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Simple Games of Complete Information
Game Theory studies strategic interactions where the outcome for each player depends on the choices of all players. A game of complete information means that all players know the rules of the game, payoffs, and strategies available to everyone.
1. Components of a Game
A strategic game consists of:
- Players (N): The decision-makers in the game.
- Strategies (S): The possible actions each player can choose.
- Payoffs (U): The rewards or consequences for each player based on chosen strategies.
- Rules: How the game is played (simultaneous or sequential moves).
A game can be represented in normal form (matrix) or extensive form (tree diagram).
2. Types of Simple Games with Complete Information
1. Simultaneous-Move Games
- Players choose their actions at the same time without knowing what others are doing.
- Best analyzed using payoff matrices.
- Solution concepts: Nash Equilibrium, Dominant Strategy.
🔹 Example: Prisoner’s Dilemma
| Player 1 / Player 2 | Cooperate | Defect |
|---|---|---|
| Cooperate | (3,3) | (0,5) |
| Defect | (5,0) | (1,1) |
- Dominant Strategy: Both players defect, even though (3,3) would be better.
- Nash Equilibrium: (Defect, Defect) → No player benefits by changing strategy alone.
2. Sequential-Move Games
- Players take turns choosing strategies.
- Represented using game trees.
- Solution concept: Backward Induction (Subgame Perfect Nash Equilibrium – SPNE).
🔹 Example: Entry Game
A firm decides whether to enter a market, and an incumbent firm chooses whether to fight or accommodate. (Incumbent’s response: Fight or Accept)\text{(Incumbent’s response: Fight or Accept)}
| New Firm | Incumbent Responds | Payoffs (New Firm, Incumbent) |
|---|---|---|
| Enter | Fight | (-1, -2) |
| Enter | Accept | (2, 1) |
| Stay Out | – | (0, 3) |
Solution (Backward Induction):
- If the new firm enters, the incumbent chooses Accept (1 > -2).
- Knowing this, the new firm enters the market.
- SPNE = (Enter, Accept).
3. Solution Concepts in Complete Information Games
1. Nash Equilibrium (NE)
A strategy profile where no player has an incentive to change strategy given the choices of others.
- Pure Strategy Nash Equilibrium: Players choose a specific action.
- Mixed Strategy Nash Equilibrium: Players randomize between strategies.
🔹 Example: Battle of the Sexes
| Husband / Wife | Opera | Football |
|---|---|---|
| Opera | (2,1) | (0,0) |
| Football | (0,0) | (1,2) |
- NE: (Opera, Opera) & (Football, Football).
- Coordination problem: Players want to match but have different preferences.
2. Dominant Strategy
A strategy that always gives the best payoff regardless of what the opponent does.
🔹 Example: Prisoner’s Dilemma (Defect is dominant strategy).
3. Backward Induction
Used in sequential games → Players reason backward from the last move.
🔹 Example: Entry Game (New firm enters, Incumbent accepts).
4. Applications of Simple Games
- Pricing Wars (Firms compete or cooperate).
- Elections (Candidates choose strategies based on voter preferences).
- Bargaining (Negotiations in business and trade deals).
- Auctions (Bidders strategize to win with the lowest bid).
5. Conclusion
- Complete information games assume all players know payoffs and strategies.
- Solutions depend on dominant strategies, Nash equilibrium, and backward induction.
- Used in business, economics, and politics to model strategic interactions.