Question

Indicate whether the below statements are True or False. If your answer is True, explain why. If your answer is False, provide a specific counter-example. (No points will be given to simply writing True or False even if the answer is correct.)

1. Every dominant strategy equilibrium is a Nash Equilibrium.
2. Every Nash Equilibrium is a dominant strategy equilibrium.
3. A 2-player game that has a strictly dominated strategy for each player is a dominance solvable game.
4. All Nash Equilibria will survive iterated deletion of strictly dominated strategies。
5. All Nash Equilibria will survive iterated deletion of weakly dominated strategies.
6. Consider the following game: Everyone in the class simultaneously chooses an integer between 1 and 100 inclusive. Goal is to choose a number that is is closest to 9/10 of the average of all of the numbers people in the class have chosen. Unique equilibrium of this game is 1.

Answer 1

In the first instance, the given statement is True as the dominant strategy of any player or individual in any game is his or her best possible strategy under all the possible contingencies involved with the outcomes or results of the particular game. Therefore, the dominant strategy provides the highest possible utility to the concerned player in a game considering all the concerned possibilities or contingencies that would influence the various results or outcomes of the game. Hence, the dominant strategies of all the players or participants in any game or the dominant strategy equilibrium of any game always constitutes a Nash equilibrium outcome of that particular game.

In this context, the second statement mentioned in the question is False as not every Nash equilibrium in any game necessarily constitute a dominant strategy as the contingencies involved with the actions and reactions/retaliations of each player or participant in a game might not necessarily lead to a dominant strategy equilibrium but can lead to Nash equilibrium/ia. It may not be possible for each player or participant in a game to always choose or select the best possible or dominant strategy simultaneously given the various contingencies and limitations associated with the actions and reactions of the respective players or participants. Therefore, Nash equilibrium outcome/s of any game might include the best possible actions of the respective players or participants in a game under the constrained circumstances or considering the contingencies and limitations of the game which may not represent the absolute best or dominant strategies of those players or participants.

The given statement here would be False as in a 2 player game the dominance solvability is found through the iterated elimination of the strictly dominated strategies and when the only strategies that are left for both players after the elimination of the dominated strategies. Therefore, in a dominance solvable 2 player game, the strictly dominated strategies of the players are basically iteratively eliminated as no player would ultimately implement or use those strategies under any contingency or circumstances considering the possible actions of the opponent player and the outcomes or results of the game.

The given statement would be False in this instance as strictly dominated strategies would not be implemented or used by the respective players or participants of a game as choosing or selecting the would generate the worst outcomes under any possible circumstances or contingencies pertaining to all the game outcomes or results and Nash equilibria represents the optimal results or outcomes of any game where all the concerned players or participants of any game is equally well-off and hence, have no rational incentive from their chosen strategies. Therefore, Nash equilibria of any game cannot practically constitute strictly dominated strategies.

In this case, the given statement would be False as some weakly dominated strategies can constitute Nash equilibria and elimination of those in any game would also consequently remove the corresponding Nash equilibria in the game. A weakly dominated strategy in a game is a strategy that provides at least the equivalent or greater amount of utility considering all the opponent player's or participant's strategy in the game and therefore, considering that Nash equilibrium/ia actually yield/s the optimal solution or outcome for all the concerned players or participants, it is possible a weakly dominated strategy can constitute Nash equilibrium.

Considering that each person in the class chooses a number between 1 and 100 with 100 inclusive, the probability of selecting each number between the given range would be 1/100 or 0.1 and the goal of the game is to choose or select a number that is closest to 9/10 or 0.9 of the average of all the numbers chosen by all the people in the class. Therefore, if the unique equilibrium outcome of the game is 1 then every participant would possibly choose the lowest number possible from the given range of numbers apprehending that choosing a large number within the given range would increasingly put them off from the target or prespecified number. Hence, in this case, choosing a significantly large number within the range would ideally constitute a weakly dominated strategy driven by the apprehension that the participants could be significantly off the target number or the prespecified goal. Hence, it is possible that the participants choose as small a number as possible as a common strategy to achieve the goal based on the same rationale.