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The ultimatum game, also known as a take-it-or-leave-it game, is a two-person, one-shot strategic interaction used to study how mutual benefits are shared. In this sequential game, the first player (the Proposer) proposes a division of a 'pie' (e.g., \$100). The second player (the Responder) can then either accept the offer, in which case the pie is split as proposed, or reject it, which results in both players receiving nothing.

Ultimatum Game

The ultimatum game is a widely used experimental tool that has been conducted around the world. Its application spans diverse groups of participants, including students, farmers, warehouse workers, and hunter-gatherers, allowing researchers to study economic behavior across different social and cultural contexts.

Global Application of the Ultimatum Game Across Diverse Groups

The experimental protocol for the ultimatum game begins with explaining the rules to all participants. To ensure impartiality, players are then randomly matched into pairs and do not know the identity of their partner. Within each pair, one person is randomly assigned the role of Proposer and the other becomes the Responder. Anonymity is maintained throughout the interaction. Participants are also aware that all players were recruited through the same process.

Setup of the Ultimatum Game Experiment

A concrete example of the ultimatum game involves a Proposer offering a \$35 share of a \$100 pie to a Responder, while planning to keep the remaining \$65. The offer is final ('take-it-or-leave-it'). Acceptance leads to the proposed \$35/\$65 split. Rejection of the offer results in both the Proposer and the Responder receiving zero.

A \$35 Offer in the Ultimatum Game

In the ultimatum game, if the Responder accepts the Proposer's offer, the resulting payoffs for both players are considered economic rents. This is because their earnings represent a gain over their next best alternative, which is to receive nothing if the offer is rejected and the 'pie' is discarded.

Economic Rent in the Ultimatum Game

In a simplified version of the ultimatum game, the Proposer is limited to two distinct choices, such as a 'fair offer' of $50 or an 'unfair offer' of $20. While a payoff matrix could display the outcomes of this game, a game tree is a more suitable representation because it effectively captures both the sequential order of decisions and the final payoffs for each path of actions.

Simplified Ultimatum Game with Two Offers

In the general version of the ultimatum game, the Proposer is not restricted to a few choices but can offer any amount within a given range, for example, any value between \$0 and \$100. This setup frames the key strategic problems for each player: the Responder must determine their minimum acceptable offer, while the Proposer must decide on the specific amount to propose, balancing potential gains with the risk of rejection.

Strategic Considerations in the General Ultimatum Game

The outcome of an ultimatum game is anticipated to be shaped by the preferences of both the Proposer and Responder, which includes their disposition towards reciprocity, as well as the social norms that are in effect.

Determinants of Ultimatum Game Outcomes

A strike over pay or working conditions provides a real-world example of an ultimatum game. In this context, a labor union (as the Proposer) can issue a demand—an ultimatum—to the employer (the Responder). The employer must then decide whether to accept the terms or reject them. A rejection leads to a strike, which is typically costly for both the union's members (who lose wages) and the employer (who loses revenue), mirroring the outcome where both players in the game receive nothing.

Labor Strikes as a Real-World Ultimatum Game

Ultimatum game experiments provide valuable insights into how individuals choose to divide economic rents in their interactions. By observing the decisions participants make, researchers can investigate their underlying motives and social preferences, including pure self-interest, altruism, inequality aversion, and reciprocity, and understand how these factors guide economic choices.

Using the Ultimatum Game to Study Social Preferences and Rent Sharing

A variation of the ultimatum game can be constructed to study the effects of competition. In this scenario, a Proposer makes an offer for a division of a \$100 pie to two Responders simultaneously, rather than to a single Responder. This setup introduces competition between the Responders, as they must compete to accept the offer.

Ultimatum Game with Competing Responders

The take-it-or-leave-it structure of the ultimatum game grants the Proposer significant bargaining power relative to the Responder. This institutional advantage is a primary reason the Proposer usually receives more than half of the total amount.

The Take-it-or-Leave-it Rule and Proposer's Bargaining Power

While the Proposer holds significant power by setting the offer, their bargaining power is not absolute. The Responder's ability to refuse or veto any offer serves as a crucial constraint, limiting the Proposer's ability to claim the entire pie.

Responder's Veto Power as a Limit on Proposer's Bargaining Power

The hiring process in the labor market can be viewed as an ultimatum game. Business owners or employers act as Proposers, as they are the ones who set the initial offer, including wages and employment conditions. Job applicants, in turn, function as Responders who can either accept or reject these terms.

The Labor Market Hiring Process as an Ultimatum Game

The ultimatum game can be evaluated using the criteria of both substantive and procedural fairness. While the experimental rules are typically seen as procedurally fair, the substantive fairness of an allocation is determined by the actual division of the 'pie'. This depends on the offer made by the Proposer and whether it is accepted by the Responder(s).

Evaluating Substantive and Procedural Fairness in the Ultimatum Game

In a one-shot ultimatum game involving a $100 prize, which of the following scenarios presents the biggest challenge to the assumption that individuals act purely out of self-interest to maximize their own financial gain?

Analyze the Responder's decision in the following scenario. Based on the principles of this interaction, what does the Responder's choice suggest about the motivations that guide human decision-making beyond pure financial self-interest?

Analyzing a Seemingly Irrational Economic Decision

A two-person game is structured as follows: Player 1 (the Proposer) is given a sum of money and must offer a portion of it to Player 2 (the Responder). The Responder can then either accept the offer, in which case the money is split as proposed, or reject it, in which case both players receive nothing. Arrange the following events of a single round of this game in the correct chronological order.

According to a model where individuals are assumed to be perfectly rational and motivated solely by self-interest, a Responder in the ultimatum game should reject any offer they perceive as unfair, even if it is greater than zero.

In a two-person, sequential bargaining scenario, one individual (the 'Proposer') is given $100 and must offer a portion to a second individual (the 'Responder'). The Responder can either accept the division, or reject it, in which case both individuals receive $0. From the Proposer's perspective, analyze the key factors they must consider when deciding what offer to make. Your analysis should explain the trade-off between maximizing their own share and ensuring the offer is accepted.

Strategic Decision-Making in a Bargaining Scenario

Match each role or outcome in the ultimatum game with its corresponding description.

In a bargaining situation, one individual (the 'Proposer') suggests how to divide a fixed sum of money with another individual (the 'Responder'). The Responder can either accept the proposed division or reject it, in which case neither individual receives any money. From the Proposer's perspective, what is the fundamental trade-off they must consider when deciding what offer to make?

Analyzing the Proposer's Strategy in a Bargaining Game

Given the following two scenarios, in which one is the Responder more likely to accept a low offer (e.g., $10 out of $100)? Justify your answer by evaluating the role of perceived intention in the decision-making process.

Evaluating the Impact of Intentionality on Bargaining Outcomes

Given the following scenario, analyze how the change in the game's structure is likely to influence the Proposer's strategy.

Analyzing the Impact of Competition on Bargaining Outcomes

In an experiment, pairs of anonymous participants play a one-time bargaining game. One participant, the Proposer, is given $100 and must offer a portion of it to the second participant, the Responder. If the Responder accepts, the money is split as proposed. If the Responder rejects, both participants receive nothing.

When this experiment was conducted in two different communities, researchers observed the following patterns:

- **Community A:** The most frequent offer made by Proposers was $50. Offers below $30 were rejected more than 50% of the time.
- **Community B:** The most frequent offer made by Proposers was $20. Offers of $20 were almost always accepted.

Based on these results, what can you infer about the likely differences in social norms regarding fairness between these two communities and how those norms influence strategic behavior in this game?

Interpreting Experimental Bargaining Results

In the ultimatum game, the 'rules of the game' or institutions explicitly define the structure of the interaction. These rules determine the total amount to be divided (the pie's size), assign the roles of Proposer and Responder, outline the Proposer's permissible actions (offering any portion of the pie), specify the Responder's choices (to accept or refuse the offer), and dictate the final allocation of the pie based on the players' decisions.

Defining the Rules of the Ultimatum Game

The set of written and unwritten rules on a pirate ship functioned as its governing institutions. These institutions served as the 'rules of the game' by establishing both constraints on behavior, such as the 8 p.m. curfew on drinking below deck, and incentives for performance, like rewarding the lookout who first spotted a prize with the best pair of pistols. In game theory terms, these rules defined the parameters of interaction: they specified who was allowed to do what, the timing of their actions, and how those actions would determine the final payoffs for each crew member.

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