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# Ultimatum: Strategy Method

## Game Description

The general structure of this game is similar to MobLab's Ultimatum game. Players are placed in pairs: one player is the Proposer, and the other is the Responder. The Proposer proposes a split of \$100. If the Responder accepts the proposal, payoffs are determined by the accepted proposal. If the Responder rejects the proposal, both earn nothing.

However, this game differs from our standard Ultimatum game in three ways:

1. Every player plays both as a Proposer and a Responder.
2. Responders indicate the smallest offer they would accept, rather than accepting or rejecting a known offer.
3. Total payoffs are the sum of payoffs as Proposer and payoffs as Responder. Each player’s offer is matched with every player’s smallest acceptable offer.

### Learning Objective 1: Backward Induction

Under standard assumptions in any subgame-perfect Nash equilibria, the Proposer allocates at most 1 to the Responder, and the Responder accepts all proposals.

### Learning Objective 2: Fairness and Altruism

Preferences for fairness may lead a Responder to reject a proposal she finds unfair, even if in doing so she loses money. Anticipating this, and perhaps also motivated by altruism or fairness, Proposers propose splits more generous than predicted by the subgame-perfect equilibria.

### Learning Objective 3: Offer and Rejection Consistency

It is likely that a relatively high fraction of players will make offers that they would not accept themselves (i.e. offer < smallest acceptable offer).

## Brief Instructions

To have all of your students in one group, make sure Group Size is larger than the number of students. When a group is incomplete (number of students < Group Size), the game ends, and players proceed to the payoff screen, when either the game reaches its Duration or you click Finish. If the number of players in a group equals Group Size, the game automatically ends when all students submit both decisions.

This is a one period game. Under the assumptions of monotonicity, and that a player gains utility only from her own payoff, the Responder accepts all offers greater than 0, and is indifferent between accepting and rejecting an offer of 0. This leads to two qualitatively similar subgame-perfect equilibria. In the first, the Responder rejects an offer of zero and accepts any other offer, and therefore, the Proposer offers 1. In the second, the Responder accepts all offers and the the proposer offers 0.

Your results will likely differ. Many Responders will reject positive, but low, offers. It is common for most offers of less than 25% of the total sum to be rejected. This is consistent with Responders gaining utility from punishing those who make offers which are considered unfair. Almost all Proposers offer a positive amount, often as high as 50%. While some of this may be due to altruism (the Proposer caring about the well-being of the Responder) or a preference for fairness, it is also consistent with anticipating the rejection of low offers.

You may get slightly fewer rejections in this game than in the standard Ultimatum game. First, knowing that she will make choices in both roles, a desire for consistency may cause an increase in offers and a decrease in smallest acceptable offer. Second, punishing a distribution of Proposers who made small offers (by rejecting their offers) may be less satisfying than punishing a particular individual who made a small offer, resulting in a decrease in smallest acceptable offer. Working in the other direction, it is possible that the warm glow from a high offer might be less in this game than in the standard game, where the matched Responder’s payoff depends entirely on the particular Proposer’s offer.

Finally, an individual offering less than she is willing to accept need not be an “irrational’ inconsistency. She may just believe her preferences for fairness differ from others. However, if a high percentage of players display this inconsistency, it may be evidence of either a lack of strategic thought or a systematic misprediction of preferences.

## Results

The results will demonstrate both empirical regularities from the Ultimatum game: the rejection of offers considered unfair, and relatively generous offers by Proposers. In addition, having each Responder’s smallest acceptable offer enables us to calculate the payoff maximizing offer(s), and having each player play in both roles allows enables us to identify those who would not accept their own offer.

The table (Figure 1) summarizes the outcomes by group, displaying average Proposer, Responder, and total payoffs. No rejections result in maximal total payoffs, and 100 minus total payoffs gives the fraction of all offers that are rejected. The last column displays the fraction of players who would not accept their own offers (i.e. offer < minimum acceptable offer).

In the first column is a radio button for each group, allowing you to choose which group’s choices to display in the subsequent charts.

The chart (Figure 2) reproduces the chart available to students at the end of the game. We calculate the resulting payoff, from 0 to 100, for every offer, given the smallest acceptable offers submitted by group members. This table will identify the offer with the highest payoff (even if not chosen by any players), and will show that the payoff-maximizing Proposer does not want to make offers that are too small, and thus, ultimately rejected.

The final chart (Figure 3) shows the distributions of both offers and smallest acceptable offers. By default, both are shown in the same graph. By clicking the checkbox next to the choice’s entry in the legend, you can control which choice is included in the graph.