GT: dominance and best response
Introduction to Dominance and Best Response
Overview of Dominance in Game Theory
- The lecture introduces the concept of dominance, a fundamental solution concept in game theory, using the prisoner's dilemma as an example.
- Player one’s strategy 'D' always yields a higher payoff than 'C', regardless of player two's choice, demonstrating that 'D' dominates 'C'.
- In this symmetric game, both players are predicted to play 'D', indicating a straightforward decision-making process for the players.
Analyzing Strategies in a Two by Three Matrix
- Player one does not have a dominated strategy; if player two plays 'X', player one should choose 'U' (5 > 3), but if player two plays 'Y', then player one should opt for 'D' (3 > 1).
- Player two's strategy analysis shows that while 'X' performs poorly, it is not dominated by either 'Y' or 'Z'; thus, no clear dominant strategy exists for player two.
Mixed Strategies and Expected Payoffs
- A mixture of strategies ('Y' and 'Z') can dominate strategy 'X', yielding an expected payoff of 5 when played against certain strategies.
- To determine the probabilities needed for this mixture to dominate, calculations show that p must be between 1/6 and 5/6 based on expected payoffs from different strategies.
Formal Definition of Dominated Strategies
- A mixed strategy sigma_i is defined as dominating another strategy s_i if it provides strictly greater expected payoffs against all possible opposing strategies.
- The distinction between strict dominance and weak dominance is clarified; this course focuses on strict dominance only.
Best Response Concept
Understanding Best Responses in Game Scenarios
- The best response is illustrated through rock-paper-scissors; if you believe your opponent will play rock, your best response would be paper.
- Best responses depend on beliefs about opponents’ strategies; multiple best responses may exist depending on these beliefs.
Defining Best Response Mathematically
- A strategy s_i is considered a best response to beliefs about other players’ strategies ( theta_-i ) if it maximizes the player's payoff given those beliefs.
Understanding Best Responses in Game Theory
Weak Inequalities and Strategy Sets
- The discussion begins with the concept of weak inequalities in game theory, where s'_i (a strategy) can include s_i , indicating that multiple strategies may be optimal.
- An example using rock-paper-scissors illustrates that various strategies (rock, paper, scissors) can all serve as best responses depending on beliefs about opponents' actions.
Player Strategies and Beliefs
- The speaker analyzes player one’s best response to a specific belief represented by probabilities (1, 0, 0), confirming that these strategies are rationalized under certain conditions.
- The focus shifts to player two's optimal choices based on their beliefs about player one’s strategies, emphasizing the importance of understanding expected payoffs.
Expected Payoffs and Best Responses
- A graph is introduced showing expected payoffs for player two's pure strategies based on varying probabilities assigned to player one's actions.
- It is noted that when probability p = 0.5 , both strategies z and y become best responses; however, strategy x never qualifies as a best response.
Dominated Strategies and Rationalization
- The notation for dominated strategies is introduced; specifically, the set of undominated strategies ( UDI ) for players is discussed.
- A key result states that for finite two-player games, the set of undominated strategies equals the set of best responses ( UDI = B_i ), which also holds true in larger games with correlated conjectures.
Correlated Conjectures and Strategy Analysis
- Correlated conjectures are explained as beliefs about other players’ actions influencing strategy choices; this correlation affects how players might respond strategically.
- Practical steps are outlined for analyzing dominated strategies: checking if a strategy is dominated, describing the set UDI , calculating best response sets given beliefs, and finding B_i .