Game theory is a branch of mathematics that deals with the study of strategic decision-making in situations where two or more individuals or groups have competing interests. One of the most fundamental concepts in game theory is the concept of a “tree”.
A tree in game theory is a graphical representation of a sequence of decisions that players make in a game. At each decision point, the player must choose one of several possible actions, which then dictates the subsequent course of the game.
Trees are commonly used to represent games with multiple stages, or games that involve a series of interrelated decisions. By using trees to model these types of games, analysts can better understand how different decision paths may influence an outcome.
Structure of a Game Tree
A typical game tree consists of nodes and branches. Each node represents a decision point in the game, while each branch represents one possible action that a player can take at that decision point.
At the topmost level of the tree, there is usually only one node representing the initial decision point. From this node, several branches will emerge, each representing a possible action that can be taken at this stage in the game.
As we move down to lower levels in the tree, additional nodes and branches will appear. Each new node represents a new decision point in the game, while each branch represents another possible action that can be taken at that point.
Let’s consider an example to illustrate how trees are used to model games. Suppose we have two players who are playing a simple card game. The objective of this card game is to get as close as possible to 21 without going over.
In this example, we’ll use a simplified version of the card game where both players are dealt only one card each and must decide whether to “hit” (draw another card) or “stay” (keep their current card). The player who gets closest to 21 without going over wins.
The game tree for this example would look something like this:
Game Tree for a Simple Card Game
- Initial Decision Point
- Player 1: Hit or Stay
- Player 2: Hit or Stay
- If both players stay, the game ends and the player with the higher card wins.
- If one player hits and the other stays, the hitting player draws another card and then the game ends.
- If both players hit, they each draw another card and then compare their totals. The player with the higher total wins.
In this game tree, the initial decision point is represented by a single node at the top of the tree. From this node, there are two branches representing the two possible actions that each player can take.
If both players choose to stay at this decision point, we can see that there is only one possible outcome – they compare their cards and the player with the higher card wins. This outcome is represented by a single node at level two of the tree.
If one player chooses to hit while the other stays, we can see that there are two possible outcomes – either the hitter will get closer to 21 than their opponent and win, or they will go over 21 and lose. These outcomes are represented by two nodes at level two of the tree.
If both players choose to hit, we can see that there are several possible outcomes depending on what cards they draw. Each of these outcomes is represented by a separate node at level three of the tree.
The Importance of Game Trees in Game Theory
Game trees are an essential tool in game theory because they allow analysts to model complex decision-making scenarios in a structured and organized way. By representing games as trees, analysts can better understand how different decisions and outcomes are related to one another.
One of the key insights that game trees provide is the idea of “backward induction”. This is the process of working backwards through a game tree to determine what decisions players should make at each stage in order to achieve their desired outcome.
By using game trees to analyze strategic decision-making scenarios, analysts can gain valuable insights into how different factors such as risk, uncertainty, and information affect outcomes. This information can then be used to develop strategies that maximize a player’s chances of success.
In summary, a tree in game theory is a graphical representation of a sequence of decisions that players make in a game. Game trees are an essential tool in game theory because they allow analysts to model complex decision-making scenarios in a structured and organized way. By using game trees to analyze strategic decision-making scenarios, analysts can gain valuable insights into how different factors affect outcomes and develop optimal strategies for success.