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Players are placed in pairs: one player is the Proposer, the other the Responder. The Proposer proposes a split of a sum of money. If the Responder accepts the proposal, payoffs are determined by the accepted proposal. If the Responder rejects the proposal, both earn nothing.
In our related game—Ultimatum (Strategy method)—each Responder indicates her minimal acceptable offer, giving a more precise measure of Responder preferences. In addition, every player plays both as a Proposer and a Responder, enabling you to identify players who would not accept their own offers.
Learning Objective 1: Backward Induction
Under standard assumptions in any subgame-perfect Nash equilibrium, the Proposer allocates at most 1 (the smallest positive amount) 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 it means losing money. Anticipating this, and perhaps also motivated by altruism or fairness, Proposers propose splits more generous than predicted by the subgame-perfect equilibria.
The default parameters (a one-shot game to split $100) will satisfy a majority of instructors. As Proposers can only choose integer splits, an instructor may want to avoid small values for the amount to be split (Total Pie).
We recommend a one-period game (Periods=1). In discussing a proposed split, we refer to the amount proposed to the Responder as the offer. 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 the Proposer offers 1. In the second, the Responder accepts all offers and the Proposers offers 0.
Your results will likely differ. First, 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 “unfair” offers. Second, almost all Proposers offer a positive amount, often as high as 50%. While some of this may be due to altruism (the proposer cares about the well-being of the responder) or a preference for fairness, it is also consistent with anticipating the rejection of low offers.
The Ultimatum game can be paired with the Dictator game to gain insights into motivations of the proposers. Because all proposals in the Dictator game must be accepted, there is no strategic element in proposer choices. Therefore the difference between offers in the two games is a measure of strategic concerns (as opposed to fairness or altruism) in the Ultimatum game.
The results will demonstrate both empirical regularities: the rejection of unfair offers and relatively generous offers by proposers
Use the Go To menu (Figure 1) to view a different period of a multi-period game. The Results display is divided into two tabs: the Graphs and Tables. Graphs (Figure 2) contain a frequency chart depicting each group's offer distribution, while Tables (Figure 3) contains key statistics about each group's decision making. If you used one of the Replay options, you can use the Comparison button to compare these linked games.
The frequency chart (Figure 2) uses a stacked bar chart to give the distribution of offers to player 2, displaying for each bin the offer frequency with accepted offers stacked on top of rejected offers. Most proposers will offer at least 30% of the total sum, and most responders will reject offers lower than 25%-30% and most will accept more generous offers.
The table (Figure 3) summarizes choices of both player types. In particular, you will likely see a reasonably high fraction of offers rejected, with the average accepted offer significantly higher than the average rejected offer.
While most instructors set Periods=1, you may choose to run a multi-period game to investigate the evolution of play and norms.
If you have played a multi-period game, the results default to the Multi-Period Summary. The table (Figure 4) summarizes for each round accepted and rejected offers. The chart (Figure 5) displays for each round not only average accepted and rejected offers, but also every accepted and rejected offer. Additionally, if you have > 4 data points per accepted/rejected category, a boxplot summarizes the results more informatively. You can use the checkboxes next to the labels in the legend to hide or reveal data categories in order to focus on a particular category across rounds.
After a multi-period game, use the drop-down menu (Figure 6) to display the Summary Table (Figure 3) and Frequency Chart (Figure 2) for a particular period.
Our robot (i.e., an automated player) strategies for ultimatum game roles are as follows:
- Proposer: Robot proposes 40 to 50 percent of their endowment.
- Responder: Robot rejects offers below 40 percent of the endowment.
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