The ⅔ Game: Predicting Human Behavior with Game Theory

Can you outsmart the crowd? Imagine a game where everyone guesses a number between 0 and 100. The winner is whoever picks the number closest to ⅔ of the average of all guesses. Sounds simple, right? But this game delves into the fascinating world of game theory, rationality, and how we anticipate the actions of others.

The Challenge: Guessing ⅔ of the Average

The rules are straightforward: Choose a whole number between 0 and 100. The goal is to guess the number that is closest to ⅔ of the average of all the numbers submitted. For instance, if the average guess turns out to be 60, the winning guess would be 40.

This game operates under the principle of "common knowledge" – everyone has the same information, and everyone knows that everyone else has the same information, and so on, ad infinitum. This creates a fascinating loop of strategic thinking.

The Logic of Rationality

Let's break down the logical approach:

Initial Assumption: The highest possible average is 100 (if everyone guessed 100).
First Iteration: ⅔ of 100 is 66.66. Therefore, no one should rationally guess higher than 67.
Second Iteration: If everyone realizes this, the new highest possible average becomes 67. ⅔ of 67 is approximately 44. Thus, no reasonable guess should exceed 44.
Continuing the Pattern: This logic can be repeated, with the highest possible logical answer decreasing with each step.

Following this line of reasoning, the logical conclusion points towards guessing the lowest possible number: 0.

Nash Equilibrium: The Theoretical Ideal

If everyone were to choose 0, the game would reach a Nash Equilibrium. This is a state where no player can improve their outcome by unilaterally changing their strategy, assuming everyone else's strategies remain the same. In this scenario, everyone has chosen the best possible strategy for themselves, given the choices of others.

The Reality of Human Behavior

However, real-world results deviate from this perfectly rational outcome. People aren't always perfectly rational, or they don't expect others to be. When this game is played in reality, the average guess typically falls between 20 and 35.

For example, when a Danish newspaper, Politiken, conducted this experiment with over 19,000 participants, the average guess was around 22, making the correct answer 14.

K-Level Reasoning: Modeling Rationality and Practicality

To understand this deviation, game theorists use k-level reasoning. "K" represents the number of times a cycle of reasoning is repeated:

K-Level 0: A player guesses randomly, without considering the strategies of others.
K-Level 1: A player assumes everyone else is playing at level 0, leading to an average of 50 and a guess of 33 (⅔ of 50).
K-Level 2: A player assumes everyone else is playing at level 1, leading to a guess of 22 (⅔ of 33).

It would take 12 k-levels to reach 0. Evidence suggests that most people operate at level 1 or 2.

Real-World Applications

K-level thinking is relevant in various high-stakes situations:

Stock Trading: Traders evaluate stocks based not only on earnings reports but also on how others perceive those numbers.
Penalty Kicks in Soccer: Both the shooter and the goalie anticipate each other's moves, creating a strategic back-and-forth.

In essence, participants must balance their own understanding of the optimal strategy with their perception of how well others understand the situation.

Beyond the Numbers: Adjusting Expectations

While most people stop at 1 or 2 k-levels, awareness of this tendency can influence expectations. Understanding the difference between the most logical approach and the most common one can lead to more informed decisions.

Game theory provides a framework for understanding strategic decision-making, but human behavior often introduces complexities that deviate from purely rational models. The ⅔ game serves as a compelling illustration of this interplay, highlighting the importance of considering both logic and psychology when predicting the actions of others.