Rationality, Dominance and Best Response

Transcription

1 Rationality, Dominance and Best Response Brett Devine ECONS Strategy & Game Theory School of Economic Sciences

2 Rationality A player is rational when she acts in her own best interests. Given a player s beliefs as to how the other players will behave, the player selects a strategy in order to maximize her payoff. Base Rationality A player can choose between A and B and is strictly better off with A then B. Then if: 1. She knows both A and B are actual options, 2. She can distinguish between A and B when choosing, 3. She knows A yields a higher payoff than B, 4. She can always control her choice perfectly, then rationality dictates she choose A over B and we expect to observe that behavior. 2/21

3 Rationality Does our definition of rationality 1. make people into selfish jerks? 2. say anything about the beliefs a person bases their actions on? Question: Rationalizing Behavior Have you ever known someone in an interactive situation that did something so stupid you were left wondering, What were they thinking? Did you try to rationalize their actions? Astonishment can occur when it is difficult to construct beliefs or a train of thought for another person that makes their behavior seem like a good idea. Their behavior is not rationalizable for a rational actor. 3/21

4 Rationality An Operational Definition of Rationality A player has strategies s 1, s 2, s 3, s 4. Given the strategies S selected by other players, if V(s i ) < V(s j ) then rationality means the player will choose s j over s i. Why? They know both are distinct, feasible strategies and they know they are better off with s j than s i. Question: What if, for any strategies chosen by other players, strategy s 2 always has a lower payoff than s 1, s 3, s 4? What does rationality predict about our use of s 2 in the game? 4/21

5 Example For an offensive game of over-generlized stereotypes, a jock and a sorority girl are matched together for a team project for their class. Each must choose a level of effort to put into the project. Game: Team-project Sorority Girl Low Moderate High Low 3, 0 4, 1 5, 2 Jock Moderate 2, 2 3, 4 4, 3 High 1, 6 2, 5 3, 4 5/21

6 Example For each choice of effort by the sorority girl, the Jock has 3 strategies. The Jock s rationality will not permit him to select High as a strategy. Game: Team-project Sorority Girl Low Moderate High Low 3, 0 4, 1 5, 2 Jock Moderate 2, 2 3, 4 4, 3 High 1, 6 2, 5 3, 4 6/21

7 Example Regardless of Sorority girl s choice, the Jock will not give High effort. Since rationality precludes High being chosen, why not remove it? Game: Team-project Sorority Girl Low Moderate High Low 3, 0 4, 1 5, 2 Jock Moderate 2, 2 3, 4 4, 3 7/21

8 Example Rationality dictates a similar story for the Jock supplying Moderate effort. The Jock knows both Low and Moderate are feasible and that Low is preferred to Moderate for every strategy of Sorority girl. Game: Team-project Sorority Girl Low Moderate High Jock Moderate Low 3, 0 4, 1 5, 2 2, 2 3, 4 4, 3 8/21

9 Example Since rationality won t let the Jock choose Moderate either, we might as well delete it from his strategy list. Game: Team-project Sorority Girl Low Moderate High Jock Low 3, 0 4, 1 5, 2 Regardless of Sorority girl s strategy, the Jock has better choices than High and Moderate. A rational Jock, will not choose Low or Moderate. 9/21

10 Dominance Strictly Dominated Strategies A strategy s i is STRICTLY dominated by another strategy s i if s i does strictly better than s i against every strategy of the other players. V i (s i, s i) > V i (s i, s i) for all s i S i Lesson Rational players NEVER play strictly dominated strategies. Since we can delete a rational player s dominated strategies, can we use rationality and deletion to solve games and make outcome predictions? Almost, we need some more assumptions first. 10/21

11 Common Knowledge of Rationality We know rational players will never play strictly dominated strategies. Okay great, that is all well and fine, but are the other players rational? Or is it only me? If we know other players are rational, then we might be able to predict what they won t do. We can delete any strictly dominated strategies from their strategy list. We assume the structure of the game and all players payoffs are common knowledge. What if I don t know they are rational? 11/21

12 Example Sorority girl s best outcome occurs with strategy profile (High,Low). Does Sorority girl know that the Jock is rational? If yes, then she can (must) delete strategies High and Moderate. Her own rationality causes her to delete Low and Moderate. Game: Team-project Sorority Girl Low Moderate High Low 3, 0 4, 1 5, 2 Jock Moderate 2, 2 3, 4 4, 3 High 1, 6 2, 5 3, 4 12/21

13 Game: Doping Game Steroids Floyd No Steroids Bernhard Steroids 2, 3, 3 3, 1, 5 No Steroids 1, 4, 5 5, 2, 6 Lance chooses steroids Floyd Steroids No Steroids Bernhard Steroids 3, 4, 1 4, 2, 2 No Steroids 5, 5, 2 6, 6, 4 Lance chooses NO steroids 13/21

14 Game: Doping Game Floyd Steroids No Steroids Bernhard Steroids 2, 3, 3 3, No Steroids 1, 4, 5 5, 1, 5 2, 6 Lance chooses steroids If Floyd believes Lance is rational, then No Steroids is strictly dominated for Floyd. However, Bernhard still doesn t know what to do. 14/21

15 Game: Doping Game Floyd Steroids Bernhard Steroids 2, 3, 3 No Steroids 1, 4, 5 Lance chooses steroids If Bernhard believes that Lance and Floyd are rational AND he believes that Floyd believes Lance is rational, then he chooses Steroids. Common knowledge of rationality leads everyone to choose steroid use! 15/21

16 First Solution Method Iterated Deletion of Dominated Strategies The process we are engaging in is iterated deletion of dominated strategies (IDSDS). It requires rationality of all players to be common knowledge. Doesn t always provide a solution or even help (see textbook examples), but... When it provides a solution (unique prediction), it can be quite powerful. Question: Is common knowledge of rationality in every day games important to every day life? 16/21

17 Dominance Strictly Dominant Strategies A strategy s i is STRICTLY dominant if it strictly dominates all other strategies for player i. More precisely, V i (s i, s i) > V i (s i, s i ) for all s i s i and for all s i S i Lesson Rational players ALWAYS play strictly dominant strategies. Dominant strategies can be powerful predictors of player behavior. Reduce issues of strategic complexity. What about imperfect implementation? 17/21

18 Best Response Strict domination is a demanding condition to meet. Rationality also dictates, that a person maximizes their payoff by matching the strategy of an opponent with an optimal response. A strictly dominated strategy is NEVER a best response. A strictly dominant strategy is ALWAYS a best response. Usually, strategies are sometimes a best response and other times not. 18/21

19 Best Response Game: Diane x y z a 1, 1 2, 1 2, 0 Jack b 2, 3 0, 2 2, 1 c 2, 1 1, 2 3, 0 19/21

20 Best Response Game: Diane x y z a 1, 1 2, 1 2, 0 Jack b 2, 3 0, 2 2, 1 c 2, 1 1, 2 3, 0 20/21

21 Best Response Game: Diane x y z a 1, 1 2, 1 2, 0 Jack b 2, 3 0, 2 2, 1 c 2, 1 1, 2 3, 0 21/21