To share or not to share:

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To share or not to share
that is the question!

Sorin Solomon
and Gur Yaari
Racah Institute of Physics, Hebrew University ,
Jerusalem, Israel. Lagrange Complexity Lab , ISI
Foundation, Turin, Italy.
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
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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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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 )
LOYAL
DEFECT
1
0
22
LOYAL
-2
0
-1
-2
DEFECT
1
-1
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Reality has larger gains and worse risks EXTREME
EXAMPLE each agent is undergoing, at random times
the following dynamics

i 1..N
0 with probability 1/2
Xi(t1)
3Xi(t) with probability 1/2
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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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
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
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Let us consider the more general stochastic
process

i 1..N
aXi(t) with probability p
Xi(t1)
bXi(t) with probability q1-p
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naïve calculation
ltXi(t1)gti(pa(1-p)b)ltXi(t)gti

FArithmetic mean
Which leads to
ltXi(T)gti(FT)ltXi(0)gti
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naïve calculation
ltXi(t1)gti(pa(1-p)b)ltXi(t)gti

FArithmetic mean
Which leads to
ltXi(T)gti(FT)ltXi(0)gti
IS THAT SO ???
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...Only for exponentially large number of
agents i.e. M eAT For example pq0.5,
a2, b1/3, T100

time evolution
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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)

wW/T lL/T
Xi(T)aWbLT-WXi(0) Xi(0)eWlog(a)Llog(b)
Xi(0)ewlog(a)llog(b)T ?Xi(0)eplog(a)
qlog(b)T Xi(0)(apbq)T

T?8
rGeometric mean
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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(0)eplog(a)qlog(b)T
U plog(a)qlog(b) Utility function
Morgenstern and Von Neumann Bernoulli, Saint
Petersburg
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Partial summary If one takes the limit N?8 then
the arithmetic mean determine the mean behaviour
of the system. If the limit T?8 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 gt1gt geometric mean
which brings the problem to a matter of life and
death !!
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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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Partial summary In cases discussed before
(arithmetic mean gt1gt geometric mean) There exist
Ncrit Which distinguishes between LIFE and
DEATH This make this altruistic meme a stable
evolutionary strategy!!

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18


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What would have J.Kelly though about all this?
The Gambler problem invest 1 and in case of
winning one gets (d1). In case of loosing- null.
(1fd)Xi(t) with probability p
Xi(t1)
(1-f)Xi(t) with probability q1-p
When f1 , the geometrical mean is 0 Kelly
introduced myopic (static) strategy Do not risk
all you have....(f?1)
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P0.55, d1
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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?tN Calculating the
growth rate for Ngtgt1 A-synchronous sharing
individuals gives us result that is actually
BETTER than the arithmetical mean!!
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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(t1)-Xi(t)DltXi(t)gti-Xi(t)

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Limited Generosity If one decides to share only
a fraction D out of the difference between it's
wealth and the average wealth, for NgtNcrit there
exist Dcrit !


pq0.5, a2, b1/3, N4
Dcrit
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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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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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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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Blagodarya
Multumesc
Dziekuje

Grazie mille
NAGYON KÖSZÖNÖM
Gracias
Thank You very much
Tika hoki
Spasibo
dunke schoen
???? ???
Shukriya
Arigato
merci beaucoup
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Caricature of the paradox an agent is foll a3 ,
b0 , p1/2 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
21/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.