🆚Game Theory and Economic Behavior Unit 3 – Dominant Strategies & Iterative Elimination

Key Concepts

• Dominant strategies are the best responses for a player regardless of the strategies chosen by other players
• Iterative elimination involves removing dominated strategies from a game to simplify the analysis
• Nash equilibrium is a set of strategies where no player has an incentive to deviate given the strategies of other players
• Payoff matrices represent the outcomes for each combination of strategies chosen by players in a game
• Strategic dominance occurs when one strategy dominates another for a player, while strict dominance requires the dominating strategy to have a higher payoff in all cases
• Weakly dominated strategies may have equal payoffs in some cases but are never the best response
• Rationality assumes that players will always choose their best response given the available information

Defining Dominant Strategies

• A dominant strategy provides a player with the highest payoff regardless of the strategies chosen by other players
• Dominance can be determined by comparing the payoffs for each strategy across all possible combinations of opponent strategies
• In a two-player game, a strategy is dominant if it yields a higher payoff than all other strategies for every possible strategy of the opponent
  • For example, in the Prisoner's Dilemma, confessing is a dominant strategy for both players as it always results in a shorter sentence regardless of the other player's choice
• Dominant strategies, when they exist, simplify the decision-making process for players
• Games with dominant strategies often have a clear Nash equilibrium where all players choose their dominant strategy

Types of Dominance

• Strict dominance occurs when a strategy provides a strictly higher payoff than another strategy for all possible opponent strategies
  • If strategy A strictly dominates strategy B, a rational player will never choose strategy B
• Weak dominance arises when a strategy provides a payoff at least as high as another strategy for all possible opponent strategies, with a strictly higher payoff in at least one case
  • Weakly dominated strategies may be rational to play in some cases, particularly when opponents are not expected to play their dominant strategies
• Very weak dominance happens when a strategy provides a payoff at least as high as another strategy for all possible opponent strategies, but there is no case where it provides a strictly higher payoff
• Dominance solvability refers to games where the iterative elimination of strictly dominated strategies leads to a unique Nash equilibrium

Iterative Elimination Process

• Iterative elimination involves removing dominated strategies from a game to simplify the analysis and identify Nash equilibria
• The process begins by identifying and removing any strictly dominated strategies for each player
• After the first round of elimination, the process is repeated with the reduced game until no more strictly dominated strategies remain
  • This may require multiple rounds of elimination, as the removal of one player's dominated strategy can cause previously undominated strategies to become dominated for other players
• If iterative elimination results in a single strategy remaining for each player, that combination of strategies is the unique Nash equilibrium of the game
• Iterative elimination can be applied to games with any number of players and strategies, although the process becomes more complex as the size of the game increases

Applications in Game Theory

• Dominant strategies and iterative elimination are fundamental concepts in game theory, with applications in various fields such as economics, political science, and psychology
• In the Prisoner's Dilemma, the dominant strategy of confessing leads to a suboptimal outcome for both players, highlighting the potential for cooperation to improve overall welfare
• In the game of Chicken, there are no dominant strategies, but iterative elimination can be used to identify the Nash equilibria and analyze the strategic considerations of the players
• Dominant strategies can help explain the behavior of firms in oligopolistic markets, such as the incentive to engage in price competition or quantity competition
• Iterative elimination can be used to solve complex games and predict the outcomes of strategic interactions, such as in auctions or bargaining situations

Common Mistakes and Pitfalls

• Failing to consider all possible strategies of the opponent when determining dominance
• Confusing strict dominance with weak dominance, which can lead to incorrect conclusions about the rationality of strategies
• Applying iterative elimination to games with no strictly dominated strategies, which will not result in any simplification of the game
• Assuming that players will always choose their dominant strategy, even when other factors such as social norms or bounded rationality may influence their decisions
• Neglecting the potential for multiple rounds of elimination when applying the iterative elimination process
• Overrelying on dominant strategies and iterative elimination without considering other important aspects of the game, such as payoff structures and player preferences

Real-World Examples

• In the game of rock-paper-scissors, there are no dominant strategies, as the optimal choice depends on the opponent's strategy
• In a sealed-bid first-price auction, the dominant strategy is to bid slightly above the second-highest bidder's valuation to maximize the chances of winning while minimizing the price paid
• In the game of chess, there are no dominant strategies due to the vast number of possible moves and the complexity of the game tree
• In a two-candidate election with a majority voting rule, the dominant strategy for voters is to vote for their preferred candidate, as voting for the less-preferred candidate cannot improve the outcome
• In the game of poker, bluffing can be a dominant strategy in certain situations, such as when the potential payoff of a successful bluff outweighs the cost of being called

Practice Problems

1. Consider the following payoff matrix for a two-player game:

   	Left	Right
   Up	(2,1)	(0,0)
   Down	(1,2)	(3,3)

   Determine whether any player has a dominant strategy.

2. In a three-player game, Player 1 has strategies A and B, Player 2 has strategies C and D, and Player 3 has strategies E and F. The payoffs for each combination of strategies are as follows:

   • (A, C, E): (3, 2, 1)
   • (A, C, F): (2, 1, 3)
   • (A, D, E): (1, 3, 2)
   • (A, D, F): (4, 4, 4)
   • (B, C, E): (2, 3, 1)
   • (B, C, F): (1, 2, 3)
   • (B, D, E): (3, 1, 2)
   • (B, D, F): (0, 0, 0)

   Use iterative elimination to solve the game and find the Nash equilibrium.

3. Suppose two firms, Firm A and Firm B, are deciding whether to invest in a new technology. The payoff matrix for their decisions is as follows:

   	Invest	Don't Invest
   Invest	(2, 2)	(1, 3)
   Don't Invest	(3, 1)	(0, 0)

   Analyze the game using the concept of dominant strategies and determine the Nash equilibrium.