In a society of completely selfish individuals where everybody is only interested in maximizing his own payoff, does any equilibrium exist for the society? John Nash proved more than 50 years ago that an equilibrium always exists such that nobody would benefit from unilaterally changing his strategy. Nash Equilibrium is a central concept in game theory, which offers a mathematical foundation for social science and economy. However, it is important from both a theoretical and a practical point of view to understand game playing where individuals are less selfish. This paper offers a constructive generalization of Nash equilibrium to study n-person games where the selfishness of individuals can be defined at any level, including the extreme of complete selfishness. The generalization is constructive since it offers a protocol for individuals in a society to reach an equilibrium. Most importantly, this paper presents experimental results and theoretical investigation to show that the individuals in a society can reduce their selfishness level together to reach a new equilibrium where they can have better payoffs and the society is more stable at the same time. This study suggests that, for the benefit of everyone in a society (including the financial market), the pursuit of maximal payoff by each individual should be controlled at some level either by voluntary good citizenship or by imposed regulations.
Deep Dive into A Constructive Generalization of Nash Equilibrium for Better Payoffs and Stability.
In a society of completely selfish individuals where everybody is only interested in maximizing his own payoff, does any equilibrium exist for the society? John Nash proved more than 50 years ago that an equilibrium always exists such that nobody would benefit from unilaterally changing his strategy. Nash Equilibrium is a central concept in game theory, which offers a mathematical foundation for social science and economy. However, it is important from both a theoretical and a practical point of view to understand game playing where individuals are less selfish. This paper offers a constructive generalization of Nash equilibrium to study n-person games where the selfishness of individuals can be defined at any level, including the extreme of complete selfishness. The generalization is constructive since it offers a protocol for individuals in a society to reach an equilibrium. Most importantly, this paper presents experimental results and theoretical investigation to show that the indi
John Nash has proved in 1950 [1] using Kakutani fixed point theorem that any n-player normal-form game [2] has at least one equilibrium. In the game, each player has only a finite number of actions to take and takes one strategy at action playing. If a player takes one of the actions in a deterministic way, it is called a pure strategy. Otherwise, if a player takes anyone of the actions following some probability distribution defined on the actions, it is called a mixed strategy. At a Nash equilibrium, each player has chosen a strategy (pure or mixed) and no player can benefit by unilaterally changing his or her strategy while the other players keep theirs unchanged.
Nash Equilibrium is arguably the most important concept in game theory, which has significant impacts on many other fields like social science, economy, and computer science. It is an important theory for understanding a common scenario in game playing.
In a Nash equilibrium, each player’s strategy is completely selfish because the player is only interested in maximizing his own payoff. Only the best action(s) to each player is accepted by the player, sub-optimal actions are not considered at all. The best action is defined as the one with the highest payoff. As a consequence of the selfishness, even if the payoff of a sub-optimal action is slightly less than the best one, the probability of picking this sub-optimal action by the player is still zero.
However, many cultures teach people to be less selfish in a society. Also, the scenario of less-selfish players may be closer to reality, such as individuals in human societies or animal kingdoms. Our conventional wisdom tells us that if each of us gives away a bit more in favor of others, we could end up with more gains as return. That is, reduced selfishness leads to better payoffs for the individuals in a society. For instance, if we, as drivers, respect other drivers sharing the same road and give considerations for each other either voluntarily and/or by following traffic laws, then each of us will end up with a faster, safer drive to his/her destination than the case when everyone is only interested in maximizing his own speed to his destination.
This paper presents both experimental results and theoretical investigation to show that, if the individuals in a society reduce their selfishness by simply accepting sub-optimal actions in some degree, a new equilibrium can be reached where better payoffs and social stability are obtained at the same time.
The first key observation of this paper is that, reducing selfishness can improve payoffs. When completely selfish players at a Nash equilibrium reduce their selfishness, they will shift to a new equilibrium with payoffs possibly better than the original one. The observations will range from the classic prisoner’s dilemma, a hard game used in other game theory literatures, to computer generated games with hundreds to thousands of players. It verifies the conventional wisdom that reducing selfishness could lead to better payoffs for everyone.
The second key observation is that, reducing selfishness can also improve social stability. A society of completely selfish individuals can be very sensitive to perturbations, the accuracy at representing individuals’ utility functions, and communication errors among the individuals in the society. The smallest change in utility function or the slightest communication error could knock the individuals out of their existing equilibrium. Furthermore, a society of completely selfish individuals can have an enormous number of equilibria. The number may increase exponentially with the population of the society, The society could end up with one Nash equilibrium or another, depending on the initial conditions and sensitive to perturbations. If the individuals reduce their selfishness together, they can reduce their sensitivity to perturbations, inaccuracy in utility functions, and communication errors. At the same time, the number of equilibria tends to drop significantly so that the outcome of the society can be more predictable. When the selfishness is below a certain level, the society tends to have only one equilibrium and converges to it with any initial conditions.
In particular, this paper gives a mathematical model for describing selfishness. The level of selfishness is controlled by one parameter of the model to cover the spectrum ranging from complete selfishness to complete selfishlessness. With the parameterized selfishness model, this paper offers a generalization of Nash equilibrium together with a proof of the existence of an equilibrium given any selfishness level using a fixed point theorem. It is a generalization because this paper offers a proof to show that a generalized equilibrium at the particular case of completely selfish players falls back to a Nash equilibrium. In other words, the definition of Nash equilibrium is a special case of the generalized one. It is important to note that
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