## Nash Equilibrium- strategic decision making

June 29,2017 0 comments

If you were to judge keenly on your daily life, over 90% of your social interactions are strategic in nature. With strategic, I mean, that each of your interaction is like a game. To prove so, consider the following basic setup of a game: A game (in strategic or normal form like football) consists of the following three elements: a set of players, a set of actions (or pure strategies) available to each player, and a payoff (or utility) function for each player. The payoff functions represent each player’s preferences over action profiles, where an action profile is simply a list of actions, one for each player i.e. if you choose these actions, then this will be your pay, whereas if you did these other actions then this will be your pay. This is indeed the nature of our daily lives, where each move (maybe in making a business decision by striking a deal with an investor) is a pure strategic action. It is also true that every player tends to win the game, yet it is the player whose actions are more strategic than the other that wins the game. It is therefore clear that strategic skills are necessary tools for a good game played and won. So here below I present you the Nash equilibrium, an important piece of strategy that if a player has and the opponent lacks, then he is sure to succeed no matter the circumstances that prevail.

### The prisoner’s dilemma

Since a skill is better gained practically than through theoretical study without an application, the Nash equilibrium skill will be imparted to you in the former form and not the latter. Read the following with an open mind (I will use a correction facility example to aid in your understanding, you should participate as instructed).

A drug dealer is caught and apprehended by a police officer while in action. He is taken to the station. On the same day, a big scheme con man is also caught by other policemen and brought to the same police station. Both of them are to serve a jail term of two years. The officer in charge of investigations, however, suspects that these two gentlemen might have also been involved in a mass bank robbery that took place two days earlier. The officer being a game strategist, he makes a very important move; he decides to give the two an incentive to betray each other. The following are the conditions:

Let the drug dealer be ‘A’, and the con-man be ‘B’.

- Both of them are to serve a jail term of two (2) years for their individual crimes (i.e. if both of them also deny having participated in the mass bank robbery)
- If A confesses while B denies, then A gets a 1-year jail term whereas B gets a 10-year jail term
- If B confesses while A denies, then B gets a 1-year jail term whereas A gets a 10-year jail term
- If both of them confesses, then both will have a jail term of 3 years each.

Follow the below instructions.

Put yourself in the shoes of either of the criminals (either A or B and not any other choice). Make your optimal move (either confess or deny and nothing less or more). Retain your answer no matter what. Do not continue to the next paragraph before doing this.

### The payoff matrix

For a perfect situational analysis, game strategists use a payoff matrix. Following is the payoff matrix for our situation.

Hoping that you made your best choice, let’s now move to the right answer together. From the matrix above, people often fall for choice number 4 (red marked number). Reasons are as below.

- A can go for choice 2 but he would not likely do this for he is not sure if B will confess so that A gets 10 years while B gets 1 year. It is a highly profitable move for only one individual, but a highly unhealthy move especially where trust is absent. It is a vice versa for B too (i.e. choice 3).
- Neither of them would likely go for choice 1 because an easier choice 4 exists for both of them.
- Both of them would likely go for choice 4
- Only the strategic one would go for choice 1

### Universal vs Nash equilibrium

Choice 4 marked ‘U’ is known as the universal equilibrium. It is the choice that often more people go for. It is, however, a highly unstable state, because of the following reasons. Consider the blue arrows. If person B, after considering the prevailing circumstances (e.g. others beyond our thinking, like his home situation) decides to change his mind and confess (while A remains constant), then he will get his sentence reduced to 1 year while A will be on a serious losing end with his term increased to 10 years. The vice versa will happen if A takes the above move from the universal equilibrium. It is however unfortunate that most people often go for this choice or rather settle for 2 or 3 blindly.

Choice 1, marked ‘N’ is the Nash Equilibrium. Consider the green arrows. If A changes his mind from confessing to denying, while B remains constant, A will have his sentence increased from 3 to 10 years (an additional 7 years). It is certain that he will not make this move. If B is the one that changes his mind while A remains constant, the same destiny awaits him. Most probably he also wouldn’t go for this. Nash equilibrium is, therefore, a very stable state. A *pure-strategy Nash equilibrium *is an action profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile.

Discovered, and named after John Nash, the Nash equilibrium is a fundamental concept in the theory of games and in practical life. It is the most widely used method of predicting the outcome of a strategic interaction in the social sciences yet it is known only to a few right-minded individuals. It is a theory that revolutionised the field of economics. Nash won the 1994 Nobel Prize for this. You can easily identify a Nash equilibrium by analysing the situation and only settling for that which ensures that no single player can obtain a higher payoff by deviating unilaterally from it.

It is a concept worth having for every business person and any other normal individual in a daily interaction. I can’t stop thinking about this, but even in the technical fields, like engineering, where systems are to be developed, the chemical equilibria plus system equilibria also obey this law. It is a must have but can only help if you settle and analyse the situation at hand before making a decision.