The Nash Equilibrium

Holla everyone!

How you doin’ 😛

I wanted to write on game theory, its profound impact and immense applications. But it seems a little enlightenment on Nash equilibrium would be good enough.

Nash equilibrium is a stepping stone in decision engineering and game theory.

As interesting as it seems but it sometimes confuses the new readers as they are unable to link and understand this concept intuitively.

So first thing first lets get to it.

Nash Equilibrium

This is an important concept of the Game Theory

  • It assumes that the agents act rationally
  • That it, he always wants to maximize the consequences
  • and he will never take an action if there exists another action that has better consequences (for him)

Example

Consider a Beauty Context Game;

  • each player says a number form 1 to 100
  • the player who says the number that is closest to 2/3 of the average wins the prise
  • ties are broken randomly
  • Nash Equilibrium in this case is 1

Strategic reasoning:

  • what will other players do?
  • What should I do in response?
  • Each player best responds to the others?

Main Ideas

  • each player wants to maximize their payoff
  • so, by the best reasoning, knowing what the others may take, they pick up an action that should be the best
  • The actions taken by all players form an action profile

An action profile is a Nash Equilibrium if

  • it is stable: nobody has an incentive to deviate from their action

Seems confusing right?

Well textbook definitions are a lot boring, but let’s look at the other side lets analyse it in our real-life scenario and imagine it intuitively.

So, in real life Nash equilibrium would look something like this:

    • basically, a set of strategies, one for each player, such that no player has incentive to change his or her strategy given what the other players are doing.

Lets simply it much further in a lay man’s terms:

    • A Nash equilibrium is a law that no one would want to break even in the absence of an effective police force.

So, in this new world example the people/players would pretend that the police doesn’t exist at all. So, if the government passes any rule/law, everyone would be following it as there would be no jail time, no punishment no nothing! This law hence is in nash equilibrium as everyone would want to follow it.

Mathematically:

In a Normal Form Game, a profile   is a Nash Equilibria if

  •  is a preference relation of a player ii
  • all components except i

This needs to hold for all the players

Let’s take some real life examples to make it much more clearer:

{PS: these are some classic examples that one would encounter quite frequently during game theory)

Prisoner’s Dilemma

The situation:

    1. Two suspected criminals are arrested
    2. The police think that they were trying to rob a store-but the cops only have enough evidence to prove that they were trespassing.
      • Thus, the police need one of the criminals to rat our he other to charge them for the bigger crime.

How can the police achieve this? Well they can secretly meet each one to rat out the other. So, they will give out an offer.

The offer:

    • If no one confesses to robbery, the police can only charge the prisoners for trespassing.
      • Punishment:1 year in jail each
    • If no one confesses and the other doesn’t, the police will be lenient on the rat and severely punish the quiet one:
      • Punishment:10 years in jail for the quiet one; 0 years for the rat
    • If both confess, the police punish both of them equally.

PLAYER 2

defect

cooperate

defect

cooperate

PLAYER 1
-1,-1 -10,0
0,-10

 

-5,-5

As it can be seen from the offer, defecting is the more beneficial mode than cooperating.

Thus the conscience of any player would tell him to defect. And hence blinds him. Lets look at the above table:

    1. Lets take into consideration the case of player 1.
      • If he defects, he can either get 0 years (p2 cooperates) or 5 years (p2 defects) of jail.
      • If he cooperates he will either get 1 year(p2 cooperates) or 10 years(p2 defects) .

Thus he will tend to defect as its more beneficial

    1. Now for player 2
      • If he defects, he can either get 1 years (p1 cooperates) or 0 years (p1 defects) of jail.
      • If he cooperates he will either get 10 years(p1 cooperates) or 5 years(p1 defects)

Thus he will tend to defect as its more beneficial

Hence (defect, defect) will be the Nash equilibrium as its stable and no one will deviate.

But lets look at the profile(cooperate, cooperate), both can get away with 1 year of jail each which is much shorter than the Nash equilibrium but is not stable as P1 wants to change his mind and choose Defect and so does P2 thus winding up in (D,D).

Matching Pennies

 

Head Tail
Head (1, -1) (-1, 1)
Tail (-1, 1) (1, -1)

In this game there’s no Nash Equilibrium:

  • if P2 knows that P1 plays H he will play H
  • then if P1 knows that P2 plays H, he will play T
  • so there’s always an incentive to deviate to other alternative

The Battle of the Sexes

In this case there are two equilibrium: (B,B) and (F,F)

wife 
husband 
B F
B (2, 1) (0, 0)
F (0, 0) (1, 2)

Equilibria

  • consider a traffic game
    • 2 cars are on crossing perpendicularly.
    • they can go or yield another car

      P1 rows, P2 cols

P2 
P1 
Go Stop
Go (-5, -5) (1, 0)
Stop (0, 1) (-1, 1)
    • not stable, players may mis coordinate
    • we place a traffic light
    • so, by putting a fair randomizing device that

tells players whether to go or wait

  • the same can be applied to Battle of the Sexes
  • benefits
    • we avoid negative outcomes
    • fairness is achieved
    • the total sum can exceed the NE
  • correlated equilibrium
    • a randomized assignment of action recommendation to agents, such as nobody wants to deviate

So that’s all about Nash equilibrium, hope I reached out to you properly. Any queries, question or discussions do reach out to me.

Question: why have I included the domino pic in the start? How is it related to Nash equilibrium?

Share your ideas with me! Stay tuned for the next writeup on the very interesting game theory.

Reach me at :: sumitpradhan.kumar@gmail.com

 

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