Predicting Behavior in Dominance-Solvable Games

The Beauty Contest Game: A Paradox of Rationality

Let's begin with a famous game that starkly illustrates the limits of pure logic. It's often called the p-beauty contest, based on an analogy by the economist John Maynard Keynes about picking stocks.

The rules are simple. A group of players each choose a number from 0 to 100. The winner is the person whose number is closest to a fraction p of the average of all numbers chosen. We will start with the case where $p=1/2$.

What should you choose? Your guess depends on what you think the average will be. But everyone else is thinking the same thing. This leads to a fascinating spiral of reasoning.

As the video demonstrates, if all players are perfectly rational and believe all other players are also perfectly rational, the only possible outcome is for everyone to choose 0. Any choice greater than 0 is suboptimal if you believe everyone else has reasoned this far. This is the game's unique Nash Equilibrium.

Yet, when this game is played in classrooms or online, the average is almost never 0. Why do people consistently fail to play the equilibrium strategy in a game that seems so straightforward to solve with logic? The answer lies in bounded rationality.

The Level-k Model of Thinking

The level-k model provides an intuitive answer to this puzzle. It assumes that players differ in their depth of strategic thinking, and it organizes them into a hierarchy.

To summarize and formalize the hierarchy:

• A Level-0 (L0) player is non-strategic. They don't reason about the game at all and are often modeled as choosing a number uniformly at random from the available options.
• A Level-1 (L1) player believes all other players are Level-0. They play their best response to the anticipated random behavior of L0 players.
• A Level-2 (L2) player believes all other players are Level-1. They play their best response to the anticipated behavior of L1 players.
• In general, a Level-k (Lk) player assumes all other players are Level-(k-1) and plays the best response to that belief.

Applying the Level-k Model to the Beauty Contest

Let's see how this model predicts behavior in the Beauty Contest game where we're trying to guess 1/2 of the average.

Let's walk through the calculations together. This is the kind of step-by-step logical deduction you're familiar with from mathematical proofs.

1. Level-0: Players choose randomly from [0, 100]. The expected average of these random choices is 50.

2. Level-1: An L1 player assumes everyone else is an L0 player. They expect the average to be 50. Their best response is to choose $\frac{1}{2}\times 50=25$.

3. Level-2: An L2 player assumes everyone else is an L1 player. They expect everyone else to choose 25, so the average will be 25. Their best response is to choose $\frac{1}{2}\times 25=12.5$.

4. Level-3: An L3 player assumes everyone else is an L2 player. They expect the average to be 12.5. Their best response is to choose $\frac{1}{2}\times 12.5=6.25$.

Notice that as $k$ increases, the predicted choice gets closer and closer to the Nash Equilibrium of 0. The Nash Equilibrium player can be thought of as a "Level-infinity" player.

The crucial insight from the experimental data shown in the video is that most people are Level 0, 1, or 2. Very few people engage in more than two or three steps of this iterative reasoning. This is why the winning numbers in these games are almost never 0, but often fall in the 15-25 range.

To reinforce this, let's look at data from a slightly different version of the game, where players must guess 2/3 of the average.

In the p=2/3 game:

• L1 players guess $\frac{2}{3}\times 50\approx 33$.
• L2 players guess $\frac{2}{3}\times 33\approx 22$.
  The video shows large spikes in guesses right around 33 and 22, providing powerful evidence for the level-k model. It's a far better predictor of actual behavior than the Nash Equilibrium of 1.

Going Deeper: The Cognitive Hierarchy (CH) Model

For someone with your interest in research, it's worth knowing that the simple level-k model has a more sophisticated successor: the Cognitive Hierarchy (CH) model.

A key, and perhaps unrealistic, assumption of the simple model is that an L2 player thinks everyone is an L1 player. A more plausible belief would be that the world contains a mix of L0 and L1 players.

The CH model formalizes this. It assumes:

• An Lk player best responds to a distribution of lower-level thinkers (i.e., levels 0, 1, ..., k-1).
• The frequency of players at each level, $f(k)$, is often modeled by a Poisson distribution: $$f(k)=\frac{e^{-\tau }{\tau }^k}{k!}$$ Here, $\tau$ (tau) is a single parameter representing the average number of thinking steps in the population. You may have encountered the Poisson distribution in physics or statistics; it's used to model the number of events occurring in a fixed interval of time or space.

This model is more complex, but it often provides a more accurate fit to experimental data. Let's see how an L2 player's thinking changes. Suppose we estimate from data that $\tau \approx 1.5$.

1. We can calculate the proportion of L0 and L1 players in the population:

   • $f(0)=\frac{e^{-1.5}(1.5{)}^0}{0!}\approx 0.22$
   • $f(1)=\frac{e^{-1.5}(1.5{)}^1}{1!}\approx 0.33$

2. An L2 player believes the population consists only of L0 and L1 players. They normalize these frequencies to form their belief:

   • Believed proportion of L0s: $g_2(0)=\frac{f(0)}{f(0)+f(1)}=\frac{0.22}{0.22+0.33}\approx 0.4$
   • Believed proportion of L1s: $g_2(1)=\frac{f(1)}{f(0)+f(1)}=\frac{0.33}{0.22+0.33}\approx 0.6$

3. The L2 player now calculates the expected average guess. L0s guess 50 on average, and L1s guess 25.

   • Expected Average = $(0.4\times 50)+(0.6\times 25)=20+15=35$

4. The L2 player's best response is $\frac{1}{2}\times 35=17.5$.

This is different from the 12.5 predicted by the simple level-k model. By accounting for the continued presence of naive L0 players, the CH model often predicts higher (and more empirically accurate) guesses for higher-level thinkers.

Conclusion

Today, we explored a powerful alternative to classical game theory for explaining behavior. Instead of modifying players' preferences, we bounded their rationality.

Key Takeaways:

• Bounded rationality suggests that people have limited cognitive ability and cannot perform infinite steps of strategic reasoning.
• Level-k models formalize this by creating a hierarchy of players. Level-0 players act randomly, and Level-k players best-respond to the belief that others are at a lower level.
• In dominance-solvable games like the Beauty Contest, level-k thinking explains why observed behavior clusters around specific numbers (e.g., 33 and 22 in the p=2/3 game) and deviates significantly from the Nash Equilibrium.
• The more advanced Cognitive Hierarchy (CH) model refines this by assuming Level-k players respond to a distribution of all lower levels, often modeled with a Poisson distribution. This provides a richer and often more accurate picture of strategic thinking.

Preview of the Next Lesson:
We have now seen two major branches of behavioral game theory: one focusing on social preferences (fairness) and one on cognitive limits (bounded rationality). In our next lesson, we will look at how specific, well-documented cognitive biases, such as framing effects and anchoring, can also systematically influence strategic choices, further diverging behavior from the purely rational ideal.