Quote from: twiifm on April 10, 2015, 06:00:05 PM

Nash equilibrium isn't some reversion to the mean or anything like that. Nash equilibria is the most probable move given the rules and each players strategy

The Nash equilibrium is also the strategy that is KEPT if the game is played over and over, when entities learn about others' strategies. This is why it is called an equilibrium. If the game is played over and over again, and you assume that the others' strategies are those they took in the last round, and everybody, using this information, updates his own strategy for maximum gain, only if a Nash equilibrium is reached, the strategies will remain the same from round to round.

Now of course that makes the hypothesis of steady state of all the game conditions, which is not true in the real world, and of course you can and will have evolution. But we make the simplification of steady state game conditions.

The Nash equilibrium is to games, what thermodynamic equilibrium is to a physical system with many degrees of freedom. Of course, if the external conditions change, the thermodynamic equilibrium will shift. And if the dynamics is fast enough, you will be off the equilibrium.

But we are considering here "models of society" which comes down to "sets of strategies adopted by individual entities".

If a whole set of individuals makes choices of strategies in a given context, some individuals will find that they gain, others will find that they will lose, and "the next round" people will adapt their strategies as a function of the observations of the previous round and previous people's strategies. As such, the set of strategies will change at each round.... unless it reaches a Nash equilibrium. At that point, every individual will find out, for himself, that he should adopt the same strategy as the previous round, because (assuming that others also act that way), any OTHER choice would bring him less advantage.

Simplistic example with Prisoners Dilemma: Joe and Jack are interrogated. Suppose Joe and Jack first don't betray each other. Both of them are sentenced 1 year. Joe now thinks "damn, I should have betrayed Jack, he didn't betray me, so I would have been free". Jack now thinks the same. Next time they are interrogated, they betray each other. They both take 2 years. And now Joe thinks: "ah, Jack betrayed me. I have nothing to win by not betraying him. Next time I'll betray him again". And Jack thinks the same. ---> Nash equilibrium reached ! If you play it over and over again, Joe and Jack will continue betraying each other.

It is very funny, because in the beginning, they were actually applying an Utopia Optimum (if I can call it that way). But the Utopia Optimum (the lowest sentence for both) was not stable against "personal improvement" of each individual. They ended up in a Nash equilibrium, that is below the Utopia Optimum, but there is no incentive any more for each of them to go away from it.

Life is worse in the Nash equilibrium than on the Utopia Optimum, but it wasn't stable against "egoistic optimisation" which turned out not to be one.

This is exactly the same problem of Communism and Socialism, btw. If everybody would work together for the best of common good, then we would have a higher economic production than in a free market (where resources are "wasted" to competition). Everybody would be happier and richer than in the case of competitive free markets.
If everybody works together to plant potatoes and elevate chickens to the best of his capacity, there would be a lot of food for everybody.
But the problem is that everybody can change strategy individually. Joe, in his Colchoze, will find out that if HE lies in the sun while his fellow comrades plant potatoes, he will have about just as much food as before, and life is much nicer lying in the sun than planting potatoes. Jack will think the same. Mary too.
And in the end, everybody lies in the sun, and then everybody is hungry. But there's no reason for any of them to change strategy. There's nothing personal to be won to be the only one NOT lying in the sun and planting potatoes for everybody. So the Nash equilibrium of Communism is reached: everybody lies in the sun and does nothing, while everybody is hungry.
This is more or less the real story behind colchozes btw. The 5% private land produced as much as the 95% communist land in the end.
In social democraties, you get a similar problem with social redistribution and things like unemployment payments and so on, if they are too generous.