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Beauty Contest (a.k.a. Keynes' Guessing Game)

Game Description

In the beauty contest game, students simultaneously choose an integer smaller than 101 with the goal of picking the number closest to a portion (P=⅔ by default) of the average of all choices. If multiple players have the closest choice, the prize is split evenly among the winners.

Learning Objective 1: Strategic Thought

Strategic thought requires making conjectures about the reasoning and choices of others whose actions affect the payoffs of my choices.

Learning Objective 2: Iterated Dominance

Consider P=⅔. As the highest possible target is 100 × ⅔ = 66.7, 67 dominates any larger integer. Assuming nobody chooses a dominated choice, the highest possible target is 67 × ⅔ = 44.7, and thus 45 dominates any larger integer. Continuing this logic of iterated dominance, the highest possible target is 45 × ⅔ = 30.0, and thus 30 dominates any larger integer …

Keynes' Beauty Contest is often presented as one (simple) model of stock markets. When making an investment, some investors are not necessarily trying to identify desirable companies, but rather are attempting to identify those assets that others will consider to be desirable (and purchase these assets before their prices are driven up by the masses). A great example of strategic thought requiring conjectures about the choices of others!

Brief Instructions

Most instructors will choose to have all students in the same group. In this case, set Group Size to a number larger than the number of students.

However, if the number of players is less than Group Size, you can only run a multi-round game if you add robots. (This is because a group only proceeds to the next round when the group is complete and all group members have submitted a choice.) To explore learning, run a series of one-period games. When a game ends—either because it reaches its Duration or because you click Finish—each student's screen will display the distribution of choices, including her choice, the average choice, and the average choice times the multiplier. At this point, you can Replay your original game.

Note: When players can choose any non-negative real number, the unique Nash equilibrium (for P (Fraction of Avg.) < 1) is all players choosing 0. However, with integer submissions (as in MobLab), this need not be the case.

With players restricted to integer submissions, coordinating on any integer less than or equal to 1/(2(1-P)) is a Nash equilibrium. For example with P=0.8, there are three pure-strategy Nash equilibria: all choosing 0, all choosing 1, and all choosing 2 (if everyone chooses 2, then the target is 1.6, and the closest integer is 2).

To ensure a unique Nash equilibrium when P<1, set the Minimum Value equal to [1/(2(1-P))] rounded down to the nearest integer. For example, with P=0.667, set Minimum Value equal to 1. Alternatively, to keep the Minimum Value at 0, choose P<½. Note that with P>1, the unique equilibrium has all choosing the Maximum Value.

Results

You can discuss results by considering the level of strategic thinking displayed. Consider P=⅔. Giving no thought to the strategic environment, a zero-level thinker chooses at random. A level-one thinker sees that when playing a group of zero-level thinkers, two-thirds of 50 is 33.333, and thus chooses 33. A level-two thinker, believing that her group members are level-one thinkers (who in turn believe they are playing zero-level thinkers), chooses two-thirds of 33, or 22. While this logic leads to the Nash prediction, more broadly it suggests that a person's choice reflects how deeply she believes others are thinking.

In fact, in looking at your students' choices, you are likely to see a spike at 0. (This might be a good time to remind students that equilibrium play is generally only a best response when others are equilibrium players!) Additionally, there are often spikes around 22 and 33, consistent with level-one and level-two thinking.

Figure 1: Summary Table

The Results page will summarize the data both in a table and in a figure. The table (Figure 1), highlights the average choice and the target for each group (i.e., P × Avg. Choice).

Figure 2: Distribution of Choices

The distribution of choices is plotted for each group (Figure 2). Note that you can choose which elements to display (all choices; the average choice; and the winning choices) by clicking the checkbox next to the labels in the legend.

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