Maurício Pinheiro
Abstract
The ultimatum game is one of the most famous experiments in game theory because it reveals something deeply uncomfortable about human decision-making: people do not always choose money over fairness.
In this simple game, one player proposes how to divide a sum of money, while the other player can either accept or reject the offer. If the offer is accepted, both players receive the proposed amounts. If the offer is rejected, both receive nothing.
Classical game theory predicts that a rational responder should accept any positive offer, no matter how small, because receiving something is better than receiving nothing. But real humans frequently reject low offers. A person may refuse $10 out of $100, even though rejection means losing the $10, simply because the offer feels unfair, insulting, or exploitative.
This article explains why the ultimatum game is not outside game theory, but one of its most important laboratories. It explores the history of the game, its mathematical structure, the classical equilibrium prediction, the behavioral evidence, the role of fairness, and its connections with neuroscience, evolution, politics, economics, and artificial intelligence.
In the age of AI agents, automated negotiation, algorithmic pricing, machine decision-making, and autonomous systems, the ultimatum game becomes more than an academic experiment. It becomes a warning: intelligence without fairness may optimize outcomes while destroying trust.
Table of Contents
- Introduction: A Simple Game That Broke the Myth of Pure Rationality
- The Birth of the Ultimatum Game
- Why the Ultimatum Game Belongs Inside Game Theory
- The Classical Game-Theory Solution
- The Human Result: “No Deal” as Punishment
- A Historical Example: Bargaining, Power, and Revolt
- Cross-Cultural Evidence: Fairness Is Universal, but Not Uniform
- Back to math: Fairness in the Utility Function
- Fehr–Schmidt and Inequity Aversion
- The Brain in the Ultimatum Game
- The Evolutionary Puzzle: Why Would Fairness Survive?
- The Ultimatum Game and Artificial Intelligence
- LLMs and the New Experimental Economics
- Reinforcement Learning: Can Fairness Emerge?
- The Alignment Problem Hidden Inside the Game
- A Concrete AI Example: Salary Negotiation
- A Political Example: AI and Resource Allocation
- What the Ultimatum Game Teaches Us About Intelligence
- Conclusion: The Small Game at the Center of a Large Future
- References for Further Reading
1. Introduction: A Simple Game That Broke the Myth of Pure Rationality
Imagine that someone gives two people $100.
One of them, called the proposer, must decide how to divide the money. The other, called the responder, can either accept or reject the offer. If the responder accepts, both players receive the proposed amounts. If the responder rejects, both players receive nothing.
That is the ultimatum game.
At first glance, it seems almost too simple to matter. One person makes an offer. The other says yes or no. Yet this tiny experiment has become one of the most important games in economics, psychology, neuroscience, evolutionary theory, political philosophy, and now artificial intelligence.
Why?
Because classical game theory makes a very clear prediction. If both players are perfectly rational and care only about their own monetary payoff, the responder should accept any positive offer, even $1 out of $100. After all, $1 is better than $0. Knowing this, the proposer should offer the smallest possible amount and keep the rest.
But real humans often refuse.
A responder may reject $10 out of $100, even though rejection means getting nothing. Economically, this looks irrational. Psychologically, it feels obvious. The responder is not merely calculating money. The responder is judging fairness, dignity, insult, exploitation, and social meaning.
The ultimatum game is therefore not a curiosity outside game theory. It is one of game theory’s most revealing laboratories. It shows the gap between the mathematical logic of strategic interaction and the richer reality of human behavior. It forces us to ask a deeper question: what exactly is a rational decision when humans care not only about outcomes, but also about fairness?
In the age of artificial intelligence, this question becomes urgent.
AI systems are no longer just passive tools that answer questions. They are increasingly becoming agents that negotiate, recommend, price, rank, allocate, moderate, and decide. If AI agents negotiate salaries, allocate medical resources, moderate social platforms, distribute public benefits, or recommend economic opportunities, should they behave like pure payoff maximizers?
Should they learn human norms of fairness?
Should they punish unfairness?
Should they imitate moral emotions?
Or should they remain coldly efficient?
The ultimatum game is small enough to solve with simple equations, but deep enough to expose the moral architecture of intelligence.
2. The Birth of the Ultimatum Game
The ultimatum game was introduced experimentally in 1982 by Werner Güth, Rolf Schmittberger, and Bernd Schwarze in their paper “An Experimental Analysis of Ultimatum Bargaining.”
Their goal was to study bargaining behavior in a simple setting where one player makes a take-it-or-leave-it proposal and the other player has no opportunity to negotiate further.
This structure matters. In many real-life negotiations, people bargain back and forth. They make counteroffers, signal intentions, build reputations, and adjust their expectations. The ultimatum game removes all that complexity. There is one offer and one response.
That is why it is called an ultimatum.
The responder does not say, “Could you improve the offer?” The responder does not make a counterproposal. The responder says only: accept or reject.
This minimalism is precisely what makes the experiment powerful. It strips bargaining down to its strategic skeleton. The proposer has power, but the responder has veto power. The proposer controls the division, but the responder controls whether the division exists at all.
The game is not a prisoner’s dilemma. It is not a coordination game. It is not a zero-sum game in the usual sense. It is a sequential bargaining game with perfect information.
The responder sees the offer before deciding. The proposer knows that the responder will see the offer. Therefore, the proposer must anticipate the responder’s reaction.
This is exactly the kind of situation game theory was built to analyze: strategic interaction under rules, incentives, expectations, and consequences.
3. Why the Ultimatum Game Belongs Inside Game Theory
Some people mistakenly treat the ultimatum game as if it were a refutation of game theory.
It is not.
The ultimatum game is deeply inside game theory. What it refutes is not game theory itself, but an overly narrow version of rational choice in which human beings care only about immediate monetary payoff.
Game theory studies how agents make decisions when the outcome depends not only on their own choice, but also on the choices of others. The ultimatum game fits this definition perfectly.
The proposer’s payoff depends on the offer and on the responder’s decision. The responder’s payoff depends on the proposer’s offer and on whether the responder accepts or rejects.
The classical solution uses backward induction. The responder’s final decision is analyzed first. Then the proposer’s initial decision is analyzed in light of what the responder is expected to do.
This leads to a subgame-perfect equilibrium, one of the central concepts of non-cooperative game theory.
So the ultimatum game does not destroy game theory. Instead, it asks game theory to become more realistic. It asks us to model not only money, but also fairness, resentment, status, reputation, dignity, and social norms.
That is the beauty of the game. It begins as a mathematical puzzle and becomes a theory of human society.
4. The Classical Game-Theory Solution
Let the total amount of money be:
M > 0
Here, M is the total amount to be divided between the two players.
The proposer chooses an offer:
x
The variable x represents the amount offered to the responder.
The proposer keeps:
M − x
If the responder accepts the offer, the proposer receives:
πP = M − x
and the responder receives:
πR = x
If the responder rejects the offer, both receive zero:
πP = 0
πR = 0
Under the classical assumption of purely self-interested rational agents, each player wants only to maximize their own monetary payoff. The responder compares accepting x with rejecting and receiving zero.
If the offer is positive:
x > 0
then accepting is better than rejecting, because receiving something is better than receiving nothing.
Therefore, the responder should accept any positive offer.
Knowing this, the proposer should offer the smallest possible positive amount. If the money is divisible into cents and the total is $100, the proposer might offer just $0.01 and keep the remaining $99.99.
More generally, game theorists represent the smallest possible positive amount by the symbol ε (epsilon). In the classical model, the optimal offer is therefore:
x* = ε
where in game theory, x* means “the optimal predicted choice” or “the equilibrium value of x.”
While the proposer keeps:
M − ε
This outcome is known as the subgame-perfect equilibrium of the ultimatum game.
The logic is straightforward: since the responder is assumed to care only about maximizing monetary payoff, accepting even a tiny amount is preferable to receiving nothing.
As a result, the proposer has no incentive to offer more than the minimum possible amount.
This is the subgame-perfect equilibrium prediction.
The logic is elegant, brutal, and fragile.
It is elegant because it follows directly from backward induction. It is brutal because it says fairness should not matter. It is fragile because human beings often refuse to behave this way.
The surprising result, however, is that real human beings rarely behave this way.
In laboratory experiments, many responders reject extremely low offers, even when rejection leaves them with nothing.
This suggests that people value not only money, but also fairness, dignity, reciprocity, and the willingness to punish behavior they perceive as exploitative.
5. The Human Result: “No Deal” as Punishment
In laboratory experiments, responders frequently reject offers they perceive as unfair. Offers below 20% or 30% of the total are often rejected. Offers around 40% or 50% are far more likely to be accepted. Proposers, anticipating this, commonly offer something closer to an equal split than the classical model predicts.
The key point is not that humans are irrational in a random way. They are predictably “irrational” only relative to a narrow model of self-interest.
A responder who rejects $10 out of $100 sacrifices money, but sends a message: “I will not accept humiliation.”
This is not merely a financial decision. It is a social act.
In the ultimatum game, rejection functions as punishment. The responder pays a cost to punish the proposer for making an unfair offer.
This is sometimes called costly punishment.
It is one of the most important ideas in the study of cooperation.
A society in which no one punishes unfairness may become efficient in the short term but exploitative in the long term.
A society in which people are willing to punish unfairness may sustain norms of cooperation, even when punishment is costly.
The ultimatum game therefore reveals a hidden mechanism of civilization: fairness is not only a preference; it can be enforced.
6. A Historical Example: Bargaining, Power, and Revolt
The ultimatum game has a simple laboratory structure, but its logic appears throughout history.
Consider a ruler, employer, empire, corporation, or elite group that controls resources. It proposes a division of wealth, rights, food, land, wages, or political power. The weaker side may have no ability to propose an alternative. It can only accept the arrangement or reject the legitimacy of the system itself.
At first glance, the weaker side should accept any improvement over starvation, poverty, exclusion, or repression. That is the logic of the classical model. Something is better than nothing.
But history repeatedly shows that humans do not evaluate social arrangements only in absolute terms. They evaluate them comparatively and morally.
A population may revolt not when conditions are at their absolute worst, but when the perceived gap between contribution and reward, sacrifice and dignity, promise and reality, becomes unbearable.
This does not mean every revolt is literally an ultimatum game. Historical societies are vastly more complex. They include ideology, institutions, violence, propaganda, memory, leadership, class structure, and international pressures.
But the analogy is useful.
When the powerful make offers that preserve too much for themselves and too little for others, the disadvantaged may reject the entire arrangement, even at enormous cost.
This is why fairness matters politically. A deal can be materially positive and still be socially unstable. The exploited party may prefer mutual loss to continued humiliation.
The ultimatum game compresses this political truth into one move.
7.Cross-Cultural Evidence: Fairness Is Universal, but Not Uniform
One of the most important discoveries in ultimatum-game research is that fairness norms vary across societies. The desire for fairness may be widespread, but the meaning of fairness is not identical everywhere.
What one culture interprets as a fair offer, another may interpret through a completely different moral, political, or social framework.
In many Western university experiments, offers close to 50% are common, and very low offers are often rejected.
This result fits well with a modern Western moral language centered on individual rights, equality before the law, autonomy, and the idea that a fair deal should treat both sides as roughly symmetrical individuals.
In this view, a 50–50 split appears almost naturally fair because both players are imagined as independent agents entering a bargain on equal moral footing.
But real societies are more complicated than this idealized picture.
Even in the West, people often accept “crumbs” when they lack bargaining power.
A worker may accept a bad wage because rent is due.
A patient may accept an opaque insurance decision because there is no realistic alternative.
A user may accept unfair platform rules because the platform has become socially unavoidable.
In theory, liberal societies celebrate individual choice.
In practice, many choices are made under pressure.
The ultimatum game exposes this contradiction with unusual clarity. A person may reject an unfair offer in a laboratory because the cost of refusal is limited.
But the same person may accept an unfair offer in real life when saying no would threaten employment, family stability, access to services, or future opportunities.
The difference is not simply psychological. It is institutional.
This becomes even more visible in societies shaped by stronger hierarchies, collective discipline, or centralized power.
In socialist and post-socialist societies, the official language often emphasizes equality, solidarity, and collective welfare.
Yet the actual experience of ordinary people may involve scarcity, waiting, limited options, bureaucratic dependence, and acceptance of small concessions from institutions that hold far greater power.
People may receive “crumbs” and still accept them, not because they believe the offer is fair, but because rejection seems useless, dangerous, or socially disruptive.
China offers a particularly interesting contrast for the ultimatum game.
Contemporary China is not simply traditional Confucianism, nor is it simply communism.
It is a hybrid civilization in which older Confucian ideas of hierarchy, order, duty, family, discipline, education, respect for authority, and social harmony have been partially adapted to a modern socialist and state-capitalist system.
In a Confucian moral universe, society is not imagined primarily as a collection of isolated individuals bargaining as equals.
It is imagined as a network of roles and obligations: ruler and subject, parent and child, elder and younger, teacher and student, family and state.
Harmony is often valued more than open confrontation, and social stability may be treated as a moral good in itself.
When this older cultural background is combined with communist political organization, the result is a society in which many people may tolerate unequal arrangements, restricted choices, or limited concessions without translating dissatisfaction into large-scale revolt.
This does not mean people are passive, irrational, or incapable of resentment.
It means that rejection does not always take the Western liberal form of public confrontation, individual protest, or open institutional challenge.
In the language of the ultimatum game, the responder may perceive the offer as unfair and still accept it.
Why?
Because the real-world payoff is not only money.
It also includes safety, stability, family interest, reputation, social harmony, fear of punishment, hope for future improvement, and the belief that disorder may be worse than injustice.
This point is crucial. The ultimatum game is simple because the responder can reject without further consequences.
Real societies are not like that. In real life, rejecting an offer may mean losing a job, damaging one’s family, attracting surveillance, being excluded from opportunities, or creating instability.
Under those conditions, people may accept what looks like an unfair deal because the cost of rejection is too high.
This does not mean that Western societies are perfectly fair or that Eastern societies naturally accept unfairness; that would be too simplistic.
Liberal capitalist societies are built around individual choice, private initiative, competition, property rights, and social mobility, allowing unfair offers to be challenged over time through innovation, education, entrepreneurship, and institutional reform.
Socialist societies may speak the language of equality while producing hierarchy, bureaucracy, and privilege.
Confucian-influenced societies may value harmony, but harmony can sometimes become a moral language for suppressing conflict.
The deeper lesson is this: fairness is never just arithmetic.
A 50–50 split may seem obviously fair in one culture, while another society may interpret fairness through hierarchy, duty, seniority, contribution, family obligation, national stability, or collective sacrifice. In one society, rejecting an unfair offer may be seen as dignity. In another, it may be seen as disorder. In another, it may be seen as politically dangerous.
This matters because it undermines two simplistic interpretations.
The first simplistic interpretation is that humans are purely selfish everywhere. The evidence does not support that. People often care about fairness, reciprocity, reputation, dignity, loyalty, and social norms.
The second simplistic interpretation is that fairness means the same thing everywhere. That is also false. What counts as a fair offer depends on culture, institutions, history, power, ideology, family structure, religion, class, and the perceived cost of saying no.
The ultimatum game therefore teaches an important lesson for economics, politics, and artificial intelligence: human values are not just private preferences stored inside isolated minds.
They are shaped by the society in which people live.
A universal model of human rationality is too thin.
A serious model of human decision-making must include local norms, power relations, fear, dignity, hierarchy, and culture.
And any artificial intelligence designed to negotiate, allocate resources, or make decisions among humans must understand that fairness is not a single formula.
It is a cultural language.
Sometimes, under pressure, even people who know they are receiving crumbs may still accept the offer.
8. Back to math: Fairness in the Utility Function
Classical game theory is often misunderstood. The problem is not game theory itself. The problem is a narrow utility function.
By shifting from a narrow definition of self-interest to a richer utility function, behavioral economics proves that rejecting low offers in the Ultimatum Game is entirely rational, mathematical, and predictable.
This is the philosophical core of behavioral game theory
If we assume that utility equals only money, then rejection of a positive offer seems irrational. But if utility includes fairness, status, anger, dignity, reciprocity, or norm enforcement, then rejecting unfair offers can become rational.
The simplest classical payoff for the responder is:
UR = x
This means that the responder’s utility is just the amount of money received.
In that case, the responder accepts any offer above zero:
x > 0
But now suppose the responder dislikes unfair inequality. A simple fairness-adjusted utility can be written as:
UR = x − α(M − 2x)
Here, UR is the responder’s utility.
The variable x is the money offered to the responder.
The variable M is the total amount of money.
The parameter α measures how strongly the responder dislikes unfairness.
The term below measures how unequal the offer is when the proposer receives more than the responder:
M − 2x
The responder accepts the offer only if utility is not negative:
UR ≥ 0
Substituting the fairness-adjusted utility:
x − α(M − 2x) ≥ 0
Expanding the expression:
x − αM + 2αx ≥ 0
Grouping the terms with x:
x + 2αx ≥ αM
Factoring x:
x(1 + 2α) ≥ αM
Dividing both sides by 1 + 2α:
x ≥ αM / (1 + 2α)
This gives us the minimum acceptable offer.
For example, suppose:
M = 100
and:
α = 0.5
Then:
x ≥ (0.5 × 100) / (1 + 2 × 0.5)
Since:
2 × 0.5 = 1
we get:
x ≥ 50 / 2
Therefore:
x ≥ 25
So the responder rejects offers below $25.
Now the rejection of low offers is no longer irrational. It is rational under a richer utility function.
This is the crucial move made by behavioral game theory. It does not abandon mathematics. It expands what the mathematics is allowed to represent.
9. Fehr–Schmidt and Inequity Aversion
A more formal model of fairness was proposed by Ernst Fehr and Klaus Schmidt in their theory of inequity aversion. The idea is that people dislike unequal outcomes, especially when they are worse off than others.
A simplified version of the Fehr–Schmidt utility function can be written as:
Ui = xi − αi max(xj − xi,0) − βi max(xi − xj,0)
This looks complicated at first, but the idea is simple.
The term xi is the payoff of player i.
The term xj is the payoff of the other player.
The parameter αi measures how much player i dislikes being worse off than the other player.
The parameter βi measures how much player i dislikes being better off than the other player.
Usually, the assumption is:
αi > βi
In plain English, people usually dislike being treated unfairly more than they dislike benefiting from an unfair advantage.
In the ultimatum game, this model helps explain two facts at once.
First, responders reject low offers because the psychological cost of inequality outweighs the monetary gain.
Second, proposers often offer more than the minimum because they anticipate rejection, dislike appearing unfair, or have fairness preferences themselves.
The mathematics therefore becomes more realistic not by becoming less rigorous, but by becoming more psychologically informed.
10. The Brain in the Ultimatum Game
The ultimatum game also became important in neuroeconomics, the field that studies the neural basis of economic decision-making.
In a famous study, participants played the ultimatum game while researchers observed brain activity. Unfair offers activated brain regions associated with both cognitive control and emotional response. In particular, activity in the anterior insula was linked to the experience of unfairness and the likelihood of rejecting unfair offers.
This is fascinating because it suggests that the responder is not simply doing a cold calculation. The brain appears to register unfairness as a kind of aversive social signal.
An unfair offer is not processed merely as “some money.” It is processed as an insult, a violation, or a threat to social norms.
The ultimatum game therefore challenges the old opposition between reason and emotion. Emotion is not always the enemy of rationality. In social life, emotion can carry information. Anger at unfairness may be one mechanism by which groups enforce cooperation.
But there is also danger. If fairness emotions become excessive, people may reject mutually beneficial compromises. If they are absent, people may tolerate exploitation.
Human cooperation requires a delicate balance between calculation and moral emotion.
11. The Evolutionary Puzzle: Why Would Fairness Survive?
From a narrow evolutionary perspective, rejecting free money seems strange.
Why would organisms evolve to sacrifice resources in order to punish unfairness?
One answer is that humans did not evolve in anonymous one-shot laboratory games. We evolved in social environments full of repeated interactions, reputation, coalition-building, kinship, memory, and punishment.
In such environments, accepting every unfair offer may signal weakness. Rejecting unfairness may protect one’s reputation. Punishing exploiters may discourage future exploitation. Groups that enforce fairness norms may become more stable and cooperative.
The ultimatum game is formally one-shot, but human psychology may be adapted to a world in which few interactions are truly one-shot. Our brains may carry ancient assumptions: people remember, reputations spread, and unfairness today predicts danger tomorrow.
This also explains why the laboratory result is not a failure of intelligence. It may be the shadow of social intelligence.
Humans are not merely utility maximizers.
They are norm-sensitive agents.
12. The Ultimatum Game and Artificial Intelligence
The ultimatum game is now becoming important again because artificial intelligence systems are increasingly expected to act as agents.
A chatbot that answers questions is one thing. An AI agent that negotiates, hires, prices, allocates resources, moderates content, recommends loans, distributes benefits, or coordinates with other agents is something else entirely.
Such systems will not merely describe the world. They will participate in strategic environments.
This raises a serious question: what kind of game-theoretic agents should AI systems become?
If an AI is trained only to maximize immediate reward, it may behave like the classical proposer in the ultimatum game: offer the minimum, exploit bargaining power, and treat acceptance as the only measure of success.
That might be efficient in a narrow sense, but socially destructive.
If an AI is trained to imitate human behavior, it may learn fairness norms. But which norms? From which culture? Under which institutional conditions? And how will we know whether it understands fairness or merely reproduces patterns from training data?
If an AI is trained through reinforcement learning in multi-agent environments, fairness may emerge as a strategy when agents interact repeatedly, learn from rejection, and adapt to social consequences.
In that case, fairness is not simply programmed from above. It emerges from the dynamics of interaction.
This is one of the most promising connections between game theory and AI. The ultimatum game becomes a test chamber for artificial social intelligence.
An AI that cannot understand the ultimatum game cannot truly understand negotiation.
13. LLMs and the New Experimental Economics
Large language models can now be placed inside economic games as simulated agents. Researchers can ask them to play as proposers or responders, to simulate different personalities, or to predict human behavior.
This creates a new form of experimental economics.
Instead of testing only humans in laboratories, we can test artificial agents in controlled social environments.
We can ask whether LLMs make fair offers.
We can ask whether they reject unfair offers.
We can ask whether they change behavior when told they are playing with a human rather than another AI.
We can ask whether they behave differently under different cultural prompts.
These questions are not trivial. LLMs are not humans. They do not have hunger, anger, childhood, social status, or biological emotion. But they have absorbed enormous quantities of human text. They may reproduce human moral vocabulary without possessing human moral experience.
This makes the ultimatum game an ideal diagnostic tool.
It is simple, measurable, and revealing. It can expose whether an AI system behaves like a rational optimizer, a moral imitator, a social simulator, or an inconsistent text generator.
14. Reinforcement Learning: Can Fairness Emerge?
In reinforcement learning, an agent learns by taking actions and receiving rewards. In a simple ultimatum-game environment, an AI proposer might try different offers and observe whether they are accepted or rejected.
Over time, it learns that extremely low offers often fail, while fairer offers produce successful deals.
Suppose the proposer’s action is an offer:
x
If the offer is accepted, the proposer receives:
RP = M − x
If the offer is rejected, the proposer receives:
RP = 0
Here, RP is the reward of the proposer.
Now the AI does not only care about how much it keeps. It must also estimate the probability that the offer will be accepted.
That probability can be written as:
P(accept | x)
This means: the probability of acceptance given the offer x.
The expected reward of the proposer is:
E[RP(x)] = P(accept | x)(M − x)
This equation is central.
It shows that the proposer should not simply minimize the offer. The proposer must consider how the offer affects the probability of acceptance.
For example, suppose:
M = 100
If the proposer offers $10, and the probability of acceptance is only 20%, we write:
P(accept | 10) = 0.2
The expected reward is:
E[RP(10)] = 0.2 × (100 − 10)
So:
E[RP(10)] = 0.2 × 90
Therefore:
E[RP(10)] = 18
Now suppose the proposer offers $40, and the probability of acceptance is 95%:
P(accept | 40) = 0.95
The expected reward is:
E[RP(40)] = 0.95 × (100 − 40)
So:
E[RP(40)] = 0.95 × 60
Therefore:
E[RP(40)] = 57
The fairer offer gives the proposer a higher expected payoff.
This is a powerful lesson. Fairness can be instrumentally rational when agents must deal with other agents who have standards, emotions, memory, or veto power.
In repeated or multi-agent environments, fairness may become not a moral luxury, but a survival strategy.
The graph makes visible the central logic of this section: the proposer should not consider only how much money they keep if the offer is accepted, but also the probability that the responder will accept the offer. As the offer x increases, the probability of acceptance P(accept | x) also rises, especially near the equal split of 50%, showing that offers perceived as fair carry a much lower risk of rejection. At the same time, the larger the offer, the smaller the amount kept by the proposer, that is, 100 − x. The expected reward curve combines these two forces: E[RP(x)] = P(accept | x)(100 − x). The result shows that an extremely low offer may look attractive from a purely selfish perspective, but becomes weak once the risk of rejection is included. Thus, the graph visually illustrates the main lesson of reinforcement learning applied to the ultimatum game: in social environments, fairness can emerge not only as a moral value, but also as a rational strategy for maximizing expected rewards.
The curve is not linear for low offers because human rejection is not a smooth mechanical response to money. Very low offers are interpreted not only as small payments, but as signals of disrespect, exploitation, or humiliation. Therefore, increasing the offer from 0% to 10% or from 10% to 20% does not simply add value in a linear way; it changes the social meaning of the proposal. The sharp jump near 50% occurs because the equal split is a powerful fairness threshold. Around that point, the offer stops being perceived as an insult and begins to be interpreted as legitimate, respectful, or socially acceptable. Above 50%, the acceptance probability becomes almost flat because the offer is already more than fair from the responder’s perspective. Once the responder is receiving at least half of the total, additional increases do not change the decision very much: most people are already willing to accept. This is why the expected reward then declines: acceptance remains high, but the proposer keeps less and less of the total amount.
15. The Alignment Problem Hidden Inside the Game
The ultimatum game also reveals a miniature version of the AI alignment problem.
A poorly aligned AI may optimize a formal objective while violating human expectations. In the ultimatum game, the formal objective might be:
maximize monetary payoff
But humans also care about legitimacy, fairness, trust, and social stability.
This mismatch between formal reward and human value is exactly what makes AI alignment difficult.
If we tell an AI:
maximize profit
it may discover strategies that humans perceive as exploitative.
If we tell it:
be fair
we must define fairness.
If we define fairness mathematically, we must choose among competing concepts: equality, proportionality, need, merit, reciprocity, envy-freeness, maximin, or social welfare.
If we train it on human preferences, we inherit human inconsistency, cultural variation, and bias.
The ultimatum game is therefore not just a game.
It is a warning.
A society increasingly mediated by AI cannot rely on optimization alone.
Thus, the AI race between West and East is not only a race for faster chips, larger models, better data centers, or more powerful algorithms.
It is also a race between different visions of human organization.
In the West, AI is more likely to be shaped by the language of individual rights, market competition, free enterprise, privacy, transparency, and personal autonomy. In China and other Eastern or state-centered systems, AI may be shaped more strongly by collective stability, social harmony, centralized planning, national strategy, and the idea that individual preferences must often be subordinated to broader social order.
The danger is that each civilization may train artificial intelligence to reproduce its own deepest assumptions. A Western AI may become brilliant at optimizing choice, competition, personalization, and consumer preference, but may struggle with social cohesion and long-term collective responsibility.
An Eastern or state-centered AI may become powerful at coordination, surveillance, planning, and stability, but may struggle with dissent, individual freedom, and the moral value of refusal.
The ultimatum game helps us see why this matters.
The central question is not only whether an AI system can calculate the optimal offer.
The deeper question is what the system has been trained to recognize as fair.
Does fairness mean maximizing total welfare?
Does fairness mean protecting individual rights?
Does fairness mean preserving social harmony?
Does fairness mean obeying the state?
Does fairness mean reducing inequality?
Does fairness mean respecting the right to say no?
Different civilizations may answer these questions differently.
And as AI systems become embedded in courts, schools, hospitals, financial markets, social platforms, military systems, and public administration, these answers will no longer remain philosophical abstractions.
They will become code, policy, ranking systems, recommendation engines, risk scores, and automated decisions.
That is why the ultimatum game is so important.
It shows that intelligence cannot be separated from values.
A machine that knows how to optimize but does not know what humans consider legitimate may produce decisions that are efficient, but socially explosive.
The future of AI will not be decided only by who builds the most powerful model.
It will also be decided by who defines fairness, who audits it, who benefits from it, and who has the power to reject the offer.
16. A Concrete AI Example: Salary Negotiation
Imagine an AI system used by a company to negotiate salaries with job candidates.
A purely self-interested corporate AI might offer the lowest salary it predicts a candidate will accept. If the candidate accepts, the system counts the negotiation as successful.
This resembles the classical proposer in the ultimatum game.
But the long-term consequences may be damaging. Candidates may feel exploited. Reputation may suffer. Employee loyalty may decline. Social media may expose unfair patterns. Regulators may investigate discrimination. The company may “win” the negotiation and lose trust.
A more socially intelligent AI would not ask only:
“What is the lowest offer this person will accept?”
It would ask:
“What offer is efficient, fair, sustainable, explainable, and consistent with institutional values?”
That is a very different optimization problem.
The ultimatum game teaches that acceptance is not the same as legitimacy. A person may accept an unfair offer because they need the money, but still experience resentment.
A society built on such acceptances becomes fragile.
17. A Political Example: AI and Resource Allocation
Now imagine an AI system used to allocate scarce public resources: medical appointments, school places, housing support, disaster relief, or welfare benefits.
If the system maximizes a narrow efficiency metric, it may produce outcomes that are statistically optimal but socially unacceptable. Some groups may perceive the allocation as unfair.
Even if the algorithm improves average performance, people may reject the system if they believe it violates basic norms of justice.
The ultimatum game helps explain why.
People do not evaluate systems only by final payoff.
They evaluate procedures, intentions, comparisons, and dignity.
A system that gives people “something” may still be rejected if it appears arbitrary, biased, opaque, or insulting.
This is one reason explainability matters in AI. People are more likely to accept unfavorable outcomes when they believe the process was fair.
In human society, legitimacy is part of the payoff function.
18. What the Ultimatum Game Teaches Us About Intelligence
The ultimatum game reveals that intelligence is not just calculation.
A purely calculative agent accepts any positive offer. A socially intelligent agent asks what the offer means.
Is it fair?
Is it exploitative?
Will accepting it invite future abuse?
Will rejecting it enforce a norm?
What precedent does this create?
How will others interpret this decision?
These questions are not irrational. They are part of strategic life among social beings.
The game also teaches that rationality depends on the model. If the model includes only immediate money, rejection looks irrational. If the model includes reputation, fairness, inequality, social norms, emotion, and future interaction, rejection can become rational.
This is a crucial lesson for AI.
The danger is not that machines will be irrational.
The danger is that they may be rational inside the wrong model.
19. Conclusion: The Small Game at the Center of a Large Future
The ultimatum game is one of the most beautiful experiments in game theory because it is both mathematically simple and philosophically explosive.
It begins with a tiny bargaining problem. One person divides money. The other accepts or rejects. Classical theory predicts that any positive offer should be accepted. Human beings disagree.
They reject insult.
They punish unfairness.
They sacrifice payoff for dignity.
They reveal that economic life is not merely about quantities, but about relationships.
This does not refute game theory. It deepens it.
The ultimatum game shows that the real challenge is not to abandon mathematical models, but to build better ones: models that include fairness, emotion, culture, learning, and social meaning.
For artificial intelligence, the lesson is urgent.
AI agents will increasingly participate in social and economic games. If they optimize too narrowly, they may become efficient exploiters. If they imitate humans blindly, they may reproduce our biases. If they learn in multi-agent worlds, they may discover fairness as a strategy — but not necessarily the fairness we want.
The ultimatum game is therefore more than a classroom experiment.
It is a miniature mirror of civilization.
It asks a question that every society, and every intelligent system, must eventually answer:
What kind of offer is worth accepting?
And perhaps even more importantly:
What kind of intelligence knows when to say no?
#UltimatumGame #GameTheory #AI #ArtificialIntelligence #AIEthics #Fairness #BehavioralEconomics #AIAlignment #HumanRationality #ReinforcementLearning #LargeLanguageModels
References for Further Reading
Güth, W., Schmittberger, R., & Schwarze, B. (1982). “An Experimental Analysis of Ultimatum Bargaining.” Journal of Economic Behavior & Organization, 3(4), 367–388.
Nash, J. F. (1950). “Equilibrium Points in N-Person Games.” Proceedings of the National Academy of Sciences, 36(1), 48–49.
Nash, J. F. (1951). “Non-Cooperative Games.” Annals of Mathematics, 54(2), 286–295.
Fehr, E., & Schmidt, K. M. (1999). “A Theory of Fairness, Competition, and Cooperation.” The Quarterly Journal of Economics, 114(3), 817–868.
Bolton, G. E., & Ockenfels, A. (2000). “ERC: A Theory of Equity, Reciprocity, and Competition.” American Economic Review, 90(1), 166–193.
Nowak, M. A., Page, K. M., & Sigmund, K. (2000). “Fairness versus Reason in the Ultimatum Game.” Science, 289(5485), 1773–1775.
Camerer, C. F. (2003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press.
Sanfey, A. G., Rilling, J. K., Aronson, J. A., Nystrom, L. E., & Cohen, J. D. (2003). “The Neural Basis of Economic Decision-Making in the Ultimatum Game.” Science, 300(5626), 1755–1758.
Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., Gintis, H., et al. (2005). “‘Economic Man’ in Cross-Cultural Perspective: Behavioral Experiments in 15 Small-Scale Societies.” Behavioral and Brain Sciences, 28(6), 795–815.
Lin, P.-H., Brown, A. L., Imai, T., Wang, J. T.-Y., Wang, S. W., & Camerer, C. F. (2020). “Evidence of General Economic Principles of Bargaining and Trade from 2,000 Classroom Experiments.” Nature Human Behaviour, 4, 917–927.
Sreedhar, K., & Chilton, L. (2024). “Simulating Human Strategic Behavior: Comparing Single and Multi-Agent LLMs.” arXiv preprint arXiv:2402.08189.
Zheng, G., Zhang, J., Ou, X., Deng, S., & Chen, L. (2025). “Decoding Fairness: A Reinforcement Learning Perspective.” Physical Review Research, 7, 023226.
Araujo, D. K. G., & Uhlig, H. (2026). “How Does AI Distribute the Pie? Large Language Models and the Ultimatum Game.” Becker Friedman Institute Working Paper No. 2026-29.
Copyright 2026 AI-Talks.org
