Game Theory Explained
Game theory is the study of mathematical models and player strategic interactions. It has usage in several fields that strictly studied and applied, such as; Econmics, Politics, Sports, Computer Science, and Logic. Modern game theory began with John von Neumann’s proof of mixed-strategy equilibria in a two-person zero-sum game.
“Game Theory is the study of decision making under conditions of uncertainty over time”
Hungarian-American mathematician John von Neumann published a book titled “Theory of Games and Economic Behavior”. This foundational book proposes solid solutions for two person games.
Several researchers worked intensively on game theory in the 1950s. It was formally applied to evolution in the 1970s, but analogous processes may be traced back to at least the 1930s. Game theory has long been acknowledged as a valuable tool in a variety of areas. Later on, it was applied in personal relations, family situations, biology and parenting.
The game theory consists of mostly three components:
- Players: The primary component of game theory is a group of players. Business executives and corporations are seen as participants.
- Strategies: The many tactics that players might employ in the game. This also contains game rules that are organised in a hierarchy to indicate the greatest feasible tactics and behaviours.
- Results: These are the potential methods and projected payoffs based on the outcomes. It is presumed that all players are aware of the paybacks.
Some major features, the most evident of which is the number of participants, may be used to categorise games. Consequently, a game can be classified as a one-person, two-person, or n-person (with n higher than two) game, with games in each category having unique characteristics.
A few explanations of terms concering the Game Theory:
- Zero-sum game: a game where the outcome is an advantage for one side and a loss for the other.
- Strategies per players: Each participant in a game choose from a set of viable actions known as pure strategies. It is noted here if the number is the same for all players.
- Constant sum: If the sum of the payoffs to each player is the same for every single combination of tactics, the game is a constant sum game. In these games, one player wins if another player loses. A constant sum game can be turned to a zero sum game by deducting a set value from all payoffs while maintaining their relative order.
Let me give couple of examples of the Theory
The most known one is the Prisoner’s Dilemma
Rules go like this:
- Two bank robbers, Robber A and Robber B, have been arrested and are being interrogated in separate rooms.
- The authorities have no other witnesses, and can only prove the case against them if they can convince at least one of the robbers to betray their accomplice and testify to the crime.
- Each bank robber is faced with the choice to cooperate with their accomplice and remain silent or to defect from the gang and testify for the prosecution.
- If they both co-operate and remain silent, then the authorities will only be able to convict them on a lesser charge resulting in one year in jail for each (1 year for Robber A + 1 year for Robber B= 2 years total jail time).
- If one testifies and the other does not, then the one who testifies will go free and the other will get five years (0 years for the one who defects + 5 for the one convicted = 5 years total).
- However, if both testify against the other, each will get three years in jail for being partly responsible for the robbery (3 years for Robber A + 3 years for Robber B= 6 years total jail time).
In this instance, regardless of the decision made by the other robber, each robber has an incentive to defect. From Prisoner A’s perspective, if Prisoner B maintains silent, Prisoner A has the option of cooperating with Prisoner B and serving a year in prison, or defecting and walking free. In this instance, she would obviously be better off betraying Prisoner B. If, on the other hand, Prisoner B defected and testified against Prisoner A, she will have to choose between being silent and serving five years in prison or speaking up and serving three years. Again, she definitely prefers three years over five years.
In both scenarios, whether Prisoner B cooperates with Prisoner A or defects to the prosecution, Prisoner A will be better off if she defects and testifies. Now, because Prisoner B has the same set of options, he will always be better off defecting.
The paradox of the prisoner’s dilemma is that both robbers can minimise the total jail time that the two of them will do if they both cooperate and remain silent (two years total), but the benefits that they each face individually will always drive them each to defect and wound up doing the maximum total prison sentence between the two of them of six years total.
The famous Rock-Paper-Scissors
Everyone knows how to play Rock-Paper-Scissors, and it is frequently employed to make a “random” decision. Yet the fundamental question is whether the game Rock-Paper-Scissors is truly random. In principle, the game has three outcomes: win, defeat, or tie. Paper defeats rock, scissors defeat paper, and rock defeats rock. Rock-Paper-Scissors players have an equal chance of winning if both players select alternatives fully at random. This, unfortunately, is not the case. Rock-paper-scissors may not be the most reliable method of determining anything really random.
Zhijian Wang, a Chinese mathematician, ran an experiment in which 72 students played 300 rounds of rock-paper-scissors. He discovered that the student’s choices during the game went towards the Nash Equilibrium, which would be selecting rock 1/3 of the time, scissors 1/3 of the time, and paper 1/3 of the time. Notwithstanding the Nash Equilibrium, he saw a pattern in the way the games were played. Winners tended to continue with their original plan, while losers switched to another. This would recur for every time a player lost, which he terms “persistent cyclic flows”.
In 1950, mathematician John Nash demonstrated that in every game with a finite number of players and a limited number of alternatives, such as Rock-Paper-Scissors, a mix of tactics exists in which no single person can perform any better by altering their own strategy alone. The theory underlying such stable strategy profiles, dubbed “Nash equilibria,” revolutionised game theory, affecting the path of economics and transforming the way everything from political treaties to network traffic is researched and evaluated. It also earned Nash the Nobel Prize in 1994.
The mathematician — subsequently portrayed in the book and film “A Beautiful Mind” — authored a two-page study that changed economic theory. His key, though totally basic, theory was that each competitive game contains a concept of equilibrium: a set of strategies, one for each participant, such that no player can win more by switching to a different strategy unilaterally.
I will be writing more about the Game Theory in the future, maybe how to use in day-to-day life.
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