Game Theory 11 – Nash Equilibrium (4)

Part 11. Nash Equilibrium (4) – Correlation between Nash Equilibrium and Public Goods

At the last post, we saw the important correlation between Nash Equilibrium and Guaranteed The Lowest Price. However, Nash Equilibrium has various examples. Today, I introduce another correlation which is the relationship between Nash Equilibrium and Public Goods.

Before starting this relationship, what is the meaning of public goods? A dictionary meaning of “public goods” is “services such as national defence, law enforcement, and road building, that are for the benefit of, andavailable to, all members of the public.” Most of the free highways, ordinary roads (except toll roads), air (oxygen) and so on. Unlike the United States, public goods of the other countries, which do not privatize public goods services, include water supply and drainage services, electricity and railroad systems.

The major features of public goods are that all people can use public goods, nobody can restrict using public goods, and most of the people want to use public goods, but they do not want to pay expenses.

---

Two towns, whose names are Santa and Claus, are planning to build the free four-lane road that connects these two towns. The residents of two towns have to spend three hours on the road to go to the other town because the current road is unpaved and very narrow.

However, the federal government and the state government refused to support the cost of the construction because the federal and state parliament cut the budget to fund this construction. For this reason, the residents of the two towns decided to build their four-lane free road without funds of the federal and state government. The inhabitant of Town Santa is richer than those of Town Claus because Town Santa has car and cement factories.

The total expense of the construction is five million dollars. The richer town, Town Santa, can pay until four million dollars (the maximum). Also, the other town, Town Claus, can absorb until two million dollars (the maximum). However, two towns conflict with the expense because it is too much a big burden to two towns.

If they do not build the road, the benefit will be 0 (zero). If they agree to build the road and share the expense, Town Santa will pay K amount of the expense, and that of Town Claus will burden 5 million-K amount of that. If only one town agree to build the road, the town will pay all of the expense.

This is a grid of their benefits of this construction.

Town Claus Town Santa	Joining the construction	Not Joining
Joining the construction	(4M-K, K-M)	(-1M, 2M)
Not Joining	(4M, -3M)	(0,0)

(*M means a million dollar)

According to this grid, K is bigger than 0 (zero), but is smaller than 5 million dollars. Therefore, the best way to get maximum benefit is “Not Joining to the construction.” Two towns select strategic dominance, so the residents of two towns cannot use a four-line well-made paved road.

Game Theory 9 – Nash Equilibrium (2)

Part 9. Nash Equilibrium (2) – The History of Nash Equilibrium

It is definitely true that Nash Equilibrium was named after John Forbes Nash, but the idea of Nash Equilibrium was based on Antonie Augustin Cornet, who was a French mathematician and philosopher. His oligopoly theory did an important role to make the basic concept of Nash Equilibrium. Nash referred Cornet’s oligopoly theory, but Nash Equilibrium was covered larger academic areas. Cornet’s oligopoly theory was focused the result when each companies’ benefit was maximized. His oligopoly thinking was only limited to economics; however, Nash Equilibrium covers not only economics, but also other academic parts. The other famous figure in the game theory, John Neumann, and Oskar Morgenstern mentioned mixed strategy Nash Equilibrium, but they limited to zero-sum games. In other words, they only introduced the concept of Nash Equilibrium at zero-sum game situations. The most important part of Nash’s role is that he proved his own definition (extended definition) of Nash Equilibrium. He used Brouwer fixed point theorem to identify mixed strategies that existed outside of zero-sum games, which means he thought that Nash Equilibrium could apply other situations.

Other game theorist found that Nash Equilibrium could make wrong predictions in some situations (specific situations). Therefore, they introduced refinements of Nash Equilibrium, which is the improved version of Nash Equilibrium, to solve perceived flaw points of Nash’s theory. In 1965, Reinhard Selten introduced subgame perfect equilibrium to solve the uncertain possibilities, which was that Nash Equilibrium based on incredible threats. Also, other theorist focused the repetition of a game and absence of perfect information. However, subsequent refinements and extensions of the Nash equilibrium concept share the main insight on which Nash’s concept rests.

Nash Equlibrium cannot be a perfect theory to explain non-cooperative situations of the game theory. However, it took the important role to identify what is non-cooperative games and how to prove them. Therefore, if the game theory will exist in the world, the basic concept of Nash Equilibrium is not able to be changed.

Game Theory 8 – Nash Equilibrium (1)

Part 8. Nash Equilibrium (1) – What is Nash Equilibrium?

I introduced various people and theories about Game Theory. However, if people did not know about Nash Equilibrium, they missed many and very important parts of Game Theory. Nash Equilibrium is a kind of non-cooperative game, which was made by John Forbes Nash. Nash Equilibrium is that when other gamer keeps his or her strategy, all the players in this game will not change their strategies, maintaining complementary cooperative relationships. There are two example to explain what is Nash Equilibrium and how to apply it.

---

1. Pursuing Lower Benefits

	Player	Steve	Steve
Player	Selections	1	2
Gina	1	10 / 10	50 / 0
Gina	2	0 / 50	0 / 0

(The Benefit of A  / The Benefit of B)

In this situation, the maximum profit of each player is 50. However, unfortunately, only one player can get 50 in this situation. If one player in this game take 50 benefits, the other player cannot receive any benefits, which means zero – the red situations-. I assume that Steve and Gina is rational people and hate to fight each other to get more benefits. Adam Smith, who wrote the Wealth of Nations and organized modern economics, and his followers would say that Steve want to choose Selection 2 (red-colored) and Gina want to  choose Selection 1 (red-colored) because total benefits are maximum (50) in this time. However, in real world, they do not choose red-colored choices because this is non-cooperative game. According to Nash Equilibrium, they will choose Selection 1 (blue-colored) because all players in this game can get benefits (10) rather than losing everything (green-colored).

---

2. A Beautiful Blonde Woman – Based on Beautiful Mind (2001)

	Player	Man A	Man A
Player	Selections	1	2
Man B	1	10 / 10	50 / 0
Man B	2	0 / 50	0 / 0

                                                                                (The Benefit of A / The Benefit of B)

According to Beautiful Mind, characters in the movie talks about a blonde woman. All of the men in this situation want to date with that beautiful blonde woman. However, if Man A and B choose the blonde woman (red-colored), they cannot date with that woman (green-colored). Also, other women do not want to date with these men, because other women think that these two men choose non-blonde women instead of the beautiul woman. Therefore, they will choose other women (non-blonde women) to get small, but stable benefits.

Game Theory 2 – The History

Part 2. The History – The Change of Game Theory

Many people think that Game Theory was started in twentieth century. However, the notion of Game Theory was appeared in 1713 by James Waldegrave. Through his letter, he introduced the card game for two players; this card game had a basic principle of game theory. Though, the most important person in the game theory is John von Newmann. Before Newmann, people can say that the Game Theory did not really appear in the world; Newmann established the concept of the game theory.

John von Newmann published a paper in 1928. In his book Theory of Games and Economic Behavior, he introduced the solutions of zero-sum games (two people type). At that time, game theory was based on groups of individuals and cooperative game theory (I will introduce types of games later).

Game theory was popularized in 1950s. The prisoner’s dilemma, as I mentioned the before post, was discussed at this time. John Nash introduced the notion of mutual confidence of each player which is known as Nash equilibrium. Thanks to Nash equilibrium, another type of game theory was appeared in this era of which name is non-cooperative game theory. Beyond just researching the game theory, the game theory applied to other academic studies (such as philosophy, economics, biology and so on).

Because of application of the game theory, various academic studies can reach great results to change paradigm. Reinhard Selten introduced the developed version of Nash equilibrium which was called subgame perfect equilibria. Also, Selten, Nash, and John Harsanyi awarded Nobel Prize in Economics because they contributed economics by using the game theory. It was a great chance to spread out the game theory, many experts and product planners used the game theory to improve their researches or sales volume.

The game theory was expended to other academic subjects in 1970s, and many experts improved the game theory. Also, Thomas Schelling and Robert Aumann became Nobel Laureates in 2005. Another game theory researcher, Leonid Hurwicz, got Nobel Prize in Economics in 2007, and Alvin E. Roth and Lloyd S. Shapley became Noble Laureates in Economics in 2012.

Game Theory was developed several hundred of years; however, the true development of Game Theory was started in twentieth century. Nowadays, Game Theory applied various kinds of academics studies, business, and marketing; therefore, people’s lives closely connected with Game Theory.