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Q. Lewis, what is a simple, and easy explanation of a Nash equilibrium?

A. A Nash equilibrium is a way of explaining the idea that if there is a set of strategies for a game and it soon becomes clear that no player can benefit by changing his or her strategy, while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute a Nash equilibrium.

One of the things that virtually all game thinkers, including chess players, video gamers, strategists, and professional athletes, business competitors seek to do, maximize their potential at the lowest possible cost. This seems easier to do than it actually is. This is because, in a closed, or game-based environment, one must create an optimum strategy based on the information available.

To do this would require a scenario where you:

  • know the rules of the game,
  • know who all of the players are,
  • know your own strengths and weaknesses,
  • knowing the strengths and weaknesses of the other players,
  • understanding the environment in which interactions are likely to take place (the game space).

An example of this would be, the Prisoners’ Dilemma, and Rock, Paper, Scissors”, but not “Chess”. Why is this? In “Chess”,  there isn’t really an optimum ( Nash equilibrium) strategy. If you could solve the entire game, every position would have moves that either lead to a win, loss, or a draw; given the best play.

 

Generally, investing is not a Nash Equilibrium because every move creates a change in strategy and many more options..

 

In real life, a loosely exhibited  “Nash equilibrium”,  might be all the things you might do to keep from catching Covid 19;  assuming even if what we have been told about transmission is not completely accurate. Still, here, wearing the best face mask, social distancing, wearing gloves, washing your hands with soap, whenever entering from outside,  and living a lifestyle that would increase your immunity,  would likely be the best strategy to protect yourself. These actions would be considered the “optimum strategy”.

 

Just to be clear here, when I use the term game space,  I am referring to a specific physical or conceptual environment where the game is taking place. Examples might include a chessboard, a basketball court, the agreements to a marriage, or the formal rules laid down by the Papacy on what acceptable behavior might be for a Roman Catholic. Here, the Catholic Church Would be the Game Space).

What we often find in life is that we find ourselves in a situation where no such optimum strategy existed, or it did,  was previously defined and/or codified.  This is where John Nash and his ideas come into play. You might know of Nash from the book, and the Academy Award Winning Biopic movie based on his life, “A Beautiful Mind”. Nash developed what is now known as the “Nash Equilibrium”. The Nash equilibrium is such an important concept in game theory that Nash was awarded the Nobel Prize in economics for his work in the area.

For the AskLewis Team, life really is a game that can be won or lost. There are exceptions of course. One doesn’t need to think like a game to explore the nature of truth, the sacred, the divine, the art, culture, or beauty. Still, The concept of Nash Equilibrium can serve us by helping us to define a kind of “optimum” strategy for games, game-based scenarios, and the game of life.  What many of us never realize is that even board games like “Chess” and “Go,” and in video games like “Astroneer”,  no such optimum strategy was previously defined.

Just in case you are getting a little lost, here is an easy to understand, simple and basic definition of a Nash Equilibrium that you can apply to daily life?: “If there is a set of strategies for a game with the property that no player can benefit by changing his strategy, while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute a Nash equilibrium.”

This definition applies to games of two or more players, thus it would not apply to a game like solitaire.  It isn’t that optimum strategies didn’t exist before Nash. It is just that Nash showed that the various definitions of “solutions” for games that had been given earlier, all yielded Nash equilibria automatically in the past, and would need to be met now by all the players are:

  1. Each player believes all other participants are rational.
  2. The game correctly describes the Payoff – The winnings (payout) any of the players in the game receive from arriving at a particular outcome. The payout can be in any quantifiable form, from dollars to goods, services, or measurable resources. In HAGT (Harrison’s Applied Game Theory), we define this as any of 24 Core Resources that most humans need to survive and prosper.
  3. The players are flawless in execution.
  4. The players have sufficient intelligence to deduce the solution.
  5. Each player is rational

 

The reasoning behind this realization of the NE, is that the first four conditions make playing the NE strategy optimal for each player and that since the fifth condition identifies each player as an optimizing agent, each will take the personally optimizing strategy – a set of strategies for a game where no player can benefit by changing his or her strategy while the other players keep their strategies unchanged!

 

Again, to clarify what is being said here;  If a game has a unique Nash equilibrium, it needs to involve:

  • Completely rational players (who know that all players are completely rational),
  • Players, who being rational, will choose the optimum strategies.

 

What makes such a game an equilibrium?  All the players in the game are applying optimum strategy.

 

A final thought:  In any game-based scenario there may be more than one Nash equilibria, or even none. As has been said, “there are many roads to Rome”.  Nash was able to prove that:

·   if we allow mixed strategies ( where players choose strategies randomly according to preassigned probabilities), i.e., plan A, plan B, plan C, etc.

·   then every n-player game (a game which is well defined for any number of players), in which every player can choose from finitely many strategies  (i.e., plan A, plan B, plan C, etc.)

·   admits at least one Nash equilibrium of mixed strategies (at least one optimal strategy).

 

Recognizing a Nash equilibrium is of utmost importance to any strategist who wishes to both survive and prosper.

 

Speaking of gamer psychology and game theory, I know and my game theorist friends use sports analytics to predict winners and losers in Fantasy sports. Here is a link to a free video and information about Sports Betting and FIFA Fantasy Football. This is an introduction to one of my favorite systems. If you want to use predictive skills to explore this Complete Analytics-Driven Sports Fantasy Football and Sports Betting System. It’s based on the ideas presented in the book and movie Moneyball. Press on the URL below or the photo of the soccer players  below.

https://df34ebjhn5jgsqae7ercxjpmsz.hop.clickbank.net/?tid=WEBPAGE9

 

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