GT: strictly competitive games
Introduction to Strictly Competitive Games
Definition and Characteristics
- A strictly competitive game is defined as a two-player scenario where one player's gain results in the other player's loss. If player one has a higher payoff under strategy s than under s' , then player two must have a lower payoff under s than under s' .
- This property indicates that players have opposite rankings over outcomes, emphasizing the zero-sum nature of these games.
Context and Importance
- While strictly competitive games are important to understand, they are not the primary focus in economics, which often deals with scenarios allowing for joint gains from trade. Most strategic situations do not fall into this category.
- An example of strictly competitive games includes zero-sum games like "match and pennies," where total payoffs sum to zero, confirming that one player's increase corresponds with another's decrease.
Examples of Strictly Competitive Games
Rock-Paper-Scissors Analysis
- In rock-paper-scissors, if player one's payoff increases by changing strategies (e.g., from rock to paper), player two's payoff decreases correspondingly, illustrating the strict competition principle.
- Ties in this game (e.g., both players choosing rock) result in unchanged payoffs but still align with the definition of strictly competitive games.
Alternative Example: Payoff Structures
- Another example involves players choosing between strategies A or B; while their payoffs do not sum to zero, they still exhibit strict competition as player one's highest payoff occurs when player two receives their lowest possible outcome.
- The analysis shows how different strategy profiles yield varying payoffs for both players while maintaining opposite outcomes based on choices made.
Security Strategies in Strictly Competitive Games
Concept of Worst Payoff
- The worst payoff for a player when selecting a strategy is denoted as w_i(s_i) . Players may choose strategies that minimize their potential losses based on what they anticipate their opponent will do. This reflects a cautious approach to gameplay dynamics.
- For instance, if Player One plays strategy A and receives a minimum payoff of 2, while playing B yields only 1, Player One would prefer strategy A as it maximizes their security level despite being potentially paranoid about Player Two's actions.
Security Strategy Definition
- A security strategy is defined as one that maximizes the worst-case scenario (security level) for a player within strictly competitive contexts; thus ensuring minimal loss regardless of opponent moves. In this case, Player One’s security strategy is identified as playing A with a corresponding security level of 2.
Max-Min Strategies vs Security Strategies
Distinction Between Strategy Types
- Max-min strategies involve optimizing expected outcomes rather than certainties; however, this lecture focuses primarily on security strategies due to their straightforward application in strictly competitive settings without delving into mixed strategies complexities.
Implications for Game Theory
- Understanding these distinctions helps clarify how different strategic approaches can influence decision-making processes within game theory frameworks and highlights why certain strategies may be dominated or less effective compared to others based on context-specific rules and expectations around payoffs.