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NASH EQUILIBRIUM

Nash Equilibrium, Fermat Principle and Governing Dynamics

Abstract -- Nash equilibrium in game theory involves multiple

players where each player knows the strategies of the other player.

The decision to change one’s strategies is based upon the

strategies of the other player. Under equilibrium, the players

cannot change their strategies without changing the strategies

of the other players. An important implication of the theory is

that the optimum output decided by each player that maximizes

the profit is not the best choice for the individual but for the

whole group.

While dealing with such a concept which has shown immense

potential for application in modern economics, one realizes that

a similar principle is encountered in physics in the name of

Fermat principle in optics or the principle of least action in the

theory of fields.

In case of Fermat principle, a light ray traveling in different

medium follows a trajectory such that it is the path of least time

for the whole group of different media. The path in each medium

may not be the shortest but increase of path in one medium is

compensated by a decrease in another such that although the

journey in each medium may not be the best, the total path

becomes the least. This corresponds to an equilibrium in a

mathematical sense and any physical system or state will fall

into such a globally optimal situation. This correspondence is

shown in this paper and it is concluded that the governing

dynamics of the physical universe and our day-to-day human

affairs are linked at deeper level.

Keywords: Nash equilibrium, Fermat principle.

I. INTRODUCTION

THE game theoretic perspective of any competitive system

requiring decision at any point of time is: There is a win or lose

situation in a game played between two or more players. The

payoff table is presented in a form of a grid where loss and

gain are shown as quantities with opposite numerical signs

and loss for one contestant is equivalent to a gain for another

contestant. The game can also be played between an individual

and nature which administer the conditions of individual

surroundings. As an example consider a problem of Decision

theory [1]. A producer of certain things must decide to expand

his plant capacity now or wait at least another year. His advisors

tell him that if he expands now and economic conditions remain

good, there will be a profit of Re. 164000 during the next fiscal

year; if he expands now and there is recession there will be a

Dr. Aniruddh Singh

Department of Applied Sciences, Ajay Kumar Garg Engineering College, P.O. Adhyatmic Nagar,

Ghaziabad 201009 UP India

loss of Re. 40000: if he waits at least another year and economic

conditions remain good, there will be a profit of Re. 80000; and

if he waits at least another year and there is a recession, there

will be small profit of Re 8000. What should the manufacturer

decide to do, if he wants to minimize the expected loss in the

next fiscal year and odds are 2 to 1 that there will be a recession?

One can analyze the above problem by constructing a payoff

table where gains are represented by a negative number and

losses are represented by a positive number:

Since the probabilities that economic condition will remain good

and there will be recession are 2/3 and 1/3 the expected loss

for the current financial year is

-164000 × 1/3 + 40000 × 2/3 = - 28000

Whereas if he waits for the next year then the expected loss is:

-80000 × 1/3 + (-8000) × 2/3 = -32000

Therefore the manufacturer should wait for one year. This is a

simple example of game theory taken from a college textbook

[1] and is meant only as a precondition for appreciation of the

conceptual features of Nash equilibrium.

II. NASH EQUILIBRIUM

An account of Nash equilibrium can be found at several places.

See for example [2]. Here we only give a basic outline which

will be relevant to our purpose and suffice for the discussion

that is about to come. Nash equilibrium is a fundamental

concept in the theory of games and the most widely used

method of predicting the outcome of a strategic dealings in the

social or political sciences. Here we only discuss briefly the so

called pure strategy Nash equilibrium which is a subset of a

general mixed strategy Nash equilibrium.

A pure-strategy Nash equilibrium is an action profile with the

property that no single player can obtain a higher payoff by

Expand now Delay expansion

Economy remains good -164000 -80000

There is a recession 40000 -8000

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AKGEC INTERNATIONAL JOURNAL OF TECHNOLOGY, Vol. 3, No. 2

deviating unilaterally from this profile. Consider first a game

involving two players. Each has two possible actions, which

we call X and Y. If the players choose different actions, they

each get a payoff of 0. If they both choose X, they each get 2,

and if they both choose Y, they each get 1. This is a kind of

coordination game where the payoff is dependent on whether

the two players coordinate or not. It may be represented as

following payoff table, where player 1 decides on a row, player

2 decides on a column, and the resulting payoffs are listed in

parentheses, with the first term in the parentheses

corresponding to player 1’s payoff and the second

corresponding to 2’s payoff:

X Y

X (2,2) (0,0)

Y (0,0) (1,1)

Player 1 can choose either strategy X or Y. Depending upon

the choice of player 2, Player 1 can have the payoffs which are

shown as the first value of each parentheses. Same holds for

player 2 whose payoff is represented by the second value in

each bracket. The action profile (Y,Y) is an equilibrium, since a

unilateral deviation to X by either 1 or 2 would result in a lower

payoff for the deviating player. Similarly, the action profile (X,X)

is also an equilibrium.

In a game with pure strategy it is not necessary for an

equilibrium to be present. As an example consider a game

known as ‘matching pennies’. It is a game between two players

where each player can choose either a head or a tail. Player 1

losses one penny if the choices are the same and wins two

pennies if the choices are different. The payoff table is shown

below:

H T

H (1,-1) (-1,1)

T (-1,1) (1,-1)

As can be seen this game with pure strategies does not have

Nash equilibria. In several cases instead of choosing an action

a player’s behavior is determined by a probability distribution

attached to set of all possible actions. Such a non uniform

behavior over a period of time when the game is played

repeatedly, over a set of actions is referred to as mixed

strategies. A mixed strategy Nash equilibrium then corresponds

to that probability distribution with the property that no player

can obtain higher returns by deviating unilaterally from action

profile determined by the probability distribution.

Just as passing mention, in 1950 John Nash an American

mathematician showed that every game with a finite set of

actions available to each player has at least one mixed-strategy

equilibrium [3].

III. FERMAT’S PRINCIPLE

Fermat’s principle in optics holds a very important position in

the sense that the path or trajectory of a given light ray is

specified by this law. In this sense, the Fermat principle is one

of the most basic law of physics.

In this article I want to give an interesting connection between

Fermat’s principle and Nash equilibrium. To do so, I must

transcribe this very important principle of physics in the

language of game theory.

To start with, I must first describe the principle briefly. A more

detailed description can be found elsewhere [4].

In optics, Fermat’s principle or the principle of least time says

that the path taken between two points by a ray of light is the

path that is traversed in the least time. A beam of light traveling

in different media does not take a straight path between two

points which is the minimum spatial path but the actual path

between two points taken by the light beam is the one that is

traversed in the least time. W ith its help, one can prove the

laws of reflection and refraction (Snell’s law) [4]. To come to

the central discussion of this paper, consider Fig. 1.

Figure 1

A light ray can either start from point A or point B in the figure.

A ray starting from point A can take any arbitrary path. Three

possible paths are shown in the figure. They are labeled as

AyB, AxB and AzB. Out of the three, AyB is the path of least

time and is the actual path of the light beam.

Let us take the to and fro journey of light from point A to point

B and back to A as a sort of a ping-pong game between two

players A and B. Any transaction initiated by A is equivalent

to the light ray starting from A and ending at B. Any transaction

initiated by B is equivalent to a ray starting from B and ending

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NASH EQUILIBRIUM

at A. x, y and z are the strategies and the optical path length of

each transaction is taken as a payoff. The minimum path AyB

is taken as reference and hence Ax=0 and Bx=0 such that AyB=0.

Similarly since any other optical path (i.e. AxB or AzB or their

reverse) is longer than AxB, it is taken as -1. The value is taken

as negative as a longer optical path requires a longer time and

is less economical and hence considered as a loss.

The payoff table is shown below. The rows are strategies(x, y

or z) decided by A and columns are those by B. The first value

in the parentheses is the path length of transaction initiated

by A and the second value is that initiated by B which in our

case is equivalent to the payoff.

As can be seen from the table, the action profile (y,y) is a Nash

equilibrium which corresponds to the actual path both in the

forward journey from A to B and also the reverse journey from

B to A. This also falls into place with the principle of reversibility

in ray optics. As there are no other Nash equilibrium in this

game, this becomes important when we consider the

uniqueness of the solution of the principle of least time in a

given optical system.

IV. CONCLUSION

An interesting connection has been explored between the

principle of least time in physics and Nash equilibrium in game

theory. It has been shown that the principle of least time as

well as the principle of reversibility of ray optics finds a

supportive argument when it is transcribed in the game theoretic

language and can be taken as an independent and unique

testimony of universality of these concepts. At the same time

further study can reveal other interesting connections and

possible generalizations.

V. REFERENCES[1] Freund and Walpole: Mathematical Statistics, Fourth Edition.

Page 306-308.

[2] Osborne, Martin J., and Ariel Rubinstein. A Course in Game

Theory. Cambridge, MA: MIT, 1994. Print.

[3] Nash, John (1950) “Equilibrium points in n-person games”

Proceedings of the National Academy of Sciences 36(1):Page

48-49.

[4] Eugene Hecht: Optics, Fourth edition, Page 102-107.

Dr. Aniruddh Singh is PhD in Theoretical

Nuclear Physics from Jamia Millia Islamia, New

Delhi.

He obtained BSc Hons in Physics from Delhi

University and MSc Physics from IIT, Kanpur.

His PhD thesis is in the field of Variational Monte

Carlo methods as applied to light nuclei and

hypernuclei. He has seven years of teaching

experience at various universities and about 1.5

years research experience in industry.

Currently, he is an assistant professor with the

Department of Applied Sciences, Ajay Kumar Garg Engineering College,

Ghaziabad.

x y z

x (-1,-1) (-1,0) (-1,-1)

y (0,-1) (0,0) (0,-1)

z (-1,-1) (-1,0) (-1,-1)