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

Game Description

In the beauty contest game (Nagel; 19951), students simultaneously choose an integer smaller than 101 with the goal of picking the number closest to a fraction (P=⅔ by default) of the average of all choices. A prize is evenly split amongst those choosing the winning number.

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 30.0= 45 × ⅔, 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 have all students in the same group. In this case, set Group Size to a number larger than the number of students. When all choices have been submitted in a one-round game, you can click Finish. With multiple rounds, setting a Group Size larger than the number of students means that a round will finish when the Period Duration (seconds) elapses.

At the end of a round, each student's screen will display the distribution of choices, including her choice, the average choice, and the average choice times the multiplier.

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 round [1/(2(1-P))]. 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.

You may want to explore learning by having students play multiple rounds. With repetition, you should expect choices to move towards the equilibrium prediction.

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 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.

Figure 3: Switching between rounds (i.e., periods) in a multi-round game.

If your students have played multiple rounds, use the Go To drop-down menu to switch between rounds (Figure 3). Across rounds, you should expect choices to move towards the Nash prediction.

Robot Strategy

In round 1, a robot's choice is drawn from the uniform distribution between 0 and min[100,P*100]. In subsequent rounds, a robot chooses a noisy best-response to the previous round’s average choice. This means that when All Play Only Robots is checked, robot choices move towards the Nash prediction as rounds progress.

1 Nagel, Rosemarie. “Unraveling in Guessing Games: An Experimental Study.” The American Economic Review 85, no. 5 (1995): 1313–1326. https://www.jstor.org/stable/2950991.
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