Nash Equilibrium Trends

Game Theory Analysis

Sidney Gautrau

John von Neumann is looked at as the father of modern game theory.

Many other theorists, such as John Nash and John Smith, have advanced the discipline.

A number of practitioners of game theory have won Nobel Prizes in Economics.

Game theory is a branch of mathematics that aims to lay out in some way the outcomes of strategic situations.

Prisoner's Dilemma- Two individuals robbed a bank together for 10 million dollars. They are both captured by the police and put into two separate rooms. If one prisoner confesses, while the other doesn’t, the confessor is let off free, while the other will go to jail for twelve years. If neither confess, they will both be sentenced to one year in jail each. If both confess, their sentences are reduced for cooperating, but they still both spend eight years in jail.

Keep Quiet Confess

Keep Quiet -1,-1 -12,0

Confess 0,-12 -8,-8

Criminal 2

Criminal 1

Chicken Game- Two individuals are driving cars straight at each other. Each driver has two options, swerve, lose, and avoid a collision, or continue driving straight resulting in either winning or crashing.

Straight Swerve

Straight -10,-10 2,-2

Swerve -2,2 0,0

Driver 2

Driver 1

Matching Pennies Matching Pennies- Two players also

play this game. Each player has a penny. They both secretly turn the penny to heads or tails. If the pennies match, player one wins. If the pennies are different, player two wins.

Heads Tails

Heads +1,-1 -1,+1

Tails -1,+1 +1,-1

Player 2

Player 1

Battle of the Sexes- A married couple agreed to meet later in the day. The wife and husband only know that they are individually going to a football game or an opera. The husband wants to go to the football game more than the opera, while the wife would rather go to the opera over the football game. Both would prefer to go to the same place rather than go to different ones. Where should they go?

Opera Football

Opera 3,2 0,0

Football 0,0 2,3

Husband

Wife

A Nash Equilibrium is the outcome that isn't always the best outcome for an individual player, but it is the most beneficial for the two players.

To find if a game has any Nash Equilibriums, look at player one's first options based on one of player two's selections. Put a star by the best choices for that option. Do that for every option of every player. If a result has a star by each result, it is a Nash Equilibrium.

EXAMPLE: The Chicken GameStraight Swerve

Straight -10,-10 2*,-2* NE

Swerve -2*,2* NE 0,0Player 1

Player 2

Hiro developed a program that could find the Nash Equilibrium for us.

EXAMPLE: Prisoner's DilemmaFor player 1, #( 1, 1) = 0 #( 1, 2) = 0 #( 2, 1) = 1 #( 2, 2) = 1

For player 2, #( 1, 1) = 0 #( 1, 2) = 1 #( 2, 1) = 0 #( 2, 2) = 1

Nash equilibrium (equilibria) at (2,2)

Keep Quiet Confess

Keep Quiet -1,-1 -12,0*

Confess 0*,-12 -8*,-8*NE

Criminal 2

Criminal 1

An example I will use is the battle of the sexes game. If the guy likes football and the girl likes opera, but they both want to be together, the NE is when they are together. A way to change the NE is to change the circumstances. If the guy would rather be by himself or with a buddy at the football game, rather then going to the opera with the girl, that would change the NE. The same goes for the girl. If she would rather go by herself to the opera or with a friend, rather then going with the guy to the football game, that too would change the NE.

Or the girl likes football more.Opera Football

Opera 3*,2*NE 0,0

Football 0,0 2*,3*NE

Opera Football

Opera 3*,2 0,4*

Football 0,0 2*,3*NE

Opera Football

Opera 3*,2*NE 4*,0

Football 0,0 2,3*

GUY

GUY

GUYGIRL

GIRL

GIRL

A Zero-sum game is a game where the sum of the player one and player two's results equals zero, hence the name.

Zero-sum games can have Nash Equilibriums. There can also be multiple Nash Equilibriums.

0*,0 -6,6*

0*,0*NE 2*,-2

0*,0*NE 0*,0*NE

-4,4 0*,0

One Nash Equilibrium

Two Nash Equilibriums

A strictly competitive game is a type of zero-sum game in which one player's gains result only from the other player's equivalent losses. The sum of the results for every situation equals zero. An example is the matching pennies game. One player will win, while the other player will lose.

Strictly competitive games have NO Nash equilibriums.

Heads Tails

Heads +1,-1 -1,+1

Tails -1,+1 +1,-1

Hiro also created four other programs that made 100 random zero-sum games with either fixed or not fixed signs and a 2x2 or 3x3 matrix. The program also found every Nash Equilibrium.

Example:( 4, -4) (-10, 10)( 0, 0) ( 0, 0)Nash equilibrium (equilibria) at2 2 Here!

After running the 2x2 zero-sum game program that has fixed signs, I discovered a rule:

When there is 2 sets of zeros, and the other 2 results are player one and player two having winning chances, there is always ONE Nash Equilibrium. When the winning/losing combos are diagonally across from each other, there is a deadlock. Deadlock- The result that benefits both players the

most. A deadlock is the obvious selection every time for that game.

0,0 -6,6*

1*,-1 0*,0*NE1*,-1 0*,0*NE

-1,1* 0*,0

1*,-1 0*,0*NE

0,0 -2,2*Deadlock

Deadlock

After running the 2x2 zero-sum game program that didn't have fixed signs, I discovered a rule:

When there is 3 identical sets of numbers, there is always TWO Nash Equilibrium. If the fourth set of numbers have higher values, and

you switch its signs, the Nash Equilibriums will shift. If the fourth set of numbers have lower values, and

you switch its signs, the Nash Equilibriums will remain the same.

-6,6* -6*,6*NE

8*,-8 -6*,6*NE

-6,6* -6*,6*NE

-1*,1 -6*,6*NE

-6,6* -6*,6*NE

1*,-1 -6*,6*NE

-6*,6*NE -6*,6*NE

-8,8* -6*,61*,-1*NE 1*,-1*NE

1*,-1 -2,2*

I got my results by analyzing the multiple games simulated by the program. I found all of the Nash equilibriums and looked for similar trends. I noticed the rules I made and tested them with multiple games to make sure they were true.

Even though some of these games are not likely, one of the games could have a story. The deadlock rule would apply to a game like rock-paper-scissors minus one of the options. There is always a deadlock in rock, scissors. This research result could help us understand some of the world’s conflicts. An example is the chicken game and the Cuban Missile Crisis. Either neither country fired missiles, one of the countries threatened the other with missiles, or we blew each other up.

Appendix

While briefly looking over the 3x3 program, I noticed another trend. For 3x3 games, row 1 column 1, row 1 column 1, and row 1 column 1 are always (0,0). The reason is that in order for the 3x3 to be a real game, there has to be three draws. A great example is rock, paper, scizzors.

R P S

R 0,0 -1,1 1,-1

P 1,-1 0,0 -1,1

S -1,1 1,-1 0,0

While looking over the generated 3x3 games, I noticed a few rules. Whenever there is 5 sets of zeros, there is a Nash equilibrium. Whenever there is 6 sets of zeros, there is two Nash equilibriums. Whenever there is 7 sets of zeros, there is four Nash equilibriums. Whenever there is 8 sets of zeros, there is six Nash equilibriums.

0,0 0,0NE 0,0

-1,1 0,0 4,-4

6,-6 -7,7 0,0

0,0 0,0NE 0,0NE

-1,1 0,0 0,0

6,-6 -7,7 0,0

0,0NE 0,0NE 0,0

0,0NE 0,0NE 4,-4

0,0 -7,7 0,0

0,0NE 0,0NE 0,0NE

-1,1 0,0 0,0

0,0NE 0,0NE 0,0NE