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Nash Equilibrium
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A Nash Equilibrium is a set of mixed strategies for finite, non-cooperative games between two or more players whereby no player can improve his or her payoff by changing their strategy. Each player's strategy is an 'optimal' response (cf. optimality) based on the anticipated rational strategy of the other player(s) in the game.
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The formulation of the Nash Equilibrium shows that its solution is simply a maximum of a certain payoff function. This fact makes it solvable by basic numerical gradient based methods or gradient free methods. Both methods require that two conditions are met by the payoff function at a stationary point .
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[I]n a scenario where collaboration exists, another Nash Equilibrium is possible - one of mutual collaboration. That this equilibrium exists, and is better, is clear - Korea and Japan have a collaborative model across the value chain.
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While the concept of Nash equilibrium is central to economic theory, to date no one has provided a credible model of how economic agents come to play such equilibria. In particular, there is no model of how agents acting in their own self-interest--without any form of coordination--can learn equilibrium behavior starting from out-of-equilibrium conditions. This paper proposes a simple learning process with this property. Agents learn about their opponents by periodically testing and rejecting alternative hypotheses about their behavior. If they use sufficiently powerful tests and their responses are sufficiently close to being optimal, this process eventually leads them to play close to equilibrium a large proportion of the time.
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To get a sense of how the Nash equilibrium works, consider the following scenario. Twenty people are dining at a restaurant that offers a luxurious $20 meal and a simpler $10 meal. The diners have agreed that the group will split the bill evenly. Suppose each person, if he had to pay only his own bill, would prefer the $10 meal. Despite these preferences, it is not a Nash equilibrium for everyone to choose the cheap meal, since for each individual, switching to the expensive meal will cost him only 50 cents more. The only Nash equilibrium is for all the players to choose the $20 meal.
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The Nash equilibrium is, again, the cyan cell where each confesses. That way, if you change your strategy (from "confess" to "don't confess"), your sentence is worse: It goes from four years to five years.
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Nash Equilibrium