Some time ago, we presented an intriguing challenge to our community. The task was to guess the whole number closest to two-thirds of the average of all numbers guessed, within a range from 0 to 100. For instance, if the average guess was 60, the correct answer would be 40.
What number do you think was the correct guess at two-thirds of the average? Let’s explore the reasoning behind this. This game operates under a concept known in game theory as common knowledge. This means every player has the same information, and they all know that everyone else has this information too, creating an infinite loop of shared understanding.
If everyone guessed 100, the highest possible average would be 100, making two-thirds of that approximately 66.66. Knowing this, it wouldn’t be logical to guess higher than 67. If everyone reaches this conclusion, no one would guess above 67. Consequently, 67 becomes the new highest possible average, and two-thirds of that is 44.
This reasoning can be extended further, continually reducing the highest logical guess. Eventually, it seems reasonable to guess the lowest number possible. If everyone chose zero, the game would reach a Nash Equilibrium. This is a situation where each player has chosen the best strategy for themselves, given the strategies of others, and no one can benefit by changing their choice.
However, real-world outcomes differ. People may not be perfectly rational or may not expect others to be. When this game is played in real settings, the average guess often falls between 20 and 35. For example, a Danish newspaper conducted this game with over 19,000 participants, resulting in an average of about 22, making the correct answer 14. In our community, the average was 31.3, so if you guessed 21, you were spot on.
Economic game theorists use k-level reasoning to model this mix of rationality and practicality. The “k” represents the number of reasoning cycles. A player at k-level 0 guesses randomly without considering others. At k-level 1, a player assumes everyone else is at level 0, leading to an average guess of 50, and thus guesses 33. At k-level 2, they assume others are at level 1, resulting in a guess of 22. It takes 12 k-levels to reach 0, but evidence shows most people stop at 1 or 2 k-levels.
This understanding is valuable in high-stakes scenarios. For instance, stock traders assess stocks not just on earnings reports but also on how others value those numbers. In soccer penalty kicks, both the shooter and the goalie decide their moves based on what they think the other will do. Goalies often study opponents’ patterns, but shooters are aware of this and can strategize accordingly.
In these situations, participants must balance their own strategy with their perception of others’ strategies. While 1 or 2 k-levels isn’t a strict rule, being aware of this tendency can help adjust expectations.
Imagine playing the 2/3 game after understanding the difference between the most logical and the most common approach. What would happen? Try submitting your guess for two-thirds of the new average, and let’s see the results!
Gather in small groups and simulate the 2/3 game. Each of you should write down your guess for two-thirds of the average number. Once everyone has submitted their guess, calculate the average and determine the winning number. Discuss the reasoning behind your guesses and how they relate to the concept of Nash Equilibrium.
Research and present a real-world scenario where game theory is applied, such as stock market strategies or sports tactics. Explain how k-level reasoning might influence decisions in these contexts and discuss the implications of human behavior deviating from purely rational models.
Engage in a role-playing exercise where you assume the roles of different stakeholders in a strategic decision-making scenario, such as a business negotiation. Use k-level reasoning to predict others’ actions and decide on your strategy. Reflect on how your assumptions about others’ reasoning affected your decisions.
Create a simple game theory model using software tools like Excel or Python. Input different strategies and outcomes to see how changes in assumptions affect the Nash Equilibrium. Share your findings with the class and discuss how this exercise enhances your understanding of predicting human behavior.
Write a reflective essay on your personal experience with the 2/3 game and k-level reasoning. Consider how these concepts might apply to your field of study or future career. Discuss any insights gained about human behavior and decision-making processes.
A few months ago, we posed a challenge to our community. We asked everyone to guess the whole number closest to two-thirds of the average of all numbers guessed, given a range of integers from 0 to 100. For example, if the average of all guesses is 60, the correct guess would be 40.
What number do you think was the correct guess at two-thirds of the average? Let’s reason our way to the answer. This game is played under conditions known to game theorists as common knowledge. Not only does every player have the same information, but they also know that everyone else does, and that everyone else knows that everyone else does, and so on, infinitely.
The highest possible average would occur if every person guessed 100. In that case, two-thirds of the average would be approximately 66.66. Since everyone can figure this out, it wouldn’t make sense to guess anything higher than 67. If everyone comes to this same conclusion, no one will guess higher than 67. Now, 67 becomes the new highest possible average, so no reasonable guess should be higher than two-thirds of that, which is 44.
This logic can be extended further. With each step, the highest possible logical answer keeps getting smaller. It would seem sensible to guess the lowest number possible. Indeed, if everyone chose zero, the game would reach what’s known as a Nash Equilibrium. This is a state where every player has chosen the best possible strategy for themselves, given everyone else is playing, and no individual player can benefit by choosing differently.
However, that’s not what happens in the real world. People, as it turns out, either aren’t perfectly rational or don’t expect each other to be perfectly rational, or perhaps it’s some combination of the two. When this game is played in real-world settings, the average tends to be somewhere between 20 and 35. For example, a Danish newspaper ran the game with over 19,000 readers participating, resulting in an average of roughly 22, making the correct answer 14. For our audience, the average was 31.3. So if you guessed 21 as two-thirds of the average, well done.
Economic game theorists model this interplay between rationality and practicality using k-level reasoning. K stands for the number of times a cycle of reasoning is repeated. A person playing at k-level 0 would approach our game naively, guessing a number at random without considering the other players. At k-level 1, a player would assume everyone else was playing at level 0, resulting in an average of 50, and thus guess 33. At k-level 2, they would assume that everyone else was playing at level 1, leading them to guess 22. It would take 12 k-levels to reach 0. Evidence suggests that most people stop at 1 or 2 k-levels.
This understanding is useful in high-stakes situations. For example, stock traders evaluate stocks not only based on earnings reports but also on the value that others place on those numbers. During penalty kicks in soccer, both the shooter and the goalie decide whether to go right or left based on what they think the other person is thinking. Goalies often memorize the patterns of their opponents ahead of time, but penalty shooters know that and can plan accordingly.
In each case, participants must weigh their own understanding of the best course of action against how well they think other participants understand the situation. However, 1 or 2 k-levels is not a hard and fast rule—simply being conscious of this tendency can help people adjust their expectations.
For instance, what would happen if people played the 2/3 game after understanding the difference between the most logical approach and the most common? Submit your own guess at what two-thirds of the new average will be using the form below, and we’ll find out.
Game Theory – A branch of mathematics that studies strategic interactions where the outcome for each participant depends on the actions of all. – In our economics class, we used game theory to analyze how firms compete in an oligopoly.
Average – A statistical measure representing the central or typical value in a set of data, calculated as the sum of all values divided by the number of values. – The professor asked us to calculate the average income of the sample population for our economics project.
Guess – An estimate or conjecture made without sufficient information to be certain. – In the absence of complete data, economists often have to make an educated guess about future market trends.
Rationality – The quality of being based on or in accordance with reason or logic, often assumed in economic models to describe decision-making. – The concept of rationality is central to understanding consumer behavior in economic theory.
Equilibrium – A state in which economic forces such as supply and demand are balanced, resulting in stable prices and quantities. – The market reached equilibrium when the quantity demanded matched the quantity supplied.
Reasoning – The action of thinking about something in a logical, sensible way, often used in problem-solving and decision-making processes. – Economic reasoning helps us understand the implications of policy changes on market dynamics.
Strategy – A plan of action designed to achieve a long-term or overall aim, especially in the context of competitive situations. – Developing a pricing strategy is crucial for firms to maximize their profits in competitive markets.
Participants – Individuals or entities involved in an economic transaction or game, whose actions and decisions affect the outcome. – In the simulation, participants acted as buyers and sellers to understand market mechanisms.
Outcomes – The possible results or consequences of an economic action or decision, often analyzed in terms of utility or payoff. – The outcomes of the investment strategies were analyzed to determine the most profitable approach.
Economics – The social science that studies the production, distribution, and consumption of goods and services. – Economics provides insights into how resources are allocated in societies and the impact of policies on economic growth.
