Game theory is a field of research that is sort of distinguished in Economics but relatively unpopular in other scientific disciplines. Nonetheless, the concepts utilized in game theory could be of interest to a wider audience, including data scientists, statisticians, computer scientists or psychologists, to call just just a few. This text is the opener to a four-chapter tutorial series on the basics of game theory, so stay tuned for the upcoming articles.
In this text, I’ll explain the sorts of problems Game Theory deals with and introduce the foremost terms and ideas used to explain a game. We’ll see some examples of games which can be typically analysed inside game theory and lay the muse for deeper insights into the capabilities of game theory within the later chapters. But before we go into the main points, I would like to introduce you to some applications of game theory, that show the multitude of areas game-theoretic concepts could be applied to.
Applications of game theory
Does it make sense to vote for a small party in an election if this party may not have a probability to win anyway? Is it value starting a price battle together with your competitor who offers the identical goods as you? Do you gain anything when you reduce your catch rate of overfished areas in case your competitors simply carry on as before? Should you are taking out insurance when you imagine that the federal government can pay for the reconstruction after the following hurricane anyway? And the way must you behave in the following auction where you might be about to bid in your favourite Picasso painting?
All these questions (and plenty of more) live throughout the area of applications that could be modelled with game theory. Every time a situation includes strategic decisions in interaction with others, game-theoretic concepts could be applied to explain this case formally and seek for decisions that will not be made intuitively but which can be backed by a notion of rationality. Key to all of the situations above is that your decisions rely upon other people’s behaviour. If everybody agrees to conserve the overfished areas, you would like to play along to preserve nature, but when you think that everyone else will proceed fishing, why must you be the just one to stop? Likewise, your voting behaviour in an election might heavily rely upon your assumptions about other people’s votes. If no one votes for that candidate, your vote can be wasted, but when everybody thinks so, the candidate doesn’t have a probability in any respect. Perhaps there are numerous individuals who say “I might vote for him if others vote for him too”.
Similar situations can occur in very different situations. Have you ever ever thought of having food delivered and everybody said “You don’t must order anything due to me, but when you order anyway, I’d take some french fries”? All these examples could be applications of game theory, so let’s start understanding what game theory is all about.
Understanding the sport
Whenever you hear the word , you would possibly consider corresponding to Minecraft, corresponding to Monopoly, or corresponding to poker. There are some common principles to all these games: We at all times have some who’re allowed to do certain things determined by the sport’s . For instance, in poker, you may raise, check or surrender. In Monopoly, you may buy a property you land on or don’t buy it. What we even have is a few notion of the best way to the sport. In poker, you may have to get one of the best hand to win and in Monopoly, you may have to be the last person standing after everybody went bankrupt. That also signifies that some actions are higher than others in some scenarios. If you may have two aces on the hand, staying in the sport is healthier than giving up.
After we take a look at games from the angle of game theory, we use the identical concepts, just more formally.
A game consists of a set of players I = {1, .., n}, where each player has a set of strategies S and a utility function ui(s1, s2, … sn). The set of strategies is decided by the principles of the games. For instance, it might be S = {check, raise, give-up} and the player would have to make your mind up which of those actions they wish to use. The utility function u (also called reward) describes how worthwhile a certain motion of a player can be, given the actions of the opposite players. Every player wants to maximise their utility, but now comes the tricky part: The utility of an motion of yours is dependent upon the opposite players’ actions. But for them, the identical applies: Their actions’ utilities rely upon the actions of the opposite players (including yours).
Let’s consider a widely known game as an instance this point. In rock-paper-scissors, we have now n=2 players and every player can make a choice from three actions, hence the strategy set is S={rock, paper, scissors} for every player. However the utility of an motion is dependent upon what the opposite player does. If our opponent chooses rock, the utility of paper is high (1), because paper beats rock. But in case your opponent chooses scissors, the utility of paper is low (-1), because you’ll lose. Finally, in case your opponent chooses paper as well, you reach a draw and the utility is 0.
As a substitute of writing down the utility function for every case individually, it is not uncommon to display games in a matrix like this:
The primary player decides for the row of the matrix by choosing his motion and the second player decides for the column. For instance, if player 1 chooses paper and player 2 chooses scissors, we find yourself within the cell within the third column and second row. The worth on this cell is the utility for each players, where the primary value corresponds to player 1 and the second value corresponds to player 2. (-1,1) signifies that player 1 has a utility of -1 and player 2 has a utility of 1. Scissors beat paper.
Some more details
Now we have now understood the foremost components of a game in game theory. Let me add just a few more hints on what game theory is about and what assumptions it uses to explain its scenarios.
- We frequently assume that the players select their actions at the identical time (like in rock-paper-scissors). We call such games static games. There are also dynamic games during which players take turns deciding on their actions (like in chess). We’ll consider these cases in a later chapter of this tutorial.
- In game theory, it is usually assumed that the players can’t communicate with one another in order that they can’t come to an agreement before deciding on their actions. In rock-paper-scissors, you wouldn’t want to do this anyway, but there are other games where communication would make it easier to decide on an motion. Nonetheless, we are going to at all times assume that communication isn’t possible.
- Game theory is taken into account a normative theory, not a descriptive one. Which means we are going to analyse games regarding the query “What can be the rational solution?” This will likely not at all times be what people do in a likewise situation in point of fact. Such descriptions of real human behaviour are a part of the research field of behavioural economics, which is positioned on the border between Psychology and economics.
The prisoner’s dilemma
Allow us to change into more aware of the foremost concepts of game theory by taking a look at some typical games which can be often analyzed. Often, such games are derived from are story or scenario which will occur in the true world and require people to make your mind up between some actions. One such story might be as follows:
Say we have now two criminals who’re suspected of getting committed against the law. The police have some circumstantial evidence, but no actual proof for his or her guilt. Hence they query the 2 criminals, who now have to make your mind up in the event that they wish to confess or deny the crime. Should you are within the situation of one among the criminals, you would possibly think that denying is at all times higher than confessing, but now comes the tricky part: The police propose a deal to you. Should you confess while your partner denies, you might be considered a crown witness and won’t be punished. On this case, you might be free to go but your partner will go to jail for six years. Seems like a great deal, but remember, that the consequence also is dependent upon your partner’s motion. Should you each confess, there isn’t a crown witness anymore and also you each go to jail for 3 years. Should you each deny, the police can only use circumstantial evidence against you, which is able to lead to at least one yr in prison for each you and your partner. But remember, that your partner is obtainable the identical deal. Should you deny and he confesses, he’s the crown witness and also you go to jail for six years. How do you choose?
The sport derived from this story is known as the prisoner’s dilemma and is a typical example of a game in game theory. We are able to visualize it as a matrix identical to we did with rock-paper-scissors before and on this matrix, we easily see the dilemma the players are in. If each deny, they receive a relatively low punishment. But when you assume that your partner denies, you could be tempted to admit, which might prevent you from going to jail. But your partner might think the identical, and when you each confess, you each go to jail for longer. Such a game can easily make you go round in circles. We’ll discuss solutions to this problem in the following chapter of this tutorial. First, let’s consider some more examples.
Bach vs. Stravinsky
You and your friend wish to go to a concert together. You’re a fan of Bach’s music but your friend favors the Russian twentieth. century composer Igor Stravinsky. Nonetheless, you each wish to avoid being alone in any concert. Although you favor Bach over Stravinsky, you’ll relatively go to the Stravinsky concert together with your friend than go to the Bach concert alone. We are able to create a matrix for this game:
You select for the row by going to the Bach or Stravinsky concert and your friend decides for the column by going to one among the concert events as well. For you, it might be best when you each selected Bach. Your reward can be 2 and your friend would get a reward of 1, which remains to be higher for him than being within the Stravinsky concert all by himself. Nonetheless, he can be even happier, when you were within the Stravinsky concert together.
Do you remember, that we said players will not be allowed to speak before making their decision? This instance illustrates why. Should you could just call one another and judge where to go, this might not be a game to analyze with game theory anymore. But you may’t call one another so you simply must go to any of the concert events and hope you’ll meet your friend there. What do you do?
Arm or disarm?
A 3rd example brings us to the realm of international politics. The world can be a much happier place with fewer firearms, wouldn’t it? Nonetheless, if nations take into consideration disarmament, additionally they have to contemplate the alternatives other nations make. If the USA disarms, the Soviet Union might wish to rearm, to have the option to attack the USA — that was the pondering through the Cold War, no less than. Such a scenario might be described with the next matrix:
As you see, when each nations disarm, they get the very best reward (3 each), because there are fewer firearms on the earth and the danger of war is minimized. Nonetheless, when you disarm, while the opponent upgrades, your opponent is in the higher position and gets a reward of two, when you only get 0. However, it might need been higher to upgrade yourself, which provides a reward of 1 for each players. That is healthier than being the just one who disarms, but not nearly as good as it might get if each nations disarmed.
The answer?
All these examples have one thing in common: There isn’t any single option that’s at all times one of the best. As a substitute, the utility of an motion for one player at all times is dependent upon the opposite player’s motion, which, in turn, is dependent upon the primary player’s motion and so forth. Game theory is now desirous about finding the optimal solution and deciding what can be the rational motion; that’s, the motion that maximizes the expected reward. Different ideas on how exactly such an answer looks like can be a part of the following chapter on this series.
Summary
Before continuing with finding solutions in the following chapter, allow us to recap what we have now learned up to now.
- A game consists of players, that determine for actions, which have a utility or reward.
- The utility/reward of an motion depends on the opposite players’ actions.
- In static games, players determine for his or her actions concurrently. In dynamic games, they take turns.
- The prisoner’s dilemma is a extremely popular example of a game in game theory.
- Games change into increasingly interesting if there isn’t a single motion that is healthier than some other.
Now that you simply are aware of how games are described in game theory, you may take a look at the following chapter to learn the best way to find solutions for games in game theory.
References
The topics introduced listed below are typically covered in standard textbooks on game theory. I mainly used this one, which is written in German though:
- Bartholomae, F., & Wiens, M. (2016). . Wiesbaden: Springer Fachmedien Wiesbaden.
Another in English language might be this one:
- Espinola-Arredondo, A., & Muñoz-Garcia, F. (2023). . Springer Nature.
Game theory is a relatively young field of research, with the primary foremost textbook being this one:
- Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior.
