Master Thesis Behavioural Economics and Game Theory
The effects of framing and
group size on selfishness
With results from an altered multi-person prisoner’s dilemma experiment
Author: Supervisor:
Teddy Tan (10574778) Ailko van der Veen
15th of July, 2018 Academic year 2017/2018
Statement of Originality
This document is written by Teddy Tan who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original 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.
Abstract
The prisoner’s dilemma (PD) is an extensively analysed game that finds its application in many fields, such as business, economics, sociology, and psychology. It is often used to mimic real life social dilemmas, including overpopulation, resource depletion, and pollution. The original PD, however, does not mimic these social dilemmas correctly, which is why the payoff structure needs to be changed. This Master thesis presents an adapted PD that better reflects the aforementioned social dilemmas, and focusses on the effects of framing and group size on selfishness. With the help of an online group decision experiment I find a small framing effect. Also, in contrast to previous studies, I find an increase in social behaviour (cooperation) as group size is increased. Lastly, a negative relation between beliefs and group size is found. The adapted version of the multi person PD and the results of the corresponding experiment could be useful in future research related to decision-making in social dilemmas.
Contents
Introduction
5
Literature review
6
Main idea
10
Experimental design
13
Hypotheses
15
Experimental results
17
Discussion
22
Conclusion
25
Future research
26
Appendices
27
Reference list
39
Introduction
Social dilemmas such as overpopulation, resource depletion, and pollution are rising problems in the modern world. What all these dilemmas have in common is that each individual part of the dilemma (1) receives a higher payoff for a socially defecting choice than for a socially cooperative choice, regardless of what others in society do, but (2) all individuals are worse off if all defect than if all cooperate (Dawes, 1980). Cooperating in a social dilemma is
individually costly, therefore game theory suggests humans are better off not cooperating. Yet there is experimental evidence for humans cooperating in situations such as these. Economists and game theorists have tried to mimic social dilemmas with the help of games. One of the games that is often used to model the aforementioned social dilemmas is the prisoner’s dilemma. The prisoner’s dilemma, however, does not mimic these social dilemmas correctly, which is why the payoff structure needs to be changed. This Master thesis presents an adapted prisoner’s dilemma that better reflects certain real life social dilemmas, and focusses on the effects of framing and group size on selfishness. The research question that this thesis tries to answer is: “What are the effects of framing and group size on selfishness?” To test people’s sensitivity to framing and to find out whether cooperation truly decreases with group size, as reported by many (Axelrod & Dion, 1988; Hamburger, Guyer, Fox, 1975; Hauert & Schuster, 1997), I conducted an online group decision-making experiment.
This thesis is composed of three parts. In the first part I will review some of the related literature and give an elaborate description of the main idea. In part two, the methodology section, I will form three hypotheses and explain the setup of the experiment. In the final part the main findings will be analysed and discussed.
Literature Review
Prisoner’s dilemma
The field of game theory uses various games to study strategic interactions between two or more individuals. A well known and commonly used game is the prisoner’s dilemma (PD). Simply put, this game involves two individuals who have to choose between two actions: a cooperative (social) one, and a defective (self-interested) one. Although the self-interested choice seems more desirable (because it is dominant), the opposite is true should both individuals choose to act self-interested. The PD game can be represented in a 2x2 matrix (see table 1), which includes 4 cells with 2 entries each. The payoff of the row-player is the first entry, whereas the second entry represents the payoff of the column-player. In order for a game to be labeled as a prisoner’s dilemma the following inequalities must be satisfied: γ < ∂ < α < ß. One additional constraint that Rapoport and Chammah (1965) introduce into the model is: 2α > ß + γ. If this inequality does not hold, the players would have two forms of tacit collusion, namely (C1,
C2) and an equal mix between (C1, D2) and (D1, C2). Assuming that these inequalities do hold,
a PD has a unique and symmetric Nash equilibrium (NE): (D1, D2).
For over 60 years researchers have been conducting PD experiments. Many of these studies, including the ones from Dawes (1980), provide experimental evidence that some humans cooperate. Kiesler et al. (1996) investigated whether cooperation with a human-like computer differed from cooperation with a real human. The results have shown that subjects broke their promises to a computer more often than to a person, indicating that humans make
heterogenous commitments. The main point that can be taken from these experiments is that humans occasionally cooperate, even though theory suggests this is not optimal. What are the drivers of cooperation, and is there a way to affect the level of cooperation?
The key determinants of cooperation
Oye (1985) suggests that there are three sets of conditions that together determine the degree of cooperation and defection: (1) payoff structure: mutual and conflicting preferences; (2) the
shadow of the future: one-shot interactions and repeated games; and (3) the number of players:
two-Table 1. Payoff matrix original PD
C2 D2
C1 α, α γ, ß
Payoff structure
The payoff structure of the PD is important, because it determines the significance of cooperation. A mutual benefit exists when players prefer mutual cooperation to mutual defection, i.e. (C1C2) > (D1D2). In order to realise this mutually beneficial outcome,
cooperation is necessary. The likelihood of cooperation is affected by the relative payoffs of mutual cooperation and unilateral defection. The higher the relative payoff of mutual cooperation to mutual defection, and the smaller the relative payoff of unilateral defection, i.e. (D1C2 vs. C1D2), the higher the probability of cooperation (Oye, 1985).
The shadow of the future
The number of times that players are expected to interact with one another is the second condition that affects the degree of cooperation. First, consider a one-shot PD. Finding the best response strategies for both players gives the unique NE: (D1D2). Regardless of what the
other player does, you are always better off choosing “defect”. In other words, defect is the strictly dominating strategy. Cooperation in a one-shot PD is unlikely, because neither player can trust the other to play “cooperate”. Also, to avoid the possibility of ending up with the worst payoff, both players have incentive to play “defect”.
If the players are interacting more than once, the likelihood of cooperation increases. More specifically, the higher the frequency of interaction, the higher the likelihood of cooperation. Experimental evidence suggests that under iterated PD the occurrence of mutual cooperation increases significantly (Rapoport & Chammah, 1965). In theory, however, the only NE is still to play mutual defection as long as the number of interactions is finite. A possible explanation for the occurrence of the mutual cooperation outcome is that a potential defector compares the immediate gains from defecting to the loss of future gains as a consequence of defecting now. The incentives to cooperate and defect are the discounted stream of payoffs across current and future encounters. The size of the discount rate affects the value of future
benefits. A patient person, i.e. someone with a low discount rate, is less likely to defect than an impatient person, i.e. someone with a high discount rate. A patient person values the payoffs in the (near) future more than an impatient person, and is therefore more inclined to
cooperate.
A third situation arises if the number of iterations is infinite. If both players want to get the highest average payoff it is their best interest to always cooperate. This, of course includes the aforementioned assumption that the mutual cooperation outcome is larger than an equal mix between the unilateral defection outcomes (2α > ß + γ).
The number of players
In the original 2-player PD, each player only has to consider the other player’s behaviour. As the number of players grows it becomes increasingly difficult to anticipate others’ behaviour and to calculate the value of future payoffs. Whether or not cooperative behaviour can be sustained with a larger number of players depends on calculations of expected utility. This expected utility, which is determined by payoff structures, discount rates, and anticipated behaviour of others, is likely to vary across players. In other words, as the number of players increases, the probable heterogeneity of players increases, which decreases the probability of mutual cooperation (Oye, 1985). This theoretical reasoning is confirmed frequently in practice (Axelrod & Dion, 1988; Hamburger, Guyer, Fox, 1975; Hauert & Schuster, 1997). Although originally aimed at the field of politics, I think these three key determinants find perfect use in the field of economics as well. Similar to politics cooperation and strategies to promote and sustain cooperation are crucial in becoming successful and reaching long-term goals.
Other determinants of cooperation
Besides the three key determinants of cooperation there are some other factors that could affect the probability of players cooperating. Altruism, reciprocity, communication, anonymity, punishment, personality, study background, and cultural factors are all among these factors. Experimental evidence has shown that when subjects are allowed to
communicate with each other a higher fraction of cooperative behaviour is observed (Caldwell, 1976; Dawes, McTavish, Shaklee, 1977). Whether player’s decisions are made public or remain private also affects the degree of cooperation. As might be expected, more cooperative behaviour is observed when decisions are made public (Fox & Guyer, 1978). The inclusion of punishment opportunities in public goods games causes a large rise in the average level of contribution (Fehr & Gächter, 2000). Nurturing, e.g. personality and study background, also play a vital role in affecting the rate of cooperation. Economics students generally act more self-interested, and are thus less likely to cooperate in social dilemmas than students from other disciplines (Frank, Gilovich, & Regan, 1993). And lastly, cultural factors are highly likely to influence someone’s decisions. In individualistic societies, such as (Western) Europe and the US, experiments commonly show a lower degree of cooperation compared to socialistic societies, such as Asia (Cox, Lobel, McLeod, 2017; Parks & Vu, 1994).
Frames and framing
The PD is not the only game that has been experimented with that involves decision-making and behaviour. Many researchers over the past couple decades have been doing experiments with games about decision-making and behaviour. One of the oldest assumptions in
traditional economics is that humans are fully rational and strictly act in their own self-interest; better known as the homo economicus assumption. In the mid-nineties, more and more people began to doubt this assumption.
Daniel Kahneman and Amos Tversky, arguably the founders of behavioural economics, wrote their famous article about prospect theory in 1979 in which they find results that are in strong contrast with (expected) utility theory, and thus the homo economicus assumption. One of the effects that violates expected utility theory is identified as the framing effect. When presented two problems that are completely identical except for the way that they are framed, people often change their preference. These so called preference reversals occur in choices regarding both hypothetical and real monetary outcomes, even in questions pertaining to the loss of human lives (Kahneman & Tversky, 1981, 1984). Another research, conducted by Dufwenberg, Gächter, and Hennig-Schmidt (2011) found experimental evidence that frames affect subjects’ beliefs and contributions in one-shot public goods games.
How do these frames work you may wonder. Frames highlight certain pieces of information by increasing its salience, i.e. making information more noticeable, meaningful, or memorable to audiences. This increased salience enhances the probability that receivers will perceive and process the information, and thus store it in their memory (Fiske & Taylor, 1991, Entman, 1993).
Seemingly unimportant on a daily basis, framing actually has important everyday applications that one might be unaware of. One example is the news and media. Frames draw attention to specific items or aspects while other elements are neglected and obscured, which might lead to different perceptions. Framing plays a vital role in the exertion of political power, and the frame in a news text is the imprint of power (Entman, 1993). It is important that people are made aware of a framing and its potential power, because it can help us in understanding decisions, and it offers valuable suggestions for communication practitioners (Tankard, 2001).
Main Idea
Now that we are acquainted with the related literature, it is time for the main idea and goal of this thesis. The research question that this thesis is interested in answering is: “What are the
effects of framing and group size on selfishness?” I would like to test whether it is possible to direct
people into making more selfish or social decisions with the use of framing, and whether people are naturally more or less inclined to behave selfishly in larger groups. In order to test the effects of framing and group size on people’s selfishness in social dilemmas I designed a game that is very similar to a prisoner’s dilemma. I made a slight change to the payoffs, which makes defecting no longer strictly dominate cooperating. This payoff adjustment is necessary, as the original prisoner’s dilemma does not mimic certain real life situations correctly. The original prisoner’s dilemma suggests that, given the other player defects, the second player is always better off defecting as well. In reality, choosing the defective option is not always preferred to choosing the cooperative option. One example that this altered prisoner’s
dilemma could mimic is a group of civilians that can choose to mine a finite resource, such as gold. If just a few civilians are mining for gold they can do so without needing a permit or license. But as more and more people start to mine for gold the municipality/government demands the miners to have a license. If there are too many miners the quarry runs out of gold and all miners end up with losses because their investments outweigh their proceeds. The miners are now worse off than the non-miners. The main point is that up until a certain threshold the “defectors” are better off than the “cooperators”, but as soon as the threshold is surpassed the opposite is true.
Each subject has two choices in the game that they are going to play: a socially cooperative one (share) and a selfishly defective one (grab). To test the framing part of the research question half of the subjects will be playing the game under the frame “Winner takes all”, while the other half will be playing under the frame “Sharing is caring”. The remaining parameters, such as the name of the actions and payoffs of the game are completely identical.
Table 2. Payoff scheme player A for all possible outcomes
Action player A
No remaining group members choose grab
Exactly one other group member chooses grab
Two or more group members choose grab
Share € 30 € 0 € 5
The subjects that will be playing under the first framing condition should be lured into choosing the (risky) selfish option. As for the latter framing condition, I hope to direct my subjects into choosing socially. The payoff for player A in all possible situations is displayed in table 2.
Initially the payoffs in the right column might seem counterintuitive, because if you choose “share” and just one other group members chooses “grab” you get a payoff of €0, while you get a payoff of €5 when two or more other group members choose “grab”. In real life the payoff for a player who chooses “grab” in that situation should be negative, but for the sake of the experiment I cannot take money from my subjects.
As mentioned in the literature section there are various determinants of cooperation. By controlling as many of these determinants as possible I am able to make better predictions about the subjects’ beliefs and decisions. Each game is played just once, players are not allowed to communicate with each other, the decisions will be made anonymously, the
number of players is varied (increased), and no punishment is involved. In previous studies all these determinants have lead to reduced levels of cooperation (Caldwell, 1976; Dawes, McTavish, Shaklee, 1977; Fehr & Gächter, 2000; Fox & Guyer, 1978). I will attempt to show that varying (increasing) the number of players does not necessarily decrease the probability of cooperation (as claimed by previous studies), and that it in contrast could possibly even favour cooperation. I hope that the designed payoff structure and chosen frames will trigger and stimulate selfish behaviour in the “winner takes all” treatment, and stimulate socialism and collectivism in the “sharing is caring” treatment.
In order to find out whether there exists a relationship between selfish behaviour and group size the subjects will have to choose between “share” and “grab”. Each decision is made in one of the four different group sizes: 3, 6, 9, and 12. Group size is relevant in social dilemmas such as these, because depending on the number of players a subject could be more or less inclined to choose selfishly (e.g. in real life a city’s population could affect people’s mining behaviour). With the help of an online experiment subjects will be playing my altered version of a one-shot multi person PD. Strictly speaking it is not a PD, as the payoffs do not meet the inequalities mentioned previously.
Expected payoffs
In the original PD, regardless of the players’ subjective probabilities, the expected value of the selfish action (defect) is always higher than the expected value of the social action (cooperate). This makes sense, because defecting strictly dominates cooperating. In this game, however, a player’s subjective probability determines whether the expected payoff of the social action (share) or the selfish action (grab) is higher. The expected utility decreases as the group size increases, but the expected payoff of “grab” declines at a faster rate than the expected payoff of “share”. Hence, for every subjective probability there is a threshold group size for which the expected payoffs flip towards “share”. This is logical, because as the group size increases the probability that all remaining players choose “share” decreases. This implies that the probability of getting 30 or 50 decreases as well. Someone who chooses “share” sees its expected payoff diminish from 30 towards 5, whereas a someone who chooses “grab” sees its expected payoff diminish from 50 towards 0. The formulas that can be used to calculate the expected utility from both actions is as follows:
In words: a player who chooses “share” gets 30 with the probability that nobody chooses “grab” (i.e. all other players share as well), 0 if exactly one remaining player chooses “grab”, and 5 if two or more players choose “grab”. A player who chooses “grab” only gets 50 if (s)he is the only one, 0 in all other possible situations.
I assume the average subjective probability to be approximately 0,7. With these probabilities the expected payoff for “share” becomes higher than “grab” for groups larger than 6 people. In appendix B you can find additional information about expected payoffs and subjective probabilities.
Experimental Design
Experimental setup and procedures
To perform the experiment I am using Qualtrics, the online questionnaire software that the University of Amsterdam provides. Anyone who wishes to participate in the experiment can do so and automatically gets a chance at winning some money.
There are two games that the participants are playing in a between subject design, i.e. each participant is only part of one treatment. This is necessary to ensure no subjects are aware of the framing of the game. Everyone who wishes to participate in my experiment can do so by clicking on a shareable link. Qualtrics is programmed in such a way that half of the subjects are assigned to each of the two treatments. Once assigned, the subjects are shown an
introduction and explanation of the experiment. I include an example question to make sure the subjects are familiar with all possible outcomes and their corresponding payoffs. The subjects that answer the example question incorrectly are shown a second example question to make sure that they truly understand the game. After reading the instructions and
completing the example question(s) the real experiment starts.
The experiment consists of four rounds in which the subjects have to choose between “share” and “grab”, each time in a different group size. The group sizes that the subjects are playing the game in are 3, 6, 9, and 12. To ensure that there are no specific order effects I am using Qualtrics’ randomization function. In each round the participants are also required to state their beliefs about the other players, i.e. how many remaining group members besides themselves they think will choose “share”. It is interesting to include these beliefs, because then I can check whether the subjects’ actions are optimal given their beliefs.
After completing the experiment each participant is asked to fill in some general information, such as sex, age, highest completed education, and nationality. In the appendices I included screenshots of the experiment.
Experimental payout
After the experiment I randomly selected one of the treatments with google’s online number generator. I assigned number 1 to the “Sharing is caring” treatment, and number 2 to the “Winner takes all” treatment. The number generator picked 2, hence someone in the latter treatment was to be rewarded. Next, I assigned four numbers to the different group sizes: 1 to
group size 3, 2 to group size 6, 3 to group size 9, and 4 to group size 12. This time the number generator ended at number 3, so the group size selected for payout was 9. Then, out of all the subjects in the “Winner takes all” treatment (28) I let the number generator select 9 random numbers. After having determined which 9 subjects were selected to form a group, 7 subjects turned out to have chosen “share”. Finally, I ran the number generator one last time to determine which out the 9 subjects would be rewarded. This turned out to be a subject who chose “share”, so this subject is rewarded €5.
Hypotheses
There are a number of effects that can occur by changing some of the parameters of the experiment. Firstly, the framing of the game could affect the fraction of selfish choices. I expect that under the “Winner takes all” frame the fraction of selfish choices will be higher than under the “Sharing is caring” frame. Although the title of the game might not be something that people pay much attention to, I think that they will unconsciously behave more selfishly under the first framing condition. Similar to the result of the research by Lieberman, Samuels and Ross (2004), I expect that under the “Sharing is caring” frame the subjects will behave more cooperatively. In other words, I expect that the subjects will choose “share” more often than under the “Winner takes all” frame. This hypothesis is based on a concept called decision-induced focusing (Dufwenberg, Gächter, and Hennig-Schmidt, 2011; van Dijk and Wilke, 2000), which rests on the idea that subjects are focused on the decision they are explicitly asked to make, i.e. “share” under the “Sharing is caring” frame and “grab” under the “Winner takes all” frame.
Secondly, changing the group size could affect the results in two directions. A larger group size could lead to more selfish actions with the explanation that a decision can be made more anonymously. Also, subjects might expect that in a larger group the probability of at least one other person choosing the selfish action is larger than in a smaller group. While some subjects rather not take the risk by choosing the selfish action others might see an opportunity to getting the entire pie by choosing the selfish action themselves.
On the other hand, however, one could argue that a larger group size will result in less selfish choices, because by choosing selfishly you could negatively affect a larger number of people compared to a smaller group size. Also, the group size and expected payoff are
negatively correlated, because the probabilities of getting a payoff of €30 or €50 is decreasing with the number of players that are involved in the game. I assume that most subjects will be aware of this fact. Some subjects might even be aware that the expected payoff of the social action becomes relatively more attractive compared to the selfish action as group size is increased. The majority of the subjects will probably be risk averse, and thus go for the action with the highest expected payoff. As mentioned previously, the expected payoff depends on the subject’s subjective probabilities. I assume that most subjects will assign a subjective probability of around 0,7 to others choosing the social action. With these subjective
probabilities the expected payoff for cooperating is higher than defecting for groups smaller than 7. Therefore, I expect the fraction of “share” choices to be highest in the smallest group, and this fraction will probably decline as the group size is increased.
My third hypothesis is about the subjects’ beliefs. Although I think the frames of my games are not as imposing as the frames Dufwenberg, Gächter, and Hennig-Schmidt (2011) used for their experiment, I still expect to find higher average beliefs under the “Sharing is caring” frame. Hopefully, the “Sharing is caring” frame directs my subjects into choosing “share” more often and increase their beliefs compared to the “Winner takes all” frame. Also, I expect that the average beliefs will decrease as the group size is increased, because the subjects are in all likelihood aware of the fact that as the group becomes larger the probability of all other players choosing “share” drops.
Experimental Results
In total I was able to collect 58 valid responses (39 male, mean age 30 years; age range 17-72), of which 30 come from the “Sharing is caring” (SIC) treatment and 28 from the “Winner takes all” (WTA) treatment. 16 subjects are high school graduates or less, the remaining 42 subjects are in possession of at least a Bachelor’s degree. Two responses are excluded from the analysis, because they seemed to not understand the payoff scheme of the experiment. Some additional responses were only partially completed, and are therefore excluded from the analysis.
With the use of Stata I analysed the data, which I will be explaining extensively in this section. There are several interesting results that I found. The three main results quite match my hypotheses. Firstly, I found that the framing of the game affected the decisions of the participants. Secondly, I found that in small groups people tend to behave more selfishly compared to larger groups. And finally, I found that average beliefs decreased as group size was increased.
As can be seen from graph 1 approximately a quarter of the subjects, when they are being part of a three-person group, chose “grab”. This fraction is higher than any of the other group sizes, which is what I expected to happen. In total I recorded 15, 8, 9, and 10 “grab” decisions for the group sizes 3, 6, 9, and 12 respectively. Not in line with my hypothesis, however, is the fact that the rate of “grab” choices is fairly constant for the three largest groups. I expected to find the highest rate of “share” choices in either group size 9 or 12.
Graph 2. Average beliefs (as fractions)
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Entire sample SIC WTA
0,70 0,67 0,68 0,64 0,68 0,72 0,73 0,74 0,73 0,84 0,70 0,77
Group size 3 Group size 6 Group size 9 Group size 12 Graph 1. Average fraction of “share” decisions
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Entire sample SIC WTA
0,75 0,90 0,83 0,90 0,79 0,84 0,90 0,82 0,86 0,79 0,70 0,74
Group size 3 Group size 6 Group size 9 Group size 12
In graph 1 you can also find the fraction of “share” decisions for the treatments separately. Out of the 120 observations (decisions) coming from the SIC treatment “grab” is chosen 17 times, whereas “grab” is chosen 24 times out of the 112 observations in the WTA treatment. In other words, 14,2% of the subjects chose “grab” in the SIC treatment, whereas 21,4% of the subjects chose “grab” in the WTA treatment. Hence, the subjects did behave more selfishly in the WTA treatment. This is in correspondence with my first hypothesis. In the SIC treatment the fraction of “share” decisions in the smallest group size was clearly lower than in the three larger group sizes, and remained stable for group sizes 6, 9, and 12. In the WTA treatment a relatively constant behaviour is observed across the four group sizes. Looking at differences between males and females the results show that in the smallest group size a third of the females choose “grab”, whereas approximately a quarter of the males choose “grab”. For the three larger groups a similar fraction of the males and females chose “grab”. Interesting to see is that the fraction of males choosing “grab” is slightly increasing with the group size; exactly the opposite of what I was expecting. Females behaved much more as I had anticipated. Females only chose “grab” when they were in a group of 3. On average, 20,5% (22 out of 156) of the male and 13,2% (10 out of 76) of the female decisions were “grab”, suggesting that males behaved more selfishly.
Regarding subjects’ beliefs I distinguished between beliefs and maximum beliefs. Maximum beliefs are beliefs that assume all remaining group members choose “share”. In theory, a subject with maximum beliefs still has a chance at winning €50 if this subject were to choose “grab”. Out of the 232 observations I recorded 74 maximum beliefs, of which half (37) come from group size 3. 38,2% of the female beliefs are maximum beliefs, whereas 28,8% of the male beliefs are maximum beliefs. Despite the fact that females have maximum beliefs more often than males, the results point out that males chose “grab” more often. In the WTA treatment 30% of the male decisions are “grab”, while only 20% report to have maximum beliefs. In accordance with my third hypothesis the average beliefs across the entire sample slightly decreased as the group size increased. Remarkably, the average beliefs in the WTA treatment are higher than in the SIC treatment for all group sizes.
Although it was not my main focus, it is striking to see that only three subjects (out of 17) aged over 30 chose “grab” in any of the rounds. It seems that younger subjects were behaving much more selfishly. Appendix E contains additional graphs and tables concerning observed behavioural gender differences.
To confirm my findings and to find out the relation and significance of the variables in my model I ran the non-parametric Spearman correlation test. I tested for the correlations between the following variables: share, SIC, group, maximum, male, and age. Table 5 shows the correlations for the entire sample, and table 6 displays the correlations for the separate treatments.
Two variables (maximum and male) seemed to decrease the probability of a subject choosing “share”, while the remaining three variables (SIC, group, and age) increased the probability of a subject choosing “share”. In other words, on average subjects with maximum beliefs and males were less likely to choose “share” than females and subjects who did not have
maximum beliefs. Also, older subjects and subjects in the SIC treatment were more inclined
Table 5. Spearman correlation coefficients entire sample Share SIC Group Maximum Male Age Share1 1,000 0,083 0,224 -0,063 -0,090 0,153*** SIC2 - 1,000 0,000 -0,606 -0,086 0,091 Group3 - - 1,000 -0,372*** 0,000 0,000 Maximum4 - - - 1,000 -0,094 0,164** Male5 - - - - 1,000 -0,418*** Age6 - - - - - 1,000
1dummy variable equal to 1 if subject chooses “share”, 0 if subject chooses “grab” 2dummy variable equal to if subject is in SIC treatment, 0 if subject is in WTA treatment 3continuous variable for the four group sizes
4dummy variable equal to 1 if subject has maximum beliefs, 0 otherwise 5dummy variable equal to 1 for males, 0 for females
6continuous variable for the age of the subjects
*p < 0,10, **p < 0,05, ***p < 0,01
Table 6. Spearman correlation coefficients for separate treatments
Share Group Maximum Male Age
SIC WTA SIC WTA SIC WTA SIC WTA SIC WTA
Share1 1,000 1,000 0,070*** -0,039 -0,116 -0,029 -0,012 -0,186** 0,058 0,261***
Group2 - 1,000 1,000 -0,347*** -0,411*** 0,000 0,000 0,000 0,000
Maximum3 - - 1,000 1,000 -0,059 -0,160* 0,082 0,240**
Male4 - - - 1,000 1,000 -0,241*** -0,599***
Age5 - - - - 1,000 1,000
1dummy variable equal to 1 if subject chooses “share”, 0 if subject chooses “grab” 2continuous variable for the four group sizes
3dummy variable equal to 1 if subject has maximum beliefs, 0 otherwise 4dummy variable equal to 1 for males, 0 for females
5continuous variable for the age of the subjects
to choose “share” than young(er) subjects and subjects in the WTA treatment. Lastly, Spearman’s test found a positive correlation between group size and share, implying that subjects chose “share” more often in larger groups. The signs (directions) of the correlations for the entire sample are the same as for the separate treatments.
When comparing the correlations between the explanatory variables there are a couple of correlations that are repeatedly significant. Maximum and group are negatively correlated and are significant at the 1% level, suggesting that most maximum beliefs are found in group size 3. This is confirmed by my results. Also, the relation between maximum and age is positive and significant, indicating that old(er) subjects had maximum beliefs more often than young(er) subjects. Finally, I found a negative correlation between male and age, which simply means that young(er) subjects were more likely to be male.
In addition to the Spearman correlation test I ran a Kruskal Wallis test to determine if there is a statistically significant difference between my variables. The Kruskal Wallis test confirms the significance of the relations found with Spearman’s correlation coefficient. See appendix E for the results.
Lastly, I ran three separate regressions; one for the entire sample, and one for both treatments. I used a probit panel data model to correct for random effects and individual player effects. In table 7 you will find the coefficients and corresponding p-values from the three probit regressions.
Table 7. Regression coefficients and p-values probit regression
Variable Entire sample SIC WTA
Coefficient p-value Coefficient p-value Coefficient p-value Intercept -0,330 0,633 -0,567 0,454 0,032 0,986 SIC1 0,143 0,723 - - - -Group2 0,016 0,738 0,132 0,076* -0,126 0,079* Maximum3 -0,631 0,083* -0,404 0,408 -1,478 0,021** Male4 0,150 0,667 0,126 0,774 0,118 0,902 Age5 0,057 0,001*** 0,040 0,046** 0,112 0,031**
1dummy variable equal to 1 if subject is in SIC treatment, 0 if subject is in WTA treatment 2continuous variable for the four group sizes
3dummy variable equal to 1 if subject has maximum beliefs, 0 otherwise 4dummy variable equal to 1 for males, 0 for females
The coefficients’ direction of the variables in the regression for the entire sample confirm the correlations I found earlier. Only the direction for male is opposite to Spearman’s correlation, but the p-value is 0,667, hence insignificant. Two of the explanatory variables are significant: maximum and age. While the latter is significant at the 1%, the former is only significant at the 10% level.
The probit regression of the SIC treatment again finds the same directions for the
coefficients. Age is still significant, but only at the 5% level. Group is also significant (α=10%), which supports my findings.
According to the regression for the WTA treatment group, maximum, and age are all significant. The significance of the latter two variables is in accordance with the significance found in the regression for the entire sample. Similar to the SIC treatment group size is significant at the 10% level, but note that the sign is reversed, implying that as group size increased the fraction of “share” decreased.
In short, an elaborate data analysis has produced three main results that quite parallel my hypotheses. Firstly, framing of the game did have an effect on the rate of selfishness. Subjects in the SIC treatment chose “share” with a higher probability than the subjects in the WTA treatment. Secondly, I found a positive relation between group size and the fraction of “share” decisions, pointing out that subjects on average behaved more selfishly in smaller groups. Lastly, I found a negative relation between beliefs and group size.
Discussion
Considering the small sample size of N=58, the statistical significance and external validity of the findings are limited. According to the law of large numbers, I need a larger sample to make any valid claims about the effects of framing and group size on people’s selfishness. Furthermore, many subjects are friends of mine or economics students, which makes them fundamentally different from others who might play this game. Also, the male/ female ratio is heavily skewed towards the males. Running an artefactual field experiment of the same approximate design might be more appropriate in creating more reliable and meaningful results. Nonetheless, I hope that my experiment can serve as demonstration of technique for further research in this area.
Limitations
Beforehand it was difficult to predict what people’s expectations about other players would be. I was hoping to be quite close to this number, because this number determines the threshold group size for which the expected payoff of share/grab will flip. I expected the average subjective probability of “share” to be approximately 0,7. With this subjective probability the expected payoff of “share” becomes higher than the expected payoff of “grab” for groups larger than 6. If the subjective probability is higher than 0,7 the group size for which “share” yields a higher payoff than “grab” becomes larger. Similarly, if the subjective probability is below 0,7 the group size for which “share” yields a higher payoff than “grab” becomes smaller. Regardless of what subjects’ subjective probabilities were I was expecting to find a positive relationship between “share” decisions and the group size, because for all players the expected payoff for “grab was decreasing at a higher rate than for “share”.
Looking back at the results of the subjects’ beliefs, I have mixed feelings. On the one hand I am pleased that the average subjective probability for the entire sample turned out to be 0,72, which is very close to my expectation. On the other hand, however, with these subjective probabilities the results should have shown a clear preference reversal towards “share” for the two largest group sizes. While this is true for the SIC treatment, it is unfortunately not the case for the WTA treatment. What I did not consider before the experiment is that the subjective probabilities are different not only between subjects, but also across different group
groups than in large groups, which could have affected the expected payoff, and, thus potentially shifted the optimal decision.
Another limitation is the existence of mixed strategy equilibria. For every subjective probability and group size there is a mixed strategy equilibrium. The game is only played once, so I was unable to record if any of the subjects actually had a mixed strategy. I have only been able to see their beliefs and actual actions.
The final limitation is the model and the variables that I used for my regression analysis. I have chosen to include 5 explanatory variables in my probit regression model. Although the model is found significant, I might have forgotten to include a variable that could be
correlated with both the dependent as well as at least one independent variable. In other words, the regression could be suffering from an omitted variable bias. If this is the case my model attributes the effect of the omitted variable to the estimated effects of the currently included variables.
Explanations for observed behaviour
Two subjects did not understand the payoff scheme as they answered both example questions incorrectly. Hence, I excluded their data from the analysis. However, having analysed all other submissions there were still decisions that make me doubt that some subjects did not
understand the payoff scheme of the experiment. For instance, there is a subject who believed 7 out of the 11 remaining group members would choose “share”. Yet, this subject chose “grab”. Initially this makes no sense, because assuming that already 4 others choose “grab” this subject could never get €50 by choosing “grab”. Out of the total 232 observations 26 comparable decisions were made. Although such behaviour might seem irrational at first, it is not necessarily the case.
I have three possible explanations for why subjects chose “grab” despite not having maximum beliefs. The first possible explanation might be that subjects truly believed some other subjects would choose “grab”, but still hoped that nobody would actually do that. In that case, subjects with these beliefs and actions can be considered risk loving, as they risk to lose a potential €5 for the big prize of €50. Another possible explanation for this remarkable behaviour, which is only applicable to subjects who believed there would be exactly one player to choose “grab”, is that a subject wanted to ensure nobody got the €50. In other words, this action can be seen as a punishment. The final explanation is that subjects assigned probabilities to all possible outcomes, and chose “grab” because the expected payoff was higher than for “share”. To
illustrate this, consider the following example. Player A is asked to state his beliefs for group size 6. Player A assigns 50% probability to the state that all 5 remaining group members choose “share”, and a 50% probability to the state that 3 remaining group members choose “share”. The average of the two states is that 4 remaining players choose “share”, so that is what player A selects when he is asked to fill in his beliefs. When we analyse the expected payoffs for “share” and “grab”, player A has a 50% probability of getting €50 if he chooses “grab”, and an equal probability of getting €5 and €30 if he chooses “share”. The expected payoff for “grab” is €25, which is higher than the expected payoff for “share”, which is €17,50. Thus, someone with these beliefs did make the optimal decision, even though it might seem irrational at first. The experiment allowed the subjects to only select one number when asked for their beliefs, so I could not record if subjects assigned probabilities to different states.
As mentioned in the experimental results section I found a slight difference in the rate of “share” decisions under the two framing conditions. After the experiment I explained to some of the participants what the goal of my experiment was, and many of them stated that they could not recall the name of the game that they were playing. The fact that many subjects were unable to remember the title of the game could explain why I have not been able to find significant differences concerning the participants’ decisions.
Both the regression and correlation coefficients indicate that the variable “age” is significant, implying a negative relation between the fraction of “share” decisions and age. The majority of the subjects below the age of 30 are economics students. As mentioned in the literature section previous studies have shown that economics students tend to behave more self-interested compared to other disciplines. Hence, this could be a possible explanation for the fact that “age” is significant.
Conclusion
The two major focus points of this thesis are the effects of framing and the relation between group size on the rate of selfishness. Put another way, I was interested in testing the effects of framing and group size on people’s selfishness. To test for these effects, I conducted an online group decision experiment in which subjects played my version of an altered multi-person PD in an effort to mimic people’s behaviour in real life social dilemmas, such as overpopulation, pollution, and depleting resources. The payoff structure required a change, because the original PD does not mimic the above-mentioned social dilemmas correctly. I tested two treatments using a within-subject design, in which group sizes were varied.
Briefly, the experiment has produced three results. The fraction of “share” decisions in the SIC treatment turned out to be higher than in the WTA treatment, indicating a small effect of framing on the rate of selfishness. If the used frames would have been more
imposing the results might have been more significant.
Secondly, in line with my second hypothesis, most selfish behaviour is observed in the smallest group. In the SIC treatment there was a clear preference reversal towards “share” for the group sizes 6, 9, and 12. For the WTA treatment not so much. Thus, in contrast to previous studies, my experiment has demonstrated that the level of cooperation can actually increase as group size is increased.
Lastly, in accordance with my third hypothesis, a negative relation was observed between beliefs and group size. The average beliefs across the entire sample slightly decreased as the group size increased. Remarkable however, is that the average beliefs in the WTA treatment were higher than in the SIC treatment for all group sizes.
As for gender differences, females more often had maximum beliefs as well as higher average beliefs, but nonetheless behaved less selfishly than the males.
I am fully aware that the small sample size and subject population limit the statistical significance and external validity of the results. Nonetheless, I regard my altered version of the multi-person PD to be an improved way of mimicking social dilemmas. To that end, I hope future research will consider using my model when analysing decision-making in social dilemmas.
Future Research
The threshold for the number of players to choose “grab” in my experiment was just one. In a future research one could test what would happen if this threshold were to be increased to two or three. Would people be more willing to take risk in order to get a higher payoff ? Increasing the threshold will highly likely also change people’s beliefs.
As mentioned in the limitations section many subjects were not aware of the title of the game. I have strong reasons to believe that this might have lead to the minor differences in the results across the two treatments. In a future research the participants should be made more aware of the title. An alternative could be to ask subjects to recall the title of the game. Those unable to recall the title should then be excluded from the results. This might produce
different results.
Future research could also include the possibility of punishment by letting participants interact with the same group members more than once and pay to punish undesirable behaviour. I am also unsure if the payoff structure that I used was suitable, thus increasing rewards might yield different results.
Appendices
Appendix A. Nash equilibria
As we know, in the original multi-person PD, the only NE that exists is for everyone to defect. Due to a change in the payoff structure this is no longer the case in this altered PD. There are four possible situations that can arise, which are explained in detail below.
1. Everybody cooperates. If all players cooperate, i.e. all players are cooperators, each player will get a payoff of 30. This is not a NE, because every player has incentive to defect, which would give him/her a resulting payoff of 50.
2. Exactly one player defects. If this is the true situation the defector gets a payoff of 50, all remaining players (i.e. cooperators) get nothing. This is a NE, because nobody can do better by changing their action.
3. Exactly two players defect. The two defectors get a payoff of 0, whereas each cooperator gets a payoff of 5. This is also a NE, because neither of the defectors can increase their payoff by switching to cooperating, and a cooperator would see its payoff decrease from 5 to 0 if this player would change its action from cooperate to defect.
4. More than two players defect. All defectors get a payoff of 0, all cooperators get a payoff of 5. This is not a NE, because each defector has incentive to change its action to cooperate, as this player’s payoff would increase from 0 to 5.
Thus, two out of the four situations result in a NE. Note that as the group size increases, so do the number of equilibria. The number of possible NE can be calculated with the following formula:
With the payoff structure used during the experiment there are 3 NE in a 2-person variant of the game (see table A1): (share, grab), (grab, share), and (grab, grab).
Number of NE for n = 3:
Number of equilibria with n = 2 is 3. So number of NE for a three-player game is 3 + 3 = 6. There are 3 NE in which there is exactly one defector, and 3 NE in which exactly two players defect. Number of NE for n = 4: Table A1 Share Grab Share €30, €30 €0, €50 Grab €50, €0 €0, €0
Number of equilibria with n = 3 is 6. So number of NE for a three-player game is 6 + 4 = 10. There are 4 NE in which there is exactly one defector, and 6 NE in which exactly two players defect.
The same principle is applicable for the group sizes 5 up until 12. In table A2 you will find an overview of the number of NE that each group size has.
Appendix B. Subjective probabilities
As can be derived from the formulas the expected utility is dependent on both group size and the subjective probabilities that each individual assigns to the two actions. Because the
subjective probabilities vary across people their expected utilities differ as well. Regardless of people’s subjective probabilities the expected payoff for “grab” decreases at a higher rate than the expected payoff decrease of “share”. Assume that there are two players, for simplicity we call them players A and B. Player A assigns probability 0,6 to an other player choosing the cooperative action; player B assumes this probability to be 0,8. With these beliefs, player B has higher expected utilities than player A, but this does not imply that player B will get a higher payoff than player A. The actual payoff is ultimately determined by all players’ actions. Important to note, however, is that player B would more likely be tempted to choose “grab” in larger groups than player A. This is interesting, because the expected payoff of “grab”
decreases at a higher rate than the expected payoff for ”share”. In small groups the expected payoff “grab” is higher, but as soon as the group size reaches a certain number, the expected payoff of “share” is higher. In table B you find the group sizes for which it is more optimal to choose grab/share with the corresponding subjective probabilities that a player assigns to the two actions. Table A2 Group size Number of NE 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 47 10 57 11 68 12 80 Table B Subjective
probabilities E(G) > E(S) E(S) > E(G) p(S) = 0,5 n < 4 n > 3 p(S) = 0,6 n < 5 n > 4 p(S) = 0,7 n < 7 n > 6 p(S) = 0,8 n < 10 n > 9 p(S) = 0,9 n < 20 n > 19
Appendix D. Experiment text.
Instructions
Welcome to this online experiment about group decision-making in which you can earn money based on your decisions and the decisions of other participants. The experiment will take approximately 10 minutes. It is therefore important that you stay focussed during these 10 minutes. All of your responses will be kept confidential and will be used for research purposes only.
Experiment and payout
The experiment you are about to participate in consists of two parts: a 4-round choice task, and a small questionnaire. After all responses have been collected, one randomly formed group from one specific round will be selected. From this group one randomly selected participant will be chosen to be paid. Your payoff depends on your action and the actions of the other participants selected in that round. You can earn up to €50. Thus, you should think carefully when making each decision, as that decision could be the one that will be paid. If you wish to make a shot at winning some money, please make sure to leave your email address at the end of the questionnaire.
Sharing is caring / Winner takes all!
In each of the 4 rounds you will have to fill in your beliefs and choose between two actions: share or grab. Depending on everyone's decisions there are three distinguishable situations with corresponding payoffs that can arise, which are listed below.
1. All group members choose "share". Everyone receives a payoff of €30.
2. All group members except for one choose "share", i.e. exactly one group member chooses "grab". All group members that chose "share" get a payoff of €0, the group member who chose "grab" gets a payoff of €50.
3. Two or more group members choose "grab", the remaining group members choose
"share". All group members who chose "grab" get a payoff of €0, whereas all group members who chose "share" get a payoff of €5.
Example
You choose "share", and all other group members except for one also choose "share", i.e. one group member chooses "grab". Your payoff should you be selected at the end of the
experiment will be: €0 / €5 / €30 / €50
The answer to example question 1 was €0. Example 2
You choose "grab" together with one other player. All remaining players choose "share". Your payoff if you will be selected will be: €0 / €5 / €30 / €50
You are now in a group consisting of 3 players including yourself. How many group members besides yourself do you think will choose “share”?
0 / 1 / 2
You are now in a group consisting of 6 players including yourself. How many group members besides yourself do you think will choose “share”?
0 / 1/ 2 / 3 / 4 / 5
You are now in a group consisting of 9 players including yourself. How many group members besides yourself do you think will choose “share”?
0 / 1/ 2 / 3 / 4 / 5 / 6 / 7 / 8
You are now in a group consisting of 12 players including yourself. How many group members besides yourself do you think will choose “share"?
0 / 1/ 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11
Payoff scheme for all possible outcomes
Your action All remaining group members choose share
Exactly one other group member chooses grab
Two or more group members choose grab
Share € 30 € 0 € 5
This was the end of round 4. Please fill in some general information below. Gender: male / female
Age: …..
Nationality: ……….
What is the highest level of school you have completed or the highest degree that you have received? Less than high school degree / High school graduate / Associate degree / Bachelor’s degree / Master’s degree / Professional degree / Doctoral degree
Email address (only necessary if you wish to get a chance at winning money) ………
Appendix E. Average and maximum beliefs
Graph E2. Fraction maximum beliefs across gender
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Entire sample Males SIC Females SIC Males WTA Females WTA
0,25 0,15 0,18 0,11 0,16 0,25 0,20 0,18 0,11 0,19 0,50 0,15 0,36 0,32 0,29 0,88 0,70 0,55 0,53 0,64
Group size 3 Group size 6 Group size 9 Group size 12
Graph E1. Fraction average beliefs
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
Entire sample SIC Males SIC Females SIC WTA Males WTA Females WTA
0,84 0,64 0,70 0,70 0,65 0,67 0,68 0,80 0,69 0,72 0,64 0,64 0,64 0,68 0,85 0,70 0,74 0,78 0,69 0,73 0,73 0,94 0,80 0,84 0,73 0,68 0,70 0,77
Table E. Fraction “share” decisions and (maximum) beliefs across gender and treatments Entire sample SIC WTA
Males Females Males Females Males Females Fraction “share” total treatment1 0,795 0,868 0,853 0,865 0,738 0,908 Fraction “share” group size 3 0,795 0,631 0,737 0,636 0,850 0,625 Fraction “share” group size 6 0,821 0,947 0,895 0,909 0,750 1,000 Fraction “share” group size 9 0,795 0,947 0,895 0,909 0,700 1,000 Fraction “share” group size 12 0,769 1,000 0,895 1,000 0,650 1,000 Average beliefs total treatment 0,689 0,784 0,669 0,712 0,709 0,856 Average beliefs group size 3 0,744 0,816 0,685 0,725 0,800 0,940 Average beliefs group size 6 0,697 0,811 0,694 0,782 0,700 0,850 Average beliefs group size 9 0,670 0,704 0,645 0,636 0,694 0,798 Average beliefs group size 12 0,646 0,761 0,651 0,703 0,641 0,841 Maximum beliefs total treatment 0,288 0,382 0,276 0,318 0,300 0,469 Maximum beliefs group size 3 0,615 0,685 0,526 0,546 0,700 0,875 Maximum beliefs group size 6 0,231 0,421 0,316 0,364 0,150 0,500 Maximum beliefs group size 9 0,154 0,211 0,105 0,182 0,200 0,250 Maximum beliefs group size 12 0,128 0,211 0,105 0,182 0,150 0,250
1Total treatment is the average of all observations for the four different group sizes
As becomes clear from the table males choose “share” less often than females. The effect of group size on the fraction of “share” decisions is opposing for males in the two framing conditions. Females tended to only “grab” if the group size was 3. Concerning beliefs, males had lower average beliefs than females, regardless of the group size of framing condition. Lastly, females had maximum beliefs more often than males, but chose “share” more often nonetheless.
Appendix F. Kruskal Wallis test results
Table F. p-values Kruskal Wallis test comparing fraction of “share”
Entire sample SIC WTA
SIC1 0,2056
Group2 0,340 0,0391** 0,936
Maximum3 0,342 0,207 0,757
Male4 0,173 0,900 0,0502*
Age5 0,0016*** 0,126 0,0451**
All variables in the first column are tested against share, a dummy variable equal to 1 if subject chooses “share”, 0 if subject chooses “grab”.
1dummy variable equal to 1 if subject is in SIC treatment, 0 if subject is in WTA treatment 2continuous variable for the four group sizes
3dummy variable equal to 1 if subject has maximum beliefs, 0 otherwise 4dummy variable equal to 1 for males, 0 for females
5continuous variable for the age of the subject
I ran the Kruskal Wallis test (in addition to Spearman’s correlation) to test for differences between the fraction of “share” decisions and the 5 variables in my model. As can be seen from table F, there are four significant p-values. Both in the entire sample as in the WTA treatment “share” and age are significant, confirming the result that older subjects on average chose “share” more frequently. The variable group in the SIC treatment is also significant. This was to be expected as the fraction of “share” decisions for the three largest groups was a lot higher than in the smallest group. Finally, the Kruskal Wallis test confirms that in the WTA treatment males chose “share” significantly less often than females.
Appendix G. Stata commands Model
anova share SIC group maximum male age. The overall model is statistically significant (F = 2,47, p = 0.001). The variables maximum, male, and age are also statistically significant (F = 5,07, p = 0.0255, F = 3.21, p = 0.0747, and F = 2,74, p = 0,001, respectively).
Correlations
spearman share SIC spearman share group spearman share maximum spearman share male spearman share age spearman SIC group spearman SIC maximum spearman SIC male spearman SIC age
spearman maximum male spearman maximum age spearman male age
Differences between two variables kwallis share, by (SIC)
kwallis share, by (group) kwallis share, by (maximum) kwallis share, by (male) kwallis share, by (age) Regressions
xtset player
panel variable: player (balanced)
xtprobit share SIC group maximum male age, re vce(cluster player) xtset player
panel variable: player (balanced)
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