Recently useful to the relevant theory and algorithm of game theory, prepare several articles to introduce the game theory in detail. Later articles will cover game theory, the mathematical definition of Nash equilibrium, and case studies.
This article is the first introduction of game theory, try to use popular language can introduce clearly what game theory is.
Game theory
Von Neumann is not only the “father of the modern computer”, but also the “father of game theory”, which he developed in 1944.
Game theory, also known as game theory, is not only a new branch of modern mathematics, but also an important subject of operational research. It’s the branch of mathematics that deals with all kinds of conflict factors, or decision theory in conflict situations.
Game theory, defined in Osborne & Rubinstein’s book, is an analytical toolkit designed to help us understand the observed interactions between decision-makers. The underlying assumption of this theory is that decision-makers pursue certain external goals and consider their own knowledge or the expectations of other decision-makers’ behavior.
Game theory may sound abstract, but game theory has a wide range of applications, including in finance and securities (such as investing in stocks), economics, international relations, computer science, political science, military strategy and many other disciplines. Some classic cases, such as prisoner’s dilemma, wise pig game and normal paradigm game, etc.
Is it useful to learn game theory?
First, the conclusion: in daily life, generally not very useful.
Some data can be found that game theory is beneficial for individuals/collectives/countries to make reasonable decisions and maximize the benefits of individuals/collectives/countries.
In fact, game theory is a series of “multi-person decision models,” a very powerful theory that can help us understand many things, such as human evolution, moral evolution and so on.
But in the concrete games of life, learning game theory is not so useful.
For example, diligence clearly can obtain more results, but why everyone will have lazy?
Game theory assumes that all players are rational (I know you know I know you know, I know you know, I know you know, I know you know I’m rational…). .
There’s a Nash equilibrium, which is the assumption that given someone else’s strategy, one’s own strategy is optimal. The problem is that if someone doesn’t use Nash equilibrium, it’s usually not optimal if I use Nash Equilibrium.
Another problem is that every game we play takes into account the actions of our opponents. When it comes to specific games, what we really want to know is the opponent’s strategy, but a lot of times we don’t know, like bidding.
Your game theory analysis doesn’t make much sense. The difficulty of playing a game in life is getting information about your opponent, it’s easy to make a strategy with that information, or we’re inherently good players. Game theory is essentially an analysis of game strategy. It’s not about teaching us how to play games or how to get information about our opponents. It’s about understanding human behavior on a larger scale.
Back to the question, is there no need to learn game theory? The answer is no. Although game theory has no explicit use in life, it has subtly changed my way of thinking, such as perspective-taking. After a long time of training in game theory, we instinctively put ourselves in other people’s shoes. And one of the things that happens a lot in game theory is, if you want to make the best decision you can, you have to think about how the other person is going to make the best decision you can make.
The history of game theory
The starting point for game theory is now generally considered to be the 1944 book game Theory and Economic Behavior by Von Neumann and Morgenstern. The idea of game theory has an ancient history. At first, the game theory mainly studied the problem of winning and losing in chess, bridge and gambling. People’s grasp of the game situation only stayed on experience, but did not develop into theory. Ancient Chinese works such as Sun Tzu’s Art of War are not only military works, but also the earliest works on game theory. And cournot’s variable decision model, the standard game model, in 1838.
The modern game theory was founded by the great Hungarian mathematician Von Neumann in 1920s. In 1944, he published the great book Game Theory and Economic Behavior in cooperation with the economist Oscar Morgenstern, which marked the initial formation of modern systematic game theory.
In 1951, Nash proposed the “Nash equilibrium” and proved the existence of equilibrium. Tucker developed the “prisoner’s dilemma”. Nash and Tucker’s work laid the foundation for noncooperative game theory.
Nash’s seminal papers, Equilibrium in N-man Games (1950), Noncooperative Games (1951), etc., gave the concept of Nash equilibrium and the existence theorem of equilibrium. In addition, the work of Reinhard Selten and John Harsanyi also promoted the development of game theory. Today game theory has developed into a more complete discipline.
In the 1960s, Seldon proposed a perfect Nash equilibrium with subgames. In the 1980s, Kryptus and Wilson studied incomplete information dynamic games. By the 1990s, work on game theory was winning Nobel Prizes. Since 1994, when the Nobel Prize in Economics was awarded to three game theorists, there have been seven Nobel Prizes in economics related to the research of game theory.
Game theory mainly studies the interaction between formalized incentive structures. It is a mathematical theory and method to study the phenomenon of struggle or competition. Game theory considers the predicted and actual behavior of individuals in games and studies their optimization strategies.
Basic concepts of game theory
First, let’s look at the building blocks of game theory:
- Player: In a contest or game, each player with decision-making power becomes a player. A game with only two players is called a “two-player game”, while a game with more than two players is called a “multi-player game”.
- Strategy: In a game, each player has a practical and feasible complete action plan, that is, the plan is not the action plan of a certain stage, but a plan to guide the whole action, a feasible action plan of a player from beginning to end, called a strategy of this player. A game in which players have a total of a finite number of strategies is called a “finite game”; otherwise it is called an “infinite game”.
- Resource gain and loss: the outcome of a game is called gain and loss. Each player’s gain or loss at the end of a game is not only related to the strategy chosen by the player in the game itself, but also to a set of strategies chosen by the global player. Therefore, the payoff of each player at the end of a game is a function of a set of strategies given by all players, which is usually called the payoff function.
- There is a game outcome for the players.
- A game is about equilibrium: equilibrium means equilibrium, and in economics, equilibrium means that the relevant quantity is at a stable value. In a relationship between supply and demand, we say that there is an equilibrium between supply and demand for a good if at a certain price each person who wants to buy the good can buy it and each person who wants to sell it can sell it. Nash equilibrium, it’s a stable result of a game.
Game theory research hypothesis:
Of course, game theory assumes that under certain conditions, we make the following assumptions:
- The decision-making body is rational and maximizes its own interests;
- Perfect reason is common knowledge;
- Each participant is assumed to have formed correct beliefs and expectations about his or her environment and about the behavior of other participants.
Basic classification of games
There are different categories of games based on different benchmarks.
- Generally speaking, game can be divided into cooperative game and non-cooperative game.
The difference between a cooperative game and a non-cooperative game is whether there is a binding agreement between the parties interacting with each other, if there is, it’s a cooperative game, if not, it’s a non-cooperative game.
What economists now talk about in game theory is generally non-cooperative games, and because cooperative game theory is more complex than non-cooperative game theory, it is far less sophisticated in theory. Non-cooperative games are divided into static games with complete information, dynamic games with complete information, static games with incomplete information and dynamic games with incomplete information. Equilibrium concepts corresponding to the above four games are as follows: Nash equilibrium, Subgame Perfect Nash equilibrium, Bayesian Nash Equilibrium, Refine the Perfect Bayesian Nash equilibrium.
- From the time series of behavior, game theory can be further divided into static game and dynamic game:
A static game is a game in which players choose at the same time or not at the same time but the later actor does not know what specific action the first actor has taken;
Dynamic game refers to a game in which the actions of the players are in sequence, and the latter actor can observe the actions chosen by the former actor. Popular understanding: “prisoner’s dilemma” is simultaneous decision, belongs to the static game; But chess and card games and other decisions or actions in order, belong to the dynamic game
- The game can be divided into complete information game and incomplete information game according to the players’ understanding of other players.
A complete game is a game in which each player has accurate information about other players’ characteristics, strategy space and payoff function.
Incomplete information game refers to a game with incomplete information if participants do not have accurate information about other participants’ characteristics, strategy space and return function, or they do not have accurate information about all participants’ characteristics, strategy space and return function.
There are many different kinds of game theory. For example, there are finite and infinite games in terms of the number or duration of the game. In the form of expression can also be divided into general (strategic) or expansion, and so on.
A classic case
The Prisoner’s Dilemma is a story about two prisoners. The two prisoners were caught doing something bad together by the police and interrogated in two separate, incommunicable cells. In this case, both prisoners can make their own choices:
The situation is divided as shown in the figure above. Either give up his partner (i.e. cooperate with the police and betray his partner), or remain silent (i.e. cooperate with his partner instead of cooperating with the police). Both prisoners knew they would be set free if they kept quiet, because the police could not convict them as long as they refused to confess. But the police knew this too, so they gave the two prisoners a little incentive: if one of them betrayed, by informing on his partner, he could be set free and receive a reward. His accomplice would be sentenced to the most serious crime, and, to increase the punishment, he would be fined as a reward for the informant. Of course, if the two prisoners betray each other, both will be sentenced to the most serious crime, and neither will be rewarded.
So what were the prisoners to do? Cooperate with each other or betray each other? On the face of it, they should cooperate and keep quiet, because then they both get the best outcome: freedom. But they have to think carefully about what options the other might take. Criminal A was no fool. He immediately realized that he could not trust his accomplice not to give evidence against him to the police, and then walk out with A big reward, leaving him alone in prison. The temptation is too great. But he also realized that his partner was no fool to imagine him the same way. So criminal A came to the conclusion that the only rational choice was to betray his partner and tell the police everything, because if his partner was dumb enough to keep quiet, he would be the lucky one to get out of jail with A prize. And if his accomplice also according to this logic to the police confessed, so, criminal A must serve A sentence anyway, at least he does not have to be fined on top of this. As a result, the two prisoners, by desperate logic, got the worst of all: jail time.
summary
This concludes the basic introduction to game theory. It’s a concept that might be easy for you to understand, but game theory is actually very theoretical.
Finally, a case is introduced. In prisoner’s dilemma terminology, failure to confess counts as “cooperation” (with your fellow inmates, not with the police), while confessing counts as “betrayal.” Both cooperate best, both betray and both lose. According to this story, cooperation is impossible. Prisoner’s dilemma scenarios are common in the real world. But co-operation seems more likely than betrayal. Why is that? The possible reason is that the dilemma is not unique. The analysis will continue in a later article.
Books recommended
My new book “ETCD Working Notes: Architecture Analysis, Optimization, and Best Practices” is available throughlinkDirect, thanks for your support.