The Pirate-Game Extended
Name Nard Koeman
Student Number 6177646
Study Economics (bachelor)
Field Behavioral Economics and Game Theory
Supervisor dhr. A. (Aaron) Kamm (MSc)
Statement of Originality
This document is written by Nard Koeman who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original, except page 2, and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is
responsible solely for the supervision of completion of the work, not for the contents.
Table of Contents
1. Introduction P. 4
2. Ultimatum-Game P. 6
A. The Rules of the Ultimatum-Game P. 6
B. The Prediction of Game Theory P. 7
C. Results Behavioral Economics P. 8
3. Pirate-Game P. 9
A. The Rules of the Pirate-Game P. 9
B. The Prediction of Game Theory P. 11
4. Experiment P. 13
A. Goal P. 13
B. General Procedures P. 14
C. Predictions and hypotheses P. 15
5. Results and Discussion P. 16
6. Conclusion P. 19
7. Reference list P. 20
8. Appendix P. 21
1. Introduction
The theory of rational choice (RCT) is an important component of many models in traditional economics (Oosterbeek, Sloof et al. 2004). “Briefly, this theory states that a decision-maker chooses the best action according to his preference, among all the actions available to him” (Osbourne 2009, p.4). Although this theory assists in predicting, postdicting and prescribing our daily life, it virtually ignores or rules out all the behavior studied by cognitive and social psychologists (Thaler 2008). An experimental game that stresses the gap between
traditional economics and behavioral economics is called: ‘the Ultimatum-Game’ (UG). The typical1 UG is a 2-player bargaining game in which a proposer is paired with a receiver (Camerer 2003, p. 48). The proposer is endowed with a treasure that he can share with the receiver. The receiver can accept or decline the proposal. If he accepts the offer is distributed as the proposer suggested, but in case the receiver declines both players receive nothing. What is interesting about this game, is that while a money-maximizing receiver2 would accept any positive offer, a human-being could act differently due to its possible influence by concepts like fairness and justice.
Thanks to, inter alia, the UG there now is a considerable amount of experimental evidence indicating that people not only care about their own material well-being but also about the well-being of others (Riedl and Výrašteková 2002). “As always with an anomaly, many tests have been run to see whether the results were robust and not a consequence of a certain experimental environment” (Camerer and Thaler 1995, p. 210).Although the “usual suspects” explanations have been intensively researched, only a few experimental studies extended the UG on more than two players (Riedl and Výrašteková 2002).
The Pirate-Game (PG), which will be discussed in this thesis, is a multiplayer-version of the UG. In this game there is one proposer3 who makes an offer concerning the
distribution of a treasure found by the pirates, a fixed amount of golden coins. The players have to decide individually whether they accept or decline the offer. Each player has a vote, but the captain’s vote is decisive. If the majority of the players decline the offer, the captain is thrown overboard. Henceforth, the next pirate in line, ordered by age, becomes captain and therefore responsible for the subsequent suggestion of the redistribution of the
1 See Camerer (2003 pp.50-55) for an extended overview of all the variants of the ultimatum game that have been played.
2 Terms such as subordinate, receiver, player 2 and responder will be used interchangeably. 3 Terms such as oldest pirate, proposer, player 1 and captain will be used interchangeably.
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treasure for himself and the remaining subordinates. The rounds continue until an offer is accepted or when there are only two players left, considering that at this stage the
suggestion will always be enforced (Talbot Coram, 1998, pp. 99-100).
This bachelor thesis will provide an addition to the conducted research on the Ultimatum-Game by approaching it in two different ways. First of all, I will use rational-choice-theory as an assumption to analyze the game-theoretical-model of any multiplayer-variant of the game. Second of all, I will run a 5-player lab-in-the-field-experiment with 90 active4 players. To my knowledge both approaches to the UG have never been conducted before. However, the experiment is interesting for two primary reasons. Firstly, It is arguable that the rational outcome is not equivalent to the one played. Secondly and of more importance, it can be used to answer the main question: to what extent is the behavior of the proposer in an UG influenced by the number of receivers?
To answer this question I first briefly explain the rules of the UG and PG in a more mathematical way. In both cases I will present the money-maximizing-hypothesis that will be used as a benchmark to make sure that there is no rational reason to explain a change in behavior5. Finally I will conclude with a comparison between existing behavioral literature on the UG and the results from my own lab-in-the-field-experiment.
Mathematical Outline 𝑔𝑔 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑡𝑡ℎ𝑁𝑁 𝑡𝑡𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁 𝑔𝑔 → { 𝑔𝑔 ∈ ℤ ∥ 0 ≤ 𝑔𝑔 ≤ ∞ } 𝑐𝑐 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁𝑐𝑐 𝑐𝑐 → { 𝑐𝑐 ∈ ℤ ∥ 0 ≤ 𝑐𝑐 ≤ ∞ } 𝑊𝑊 = 𝑇𝑇ℎ𝑁𝑁 𝑡𝑡𝑁𝑁𝑜𝑜𝑁𝑁𝑐𝑐𝑡𝑡 𝑜𝑜𝑜𝑜 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜𝑜𝑜𝑁𝑁𝑁𝑁𝑐𝑐 𝑡𝑡ℎ𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁(𝑐𝑐) 𝑊𝑊 → { 𝑊𝑊 ∈ ℤ ∥ 0 ≤ 𝑊𝑊 ≤ ∞ } 𝑡𝑡𝑐𝑐𝑎𝑎0 ≤ 𝑊𝑊 ≤ 𝑔𝑔 𝑍𝑍 = 𝐼𝐼𝑐𝑐 𝑁𝑁𝑐𝑐𝑡𝑡ℎ𝑁𝑁𝑁𝑁 𝐴𝐴 𝑜𝑜𝑁𝑁 𝐷𝐷 𝐴𝐴 = 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡𝑐𝑐 𝑡𝑡ℎ𝑁𝑁 𝑜𝑜𝑜𝑜𝑜𝑜𝑁𝑁𝑁𝑁 𝑜𝑜𝑁𝑁𝑜𝑜𝑁𝑁 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 𝐷𝐷 = 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁 𝑎𝑎𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁𝑐𝑐 𝑡𝑡ℎ𝑁𝑁 𝑜𝑜𝑜𝑜𝑜𝑜𝑁𝑁𝑁𝑁 𝑜𝑜𝑁𝑁𝑜𝑜𝑁𝑁 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 ∅ = 𝐸𝐸𝑁𝑁𝑝𝑝𝑡𝑡𝑝𝑝 ℎ𝑐𝑐𝑐𝑐𝑡𝑡𝑜𝑜𝑁𝑁𝑝𝑝 i = 𝑈𝑈𝑐𝑐𝑐𝑐𝑈𝑈𝑁𝑁𝑁𝑁 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑐𝑐 → { 𝑐𝑐 ∈ ℤ ∥ 1 ≤ 𝑐𝑐 ≤ 𝑐𝑐 } 𝑡𝑡 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡 𝑁𝑁𝑜𝑜𝑁𝑁𝑐𝑐𝑎𝑎 𝑈𝑈 = 𝐴𝐴𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁 𝑝𝑝𝑁𝑁𝑟𝑟𝑁𝑁𝑝𝑝 q→ { 𝑈𝑈 ∈ ℝ ∥ 0 ≤ 𝑈𝑈 ≤ 1 } 𝑝𝑝 = 𝑜𝑜𝑡𝑡𝑐𝑐𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡 𝑎𝑎𝑐𝑐𝑐𝑐𝑡𝑡𝑁𝑁𝑐𝑐𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 𝑜𝑜 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑡𝑡ℎ𝑡𝑡𝑡𝑡 𝑡𝑡𝑁𝑁𝑝𝑝𝑝𝑝𝑐𝑐 𝑁𝑁𝑐𝑐 ℎ𝑜𝑜𝑜𝑜 𝑜𝑜𝑡𝑡𝑐𝑐𝑁𝑁 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 𝑐𝑐𝑐𝑐
4 I use this term because in some studies there exist an inactive dummy-player (Güth and Van Damme 1998). 5 I have to make sure that a change in the methodological variables does not alter the unique SPNE.
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2. The Ultimatum-Game
A. The Rules of the Ultimatum-Game (UG)
The one-shot UG is an extensive game where 2-players are bargaining over the division of 𝑔𝑔-coins. The proposer has to offer the receiver a number of coins (𝑊𝑊) where 0 ≤ 𝑊𝑊 ≤ 𝑔𝑔. If the receiver accepts (𝐴𝐴) this offer, both players will receive the proposed division. This means that the proposer gets (𝑔𝑔 − 𝑊𝑊) and the receiver gets (𝑊𝑊). However if the receiver declines (𝐷𝐷) the offer, neither will receive a pay-off (Osbourne 2009).
Players: There are 2-players.
Terminal Histories: The set of sequences (𝑊𝑊, 𝑍𝑍)
𝑊𝑊 represents the amount of coins the proposer offers the receiver.
𝑍𝑍 represents the action of the receiver and can be either (𝐴𝐴) or (𝐷𝐷).
Payoff Functions: ⎩ ⎪ ⎨ ⎪ ⎧ 𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 � 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 2𝑐𝑐𝑎𝑎 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁 𝑎𝑎𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁𝑐𝑐: 0 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 2𝑐𝑐𝑎𝑎 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡𝑐𝑐: 𝑔𝑔 − 𝑊𝑊 0 𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑅𝑅𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁 � 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 2𝑐𝑐𝑎𝑎 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡𝑐𝑐: 𝑊𝑊 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 2𝑐𝑐𝑎𝑎 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁 𝑎𝑎𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁𝑐𝑐: 0
Figure 1. A graphical representation of the UG.
Player function: 𝑃𝑃 (∅) = Proposer
𝑃𝑃 (𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐) = Receiver
For all 𝑊𝑊
B. The Prediction of Game Theory
How to find the Subgame Perfect Nash-Equilibrium (SPNE)
Assuming that both players are strictly self-interested and try to optimize their share, the SPNE can be found by using backward induction. Hence, the solving of this game starts at point 2 (see figure 1.). Here the receiver has to either accept (A) or decline (D) the proposal made by the proposer. If the receiver truly only cares about the amount of coins that he receives he will always accept any non-zero offer considering that allows for a higher level of utility in comparison to receiving nothing.
� 𝑊𝑊 > 0 𝑇𝑇ℎ𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑡𝑡𝑝𝑝𝑝𝑝𝑜𝑜𝑡𝑡𝑝𝑝𝑐𝑐 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡 (𝐴𝐴) 0
𝑊𝑊 = 0 𝑇𝑇ℎ𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑁𝑁𝑁𝑁 𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐𝑜𝑜𝑜𝑜𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡 𝑁𝑁𝑁𝑁𝑡𝑡𝑜𝑜𝑁𝑁𝑁𝑁𝑐𝑐 (𝐴𝐴) 𝑡𝑡𝑐𝑐𝑎𝑎 (𝐷𝐷)
To maximize his profit the proposer will decide to offer the lowest possible positive share ( lim𝑤𝑤 → 0 𝑊𝑊) to ensure him from acceptance (𝐴𝐴). This will result in the following payoffs:
�
𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 1𝑐𝑐𝑡𝑡 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁: ( 𝑔𝑔 − ( lim𝑤𝑤 → 0 𝑊𝑊)) − 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐
0
𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 2𝑐𝑐𝑎𝑎 𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝𝑁𝑁𝑁𝑁: ( lim𝑤𝑤 → 0 𝑊𝑊) − 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐
Other Nash-Equilibrium (NE)
There are also Nash-Equilibria that are not subgame perfect. These occur when the proposer makes a positive offer that is equal to the Minimum Acceptable Offer (MAO)6 of the receiver. If the proposer would make an offer which is higher than the MAO of the receiver, he could be better off by lowering his bid. If he makes an offer which is lower than the MAO of the receiver, his offer will be rejected. Therefore, proposing the amount equal to the MAO of the receiver is a NE. These Nash-Equilibria are based on incredible threats because if the proposer would offer a positive offer that is lower than the MAO of the receiver, it would still be in his best interest to accept the offer.
6 The purpose of this paper is not to create more clarity concerning Minimal Acceptable Offers. However If you are interested in this topic I recommend to read Camerer (2003, p.49). A MAO is the minimum amount of coins a subject wants to have in order to accept the proposal. All lower offers will be rejected.
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C. Results Behavioral Economics
The sharp prediction of the SPNE (Osbourne 2009, p.183), the simplicity of the game-theoretical-model (Camerer 2003, p.43) and the results, that are considered to be
inconsistent with the neo-classical-paradigm (Camerer and Thaler 1995, p.210), made the UG one of the most famous games in behavioral game theory. This section discusses some of the existing behavioral literature on the UG.
The first time that the UG was studied experimentally was in the late 1970s, by Güth, et al. They used 21-pairs of students to test whether subjects would behave according to the RCT and in case they did not, in which direction they would deviate (1982, p. 367). In the first round, the mean offer of the proposers was about 40% and one positive offer was declined. In the second round the mean offer had decreased to about 30% and five positive offers were declined. Both players thus take actions that are inconsistent with RCT7.
In his paper “Anomalies: the ultimatum game”, Thaler discusses why some receivers might decline a positive offer. He concludes that notions of fairness can play a significant role in determining the outcomes of negotiations (Thaler 1998, p. 205). When a responder is offered a low amount, he might feel offended and therefore decline8. Kirchsteiger agrees and states that the behavior of proposers is being influenced by their fear of having the offer declined by these envious responders (1994, p. 379).
Considering that in virtually every experiment all subjects were part of a student-population, it was not clear whether the observed deviations can provide evidence of universal patterns or that it was the individual’s economic and social environment that shaped the behavior of the participant. Henrich et al. therefore conducted a large cross-cultural behavioral study among 15 small-scale societies. They concluded that, although there was more behavioral variability than assumed by earlier cross-cultural studies, in none of the studied societies RCT was supported (2001, p.73).
Camerer states that the results are very consistent. “Modal and median ultimatum offers are usually 40-50 percent and means are 30-40 percent. There are hardly any offers in the outlying categories of 0, 1-10 and hyper-fair category 51-100.” (Camerer 2003, p. 49)
7 When conducting the experiment, if the outcome is rational, two observations should be made:
proposers should make offers in the proximity of zero and receivers should accept all offers greater than zero (Rubinstein: Thaler, 1988, pp. 196-197).
8 How would you feel when you are invited to a birthday-party but the birthday-boy eats all the cake?
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3. The Pirate-Game
A. The Rules of the Pirate-Game (PG)
The one-shot PG is an extensive game where 𝑐𝑐-players are bargaining over the division of 𝑔𝑔-coins. The proposer has to offer each receiver an individual number of coins (𝑊𝑊i) where 0 ≤ (∑.𝑊𝑊i) ≤ 𝑔𝑔 and 0 ≤ 𝑊𝑊i. The players individually and simultaneously decide on 𝑍𝑍i and wait for the responses to be collected. Then the supervisor calculates 𝑈𝑈 and informs all players whether the proposal is implemented or not using the following two formulas.
𝑈𝑈 = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑝𝑝𝑐𝑐𝑁𝑁𝑡𝑡𝑡𝑡𝑁𝑁𝑐𝑐 𝑡𝑡ℎ𝑡𝑡𝑡𝑡 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐𝑇𝑇𝑜𝑜𝑡𝑡𝑡𝑡𝑝𝑝 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑝𝑝𝑐𝑐𝑁𝑁𝑡𝑡𝑡𝑡𝑁𝑁𝑐𝑐 𝑜𝑜ℎ𝑜𝑜 𝑡𝑡𝑁𝑁𝑁𝑁 𝑡𝑡𝑝𝑝𝑐𝑐𝑟𝑟𝑁𝑁
𝑈𝑈 � 𝑈𝑈 > .5 𝑇𝑇ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑡𝑡𝑝𝑝 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑁𝑁𝑁𝑁 𝑐𝑐𝑁𝑁𝑝𝑝𝑝𝑝𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡𝑁𝑁𝑎𝑎 𝑈𝑈 = .5 𝑇𝑇ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁𝑐𝑐 𝑐𝑐𝑡𝑡𝑐𝑐𝑡𝑡𝑐𝑐𝑐𝑐𝑔𝑔 𝑟𝑟𝑜𝑜𝑡𝑡𝑁𝑁 𝑐𝑐𝑜𝑜𝑁𝑁𝑐𝑐𝑡𝑡𝑐𝑐 𝑈𝑈 < .5 𝑇𝑇ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑡𝑡𝑝𝑝 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑐𝑐𝑜𝑜𝑡𝑡 𝑁𝑁𝑁𝑁 𝑐𝑐𝑁𝑁𝑝𝑝𝑝𝑝𝑁𝑁𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡𝑁𝑁𝑎𝑎
If the majority of the players decline the offer, the captain is thrown overboard. Henceforth, the next pirate in line, ordered by age9, becomes captain and therefore responsible for the subsequent suggestion of the redistribution of the treasure for himself and the remaining subordinates. The rounds continue until an offer is accepted.
Players: There are n-players.
1 of them is the captain.
n-1 of them are subordinates.
Terminal Histories: The set of outcomes (𝑊𝑊i, 𝑍𝑍i)where (𝑊𝑊i) represents the proposal the proposer gives to player 𝑐𝑐 and (𝑍𝑍i) whether player 𝑐𝑐 accepts (A)
or declines (D).
9 Before the game starts all players are ordered by their age:P
(1) < P(2) < …. < P(n-1) < P(n) where P(1) is the youngest player and P(n) the oldest. The oldest pirate (proposer) that is alive, makes the proposal and has the casting vote in case 𝑈𝑈 = .5.
9
Player function: 𝑃𝑃 (∅) = 𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 𝑃𝑃 (𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐) = 𝑐𝑐 − 1 = 𝑆𝑆𝑁𝑁𝑁𝑁𝑜𝑜𝑁𝑁𝑎𝑎𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑁𝑁𝑐𝑐 For all 𝑊𝑊𝑐𝑐. Payoff Functions: ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎧ 𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 � 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 𝑐𝑐𝑁𝑁𝑁𝑁𝑟𝑟𝑐𝑐𝑟𝑟𝑁𝑁𝑐𝑐 ℎ𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑔𝑔𝑁𝑁𝑡𝑡: 𝑔𝑔 − ∑ 𝑊𝑊𝑐𝑐(𝑡𝑡) 0 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑁𝑁𝑁𝑁 𝑎𝑎𝑜𝑜𝑁𝑁𝑐𝑐𝑐𝑐′𝑡𝑡 𝑐𝑐𝑁𝑁𝑁𝑁𝑟𝑟𝑐𝑐𝑟𝑟𝑁𝑁 ℎ𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑔𝑔𝑁𝑁𝑡𝑡: 0 0 𝑃𝑃𝑡𝑡𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜 𝑆𝑆ubordinates ⎩ ⎪ ⎨ ⎪ ⎧ 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑡𝑡𝑝𝑝 𝑐𝑐𝑐𝑐 𝑡𝑡𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡𝑁𝑁𝑎𝑎 ℎ𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑔𝑔𝑁𝑁𝑡𝑡: 𝑊𝑊𝑐𝑐(𝑡𝑡) 0 𝐼𝐼𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑃𝑃𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑡𝑡𝑝𝑝 𝑐𝑐𝑐𝑐 𝑎𝑎𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁𝑎𝑎: ⇊ 1. 𝑇𝑇ℎ𝑁𝑁 𝑜𝑜𝑝𝑝𝑎𝑎𝑁𝑁𝑐𝑐𝑡𝑡 𝑐𝑐𝑁𝑁𝑁𝑁𝑜𝑜𝑁𝑁𝑎𝑎𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑔𝑔𝑁𝑁𝑡𝑡: 𝑔𝑔 − ∑𝑛𝑛 . 𝑖𝑖=1 �𝑊𝑊𝑐𝑐(𝑡𝑡 + 1)� 2. 𝑇𝑇ℎ𝑜𝑜𝑐𝑐𝑁𝑁 𝑜𝑜ℎ𝑜𝑜 𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑐𝑐 𝑐𝑐𝑁𝑁𝑁𝑁𝑜𝑜𝑁𝑁𝑎𝑎𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑁𝑁 𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝 𝑔𝑔𝑁𝑁𝑡𝑡: �𝑊𝑊𝑐𝑐(𝑡𝑡 + 1)�
All players get utility from seeing another pirate walking the plank but given the choice, each of them would always prefer to receive a part of the treasure. The proposer will always prefer survival over any positive payoff.
𝑡𝑡 is a variable that represents the number of the current round. For example: assume that we are in the second round. In case of acceptance plugging in 𝑡𝑡 = 2 leads to conclude that the proposer remains with the treasure minus what he promised the subordinates in the second round. In case of decline plugging in 𝑡𝑡 = 2 leads to conclude that the proposer remains with nothing. The payoff the subordinates will receive depends on next round (𝑡𝑡 + 1) and the order of age.
B. The Prediction of Game Theory How to find the SPNE.
Assuming that every player is strictly self-interested and tries to optimize their share, the SPNE can be found (using backward induction) for any number of pirates (𝑐𝑐) and for any number of coins (𝑔𝑔). I will first discuss how the SPNE can be found for (𝑐𝑐)-players and later on as an extension I will discuss how one can find the SPNE for (𝑔𝑔)-coins for completeness. The SPNE for (𝒏𝒏)-players
Assume that there are only two pirates in the game; P(1) and P(2). To maximize his profit the proposer, P(2) , will take the entire treasure (𝑔𝑔) leaving the receiver, P(1) , with nothing. Just as previously discussed in the UG the receiver will be indifferent, however this will not matter because the proposer will always accept, resulting in 𝑈𝑈 ≥ 0.5.
Now consider a situation where there are three pirates; P(1) , P(2) and the proposer P(3). The proposer considers the situation where he is already thrown overboard. He knows that in that case P(1) will be left with 0-coins. To buy the loyalty of P(1) the proposer offers him 1-coin that allows him for a higher level of utility in comparison to receiving nothing. Therefore both P(1) and P(3) will accept the proposal, resulting in implementation. P(3) will offer 0-coins to P(2) because his vote does not matter anymore. With the same reasoning the
proposer can figure out the SPNE for any number of pirates. In figure 2. this is shown.
Figure 2. Through backward induction the proposer can figure out what to propose.
The horizontal axis shows all pirates (P(1) till P(n)) in order of age, and the vertical axis shows the number of rounds. The table displays the corresponding payoff per player and round, according to the SPNE. For example: assume that the game starts with 5 pirates. Plugging in n=5 leads to conclude that the bottom round in this figure, n-4 is the first one. In this case the optimal play by the oldest pirate P(5) entails taking g-2 for himself, while bribing P(1) and P(3) with one coin (each) and offering nothing to P(2) and P(4).
The SPNE for (𝒈𝒈)-coins10 (extension)
To find the SPNE for 𝑔𝑔-coins we have to make a distinction between two cases.
1. The treasure (g) is ‘large’ enough to bribe the majority of the pirates.
What is ‘large’ enough? � 𝑝𝑝𝑡𝑡𝑁𝑁𝑔𝑔𝑁𝑁 𝑁𝑁𝑐𝑐𝑜𝑜𝑁𝑁𝑔𝑔ℎ 𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑐𝑐: 𝑔𝑔 ≥ ��(𝑛𝑛−2)2 ��
The SPNE can be found using the same method as previously described11. a. 𝑐𝑐 is an even integer and 𝑐𝑐 > 0
P(n) (accept), P(n-1) (decline), P(n-2) (accept), P(n-3) (decline), … etc.
In this N.E. all pirates who have an even number will accept and all pirates
who have an uneven number will decline.
b. 𝑐𝑐 is an uneven integer and 𝑐𝑐 ≥ 1
P(n) (accept), P(n-1) (decline), P(n-2) (accept), P(n-3) (decline), … etc.
In this N.E. all pirates who have an even number will decline and all pirates
who have an uneven number will accept.
2. The treasure (g) is ‘not large’ enough to bribe the majority of the pirates.
What is ‘not large’ enough? � 𝑐𝑐𝑜𝑜𝑡𝑡 𝑝𝑝𝑡𝑡𝑁𝑁𝑔𝑔𝑁𝑁 𝑁𝑁𝑐𝑐𝑜𝑜𝑁𝑁𝑔𝑔ℎ 𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑐𝑐: 𝑔𝑔 < ��(𝑛𝑛−2)2 ��
To test whether an insufficiently large g always causes the oldest pirate to have to walk the plank, we use an applied example which will later be generalized. Let us assume that there are 6 pirates who find a treasure existing out of 1 coin.If there are...
2 pirates the oldest pirate P(2) will take the coin (g ).
3 pirates the oldest pirate P(3) will bribe the youngest P(1) and gets to live another day. 4 pirates the oldest pirate P(4) will bribe P(2) and gets to live another day.
5 pirates the oldest pirate P(5) will be thrown overboard
6 pirates the oldest pirate P(6) will get the vote of P(5) because he knows he will die if he declines the proposal. With g P(6) is able to bribe another pirate (P(1) or P(3)) and get an 𝑈𝑈 = 0.5 Because of the casting vote P(6) and P(5) will survive. 10 All formulas in this section will be derived in appendix 1 and 2
11
This is similar to finding the SPNE for (𝑐𝑐)-players 12
In the previous example we saw that for 𝑔𝑔 = 1 there exist a possibility that 6 pirates might survive which might look counterintuitive. How do we know if we start with 𝑐𝑐 pirates and a treasure that exists out of 𝑔𝑔 coins how many pirates will stay alive?
1. 𝑥𝑥 𝑐𝑐𝑐𝑐 𝑡𝑡 𝑐𝑐𝑡𝑡𝑡𝑡𝑁𝑁𝑁𝑁𝑡𝑡𝑝𝑝 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 ≥ 0
2. 𝑀𝑀𝑡𝑡𝑥𝑥𝑐𝑐𝑁𝑁𝑐𝑐𝑀𝑀𝑁𝑁 𝑥𝑥 𝑁𝑁𝑐𝑐𝑎𝑎𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑡𝑡𝑁𝑁𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 𝑜𝑜𝑜𝑜: 2𝑥𝑥+ 2𝑔𝑔 ≤ 𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑔𝑔𝑠𝑠𝑔𝑔𝑔𝑔 3. 𝑇𝑇ℎ𝑁𝑁 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑐𝑐𝑁𝑁𝑁𝑁𝑟𝑟𝑐𝑐𝑟𝑟𝑐𝑐𝑐𝑐𝑔𝑔 𝑝𝑝𝑐𝑐𝑁𝑁𝑡𝑡𝑡𝑡𝑁𝑁𝑐𝑐: 2𝑥𝑥+ 2𝑔𝑔 = 𝑐𝑐𝑔𝑔𝑛𝑛𝑒𝑒 𝑔𝑔𝑠𝑠𝑔𝑔𝑔𝑔 Among the surviving pirates the treasure will be distributed.
1. Some pirates, who are certain of not having to walk the plank, will accept the proposal because they are being bribed. If x is an uneven number the proposer will bribe only pirates who have an even number. If x is an even number he will bribe only pirates who have an uneven number.
2. Pirates who are certain of death in case they do not accept will accept any non- negative proposal.
4. Experiment A. Goal
All subjects took part in a multiplayer variant of the Ultimatum-Game. The main goal of this experiment is to establish a baseline that tells us how much a proposer is willing to give when there are 5-players. Later on this percentage will be compared to the existing behavioral-literature in a 2-player Ultimatum-Game.
B. General Procedures12
For the experimental-design of the Pirate-Game about 250 subjects were notified that they could actively participate in my bachelor-thesis by playing a Pirate-Game13. Out of these 250
subjects 96 were willing to help me and participate voluntarily. I was able to make 18 groups that consisted out of 5 players each who played the game at multiple different locations. All
12 The general procedures are inspired by an article called ‘exploiting moral wiggle room’ by Dana and Weber. 13 Because of the complexity of this one-shot-game I decided to use the Pirate-Game-context in order to make sure that the subjects understood the rules.
13
experiments were run with 5 players present and me supervising the procedure. To my knowledge none of the participants had any background in game theory.
Upon arriving at the experiment all subjects were seated at a table14, in order of age, with an instruction-sheet15 upside down. Before the game started each of them were given 5 minutes to read the instruction-sheet, that was also read out loud. After this was
completed people got the opportunity to ask questions and were given a short quiz
containing three questions to ensure that they understood the game. The instruction sheet can be found in appendix 3, the small quiz can be found in appendix 4 and the answer sheet can be found in appendix 5.
First, the oldest pirate (participant) proposes how the treasure should be split between the five group members. Each participant is informed of the total distribution that is proposed, so each participant knows how much everyone individually in the group is offered. Next, participants write down whether they accept or decline the share that was proposed to them. This is done individually, unknowing the others’ choices. Finally, the experimenter collects the chosen moves, and notifies the group whether the split is accepted or not.
It is chosen to inform all players of the offer made in order to make the game more realistic, since in a company it is likely that employees are aware of the remuneration of their colleagues. Furthermore it allows to test whether feelings of injustice, fairness and the relative size of ones’ own share compared to the others’ plays a part in accepting or
declining the proposal. Hopefully this way more interesting behavioral facts will appear in the results.
If the proposed split is accepted by the group, the game is over and every participant receives16 the proposed share of the treasure. In case the proposed split is declined, the oldest participant is removed from the game without any share of the treasure. The next round is then played with the remaining four participants and this procedure will continue until a proposal is accepted or only two participants are left, since in that round the will of the player with the casting vote will be implemented.
14 Subjects did not play the game anonymously. I will discuss this later on in chapter 6 15The story on this instruction-sheet was written by Ian Steward.
16 Although there were no incentives in my game every subjects automatically participates in a lottery. The price of this lottery is a ‘nationale bioscoopbon’. Of course everybody also earns my eternal gratitude. The odds of winning the price did not depend on the earnings in the experiment (Pwin = 1/90). The number of the winning participant was drawn with the help of https://www.random.org/ and is called S. Scheepjens.
14
C. Predictions and hypotheses
The previously discussed literature on the UG reported robust empirical evidence that players take actions that are considered inconsistent with RCT. I have no reason to expect that a change in the game-theoretical-model would lead to different results17. To verify this I will look in my own data if the money-maximizing-hypothesis and thus the SPNE occur. Secondly and of more importance, I will compare the results that Camerer published with the results from my lab-in-the-field-experiment to test whether a proposer behaves more fair in a 2-player UG than in a 5-player PG or not.The alternative hypotheses doesn’t have a direction.
H0: The proposer plays just as fair in a PG as in an UG H1: The proposer does not play as fair in a PG as in a UG
I cannot say anything about this hypothesis without having a definition of ‘fair’. I think the fairest offer a proposer can make is when he offers the same amount (𝑝𝑝) to all players. To establish a number (𝑜𝑜) that tells us how fair a proposal is I will divide the offer the proposer made (𝑊𝑊i) with the fair-distribution (𝑝𝑝). The letter 𝑐𝑐 refers to an unique player number. We need this because every participant can be offered a different amount of coins. The 3 formulas will look as follows:
𝑝𝑝 =𝑔𝑔𝑛𝑛 𝑜𝑜𝑐𝑐 =𝑊𝑊𝑖𝑖𝑦𝑦 𝑜𝑜𝑐𝑐 ⎩ ⎨ ⎧ 𝑜𝑜𝑐𝑐 ~ (1, ∞) 𝐻𝐻𝑝𝑝𝑝𝑝𝑁𝑁𝑁𝑁 − 𝑜𝑜𝑡𝑡𝑐𝑐𝑁𝑁18 𝑜𝑜𝑐𝑐 = 1 𝑃𝑃𝑁𝑁𝑁𝑁𝑜𝑜𝑁𝑁𝑐𝑐𝑡𝑡 − 𝑜𝑜𝑡𝑡𝑐𝑐𝑁𝑁 𝑜𝑜𝑐𝑐 ~ (0, 1) 𝑁𝑁𝑜𝑜𝑡𝑡 − 𝑜𝑜𝑡𝑡𝑐𝑐𝑁𝑁 𝑜𝑜𝑐𝑐 = 0 𝑅𝑅𝑅𝑅𝑇𝑇
17 Bounded rationality, emotions, mistakes and limited information can still occur for every player. Also the unique SPNE has not been altered.
18 Although I’m aware that ‘hyper-fair’ is not necessarily the best word for this I still choose to use it. Although a hyper-fair offer is not necessarily hyper-fair for all the receivers it is for receiver 𝑐𝑐.
15
So how fair does one play in a typical 2-player19 UG? (Camerer 2003, p. 49) 1. Camerer states that in an UG mean offers are usually 30-40 percent. → Therefore the mean of 𝑜𝑜 ~ [ 3050,4050 ]
2. Camerer states that in an UG modal/median offers are usually 40-50 percent. → Therefore the modal and median of 𝑜𝑜 ~ [ 4050,5050 ]
3. Camerer states that there are hardly any offers in the outlying categories → Therefore 𝑜𝑜𝑐𝑐 > 1 and 𝑜𝑜𝑐𝑐 = 0 should hardly occur.
Because 𝑜𝑜 is measured relatively we should find similar results in both games if the proposer plays just as fair in the PG as in the UG. However if the fairness-level of the proposer was influenced by the number of active receivers we should find that 𝑜𝑜 will differ in both games. If 𝑜𝑜 is not equal in both games I will reject H0.
5. Results and Discussion
In this bachelor thesis I investigated behavior in a 5-player ultimatum-game where 4 active-receivers independently and simultaneously decide to accept or decline an offer made by a proposer. The SPNE equilibrium entails that the proposer will take 98 coins for himself, while bribing P(1) and P(3) with one coin (each) and offering nothing to P(2) and P(4). SPNE: (98-0-1-0-1)
Figure 3. The proposal that was made in every round followed by the decisions made ( 𝑍𝑍𝑐𝑐 ) All numbers in this figure are rounded up to the nearest integer.
19
In any 2-player game the fair offer would be 𝑝𝑝 = 50 percent. 16
Figure 3. indicates that there is strong evidence that proposers (most left column) try to exploit their bargaining power. In most groups studied the proposer granted himself more coins than that he offered to the receivers. More than 85% of the oldest-subordinates rejected the proposal, which might indicate that they prefer to have this bargaining power for themselves. In this sample subjects who are lower-ranked had a higher acceptance level compared to their higher-ranked subordinates. Although RTC predicts that all groups should finish in the first round, empirical results show that some groups took up to three rounds to come to an agreement. These results are not in accordance with the theorem of rationality and therefore the outcome is not that of backward induction.
Another nice game that empathizes the importance of the debate over the usefulness of idealizations such as backward induction is called: “the centipede game” (Aumann 1995, p.7). Although in this game backward induction gives a clear prediction that the first player will defect on the first round it fails to predict human play. In this game it might be advantageous (and rational) to cooperate in the initial rounds because you think your opponent has not reasoned completely through the backward induction. Or maybe your limited rationality prevents you from doing that yourself. Similar to the results of the centipede game, in my data set it seems that in the later rounds people seem to correspond more to backward induction. Maybe they feel the end is near.
In accordance with earlier literature on the UG, figure 4 shows that behavior that is considered hyper-fair or consonant with RTC hardly occurred in the PG20. In this sample hyper-fair behavior only appeared twice. In both cases the proposer successfully tried to bribe the loyalty of a subject in order to get the proposal accepted, leaving the remaining subjects with nothing.
In all rounds the average of 𝑜𝑜𝑐𝑐 (see figure 5.) was relatively similar to previously published literature on the UG21. However there is a small deviation from prediction in the first round. In this round the mode and median are not in the right interval (𝑜𝑜 ~ [ 45,55 ]). A logical explanation would be that the majority of the subjects found it easier to calculate with the number 15 (𝑜𝑜= .75) than with the numbers 16,17,18 or 19 which would result in the predicted value for 𝑜𝑜.
20
See figure 4. where 𝑜𝑜𝑐𝑐 > 1 and 𝑜𝑜𝑐𝑐 = 0 are not common. 21 In every round the mean was contained in the interval 𝑜𝑜 ~ [ 3
5, 4 5 ]. 17
Figure 4. For every 𝑊𝑊𝑐𝑐 there is a corresponding 𝑜𝑜𝑐𝑐 Figure 5. Some statistical data on figure 4.
Furthermore for this analysis I thought about Fehr and Schmidt (1999, p. 817) who wrote that if some people might care about equity this can drive the results. Similar to their results it turns out that in my results economic environment determines whether the fair or selfish types dominate equilibrium behavior. For example in ‘Game A’ selfish types dominated while in ‘Game B’ fair types dominated.
In all groups subjects knew the identity of the people they were bargaining with, and maybe even more important they knew the identity of the supervisor (me). This is a
limitation that could influence the results and therefore ought to be discussed. Although I’m aware that anonymity can guarantee that there will be no reputational consequences to behavior I was forced, by limited resources and time, to make groups that were familiar to each other. What worries me more is that subjects might adjust their behavior because they wanted to help me find the results that I was looking for22. Although all subjects were asked to play the game seriously and not aware what I was looking for, I cannot eliminate this possibility. Therefore I propose for further studies on the PG to use subjects who are not familiar with the supervisor to guarantee anonymity.
22 Especially the combination between ‘no anonymity’ and ‘low incentives’ can cause problems.
18
6. Conclusion
In this bachelor thesis I investigated behavior in a 5-player ultimatum-game where 4 active-receivers independently and simultaneously decide to accept or decline an offer made by a proposer. The results of the lab-in-the-field-experiment were fully in line with my
expectations. In none of the groups studied the Subgame-Perfect-Nash-Equilibrium occurred; leading us to believe that the money-maximizing-hypothesis is systematically violated. I would like to stress that: although my results show empirical evidence that people do not act as traditional economics predicts, I do not imply that economists should abandon rational choice theory. Furthermore in this sample there is no evidence that the behavior of the proposer can be considered less fair in the PG than that the previous literature on the UG predicted.
7. Reference list
Aumann, R. J. (1995). Backward induction and common knowledge of rationality. Games and Economic Behavior, 8(1), 6-19.
Baltz, K. (2004). Colin Camerer: Behavioral game theory, experiments in strategic interaction.
Politische Vierteljahresschrift, 45(3), 446-449.
Camerer, C., & Thaler, R. H. (1995). Anomalies: Ultimatums, dictators and manners.
The Journal of Economic Perspectives, 209-219.
Dana, J., Weber, R. A., & Kuang, J. X. (2007). Exploiting moral wiggle room: experiments demonstrating an illusory preference for fairness. Economic Theory, 33(1), 67-80.
Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. Quarterly journal of Economics, 817-868.
Güth, W. (1995). On ultimatum bargaining experiments—A personal review.Journal of Economic Behavior &
Organization, 27(3), 329-344.
Güth, W., & Van Damme, E. (1998). Information, strategic behavior, and fairness in ultimatum bargaining: An experimental study. Journal of Mathematical Psychology, 42(2), 227-247.
Güth, W., Schmittberger, R., & Schwarze, B. (1982). An experimental analysis of ultimatum bargaining. Journal of economic behavior & organization, 3(4), 367-388.
Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E., Gintis, H., & McElreath, R. (2001). In search of homo economicus: behavioral experiments in 15 small-scale societies. American Economic Review, 73-78. Kirchsteiger, G. (1994). The role of envy in ultimatum games.
Journal of economic behavior & organization, 25(3), 373-389.
Oosterbeek, H., Sloof, R., & Van De Kuilen, G. (2004). Cultural differences in ultimatum game experiments: Evidence from a meta-analysis. Experimental Economics, 7(2), 171-188.
Osborne, M. J. (2004). An introduction to game theory (Vol. 3, No. 3). New York: Oxford University Press. Riedl, A., & Výrašteková, J. (2002). Social preferences in three-player ultimatum game experiments. Tilburg University.
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. Stewart, I. (1999). A puzzle for pirates. Scientific American, 280, 98-99.
Talbot Coram, B. (1998). Robert E. Goodin, ed. Chapter 3: Second best theories.
The Theory of Institutional Design. Cambridge: Cambridge University Press. pp. 99–100. Thaler, R. H. (1988). Anomalies: The ultimatum game. The Journal of Economic Perspectives, 195-206. Thaler, R. H., & Mullainathan, S. (2008). How behavioral economics differs from traditional economics.
The concise encyclopedia of economics, 2.
Xue, Y. Q. (2013). Towards Closed-World Reasoning in Games–Ultimatum Game Revisited (Doctoral dissertation, Universiteit van Amsterdam).
8. Appendix Appendix 1
Every rational proposer will always accept his own proposal. Because he has a casting vote one other pirate doesn’t have to agree to get an acceptance level that is sufficient to remain alive for the proposer. ( 𝑈𝑈 = 0.5 ). Therefore there are (n-2) pirates remaining aboard who can be convinced that matter.
1. 𝑐𝑐𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑝𝑝𝑐𝑐𝑁𝑁𝑡𝑡𝑡𝑡𝑁𝑁𝑐𝑐 𝑜𝑜ℎ𝑜𝑜 𝑐𝑐𝑡𝑡𝑐𝑐 𝑁𝑁𝑁𝑁 𝑐𝑐𝑜𝑜𝑐𝑐𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁𝑎𝑎 𝑡𝑡ℎ𝑡𝑡𝑡𝑡 𝑁𝑁𝑡𝑡𝑡𝑡𝑡𝑡𝑁𝑁𝑁𝑁 = (𝑐𝑐 − 2)
To bribe another pirate is costly for the proposer. He therefore wishes to minimize the amount of pirates which he bribes under the restriction that he stays alive. Because the proposer has the casting vote he only needs 50% to agree to remain alive. This means that he needs to give 50% of the pirates that matter a coin.
2. �𝑛𝑛𝑛𝑛𝑔𝑔𝑛𝑛𝑔𝑔𝑠𝑠 𝑜𝑜𝑜𝑜 𝑝𝑝𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑔𝑔𝑠𝑠 𝑤𝑤ℎ𝑜𝑜 𝑐𝑐𝑠𝑠𝑛𝑛 𝑛𝑛𝑔𝑔 𝑐𝑐𝑜𝑜𝑛𝑛𝑐𝑐𝑖𝑖𝑛𝑛𝑐𝑐𝑔𝑔𝑒𝑒2 � = �𝑛𝑛−22 �
The treasure can only however can only exist out of integers; 𝑔𝑔 ~ [0, ∞). If n is a natural even number greater than zero the answer will always be an integer. However if n is an uneven natural number greater than zero there will always be a digit behind the comma. To correct for this brackets23 ( ⌈0⌉ ) that round to the upper boundary that is an integer have to be used. We choose the upper boundary because the acceptance level 𝑈𝑈 has to be at least .5 to remain alive. If the treasure is ‘large enough’ the formula looks as 3.
3. � 𝐹𝐹𝑜𝑜𝑁𝑁 𝑁𝑁𝑟𝑟𝑁𝑁𝑁𝑁𝑝𝑝 𝑐𝑐 > 0 𝑁𝑁𝑐𝑐𝑜𝑜𝑁𝑁𝑔𝑔ℎ 𝑁𝑁𝑁𝑁𝑡𝑡𝑐𝑐𝑐𝑐: 𝑔𝑔 ≥ ��(𝑛𝑛−2)2 ��
In every case where 3. does not hold the treasure is considered ‘not large enough’.
23 If you will leave these brackets, the formula will assume that you coins can be split in to smaller amounts.
21
Appendix 2
For every coin that is in the treasure the proposer can bribe one other pirate.
For every pirate that is bribed another pirate can decline so that 𝑈𝑈 = 0.5 and the proposer lives. For every coin 2 pirates can survive.
1. 2𝑔𝑔 = 𝑐𝑐
However (as we saw in the “7 pirates who will find 1 coin” example) more pirates can survive on less coins. Therefore we note that the left hand side of this equation is not yet completed.
2. … + 2𝑔𝑔 = 𝑐𝑐
with 0 coins 1 or 2 pirates can manage to survive. with 1 coins 1, 2, 3 or 4 pirates can manage to survive. with 2 coins 1, 2, 3, 4, 5 or 6 pirates can manage to survive. 3. 2 + 2𝑔𝑔 = 𝑐𝑐
Step 3 is complete for the scenario where the treasure (𝑔𝑔) is ‘large’ enough to bribe enough pirates. Let’s call these “Pirate(s) A”. However we also encountered a different scenario where the treasure (𝑔𝑔) was considered ‘not large’ enough to bribe enough pirates. Let’s call these “pirate(s) B” We need to take Pirate(s) B into account to better our formula.
Pirate(s) A Pirate(s) B
with 0 coins [1,2] 2, 4,8,16,32,64,… pirates can manage to survive. with 1 coins [1,4] 4, 6,10,18,34,66,… pirates can manage to survive. with 2 coins [1,6] 6, 8,12,20,36,68,… pirates can manage to survive.
First note that all pirates in group Pirate(s) B never receive a coin but accept to remain alive. Secondly note that the constant must be 2𝑥𝑥 where x is a natural number ≥ 0
If we correct our formula we find: 4. 2𝑥𝑥+ 2𝑔𝑔 = 𝑐𝑐
Appendix 3
The Pirate-Game
Instruction sheet
Please be quiet during the experiment!!
Story:
Five pirates have gotten their hands on a hoard of 100 gold pieces and wish to divide the loot. They are democratic pirates, in their own way, and it is their custom to make such divisions in the following manner: The oldest pirate (captain) makes a proposal about the division, and everybody votes on it, including the captain. If 50 percent or more are in favor, the proposal passes and is implemented forthwith. Otherwise the captain is thrown
overboard, and the procedure is repeated with the next oldest pirate being the captain.
Your Tasks: 𝑅𝑅𝑡𝑡𝑝𝑝𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐: � 1. 𝐷𝐷𝑁𝑁𝑐𝑐𝑐𝑐𝑎𝑎𝑁𝑁 𝑜𝑜𝑐𝑐 ℎ𝑜𝑜𝑜𝑜 𝑝𝑝𝑜𝑜𝑁𝑁 𝑜𝑜𝑜𝑜𝑁𝑁𝑝𝑝𝑎𝑎 𝑝𝑝𝑐𝑐𝑙𝑙𝑁𝑁 𝑡𝑡𝑜𝑜 𝑎𝑎𝑐𝑐𝑟𝑟𝑐𝑐𝑎𝑎𝑁𝑁 𝑡𝑡ℎ𝑁𝑁 ℎ𝑜𝑜𝑡𝑡𝑁𝑁𝑎𝑎 𝑜𝑜𝑜𝑜 100 𝑔𝑔𝑜𝑜𝑝𝑝𝑎𝑎 𝑝𝑝𝑐𝑐𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐 2. 𝑇𝑇𝑁𝑁𝑝𝑝 𝑡𝑡𝑜𝑜 𝑐𝑐𝑡𝑡𝑡𝑡𝑝𝑝 𝑡𝑡𝑁𝑁𝑜𝑜𝑡𝑡𝑁𝑁𝑎𝑎 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑐𝑐ℎ𝑐𝑐𝑝𝑝 3. 𝐺𝐺𝑡𝑡𝑡𝑡ℎ𝑁𝑁𝑁𝑁 𝑡𝑡𝑐𝑐 𝑁𝑁𝑁𝑁𝑐𝑐ℎ 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐 𝑡𝑡𝑐𝑐 𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑁𝑁 4. 𝐷𝐷𝑁𝑁𝑐𝑐𝑐𝑐𝑎𝑎𝑁𝑁 𝑐𝑐𝑜𝑜 𝑝𝑝𝑜𝑜𝑁𝑁 𝐴𝐴𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡 𝑜𝑜𝑁𝑁 𝐷𝐷𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁 𝑝𝑝𝑜𝑜𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜𝑐𝑐 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 𝑆𝑆𝑁𝑁𝑁𝑁𝑜𝑜𝑁𝑁𝑎𝑎𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑁𝑁: � 1. 𝑊𝑊𝑡𝑡𝑐𝑐𝑡𝑡 𝑁𝑁𝑐𝑐𝑡𝑡𝑐𝑐𝑝𝑝 𝑝𝑝𝑜𝑜𝑁𝑁 𝑁𝑁𝑁𝑁𝑐𝑐𝑁𝑁𝑐𝑐𝑟𝑟𝑁𝑁 𝑡𝑡 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 2. 𝑇𝑇𝑁𝑁𝑝𝑝 𝑡𝑡𝑜𝑜 𝑐𝑐𝑡𝑡𝑡𝑡𝑝𝑝 𝑡𝑡𝑁𝑁𝑜𝑜𝑡𝑡𝑁𝑁𝑎𝑎 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑁𝑁 𝑐𝑐ℎ𝑐𝑐𝑝𝑝 3. 𝐺𝐺𝑡𝑡𝑡𝑡ℎ𝑁𝑁𝑁𝑁 𝑡𝑡𝑐𝑐 𝑁𝑁𝑁𝑁𝑐𝑐ℎ 𝑐𝑐𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐 𝑡𝑡𝑐𝑐 𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑁𝑁 4. 𝐷𝐷𝑁𝑁𝑐𝑐𝑐𝑐𝑎𝑎𝑁𝑁 𝑐𝑐𝑜𝑜 𝑝𝑝𝑜𝑜𝑁𝑁 𝐴𝐴𝑐𝑐𝑐𝑐𝑁𝑁𝑝𝑝𝑡𝑡 𝑜𝑜𝑁𝑁 𝐷𝐷𝑁𝑁𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐𝑁𝑁 𝑡𝑡ℎ𝑁𝑁 𝑝𝑝𝑁𝑁𝑜𝑜𝑝𝑝𝑜𝑜𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑜𝑜𝑐𝑐 Setup:
After the proposal is announced you will have 2 minutes to consider your choice and to write down your answer (accept or decline) on your answer sheet. The answer sheet states your individual unique number and the number of the round. After each round, your response will be collected. I will spend a few minutes on gathering the results. During this time you can lean back and await our announcement as we will inform you if the proposal is implemented or not. Please do not talk during this intermezzo.
Appendix 4
The Pirate-Game
Question sheet
1. Imagine you are the captain, give an example of a proposition when there are 5 pirates (including yourself) on the ship.
2. Imagine that there are 4 pirates on the ship (including yourself), two of them accept and the other two decline the proposal. Will the proposal be implemented or not?
3. If the current captain is thrown of the boat, who will become the new captain?
Now we will start the game!
Appendix 5
The Pirate-Game
Answer Sheet
Please take the experiment seriously!! Subordinate
Your number: ………
Number of the round: …….………..
Do you either accept or decline the proposal given to you?
………... ... ………... ... Captain Your number: ………
Number of the round: …….………..
How would you like to divide the hoard of 100 gold pieces among the pirates that are alive? ………... ... ………... ... ………... ... Do you either accept or decline your own proposal?
………... ………... ………... ...
