A Guessing Game
Years pass. You are eventually joined by hundreds of other war prisoners. This Cookie Monster must be a maniac on the battlefield, you think.
Cookie Monster’s games are becoming more creative to accommodate the
multitude of players. In today's game, all of you can pick any integer from
1 to 100. Suppose you pick the integer
What number will you choose?
Congrats! You’ve won an astounding cookies! The winning number was around 13.4, which I chose pretty arbitrarily.
Well, after some thought, you may realize that it is impossible to win with any integer greater than 66, as all numbers are less than 100. Thus, every rational player should play a number less than or equal to 66. However, given this assumption, it is impossible to win with any integer greater than 44. Thus, all rational players will play under 44. If we repeat this process indefinitely, the only rational strategy will be to play the number 1.
This process is sometimes known as the iterative deletion of dominated strategies. While no strategy is unconditionally optimal, some strategies are never optimal (e.g. 67-100). Thus, we can effectively ignore these "dominated strategies" from the possible strategy set. Each time we do this, we assume an additional layer of rationality among the participant pool.
Importantly, our optimal solution of 1 relies on the fact that every player is perfectly rational as well. Experimentally, however, the degree of rationality may vary depending on the setting in which this experiment is conducted. In an undergraduate game theory lecture at Yale, students were asked to play this game directly after learning about strictly dominated strategies. The winning number was around 9. However, when the game was played among readers of an innocent-looking Danish newspaper, the winning number was around 22. It appears to be rational to assume that not all players will act rationally.
You may also notice when deleting strategies, all removed strategies were "strictly dominated," which means that there would always be a benefit to deviating from a strictly dominated strategy. Thus, by continually deleting strictly dominated strategies, we never risk excluding a Nash equilibrium. So, the solution where all players pick 1 is a unique Nash equilibrium.
However, strategies can be "weakly dominated," which means that it is never better than any other strategy in every action profile, but it may sometimes yield an equivalent payoff. Try to come up with an example of where deletion of weakly dominated strategies excludes a Nash equilibrium.