Game Theory Definition Formulas Nash Equilibrium and Solved Examples
Based on true logic and analytical decision-making, the theories and steps of Game Theory are applied to various sources in real-life. Both mathematical and non-mathematical fields make use of Game theory to acquire relevant information and make the event a success. Sounds a bit vague but let’s take a deeper look at the concept of Game theory with suitable examples in the following sections.
Game Theory - Definition and Important Pointers to Learn
Game theory - definition is quite straightforward in its meaning. It is a practical math concept on how a certain game is played using effective strategies. From every point of view, the player’s information such as timing, type of move, the direction of motion, the order of playing, etc. are some of the Game theory’s important determinants.
The Game theory is based purely on strategic playing. Balancing the diplomatic behaviour along with awareness about the rules of the event and each candidate participating is vital to winning the play.
The essence of a given game lies in its interdependence of the next move of each player.
The game theory falls under the Probability Distribution majors.
This interdependence is classified into 2 forms - Sequential and Simultaneous.
Being cautious about the other player, the person will move and arrange entities in a game within a specifically ordered fashion. This is called sequential game theory.
On the other hand, the simultaneous game theory involves playing all in 1 go, not accounting the moves or actions of the other player (s).
Game theory is referred to as the ‘science of strategies’.
The only best possible outcome is expressed only when the game theory is balanced between logic and mathematics.
This mathematical tactic deals with both conflict and cooperation among each player. Hence, rational decision-making can affect a player’s interest in their part.
What is the Theory of Zero-Sum Game?
Zero-sum game or the ‘constant-sum game’ deals with increasing or reducing the resources available during certain games. As the name says, the addition of every strategic combination will sum up to zero (0) only. Hence, the benefits gained by each player adds to zero. So, if 1 opponent wins, the other exact will lose the game.
Unlike the definition of game theory, here in a zero-sum game, there is an opposition between the interest of 2 players.
If a player wins the match but it is not probable to state if the other wins or not is termed to be the ‘non-zero-sum game’.
The Applications of Game Theory
The idea of studying sociological, psychological, and political framework or something or someone, makes the application of game theory important in the field of social sciences like the population census, psychiatry, sociology, social work, etc.
From understanding the behaviour of firms and agencies to consumers and the market, this theory is preferred in the fields of economics and certain business sectors as well.
Even a few analysis-related subject-matters are discussed inside the fields of biology and zoology.
Game theory is useful in the learning of animal and human behaviour.
Not only is the application of game theory defined in the understanding of general behaviour in elements. Even developmental, normative, legal and ethical behaviours and motives are also assessed and judged with better precision using the game theory.
Let us finally close our learning with a simple example of game theory in the context of the most classic example of history deemed the ‘Prisoner’s Dilemma.’
The Classic Instance of Game Theory
The imaginary case of the Prisoner’s Dilemma involves 2 individuals stealing a vehicle and is held up. Upon further enquiry, it is also found that both the men were involved in a bank robbery issue as well. 2-years imprisonment is going to the judgement here.
Both the persons were put in different jails. Till now, both the men are suspected for the bank robbery case. Since the prisoners are jailed in 2 separate chambers, the line of communication between the 2 is already cut.
Now, with the investigation procedures forwarded, the following 2 response situations are about to be given for the 2 prisoners:
3-years of imprisonment, if both accused of bank robbery and vehicle theft.
1-year imprisonment for the one confessing the truth behind bank robbery. The other unanswered is given the order for 10-years imprisonment.
The table below includes the possible solutions/outcomes to be expected from both.
2-CONFESS
2-DENY
1-CONFESS
Both punished 3 years
Prisoner 1 punished 1 year
Prisoner 2 punished 10 years
1-DENY
Prisoner 1 punished 10 year
Prisoner 2 punished 1 year
Both punished 2 year
The table displayed above is drawn based on Game theory. And now, the likely choice is to not accept the theft. In that scenario, both the prisoners would serve 2-years imprisonment. However, it is nearly impossible to say whether the other person will confess or not. Hence, the chances are high for both the men to get confessed and serve the 3-years jail sentence.
Conclusion
Game theory is a practical concept and is regarded as the science of strategies. Game theory involves logic and mathematics. There are 2 forms of playing a game - sequential and simultaneous. Sequential gameplay takes account of the other player’s move and simultaneously is randomly ordered. Fields such as psychology, biology, statistics, social sciences and more use game theory. The ‘Prisoner’s Dilemma’ is the famous case of interpreting the outcomes using game theory.
FAQs on Game Theory in Mathematics and Economics
1. What is game theory in mathematics?
Game theory is the mathematical study of strategic decision-making where players choose actions that affect each other's outcomes. In mathematics, it analyzes situations involving:
- Players (decision-makers)
- Strategies (possible actions)
- Payoffs (rewards or outcomes)
2. What is a Nash equilibrium in game theory?
A Nash equilibrium is a strategy profile where no player can improve their payoff by changing their strategy alone. At Nash equilibrium:
- Each player's strategy is the best response to others.
- No unilateral deviation increases payoff.
3. What is the difference between cooperative and non-cooperative game theory?
The main difference is that cooperative game theory allows binding agreements, while non-cooperative game theory does not. In detail:
- Cooperative games: Players form coalitions and share payoffs.
- Non-cooperative games: Players act independently and cannot enforce agreements.
4. What is a payoff matrix in game theory?
A payoff matrix is a table that shows the payoffs for each player for every combination of strategies. In a 2-player game:
- Rows represent Player 1's strategies.
- Columns represent Player 2's strategies.
- Each cell shows ordered payoffs (Player 1, Player 2).
5. What is a dominant strategy in game theory?
A dominant strategy is a strategy that gives a player a higher payoff regardless of what the other players do. There are two types:
- Strictly dominant: Always gives a strictly higher payoff.
- Weakly dominant: Never worse and sometimes better.
6. What is a zero-sum game?
A zero-sum game is a game where one player's gain equals the other player's loss, so the total payoff sums to zero. Mathematically:
- If Player 1 gains +x, Player 2 loses −x.
- Total payoff = 0.
7. How do you find the Nash equilibrium in a payoff matrix?
To find a Nash equilibrium in a payoff matrix, identify mutual best responses. Follow these steps:
- Step 1: For each column, mark Player 1's highest payoff (best response).
- Step 2: For each row, mark Player 2's highest payoff.
- Step 3: Cells where both players' best responses coincide are Nash equilibria.
8. What is a mixed strategy in game theory?
A mixed strategy is a probability distribution over possible pure strategies. Instead of choosing one action, a player:
- Assigns probabilities to each strategy.
- Randomizes according to those probabilities.
9. What is the Prisoner's Dilemma in game theory?
The Prisoner's Dilemma is a game where rational players choose strategies that lead to a worse collective outcome. Key features:
- Each player can cooperate or defect.
- Defection is the dominant strategy.
- Mutual defection is the Nash equilibrium.
10. What are the main applications of game theory?
The main applications of game theory include economics, business strategy, and evolutionary biology. It is used in:
- Economics: Auctions, market competition, pricing strategies.
- Business: Competitive strategy and negotiations.
- Political science: Voting systems and conflict analysis.
- Biology: Evolutionarily stable strategies.
