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Game Theory 101 (#3): Iterated Elimination of Strictly Dominated Strategies
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Game Theory 101 (#3): Iterated Elimination of Strictly Dominated Strategies

Introduction to Iterated Elimination of Strictly Dominated Strategies

In this section, William Spaniel introduces the concept of iterated elimination of strictly dominated strategies in game theory. He explains that this technique is used to solve games where one strategy is not always better for each player and players may need to change their strategies based on what the other player is doing.

The Prisoner's Dilemma and Strictly Dominated Strategies

  • In the previous video, William discussed the Prisoner's Dilemma game.
  • The solution to this game was for both players to confess because "Confess" strictly dominated "Keep Quiet".
  • It was never in a player's best interest to individually choose "Keep Quiet" as "Confess" always produced a better outcome.

Challenges with Games and Solution Approach

  • In most games, it is not always the case that one strategy is always better for each player.
  • Players may need to change their strategies based on what the other player is doing.
  • This makes it difficult to agree on a solution where one player always plays a specific strategy regardless of the other player's actions.

Example Game with Three Strategies for Each Player

In this section, William presents an example game with three strategies for each player. He demonstrates how players can use iterated elimination of strictly dominated strategies to find the optimal solution.

Analysis of Player 1's Strategies

  • If Player 2 chooses "Left", Player 1's best strategy is "Up".
  • If Player 2 chooses "Center", Player 1's best strategy is "Middle".
  • If Player 2 chooses "Right", Player 1's best strategy is "Down".

Analysis of Player 2's Strategies

  • Player 2 should never choose "Right" as it is strictly dominated by "Center".
  • Regardless of Player 1's strategy, "Center" always produces a greater payoff for Player 2.

Implications and Iterated Elimination

  • Player 2 can ignore the option of choosing "Right" and focus on the smaller game with two strategies: "Left" and "Center".
  • Player 1, knowing that Player 2 is super intelligent and would not choose "Right", can infer that he should not choose "Down".
  • This leads to further elimination of strategies, narrowing down the game to a more manageable size.

Final Solution Based on Iterated Elimination

In this section, William concludes the example game by determining the final solution based on iterated elimination of strictly dominated strategies.

Analysis of Player 1's Strategies

  • Since Player 2 would never choose "Right", Player 1 should not choose "Down".
  • This leaves only two reasonable strategies for Player 1: "Up" and "Middle".

Analysis of Player 2's Strategies

  • Knowing that Player 1 would never choose "Down", Player 2 can infer that she should not choose "Left".
  • The only sensible strategy for Player 2 is now to choose "Center".

Final Solution

  • Based on the knowledge that both players are rational and aware of each other's intelligence, the optimal solution is for Player 1 to play "Middle" and for Player 2 to play "Center".

Iterated Elimination of Strictly Dominated Strategies

This section explains the concept of Iterated Elimination of Strictly Dominated Strategies (IESDS) and its application in game theory.

IESDS Process

  • The players infer information about each other's intelligence and strategies, leading to the elimination of dominated strategies.
  • The process involves iteratively eliminating strictly dominated strategies until a solution is reached.
  • The name "Iterated Elimination of Strictly Dominated Strategies" comes from the elimination of strictly-dominated strategies such as 'Right', 'Down', 'Left', and 'Up'.
  • It is recommended to eliminate strictly-dominated strategies immediately in any game.
  • The order of eliminating multiple strictly-dominated strategies does not matter; the outcome will be the same.
  • IESDS leads to a single outcome or solution, such as knowing that players will play 'Middle-Center'.

Application and Limitations

  • IESDS is useful for games with dominated strategies but may not be applicable to most games.
  • For games without dominated strategies, other methods like the Stag Hunt or Pure Strategy - Nash Equilibrium are needed for solving them.

Timestamps are provided for each bullet point.

Video description

Game Theory 101: The Complete Textbook on Amazon: https://www.amazon.com/Game-Theory-101-Complete-Textbook/dp/1492728152/ http://gametheory101.com/courses/game-theory-101/ The prisoner's dilemma had an obvious solution because each player had one strategy that was always the best regardless of what the other players do. Most games don't have solutions that are that simple, however. What if one player's best strategy depends entirely on which strategy the other player chose? This video covers iterated elimination of strictly dominated strategies. If one strategy is always worse than another for a player, that means the other player should infer that the first player would never choose that poor strategy. But this has interesting implications. It is possible to keep removing strategies from a game based on this information, until you eventually arrive at a single solution. We go over an example in this video. As a general rule, if you ever see a strictly dominated strategy, you should always eliminate it immediately. Although there may be more strictly dominated strategies that you could eliminate first, those other strictly dominated strategies will still be strictly dominated in the reduced game. Therefore, you lose nothing by immediately eliminating a strategy.