Game Theory
Game Theory in Microeconomics
Game theory, also known as the theory of games, is a core theoretical tool in microeconomics that studies strategic interactive behaviors among rational decision-makers. It focuses on how economic agents maximize their own interests through strategic analysis when the outcome of one agent’s decision depends not only on its own choices but also on the choices of other agents. Breaking through the traditional microeconomic assumptions of "perfect competition" and "perfect information", game theory is more applicable to real-world scenarios such as oligopoly, corporate cooperation and competition.
I. Core Basic Concepts of Game Theory
To analyze any game, five core elements need to be clarified:
- Players
Refers to the rational entities that make decisions in the game, which can be individuals, enterprises, or even countries. The goal of players is to maximize their own payoffs by choosing strategies. - Strategies
Refers to the action plans or rules that players can choose during the game. Strategies are divided into pure strategies (clear single action choices, such as "cooperate" or "defect") and mixed strategies (choosing different pure strategies with certain probabilities, such as cooperating with a 60% probability and defecting with a 40% probability). - Payoffs
Refers to the utility or rewards obtained by players after choosing a specific combination of strategies, usually intuitively represented by a payoff matrix. - Information
Refers to the degree to which players understand the game rules, the strategies of other players and their corresponding payoffs. According to whether information is completely symmetric, games can be divided into games of complete information and games of incomplete information. - Equilibrium
Refers to a stable state where all players have chosen their optimal strategies and no one is willing to change their strategy unilaterally. The most core equilibrium concept in game theory is the Nash equilibrium.
II. Main Classifications of Game Theory
1. Classification by Information Symmetry
- Games of Complete Information
All players are fully aware of the game rules, the strategy sets of other players and the corresponding payoffs. A typical example is the Prisoner’s Dilemma. - Games of Incomplete Information
At least one player does not know the payoff functions or strategy choices of other players. In this case, "beliefs" and "Bayesian rules" need to be introduced for analysis, and the core equilibrium concept is the Bayesian Nash equilibrium.
2. Classification by Order of Actions
- Static Games
Players choose their strategies simultaneously, or if not simultaneously, the later movers do not know the choices of the earlier movers. The core analysis tool is the payoff matrix. The Prisoner’s Dilemma and the Boxed Pigs Game are both static games. - Dynamic Games
Players take actions in a sequential order, and later movers can observe the choices of earlier movers. The core analysis tool is the game tree (extensive-form representation), and the core equilibrium concept is the subgame perfect Nash equilibrium. A typical example is the Stackelberg model (output competition in oligopolistic markets).
3. Classification by Number of Repetitions
- One-shot Games
Players interact strategically only once. Since there is no need to consider future impacts, they tend to choose strategies that are optimal in the short term but may harm collective interests. The Prisoner’s Dilemma is most typical in one-shot games. - Repeated Games
The same game structure is repeated multiple times, divided into finitely repeated games and infinitely repeated games.- Finitely repeated games: Due to the "last-period effect", the final equilibrium result is similar to that of a one-shot game.
- Infinitely repeated games: Players can punish each other’s betrayal through strategies such as "tit for tat", thus achieving a cooperative equilibrium (e.g., oligopolistic enterprises maintaining price alliances for a long time).
III. Classic Game Models
1. Prisoner’s Dilemma
It is the most classic static game of complete information, revealing the conflict between individual rationality and collective rationality.
- Model Assumptions
Two suspects (A and B) are arrested by the police and interrogated in isolation. The police have insufficient evidence and offer the following conditions:- If one confesses (defects) and the other remains silent (cooperates), the confessor is acquitted, and the silent one is sentenced to 10 years in prison.
- If both confess, each is sentenced to 5 years in prison.
- If both remain silent, each is sentenced to 1 year in prison.
- Payoff Matrix (Prison terms are negative payoffs, unit: years)
| Prisoner A \ Prisoner B | Silent (Cooperate) | Confess (Defect) |
|---|---|---|
| Silent (Cooperate) | (-1, -1) | (-10, 0) |
| Confess (Defect) | (0, -10) | (-5, -5) |
- Equilibrium Analysis
For Prisoner A, confessing is the dominant strategy regardless of whether Prisoner B chooses to be silent or confess; similarly, confessing is also the dominant strategy for Prisoner B. The final Nash equilibrium of the game is (confess, confess), with each sentenced to 5 years. However, from a collective perspective, (silent, silent) is the optimal outcome (each sentenced to 1 year), which reflects the deviation of individual rationality from collective rationality.
2. Boxed Pigs Game
This model explains the strategic differences between large and small enterprises in oligopolistic markets, with the core being the rationality of free-riding behavior.
- Model Assumptions
There are two pigs in a pen: a big pig and a small pig. The feeding trough is at one end, and a button is at the other end. Pressing the button releases 10 units of food, but pressing the button costs 2 units of effort. If the big pig reaches the trough first, it eats 9 units and the small pig eats 1 unit; if the small pig reaches first, the big pig eats 6 units and the small pig eats 4 units; if they reach simultaneously, the big pig eats 7 units and the small pig eats 3 units. - Payoff Matrix (After deducting costs, unit: food units)
| Small Pig \ Big Pig | Press Button | Wait |
|---|---|---|
| Press Button | (5, 1) | (4, 4) |
| Wait | (9, -1) | (0, 0) |
- Equilibrium Analysis
The optimal strategy for the small pig is to wait (the payoff of waiting is higher regardless of whether the big pig presses the button or not); knowing that the small pig will wait, the big pig has to choose to press the button. The final Nash equilibrium is (big pig presses button, small pig waits), with a payoff of (4, 4). This model corresponds to the real-world phenomenon where large enterprises bear the costs of R&D and advertising, while small enterprises follow and share the profits.
3. Definition of Nash Equilibrium
In a game, if given the strategies of other players, each player chooses its own optimal strategy and no player is willing to change its strategy unilaterally, then this combination of strategies is called a Nash equilibrium.
- (Confess, confess) in the Prisoner’s Dilemma and (big pig presses button, small pig waits) in the Boxed Pigs Game are both Nash equilibria.
- Nash equilibrium does not guarantee the maximization of collective interests, but it is a stable outcome of the game.
IV. Applications of Game Theory in Microeconomics
Game theory has greatly expanded the analytical boundary of microeconomics, with main application scenarios including:
- Oligopoly Market Analysis: Used to explain output competition (Cournot model), price competition (Bertrand model), sequential competition (Stackelberg model) among oligopolistic enterprises, as well as collusion and antitrust issues between enterprises.
- Industrial Organization Theory: Analyzes enterprises’ entry and exit strategies, R&D competition, advertising strategies, etc.
- Contract Theory: Studies information asymmetry and incentive mechanism design in principal-agent problems (e.g., employer-employee, shareholder-manager relationships).
- Auction Theory: Analyzes bidders’ strategy choices under different auction rules, providing a theoretical basis for auction mechanism design.