Cooperate or Not? The Nash Equilibrium and the Shapley Value



Nash Equilibrium

"The Nash equilibrium is a stable state of a system that involves several interacting participants in which no participlant can gain by a change of strategy as long as all the other participants remain unchanged."


Make 4 dollars or 5 dollars?
I'll explain it without the traditional prisoner story.

Imagine someone approaches you, as well as a random stranger, on the street. He says to the two of you:

If you both give me a dollar, I will give each of you $5. But if only one of you gives me a dollar, I will keep that dollar, and give $5 only to the person who gives me nothing. You aren't allowed to talk to each other.

Of course, if you could talk to the other player, you would surely agree that you both will give the guy a dollar, and each walk away $4 richer than you were. But since you can't, all you can do is figure out what is best for yourself. And regardless of what the other player chooses, it is best for you to give nothing.

If the other person gives a dollar, you are better off giving nothing -- as you'd make $5 profit rather than only $4. And if he gives nothing, you'll lose nothing, rather than losing a dollar.

The problem, of course, is that both players are likely to think this way, and each give nothing. This results in a lost opportunity to make some free money.

This kind of situation is everywhere in the real world. Rational people make choices that seem (to the Prisoner's-Dilemma naive) to be non-rational. They are not being stupid, they make these choices because it is in their interest to do so.....purely due to inability to make and enforce agreements.


ECONOMICS
In economics, the Prisoner's Dilemma can be used to understand advertising.

When two businesses are competing against each other, say Starbucks and San Francisco Coffee, an essential part of the competition is advertising. The Prisoner's Dilemma is a perfect way to understand companies' incentives behind advertising. Using the numbers above, if both companies don't advertise, then they gain money, as opposed to losing money due to advertising. However, if one company decides to advertise and the other holds back, then the company who advertises profits greatly. The equilibrium option of the game is that both companies advertise, as other options are unstable.

The Prisoner's Dilemma is also involved with a simple supply-demand model. Say that a tiny village in the middle of nowhere has two water vendors. Each vendor wants the business of all of the village's potential customers, and so they compete to lower the price. It shapes up into a "Prisoner's Dilemma"-esque nightmare that benefits none but the consumer.

TREATIES AND POLITICAL RELATIONS
The Prisoner's Dilemma was originally developed to explain the Cold War, and why neither side was willing to back down.

During the Cold War, NATO and its opponent the Warsaw Pact both had the choice to arm or disarm. From each side's point of view, disarming while your opponent continues to remain armed was extremely dangerous, leading to military inferiority and potential annihilation. However, if both sides remained armed, neither side would risk an attack, but both would rack up high costs developing and maintaining a large nuclear arsenal. The equilibrium and optimal option would be for both to disarm, as war would be avoided and no one would have to pay. However, the likelihood is that one country would arm again, thus gaining the advantage and disrupting equilibrium.

So although from a bird's eyes view, the decision for both sides to disarm would have been the best overall outcome, the rational, selfish course for both sides is to increase their arsenal. This simple but fundamental tenet of human behavior defined three decades of economic and military waste that to this day, remains one of many blemishes on human history.

It's relatively simple, then, to apply this logic to any treaty, say, the Kyoto protocol. It's better overall if every country reduces their CO2 emissions, but the individual benefits of maintaining the status quo are perceived to be greater than the benefit to all countries if the emissions were changed. This is part of the reason why we're in a 'climate gridlock', and not making significant progress on the reduction of greenhouse gas emissions.

BIOLOGY
One of Darwin's biggest challenges was explaining the coexistence of altruism and evolution. If selection was really on an individual level, then altruism cannot and would not make sense. But universal selection contradicts the mathematics of game theory and is not often found in nature.

The solution to the altruism paradox can be found in the Prisoner's Dilemma. In evolutionary game theory, the Prisoner's Dilemma is analyzed as a repetitive game, so as to allow players to retaliate for bad behavior / defection in previous rounds of the game. The best competitive strategy, according to extensive study, is general cooperation with occasional retaliatory responses. This can expressed through the famous "Tit for Tat" algorithm.

The Prisoner's Dilemma can help us understand the Tit-for-Tat strategy, and thus the idea of cooperation and altruism in evolving populations.

In conclusion, the Prisoner's Dilemma is useful in a variety of fields, and can model and predict human behavior very well.


Shapley Value



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