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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.

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slide1
The case of the altruistic meme

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

slide2
The case of the altruistic meme

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

slide4

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

-2

-1

DEFECT

1

-1

slide5

Reality has larger gains and worse risks:

EXTREME EXAMPLE

each agent is undergoing, at random times the following dynamics:

The case of the altruistic meme

i = 1..N

0 with probability 1/2

Xi(t+1)=

3·Xi(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...

slide6
The case of the altruistic meme

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

slide7
The case of the altruistic meme

Let us consider the more general stochastic process:

i = 1..N

a·Xi(t) with probability p

Xi(t+1)=

b·Xi(t) with probability q=1-p

slide8
The case of the altruistic meme

naïve calculation:

<Xi(t+1)>i=(p·a+(1-p)·b)·<Xi(t)>i

{

F≡Arithmetic mean

Which leads to:

<Xi(T)>i=(FT)·<Xi(0)>i

slide9
The case of the altruistic meme

naïve calculation:

<Xi(t+1)>i=(p·a+(1-p)·b)·<Xi(t)>i

{

F≡Arithmetic mean

Which leads to:

<Xi(T)>i=(FT)·<Xi(0)>i

IS THAT SO ???

slide10
The case of the altruistic meme

...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

slide12
The case of the altruistic meme

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)

w≡W/T

l≡L/T

Xi(T)=aW·bL=T-W·Xi(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 ≡Xi(0)·(ap·bq)T

{

T→∞

r≡Geometric mean

slide13
The case of the altruistic meme

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)·e{p·log(a)+q·log(b)}·T

U= p·log(a)+q·log(b)

Utility function; Morgenstern and Von Neumann

Bernoulli, Saint Petersburg

slide14
The case of the altruistic meme

Partial summary:

If one takes the limitN→∞then thearithmetic mean

determine the mean behaviour of the system.

If the limitT→∞ is taken first, then thegeometric meansets 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 !!

slide15
The case of the altruistic meme

How could thearithmetic 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)

slide16
The case of the altruistic meme

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!!

slide19
The case of the altruistic meme

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.

(1+f·d)·Xi(t) with probability p

Xi(t+1)=

(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)

slide21
The case of the altruistic meme

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!!

slide23
The case of the altruistic meme

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)-Xi(t)=D·{<Xi(t)>i-Xi(t)}

slide24
The case of the altruistic meme

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

slide25
The case of the altruistic meme

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..

slide26
The case of the altruistic meme

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 !

slide27
The case of the altruistic meme

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..)

slide28
The case of the altruistic meme

Blagodarya

Multumesc

Dziekuje

Grazie mille

NAGYON KÖSZÖNÖM

Gracias

Thank You very much

Tika hoki

Spasibo

dunke schoen

תודה רבה

Shukriya

Arigato

merci beaucoup

slide29
The case of the altruistic meme

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.

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