Game theory provides a common language to formulate, structure, analyze, and understand strategic scenarios involving conflict and cooperation. It investigates situations where the outcome for each participant depends not just on their own actions but also on the decisions of others. Essentially, game theory deals with scenarios of interdependence, where players—whether individuals, groups, companies, or associations—make decisions that affect each other.
What is a Game?
At its core, a game in game theory involves a set of players who interact according to specific rules, with each player having a set of strategies available to them. Players choose these strategies to maximize their benefits, with their choices depending on each other.
The impact of these choices can affect not only the individual players but also the overall group. This interdependence means that game theory provides insights into how players can anticipate the actions of others and adjust their strategies accordingly.
Key Elements of a Game
A ‘game’ in game theory is an abstract representation of a strategic situation that involves interdependence among the players. A game can be defined by three main components:
- Players: Players are the decision-makers—these can be individuals, firms, countries, or any entity capable of choosing strategies. In simpler terms, they are the agents involved in making choices that affect outcomes. For example, in a duopoly scenario, the players are the two competing firms.
- Strategies: Strategies represent the possible actions or choices available to the players. In the Cournot competition, each firm’s strategy involves choosing a quantity to produce, considering what the other firm might do. In a general n-player game, each player has a strategy set, Si, consisting of various strategies. Each strategy in the set is denoted by si.
- Payoffs: Payoffs are the rewards or returns that players receive based on the strategies chosen and the outcomes of the game. In a profit-maximizing scenario, profits could represent payoffs and each player.
What is Game Theory?
Game theory is an approach to modelling situations where the outcome of one’s decisions depends on the actions of others. It studies strategic, interactive decision-making among rational individuals or organizations, where each participant’s choices are interdependent. This means that each player must consider what others might do when formulating their strategy.
Game theory is a branch of applied mathematics, that provides tools for analyzing scenarios where parties (players) make decisions that affect each other’s outcomes. These players can be individuals, nations, corporations, or teams. The core idea is that each player’s strategy depends on the possible decisions of others, and they must think strategically about how their choices will impact others and vice versa.
A solution to a game describes the optimal decisions of the players, considering their shared, conflicting, or mixed interests, and the potential outcomes from these decisions. Game theory is widely used in the business world to determine strategies for competition, market entry, pricing, and more.
History of Game Theory
- Foundations by John von Neumann (1928-1944):
- 1928: John von Neumann laid the foundation for game theory with his theory of two-person zero-sum games. This theory provided a mathematical framework to understand strategic interactions between two players where one player’s gain is equivalent to the other’s loss.
- 1944: The publication of “Theory of Games and Economic Behavior” by von Neumann and Oskar Morgenstern marked a significant development in game theory. It introduced the concept of mixed strategies and the notion that economics and strategic decision-making can be modelled as games where players anticipate the moves of others. This collaboration extended game theory beyond military strategy into economic and social behaviours, treating strategic decisions as a form of a competitive game.
- Extension by John Nash (1950s-1960s):
- In the 1950s and 1960s, John Nash made significant contributions by extending the theory to non-cooperative games. Nash introduced the concept of Nash Equilibrium—where a set of strategies for all players exists such that no player can benefit by deviating unilaterally. This concept revolutionized the understanding of rational behaviour in strategic settings, particularly in economic contexts.
- Nash’s work was crucial in developing the idea of equilibrium points, which became essential in analyzing how rational players interact under interdependence. His insights helped formalize the concept of stability in strategic decision-making.
- Recognition and Impact (1994):
- The Nobel Prize in Economics in 1994 recognized the profound impact of game theory on various fields. It acknowledged how game theory helped explain conflicts, cooperation, and competition in modern economic and social systems. The award specifically highlighted the importance of Nash Equilibrium in understanding the stability of outcomes in competitive settings.
Key Examples of Game Theory
- The Prisoner’s Dilemma: This classic example involves two suspects separated in custody. They must choose whether to betray each other or remain silent. The best individual strategy for each prisoner is to betray the other (resulting in five years of prison time for each), even though both would be better off staying silent (serving six years together). This outcome is known as a Nash equilibrium, where each player’s best strategy is the same when both are acting rationally.
- Conflict of Sales: Imagine a pharmaceutical company with a dominant drug facing aggressive pricing from a competitor. The client wants to know how to respond strategically without engaging in a price war. Using game theory, the solution was to maintain high prices and avoid competition, maximizing profits in the long term.
- Advertising War: Coke vs. Pepsi: Both companies earn $5 billion/year from cola sales. They can choose to spend $2 billion on advertising, knowing that this decision affects the other’s strategy. If one company advertises and the other does not, the advertiser captures $3 billion from the competitor. The equilibrium strategy is complex, showing how advertising can disrupt market balance.
Applications in Everyday Life
Game theory is used in various fields including business, politics, economics, sports, and personal decision-making. From negotiating deals, and setting prices, to making strategic decisions in poker or chess, game theory provides a framework to understand the interdependent actions of different agents.
Game theory in personal life: This can be seen in dating strategies, career choices, or even deciding whether to invest in stocks based on market trends. Every decision affects other players’ decisions, thus influencing one’s outcomes.
Classification of Game Theory
- Classical Game Theory: Focuses on strategic play in scenarios where the impact of a player’s decision is known, such as poker, negotiations, and military strategy.
- Combinatorial Game Theory: Involves two-player games like chess and checkers where moves change the game’s structure.
- Dynamic Game Theory: Analyzes games where players make decisions over time, such as optimal strategies in military conflicts or tracking a fleeing individual.
Types of Games
1. Normal Form Games
In this form, games are represented through a payoff matrix. Each cell in the matrix shows the payoff for each player for the various combinations of strategies they might choose. This type is useful for simple interactions like the Prisoner’s Dilemma, where each player must choose between cooperating or defecting, and the payoffs depend on the choices made by both players.
For instance, consider the Prisoner’s Dilemma:
- Player 1: Cooperate (C) or Defect (D)
- Player 2: Cooperate (C) or Defect (D)
- The payoff matrix could look like this:
| Player 2 (D) | Player 2 (C) |
| Player 1 (D) | (3, 3) |
| Player 1 (C) | (4, 1) |
Here, each pair of numbers represents the payoff for Player 1 and Player 2 respectively when they choose their strategies. For example, if Player 1 defects (D) and Player 2 cooperate (C), Player 1 gets 1 and Player 2 gets 4.
2. Extensive Form Games
Unlike normal-form games, extensive-form games are sequential, meaning players make decisions one after the other. They are represented as a game tree, where each node represents a decision point and each edge represents a possible action. The terminal nodes of the tree show the payoffs for the players based on the chosen strategies. This format is useful for complex interactions where the order of moves matters, such as in negotiation or chess.
3. Non-cooperative vs. Cooperative Games:
Cooperative Game Theory involves players forming binding agreements to reach an optimal outcome that benefits the group as a whole. These agreements are shared equally among the members. For example, two firms might negotiate joint investment in new technology. In contrast, non-cooperative game theory involves players acting independently and self-interestedly, maximizing their own utility without regard for the effects on others. For example, competing firms independently set prices without making binding agreements.
4. Games of Complete and Incomplete Information:
- In games of complete information, every player knows the payoffs, strategies, and types of other players involved. This allows players to plan their actions effectively, assuming they can anticipate the moves of others and adjust their strategies accordingly. Examples include simultaneous-move games like the Cournot competition.
- In games of incomplete information, players do not have full information about their opponents. Each player may possess private information, which influences their decisions. Auctions are a classic example, where players do not know each other’s valuations for an item. The solution concepts used here are Bayesian Nash equilibrium or Perfect Bayesian equilibrium, which take into account uncertainty and incomplete knowledge.
5. Zero-sum vs. Non-Zero Sum Games:
- Zero-sum games are those where the gain of one player is exactly equal to the loss of another, meaning the sum of payoffs is zero. A classic example is a two-player game of chess or poker, where one player’s gain is another’s loss.
- Non-zero sum games involve scenarios where players’ payoffs do not necessarily sum to zero—gains can be mutually beneficial. For example, advertising strategies lead to higher profits for multiple firms rather than one firm taking away profits from another.
Final Words
Game theory’s power lies in its ability to structure and analyze complex strategic choices, offering a clearer view of decision-making processes. As a mathematical tool for decision-makers, it provides a methodology to model scenarios, helping to anticipate outcomes and optimize strategies across various fields. Through its prescriptive application, game theory improves strategic decision-making by clarifying options, preferences, and potential reactions of others.
