Beyond Rationality: Experimental Evidence Against Classical Game Theory

What is Behavioral Game Theory?

Before diving into the experiments, let's get a clearer sense of the field's purpose. It doesn't aim to "disprove" game theory, but rather to enrich it with empirical evidence about human preferences and reasoning.

As the text highlights, BGT seeks to understand the extent to which standard theory works, and where it doesn't, to uncover the underlying psychological drivers of behavior. We'll start this process by looking at the "deviations."

Deviation 1: Surprising Cooperation in the Prisoner's Dilemma

Let's start with the most famous game of all. As you'll recall, the Prisoner's Dilemma has a single, dominant strategy equilibrium: both players defect (or "Fink"). A rational, self-interested player should always defect, regardless of the other's action. The prediction is unequivocal.

What do real people do?

The key finding, replicated across countless experiments, is that people cooperate far more often than classical game theory predicts. While cooperation rates vary, they are almost never zero.

• Classical Prediction: 100% of players choose to Defect/Fink.
• Experimental Result: A significant fraction of players choose to Cooperate/Stay Quiet.

This suggests that the simple model of payoffs might be missing something. Players may not be purely "selfish" in the way the model assumes; they might have preferences for fairness, or perhaps they simply fail to correctly identify their dominant strategy. This was one of the first major cracks in the armor of the perfect rationality assumption.

Deviation 2: The Rejection of "Free Money" in the Ultimatum Game

The Prisoner's Dilemma shows a deviation from a dominant strategy equilibrium. The Ultimatum Game reveals a similar deviation from a subgame perfect equilibrium, challenging the principle of backward induction.

Let's first understand the game and its "rational" solution.

The logic of subgame perfection is ironclad:

1. In the final subgame, the Responder (Player 2) is choosing between accepting an offer $x>0$ and getting 0. A rational player will always choose $x$ over 0.
2. The Proposer (Player 1), anticipating this, knows that any offer greater than zero will be accepted. To maximize their own payoff, the Proposer should offer the smallest possible positive amount (e.g., 1 cent out of $10).

• Classical Prediction: The Proposer offers the smallest possible positive amount, and the Responder accepts.
• Experimental Result: Now, let's see what really happens.

The experimental data reveals two major deviations:

1. Offers are much fairer than predicted. The most common offer is a 50/50 split, not the minimum possible amount.
2. Low offers are frequently rejected. Responders often choose to receive a payoff of 0 rather than accept what they perceive as an unfair offer (e.g., keeping $9 and offering $1).

This is a powerful result. It demonstrates that people are willing to pay a price (by forgoing a small gain) to punish behavior they see as unfair. This points to the existence of social preferences, where a person's utility depends not just on their own payoff, but also on the payoffs of others and the perceived fairness of the process.

Deviation 3: Limited Reasoning in the "Beauty Contest" Game

So far, the deviations could be explained by players having "social" preferences. But what if players are purely self-interested, but simply lack the infinite cognitive horsepower that classical models assume? The "Beauty Contest" game, inspired by a thought experiment by John Maynard Keynes, is a brilliant tool for measuring players' depth of strategic reasoning.

First, let's understand the game and its Nash Equilibrium.

As the video explains, if everyone is perfectly rational, and knows that everyone else is perfectly rational (and so on, ad infinitum), the process of iteratively eliminating dominated strategies leads to a unique Nash Equilibrium: everyone chooses 1.

• Classical Prediction: All players announce the number 1.
• Experimental Result: Let's look at the data from over 10,000 players.

The results are starkly different from the prediction:

1. Almost nobody chooses the equilibrium number, 1.
2. There are large spikes in the data corresponding to what you would expect from players performing only a few steps of reasoning:
   • A spike at 50 (Level-0 thinking: choosing randomly).
   • A spike around 33-35 (Level-1 thinking: 2/3 of 50).
   • A spike around 22-23 (Level-2 thinking: 2/3 of 33).

This shows that human rationality is bounded. Most people do not, and perhaps cannot, perform the infinite series of deductions required to reach the Nash Equilibrium. They think a few steps ahead and then stop. This insight is the basis for "level-k" models of strategic thinking, which we will explore later.

A Quick Look at Another Classic: The Centipede Game

To reinforce the point about backward induction failures, consider the Centipede Game. In this sequential game, two players take turns choosing to either TAKE a slightly larger share of a growing pot of money, ending the game, or PASS the decision to the other player, which causes the pot to grow.

Backward induction gives a definitive, if disheartening, prediction: Player 1 should TAKE the money on the very first move. Why? Because Player 1 assumes Player 2 would TAKE it on the last move, which means Player 1 should TAKE it on the second-to-last move, and so on, all the way back to the beginning.

But look at the experimental results:

Once again, behavior systematically deviates. Players almost never end the game on the first move. They exhibit a degree of trust and reciprocity, passing to their opponent in the hope of reaching the larger payoffs later in the game.

Conclusion

Today, we've taken our first step into behavioral game theory by identifying crucial instances where the predictions of classical theory do not match the observed behavior of real people.

Key Takeaways:

• Classical game theory models behavior based on assumptions of perfect rationality and pure self-interest.
• Behavioral game theory uses experiments to test these predictions and has found systematic, predictable deviations.
• In the Prisoner's Dilemma, people cooperate more than the dominant strategy equilibrium would suggest.
• In the Ultimatum Game, proposers make fairer offers and responders reject low-but-positive offers, violating the predictions of subgame perfection.
• In the Beauty Contest and Centipede Games, players exhibit bounded rationality, performing only a few steps of strategic reasoning rather than the infinite iteration or full backward induction required for the equilibrium prediction.

These results don't mean game theory is "wrong." They mean that to build models that are useful for "meaningful planning in a chaotic world," as you put it, we need to enrich our models to account for the realities of human psychology: social preferences and bounded rationality.

Preview of the Next Lesson:
We will begin the work of "enriching our models" in the next lesson. We will dive deeper into the Ultimatum Game, formally analyzing how concepts like fairness and social preferences can be mathematically incorporated into a player's utility function to explain why someone might reject an "unfair" offer.