To a large extent, it can be said that the most dynamic principle that governs our existence is interaction. We interact with each other within varied frameworks such as social, political, economic, military so on, and so forth. It is not just humans who interact but businesses, processes, and technology almost everything interacts with every other thing. And this inevitable interaction warrants the finest strategy and optimal decision making. Right from choosing toothpaste to choosing the right financial portfolio every one of us is a player in this grand game called life and is expected to make a sound rational decision through strategic interaction with other players. And to assist us in our endeavor and our pursuit of happiness we have a fine companion in Game theory.
As Wikipedia puts it Game theory is the study of mathematical models of strategic interaction among rational decision-makers. So, game theory provides a mathematical framework to a model wide variety of decision problems. While game theory is imperative for many disciplines, it is indispensable for the study of economics. Two competing firms having identical products in perfect competition, having to choose an equilibrium strategy to decide the pricing of the product is an example of game theory modeling in economics. In the field of politics, the decision that how should candidates position themselves along the political spectrum is a typical game theory Hotelling problem. In biology, there have been numerous incongruous behaviors for example consider the case of worker bees that slog their entire life to nourish queen bee and never mate. Evolutionary game theory has been used to explain such inconsistent behaviors. In computer science, Algorithmic game theory helps understand and design algorithms in strategic environments. From philosophy to operational research, game theory offers its coffer in full to anyone who approaches it with utmost dedication and interest.
Before we embark on our voyage in the inspiring world of game theory, let us quickly look at basic elements and assumptions in game theory:
At the heart of game theory lies a Game: Any competitive activity whose outcome depends on the strategic interaction between two or more decision-makers (better known as players) is a game. Chess, football, boxing, etc are examples of games and so are missile defense, price wars, and management negotiations. All the strategic decision-makers within the context of the game are Players. Players could be competing individuals who say as in chess and could be competing firms in a duopoly. In the context of the game, every player encounters a decision problem characterized by three fundamental questions:
1: What are the possible choices?
2: What is the result of each of these choices?
3: And how does each outcome affects the player?
These questions now help us formalize three basic concepts of; action, outcomes, and pay-offs. The set of all possible choices that a player has is called actions. Actions result in possible consequences known as outcomes. Each player receives a payout from every outcome, which is known as the payoff. A payoff function assigns quantifiable value to ordinal payoffs ranging from money to utility.
A very interesting example in Game theory is the game of matching pennies played between two players, Player 1 and Player 2. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), then Player 1 keeps both pennies, so wins one from Player 2 (resulting in +1 for Player 1, −1 for Player 2). If the pennies do not match (one head and one tail) Player 2 keeps both pennies, so receives one from Player 1 (−1 for Player 1, +1 for Player 2).
In this game, each player has two choices and thus action set has two elements either they can play heads or tails. The interaction of two players’ choices results in four outcomes which can be represented by the following matrix. HH, HT, TH, & TT. If it HH or TT player 1 win and HT or TH player 2 wins. If the outcome is HH then the payoff of players 1 and 2 are +1,-1 respectively and in the same way we can write payoffs for all the outcomes.
Now before moving forward, let us examine some assumptions in game theory.
Rational Choice assumption: Rational choice assumption asserts that a player’s choice in choosing between potential actions will choose the action that gives him the highest possible payoff. So, the rational choice assumption in the case of matching pennies imposes the condition that both players know all possible actions, all possible outcomes, and the possible payoffs from each outcome.
Another way to represent matching pennies game is by using a decision tree using branches and nodes.