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The question we'll discuss is less general then could be expected, we shall not consider psychological or theological issues, rather we will discuss problems such as the famous "prisoner dilemma" or more common situation such as
 

Where player 2 can co-operate with player one , in which case he'll pay the small amount of b and player 1 will get the a much bigger prize a. Such situation are numerous for example player 2 asked player 1 for the time , player 2 decided to waste time commenting to player 1 that his shoe laces are open etc. Those situation appear in economical , biological and sociological issues.

It is quite obvious that whenever an individual is encountered with such situation natural selection should favour individuals who choose defect over those choosing co-operation (by natural selection I do not only mean it in the Darwinian meaning , but also in economical and sociological meaning (i.e. more people will imitate the more successful method that is the non-cooprative one.)
yet, it is quite clear that altruism do exist in nature. We see it in both economical world ,  where companies form cartel instead of competing in perfect competition, in biological world for example where other birds raise cockoo hatchlings, in every day to day life and in numerous other examples.

Common answers to the altruism question include

1. kin selection - selfish gene (W.D. Hamilton)
2. group selection - Evolution in structured population , deme structure. (Eshel 1972 , S. Wilson)
3. reciprocal altruism , repeated games (Treeverse Axelrod)
4. Non Altruism

Non Altruism

None altruism refer to cases where it may seem like there is an altruism but the situation is either quite the opposite or can be explained in other examples...
(this , in quite humorous way , may include politician claming they helped the needed while requesting people to vote for them.... ;-p)

Examples include raising other cockoo hatchling by other birds not caused by the other bird love for cockoos but rather its own stupidity and lack of ability to recognize their own eggs.
Infect it is seen that if an egg is recognized as a cockoo egg it will be thrown away from the nest and cockoos are subject to evolution of their eggs making them more similar to the host eggs.

Another explanation offered by A. Zehavi is adding altruism as an handicap in the "handicap principal" theory (for example if i choose to teach you , its not because I am an altruist rather , I wise that other individuals , having to choose between me and you will prefer me as the teacher.)

There are many other situation were seemingly altruism is proved to be egotism or stupidity however
this doesn't seem to cover all the cases. (I hope)

Kin Selection

Lets look at parents helping their children, this may be considered altruism however it is much more egotistic from the gene point of view then meets the eye.

When a parents is giving food to his child he may be depriving himself from that food but he gives this food to his children who have 50% percent of holding the same genes as he do (other 50% came from other parent)

Same argument applies to helping brothers and/or sisters (they also have 50 % of one own genes), to bees and their queen and to numerous other situation a gene might run into. i.e. a gene causing me to sacrifice myself to help sibling and\or kins will prosper due to the fact that it has very high probability to appear in the ones I am helping. for examples if a gene says if you see A drawning after jumping to the water to save B , jump to the water to save A will have very good chances to survive in the genetic pool. That argument deals with individual as machines totally controlled by genes only wishing to reproduce themselves. That kind of thinking deal with the fact "inclusive fitness" i.e. not only my own fitness and chances of survivel but the inclusive fitness of all duplicate of the gene. While those action may appear as altruism again this is not altruism from the gene point of view and though this is very interesting arguement it is not one I wise to deal with. A good source to read about the selfish gene theory is Richard Dawkins book the selfish gene.

Reciprocal altruism

This theory was offered by Axelrod and it deals with situation where the same individuals meet over and over having to take action lets say I am about to play unknown finite amount of games (those games will be simple case of the prisoner dilemma) with somebody (infinite amount of games will yield unwanted consequences of infinite sum, moreover the amount of games has to be unknown otherwise , the last game is trivial (can be treated as a single game) making the one before him trivial and so on and so forth)

this games can be described using the following matrix
 

side 1  \ side 2	co - operation	defect
co-operation	(3 , 3)	(0 , 5)
defect	(5 , 0)	(1 , 1)

no since we play more then one game , it can be advisable for you to "trust me" just once.
and even "forgive me" if i defected you once or twice (though no more then that!) this will yield
bigger prise for the both of us. a good strategy in that field is "Tit for tat" always co-operate at first , afterward do just what your oponent did the round before. (this strategy is both "nice" and "forgiving") though this strategy may very well lose to strategys such as always defect , when meeting other copies of itself (and other "nice" strategys) it will prosper
 

this explains why it is common to sometimes we favour those we know (and those we know that will return the favour) over those we don't know.
 

Group selection

This does not refer to selection of species, but rather this refer to selection between groups of the same spicy.

This means the population isn't "well mixed" and the individual doesn't meet other individuals at random, rather they tend to meet the same individuals over and over again, this gives altruist higher odds to meet other altruist and thus altruists prosper.

An Example to this is "deme structure" population where individuals belong to specific deme and are more likely to meet other individuals from the same deme they belong too rather then individuals from other demes.

This allows altruists gain from meeting other altruist making them favourable in some situation

My Model

My model deals with special case of deme structure.
In my model Individuals leave in their deme and get certain utility which is affected by the following factor
 

• their level of recognition with the deme (are they familiar with the place or are they newcomers? newcomer gain less utility from many reason from xenophobia to lack of familiarity with their surrounding. newcomers become familiar with their surrounding after time an exponential amount of time.)
• the amount of altruists in the deme
• the amount of egotist in the deme
• being or not being an altruist

the demes receive new immigrants all the time , however , when a deme reaches certain size it explodes to two small demes.

notes

• nobody dies or born in the model all evolution is done by persuading.
• everything takes place on continues time. no two action happen at the same time.
• the current action (changing familiarity status , leaving a deme , arriving to deme , changing ones opinion (from egotist to altruist) , etc. ) is chosen at random from all possible action. each action has its passion power , each action has the probability of its own passion power divided by the sum of passion powers.