Nash Equilibrium Formal Definition Essay

Let denote the set of strategies for the th player, and denote the set of strategy profiles. This means that the elements of are all possible combinations of individual strategies. Let denote the payoff to player when evaluated at strategy profile ; note that the payoff to an individual player depends on the strategies of the other players as well.

An individual mixed strategy is a probability distribution on the set of available strategies. For example, selecting one of "rock", "paper", or "scissors" uniformly at random is an example of a mixed strategy. There can also be a choice of weighting so that strategies are picked with different probabilities. A pure strategy is one that does not involve randomization at all, instead choosing a particular strategy all of the time. Note that, importantly:

A pure strategy is simply a special case of a mixed strategy, in which one strategy is chosen 100% of the time.

Which means that the same methods used to calculate mixed strategies are equally useful in detecting pure strategies.

A Nash equilibrium is a strategy profile with the property that

for all , where denotes a strategy other than available to player .

In the event that this inequality is strict; i.e.

for all , the profile is called a strict Nash equilibrium. Otherwise, is called a weak Nash equilibrium.

Nash's existence theorem guarantees that as long as

Probability >

A Nash Equilibium in game theory is a collection of strategies, one for each player in a social game, where there is no benefit for any player to switch strategies. In this situation, all players the game are satisfied with their game choices at the same time, so the game remains at equilibrium. Following this, where the natural world is governed by the laws of physics, the social world is governed by the Nash equilibrium. The concept is named after the American Mathematician John Nash, who won the 1994 Nobel Memorial Prize in Economic Sciences for his work on game theory.

In any equilibrium, all sides are in a state of no-change. For example, in a chemical equilibrium, quantities of chemicals don’t change. When equilibrium is reached in a game, whatever “items” are in the game are also in a state of non-change, as all players have reached individual points of maximum benefit. Following this, when there is a mutual equilibrium, there isn’t any incentive for any of the game players to change strategies. As long as all players are satisfied with their place in the game and their strategy. The quantities involved remain stable as long as none of the other players change strategy.

For modeling any social situation, it works like this:

  • Identify a social situation you want to find an equilibrium for.
  • Devise a game with x players that behaves the same way as the social situation.
  • Find the equilibrium point for the game (every game has one), and then apply it to the social situation.

Every multiple player game has a Nash equilibrium point as long as the number of players isn’t infinite. It’s important to realize that the Nash equilibrium only describes a point of mutual equilibrium. It doesn’t predict how people will behave. Nor does it correspond to outcomes that are most efficient. In fact, alternative outcomes might exist that are equally feasible and preferred by all players in the game (Sethi, 2008).

Simple Example #1: The Crumpled Map

Crumpled Map of Rome. Martijn van Exel | Flickr CC

Siegfried (2006) offers a simple example that skirts around the dense mathematics required to understand the proof of the game. Take a map (any map will do, but let’s say you have a map of Rome). Place the crumpled paper anywhere in Rome, and there will be one point on the map that matches the exact location on the crumpled map.

Simple Example #2: The Prisoner’s Dilemma

In the real world, people don’t always reach that mutually beneficial point. Because of the prisoner’s dilemma is a famous example of why two completely “rational” individuals fail to reach an equilibrium point. This is even if it appears that it is in their best interests to do so. Following this, the prisoner’s dilemma is about two accomplices (let’s call them Reggie and Ronnie) who are caught for a crime. The police have enough evidence to convict on a lesser charge — let’s say it’s manslaughter. But the police know (but can’t prove) that the pair committed murder.
They have a choice: confess or remain silent.

  • Say if one confesses and the other remains silent, the one who confesses is let go, while the other is convicted of murder.
  • Or if they both confess, they both serve time for the lesser charge.
  • If they both remain silent, they will both serve time for the lesser charge.

The dilemma faced by each of the prisoners is obviously: which is the best option? But forgetting the philosophical arguments here of which is “best” (you can read them over on the Stanford Encyclopedia of Philosophy if you’re interested). Following this, the Nash equilibrium is at the point where neither Ronnie or Reggie will benefit from changing strategy. If they stay silent, they remain in the dilemma, as there is benefit to be had by confessing. But if both confess, then there’s no benefit in changing strategy (keeping silent again), so the equilibrium point for the Prisoner’s Dilemma is that both prisoners confess.

Formal Definition

Formally, the Nash equilibrium is defined in terms of a n-player game where:

  • i = {1,…,n} players,
  • Si = the set of player strategies i ∈ I
  • gi = the set of goal functions S1 x … x Sn → ℝ.

Payoff functions:

(Also called goal functions) are the preferences of the ith player over the strategies chosen by all players (called strategy profiles or n-tuples). If one player prefers a particular strategy profile to another, then that strategy profile has a higher goal function value or payoff.
Now suppose that:

  • S = the set of strategy profiles 1 X … X Sn with generic element s,
  • (ti, s_i) = the strategy profile (s1,…,si-1, ti, si+1,…sn) from generic element s by player i switching strategy to ti ∈ Si while all other strategies remain unchanged.

The the equilibrium point is where s* ∈ S, for each player i and each strategy ti ∈ Si:

gi(s*) ≥ gi (ti, s*_i).
According to Sethi & Weibull The game has a finite set of alternatives (called pure strategies) which are drawn from probability distributions called mixed strategies. With this in mind, a pure-strategy Nash equilibrium is a list of actions where no player can get a higher payoff by switching from their profile choice. Nash proved that each game has at least one equilibrium point in mixed strategies, given a single restriction on preferences . The restriction is rather dense, and involves completeness and consistency conditions initially laid out by John von Neumann and Oskar Morganstern in 1944.

References:
Nash, John F. 1950. Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences 36 (1): 48–49
Sethi, R. (2008). Nash Equilibrium. In International Encyclopedia of the Social Sciences, 2nd. Ed.
Sethi & Weibull (2016) What is…Nash Equilibrium? Notices of the AMS, Vol 63, Number 5. Retrieved August 29, 2017 from: http://www.ams.org/journals/notices/201605/201605FULLISSUE.pdf (full issue pdf)
Siegfried, T. (2006). A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. National Academies Press, Sep 21, 2006
Von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

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