The Ultimatum Game: Fairness and Social Preferences

Introduction

Hello! In our last lesson, we surveyed several key experiments in behavioral game theory that revealed systematic deviations from the predictions of classical models. We saw that in the Ultimatum Game, Proposers make surprisingly fair offers, and Responders often reject offers that are positive but perceived as "unfair." This behavior directly contradicts the subgame perfect Nash equilibrium, where a rational, self-interested Responder should accept any positive amount.

This observation suggests that human utility isn't solely derived from personal monetary gain. Today, we will formalize this intuition. Our goal is to analyze the Ultimatum Game to explain how fairness concerns and social preferences affect outcomes. We will move beyond simply observing the deviation and instead build a mathematical model that incorporates a preference for fairness, or more precisely, an aversion to inequity. By modifying the players' utility functions, we can explain why rejecting "free money" can be a rational choice and why offering a generous slice of the pie is a smart strategy.

The Ultimatum Game: A Quick Refresher

Let's begin with a quick recap of the game's structure, the classical prediction, and the real-world results.

The Ultimatum Game | The Greatest Example of Human "Irrationality"

This short video provides an excellent summary of the Ultimatum Game, contrasting the predictions of traditional economic theory with observed human behavior.

Please watch the entire video (it's less than 3 minutes). Focus on the core conflict it highlights: the 'rational' self-interested prediction versus the actual, much fairer, outcomes driven by human psychology.

As the video explains, the paradox of the Ultimatum Game is that people are willing to sacrifice their own payoff to punish what they perceive as unfair behavior. This implies their utility function contains more than just their own cash. How can we model this?

Modeling Social Preferences: Inequity Aversion

The key insight of behavioral economists Ernst Fehr and Klaus Schmidt was to propose that people's utility depends not only on what they get, but on what they get relative to others. They developed a now-famous model of inequity aversion.

The central idea is that people experience disutility (a loss of utility) from unequal outcomes. Crucially, they proposed that people dislike being behind more than they dislike being ahead.

To understand this model, let's watch a video that breaks down the components of the utility function.

Fehr Schmidt Inequality Aversion Utility Model

This video from economist Ashley Hodgson explains the Fehr-Schmidt model of inequality aversion step-by-step. It's a great conceptual walkthrough before we dive into the math.

Watch from 00:28 to 04:34. Focus on understanding the three main parts of the utility function: one's own payoff, the penalty for being worse off than others, and the penalty for being better off than others.

Now, let's formalize this. The Fehr-Schmidt utility function for a player $i$ in a game with $n$ players is given by:

U_{i} (x) = x_{i} - \frac{α _{i}}{n - 1} j \neq = i \sum max {x_{j} - x_{i}, 0} - \frac{β _{i}}{n - 1} j \neq = i \sum max {x_{i} - x_{j}, 0}

Inequity Aversion Theory Mathematical Formulation — The general form of the Fehr-Schmidt utility function. Here, $x_i$ is the player's own monetary payoff.

For a two-player game like the Ultimatum Game ( $n = 2$ ), the formula simplifies considerably:

U_{i} (x) = x_{i} - α_{i} max {x_{j} - x_{i}, 0} - β_{i} max {x_{i} - x_{j}, 0}

Let's break down the parameters:

$x_{i}$ is player $i$ 's monetary payoff. This is the standard "selfish" part of utility.
The second term, $α_{i} max {x_{j} - x_{i}, 0}$ , captures disadvantageous inequality. It's only active when player $j$ has more than player $i$ . The parameter $α_{i} \geq 0$ measures player $i$ 's "envy" or sensitivity to being behind.
The third term, $β_{i} max {x_{i} - x_{j}, 0}$ , captures advantageous inequality. It's only active when player $i$ has more than player $j$ . The parameter $β_{i} \geq 0$ measures player $i$ 's "guilt" or sensitivity to being ahead.

The model typically assumes $α_{i} \geq β_{i}$ (envy is stronger than guilt) and $0 \leq β_{i} < 1$ (people don't like being ahead, but not so much that they'd throw away their own money to reduce the gap).

Explaining the Responder's Choice

Now, let's use this utility function to solve the puzzle of why a Responder would reject a low-but-positive offer. Let the total pie be 1, and the Proposer offers a share $s$ to the Responder (Player R). The Proposer (Player P) keeps $1 - s$ .

If the Responder Rejects:
The outcome is (0, 0). The Responder's utility is:

U_{R} (Reject) = 0 - α_{R} max {0 - 0, 0} - β_{R} max {0 - 0, 0} = 0

If the Responder Accepts:
The outcome is ( $1 - s$ for the Proposer, $s$ for the Responder). Let's focus on the interesting case where the offer is "unfair", meaning $s < 0.5$ . In this situation, the Proposer has more money, so the Responder experiences disadvantageous inequality.

The Responder's utility is:

U_{R} (Accept) = s - α_{R} max {(1 - s) - s, 0} - β_{R} max {s - (1 - s), 0}

Since $s < 0.5$ , the second term is positive and the third term is zero. So, this simplifies to:

U_{R} (Accept) = s - α_{R} (1 - 2 s)

The Decision:
A rational Responder with these social preferences will reject the offer if the utility of accepting is less than the utility of rejecting:

U_{R} (Accept) < U_{R} (Reject)

s - α_{R} (1 - 2 s) < 0

Now, we can solve for the offer $s$ . This is the kind of algebraic manipulation that should feel familiar from your physics and math studies.

s < α_{R} (1 - 2 s)

s < α_{R} - 2 s α_{R}

s + 2 s α_{R} < α_{R}

s (1 + 2 α_{R}) < α_{R}

s < \frac{α _{R}}{1 + 2 α _{R}}

This final inequality is the key. It defines the rejection threshold. A Responder will reject any offer $s$ that is less than this value. Notice that if $α_{R} = 0$ (the player has no envy), the threshold is 0, and they will accept any positive offer—recovering the classical result. But if the Responder has a positive $α_{R}$ , say $α_{R} = 1$ , they will reject any offer less than $1/ (1 + 2) = 1/3$ .

This mathematically explains why people reject "unfair" offers: the monetary gain from $s$ is not enough to compensate for the psychological disutility from the perceived inequity, which is weighted by $α_{R}$ .

Explaining the Proposer's Choice

So, why do Proposers make fair offers? Because they anticipate the Responder's reaction.

Consider a purely self-interested Proposer ( $α_{P} = β_{P} = 0$ ). Their goal is simply to maximize their monetary payoff, $1 - s$ . They know (or have a belief about) the Responder's $α_{R}$ . They know that if they offer less than the rejection threshold $\frac{α _{R}}{1 + 2 α _{R}}$ , the offer will be rejected, and their payoff will be 0. To maximize their payoff, they must make an offer that will be accepted. The optimal strategy is therefore to offer the smallest amount they believe the Responder will accept.

If the Proposer believes the Responder has $α_{R} = 1$ , they will offer slightly more than $1/3$ of the pie. This explains why offers are rarely minimal.

What if the Proposer also has social preferences? Suppose the Proposer has a "guilt" parameter, $β_{P} > 0$ . Their utility for an accepted offer $s < 0.5$ is:

U_{P} (Accept) = (1 - s) - β_{P} ((1 - s) - s) = (1 - s) - β_{P} (1 - 2 s)

A high $β_{P}$ means the Proposer feels significant disutility from being too far ahead. For instance, if $β_{P} > 0.5$ , the Proposer's utility actually increases as $s$ gets closer to 0.5. Such a Proposer would offer $s = 0.5$ not just out of fear of rejection, but because they genuinely prefer a fair outcome.

This combination—Proposers' fear of rejection and their own potential aversion to advantageous inequality—explains why experimental offers are not only non-minimal, but often cluster around the 40-50% mark.

For a deeper dive into the formal derivation, you can explore the original paper by Fehr and Schmidt.

A THEORY OF FAIRNESS, COMPETITION, AND COOPERATION

The following is an excerpt from the seminal 1999 paper in the Quarterly Journal of Economics that introduced this model. Reading it will show you how these ideas are presented in formal academic research.

Read the section 'A. The Ultimatum Game'. You don't need to read the whole thing in minute detail. Focus on understanding Proposition 1 and the logic in its Proof. The proof formalizes the derivation of the Responder's acceptance threshold, s < a_2/(1+2a_2), which we just walked through (they use 'a' instead of \alpha). Also, note how it describes the Proposer's optimal offer s^*.

Conclusion

Today we moved from observing a behavioral anomaly to explaining it with a rigorous mathematical model. By augmenting the standard utility function to include social preferences, we can account for behavior that seems "irrational" from a purely self-interested perspective.

Key Takeaways:

The rejection of low offers in the Ultimatum Game can be explained by inequity aversion—a dislike for unequal outcomes, especially when one is at a disadvantage.
The Fehr-Schmidt utility function formalizes this by adding terms for disadvantageous inequality (weighted by $α$ , or "envy") and advantageous inequality (weighted by $β$ , or "guilt").
A Responder will reject an offer $s$ if the monetary gain is outweighed by the psychological cost of the inequality. This occurs if $s < \frac{α _{R}}{1 + 2 α _{R}}$ .
A rational Proposer anticipates this and will offer at least the Responder's rejection threshold. The Proposer's own fairness concerns ( $β_{P}$ ) can push the offer even higher, explaining the prevalence of offers near a 50/50 split.

Preview of the Next Lesson:
We have successfully used social preferences to explain the outcomes of the Ultimatum Game. However, fairness doesn't explain the results of the Beauty Contest game we saw in the last lesson. That requires a different kind of deviation from classical theory: bounded rationality. In our next lesson, we will explore "level-k" models of thinking, which formalize the idea that people can only perform a limited number of steps of strategic reasoning.

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