t)}
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Propagation Mechanisms in Markets
The case of the altruistic meme 24
Limited Generosity:
If one decides to share only a fraction D out of the difference between it's wealth and the average wealth,
for N>Ncrit
there exist Dcrit
!
p=q=0.5, a=2, b=1/3, N=4
Dcrit
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Propagation Mechanisms in Markets
The case of the altruistic meme 25
Limited Generosity:
What happens when you have politicians that are taking a fraction of whatever one intended to give to
others ??
An optimal level of generosity appears (≠1):
Tell me the level of corruption in society and I'll tell you the level of generosity you have to have..
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Propagation Mechanisms in Markets
The case of the altruistic meme 26
Conclusions:
We offer a solution to the altruism stability paradox by noticing that it is in the selfish interest of oneself to donate to his/her peers and by this to ensure that he will not be alone in this hostile environment called life.
This “Selfish Altruism” is not distinguishable behaviourally from “pure” altruism and as such could be evolutionary stable !
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Propagation Mechanisms in Markets
The case of the altruistic meme 27
Scope:
Show that Altruism could be also an EMERGENCE property of life by simulating a system with genetic information of level of altruism (D) and desirable group size (N) and show that free riding will not pay off as it will destroy the group: Group Selection (back to Darwin..)
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Propagation Mechanisms in Markets
The case of the altruistic meme 28
Thank You very much
Grazie mille
תודה רבה
NAGYON KÖSZÖNÖM
dunke schoen
BlagodaryaDziekujeMultumesc
Gracias
Shukriya Arigato
Tika hoki Spasibo
merci beaucoup
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Propagation Mechanisms in Markets
The case of the altruistic meme 29
Caricature of the paradox:
an agent is foll
a=3 , b=0 , p=1/2
a single agent would decay to zero in 2 steps on average...
for 10 sharing individuals it will take 1000 steps on average...each additional agent will contribute a factor of
2=1/p to the average life time.
Could one do better?? YES-
When the time updating is A-synchronous even TWO sharing individuals will NEVER get to zero.