GAME THEORY — The Science of Strategy
INTRODUCTION
Game theory is known as the science of strategy. By strategy, I mean the optimal decisions taken by the rational players, everywhere and every time in their life. It includes mathematical models to solve the competitive decision-making problems between the people (rational players).
It has applications in many fields like social science, behavioral economics, biology, computer science, philosophy, etc. The key pioneers of it were mathematician John Von Neumann and economist Oskar Morgenstern. They published the book in 1944 called Theory of Games and Economic Behavior.
Let me phrase a simple sentence to determine how game theory works. “In a competitive situation, using game theory one can determine the best action or best decision a rational player should choose which depends upon the expectations about what best action another rational player will choose and whose best decision again depends upon the expectations about the best action of initial rational player”.
GAME
Every game consists of a bunch of rules and I am sure you all must be aware of some, like cricket or football or tennis, etc. Every game has the players and three outcomes i.e., win, lose and draw and which we referred to as payoff (refer to my article on “expectations”) as every player gets some utility with each outcome. Now, comes the role of strategies or optimal decisions about how to play the game. In the end, we get the results (solutions or equilibria) of the game. Since Nobel Laureate Mathematician John Nash did the first significant extension of the John Von Neumann and Morgenstern’s pioneering work, the above-specified solutions/outcomes/equilibria of the games are known as Nash Equilibrium. At Nash Equilibrium, a player can’t improve his payoffs by changing his own decision given the strategies or decisions of the other player in the game.
TYPES OF GAME
1.Simultaneous Games: Games where players move simultaneously or they move together without knowing the decisions or strategies played by the other players at the same time.
2.Sequential Games: Games where players move in sequence i.e., one after another, and the later players have either the perfect or imperfect information about the decisions or strategies played by the initial players.
3.Cooperative Games: Games where players agreed on binding commitments like forming a coalition, taking grouped decisions, and getting collective payoffs.
4.Non-Cooperative Games: Games where players can’t form alliances and deals with each other on an individual basis to achieve their goals.
5. Symmetric Games: Games where we can interchange the player’s identity without changing the respective payoff and still get the same game or it doesn’t matter who is playing the game but what strategy is being played in the game.
7.Zero-Sum Games: Games where the total payoff or sum of the payoff of all players is equal to zero in every combination of strategy sets. For ex-
Total payoff at {A, A} = -1+1 = 0
Total payoff at {A, B} = 3+(-3) = 0
Total payoff at {B, A} = 0+0 = 0
Total payoff at {B, B} = -2+2 = 0
APPLICATIONS
Game theory helps to formulate exhaustive models to define a decision-making process by the rational players and the best outcomes in most of the real-world scenarios.
In businesses, where firms are facing competition from the other firms in the market, in terms of product pricing, product quantity, quality, marketing, and many more. Game theory emerges as a way out of these competitive situations by providing the best decision model for the firms to play their best possible decisions or strategies considering the best possible decisions of the other firms. It helps in understanding and modeling all the monopolistic, duopolistic, oligopolistic behaviors of the firms.
Game theory can be applied in many real-world scenarios but after talking about businesses, I would like to talk about one interesting application of it, which is in biology.
Biologists used game theory to make some models to understand the evolution and behavior of different species towards each other. One most popularly used game is known as Game of Chicken or Hawk- Dove Game.
X ~ die, W ~ Win, L ~ Lose, T ~ Tie
It can be seen that if two hawks confront each other either competing for a mate or for some disputed territory, they start fighting till either anyone dies or both die due to fighting, which implies a high level of casualty rate. In the case of two doves, they start cooperating but if we consider an invasion of a hawk then they both become vulnerable and the hawk wins and dove loses as shown in the above matrix. Hence, species consisting of either complete hawks or complete doves are vulnerable.
FAMOUS EXAMPLES OF GAME
1.Prisoner’s Dilemma: Suppose police caught two people A and B as suspects of committing a robbery together. Two were being isolated and asked to confess by the police. Police made the following offer to both of them.
· If both confess their crime then they will go to jail for 5 years.
· If no one confesses then they will go to jail for 1 year.
· If one confesses while the other doesn’t then the confessor will leave free and the silent one goes to jail for 10 years
We can show the above game using the following matrix.
Since both prisoners are not allowed to talk to each other, no one knows about the decision or strategy that another prisoner is going to play and both prefer a less sentence in jail. As can be seen in the above matrix, the best decision for every prisoner is to confess irrespective of what another prisoner is choosing. Suppose prisoner B chooses to confess then for prisoner A it is optimum to also choose confession, otherwise, he will go to jail for 10 years instead of 5 years and if prisoner B choose to remain silent then again prisoner A will choose to confess, otherwise, he will go to jail for 1 year instead of going free. Similarly, for Prisoner B, it will be optimum or best to choose confession irrespective of what prisoner A is going to choose.
The learning outcome is that when both of them confesses, as shown above, both will end up with 5 years in jail but if they would have allowed to talk or made an agreement to cooperate by remaining silent or acting selflessly then they would have to go to jail only for 1 year.
2.Stag–Hunt: There are two hunters in a jungle and they have two choices for hunting i.e., either a stag (more meat) or a hare (less meat). Following information is given.
· Both prefer a stag than a hare
· To get a stag, they both have to do hunting together or cooperate while hunting a stag and will divide the stag equally and get the utility or payoff of 2 each.
· If both go for a hare, they both will get a hare and payoff of 1 each.
· If one goes for a hare while another for a stag, the initial one will get a hare with a payoff of 1 but the latter will get nothing with a payoff of 0.
We can show the above game using the following matrix.
Here, the best decision of the hunters is to cooperate and go for hunting together to get the best and optimum payoffs i.e., (2,2) but if each hunter decides individually or without knowing about another hunter’s strategy and think that another hunter will go for a hare, then it will be best for the initial player to also go for a hare, otherwise he will get a payoff of 0 instead of 1. Similarly, we can show these same sets of strategies from the second player’s point of view.
