Status Quo Bias in Strictly Competitive Games
Master’s Thesis
Program: Master of Science in Economics
Track: Behavioral Economics and Game Theory
Author: Yunrui Zhang Student Number: 11736410 Date: 09-07-2018
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State of Originality
This document is written by Yunrui Zhang 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 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.
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Abstract
This paper aims to study the possible existence of status quo bias in strictly competitive games, where players are theoretically predicted to play randomly and independently, and the effects of default options on choice dependency. By means of a uniquely designed online experiment, data of individual decision-making in a repeated constant-sum game was collected and analyzed with measurements of conditional probabilities, number of runs, earnings and logit equations. Statistical tests revealed the differences between participants with and without default treatment which set their previous choice as default option. Results showed that there was general status quo bias for players both with and without default options and the differences between them were not significant in all dimensions. Though with limitations of computerization issues and sample selection, this study contributes to the extant literature about status quo bias in games in the way that it turned to a strictly competitive game setting and confirmed inertia behavior under competitive circumstances. In practice, status quo bias can make competitive players suffer from exploitation and incur tremendous opportunity costs. So, the power of status quo bias or default option should not be ignored.
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Table of Contents
State of Originality ... 2 Abstract ... 3 1. Introduction ... 5 2. Literature Review... 6 3. Methodology ... 8 3.1 Experimental Design ... 8 3.2 Procedures ... 10 3.3 Hypotheses ... 11 4. Results ... 12 4.1 Conditional Probabilities ... 12 4.2 Runs Analysis ... 15 4.3 Earnings Analysis ... 17 4.4 Logit Equations ... 20 4.5 Strategies Analysis ... 225. Conclusion and Discussion ... 25
5.1 Main Findings ... 25
5.2 Methodological Issues and Limitations ... 26
5.3 Possible Further Research ... 28
References ... 30
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Status Quo Bias in Strictly Competitive Games
1. Introduction
Status quo bias, the propensity to stay with the current situation, has been proved prevalent in various decision-making situations. Given status quo or default option, decision-makers have a large tendency to stick to the current choice. Status quo bias could also exist in games, especially the competitive games where players should play mixed strategies to reach equilibrium. The optimal way to get higher payoff in a repeated competitive game is to play as randomly as possible. Status-quo-biased players, however, might stay with their previous choices, either hoping to win another round or gambling to see things turn, which leaves them vulnerable to exploitation by the opponents. Nevertheless, researches about strictly competitive games suggested the completely opposite behavioral pattern: players changed options too often. There could be a combination of status quo and strictly competitive games to further explore inertia behavior under competitive circumstances. With treatment of default option, players are supposed to show behavioral patterns of status quo bias. If status quo bias does exist in real competitive economy, then the party with this inertia behavior would easily be exploited and suffer incessant losses. This paper therefore delves into the question about whether status quo bias exists in strictly competitive games. Using an online experiment of iterated constant-sum game with default treatment, this research through multiple tests confirms the existence of status quo bias and the influence of default option.
This paper first introduces the concept of status quo bias and its impact on economic games as well as the research methods used in previous literature, followed by the methodology part that illustrates the design of the experiment and explains its implementation in detail. Then the results part analyzes data with multiple measurements, both quantitively and qualitatively. Conditional probabilities, runs tests, earnings analysis, logit equations and strategies analysis serve for the conclusion in different dimensions. Finally, the paper ends with a summary of main findings, discussion about limitations and potential further research.
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2. Literature Review
Status quo bias was first introduced by Samuelson and Zeckhauser (1988), who designed questionnaires in which decision-making subjects faced different tasks with and without a predetermined option. When given status quo options, subjects were inclined to stay with the status quo. In public policies, positive applications of status quo bias such as default options can nudge people into better decisions. For instance, automatic enrollment substantially increased participation rates of 401(k) saving plan in the US (Madrian & Shea, 1999); the default fund in Swedish social security privatization plan performed better than the actively chosen portfolio (Thaler & Sunstein, 2009); countries implementing presumed consent for organ donation had significantly more potential donors than those with explicit consent (Johnson & Goldstein, 2003).
Status quo bias can also affect strategic behavior in economic games. Cox et al. (2013) conducted status quo treatment in fairness games and found that acts of commission overturning the status quo elicited more reciprocity from the responder. Kagel and McGee (2016) through their experiment of finitely repeated prisoner’s dilemma game found that subjects exhibited status quo bias when deciding the starting round for defection. When subjects could buy or sell the right to play a standard two-player game, Willingness-to-Accept was higher than Willingness-to-Pay for most games and for most subjects, indicating status quo bias in the ownership of games (Drouvelis & Sonnemans, 2017).
In strictly competitive games with Nash equilibrium in only mixed strategies, imperfect randomness of subjects’ strategies implied choice dependency. O'Neill (1987) designed a 4x4 zero-sum matrix for a card game experiment, in which one of the four cards should be played with probability of 0.4 and the other three with equal probability of 0.2. The unusually high number of runs (i.e., unbroken strings of the same card) indicated that subjects switched too quickly between types of cards and displayed gambler’s fallacy, which means they avoided the choice that had just won and thus their behavior went against status quo bias. As subjects played the game face to face, there
7 was sufficient involvement in competition, ensuring the naturality of the observed behavior. However, the connotation of cards like ace or joker might have attracted subjects to present them more often, or to switch from them more rapidly. And the author did not perform other statistical tests than the number of runs to verify subjects’ dependency upon previous choices. Brown & Rosenthal (1990) reexamined the data set of O’Neill experiment and confirmed by significance tests of logit equations that subjects’ choices were significantly correlated to their previous choices. Rapoport & Budescu (1992) also found that subjects alternated their choices too often in their 2x2 zero-sum game experiment with (0.5, 0.5) being the mixed NE strategy. Apart from runs analysis, they also applied sequential dependencies and conditional probabilities to figure out whether subjects could actually play random. Subjects in this card game also received direct feedback when playing face to face, and they were endowed with initial tokens to enable losing as well as enhance competition. However, due to limited memory, subjects might have only focused on last round’s result, leading to their quick switching. Moreover, the absolute symmetry of payoffs might have caused naïve playing, namely switching between cards too often. Coricelli (2001) applied O'Neil’s experimental framework and let subjects play against preprogrammed computers. Subjects who played against computers that were assigned over-alternation showed less variation in their choices. This was explained by strategic dependence, meaning that subjects were able to find counterstrategies against the computer.
Combining strengths of the above literature and avoiding weaknesses, this research designed an asymmetric game matrix so that subjects could not easily find out the optimal strategy. Subjects played against preprogramed computer that seemingly played random while responding to subjects’ decisions, so that the observed game-playing behavior would be more natural. Also, subjects were able to see results of all previous rounds on the game page; in this way, the feeling of competition could be strengthened. Furthermore, to elicit status quo, default option was applied as treatment, which adds to the current literature concerning inertia behavior in strictly competitive games. As to methodology, this research combined the above-stated analysis of
8 conditional probabilities, runs tests and logit regression to obtain a profound knowledge of status quo bias behavior.
3. Methodology
3.1 Experimental Design
An online experiment powered by CREED was designed to study the effects of status quo bias on strictly competitive games. The experiment was basically a repeated game with only mixed Nash Equilibrium strategies. In order to avoid monetary loss from the side of participants, the game was designed to be constant-sum. The game matrix is shown as follows:
C D
A 4, 0 0, 4
B 1, 3 3, 1
Participants chose between option A and option B, while the computer chose between option C and option D. In all situations, the sum of payoff was 4, which means that either the participant or the computer won 4, or the participant and the computer divided by 1-3 or 3-1. The mixed Nash Equilibrium for the game was (1/3, 2/3) for the participant and (1/2, 1/2) for the computer. The computer was programmed to respond to participants’ previous choices and play best against them. Specifically, the computer memorized the last 6 periods’ decisions of the participant and calculated the payoffs if it would have played C ($profC) and D ($profD). Then it chose C with probability $profC/($profC+$profD) and D with probability $profD/($profC+$profD). The computer strategy has the advantage of presenting seemingly randomness while reacting to the participant. The computer approximately plays Nash when the participant plays Nash, and it also corresponds to extreme behavior: the computer plays D for sure after 6 runs of A and plays C with 75% probability after 6 runs of B. The first 6 periods’ choices of the computer were fixed as CCDCDD, so that any differences between participants were caused by themselves instead of luck. The game was repeated for 90 periods.
9 In terms of default options, participants were randomly assigned to two groups: treatment or control. The treatment group in each period after the first saw their choice in the last round pre-selected. As for the control group, participants had to actively select A or B in each period. Figure 1 and Figure 2 displayed screenshots of the game interfaces for control and treatment groups respectively.
Figure 1
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Figure 2
Game Interface for Treatment Group
3.2 Procedures
After testing, the experiment link was sent online to recruit participants. 40 participants in total completed the experiment. Participants who went to the experiment page would get a number. Those with odd numbers were assigned to treatment group and those with even numbers to control group. Among those that completed the experiment, 22 participants faced default treatment and 18 were in the control group. Participants were informed before taking part in the experiment and on the first instruction page as well to open the link with laptop or computer (see A1), because the interface was not made adaptive to mobile phones. But there could still be participants that did the experiment on phones, since some participants’ data saw missing periods, which might be caused by refreshing when the page did not go on smoothly. Another reason could be that some browsers had difficulty loading the page, which remained undetected during testing. Missing periods where participants made no decisions were deleted and so some
11 participants had less than 90 periods’ decisions1.
Before starting to play the game, participants needed to read the instructions and answer two questions correctly to ensure their understanding of the game. It was stated in the instructions that two participants would be randomly selected to get paid according to their total earnings in the game; specifically, 10 points equal 1 euro. Therefore, participants were incentivized to play the game. During the 90-period game, participants clicked the button beside A or B and then clicked “submit” to enter next period. On the right side of the page, participants could see previous choices and earnings of themselves and the computer in a table, on top of which their total earnings and the computer’s earnings were highlighted, as shown in Figure 1 and 2. Two participants, one in treatment group and the other in control group went through unclosed loop and continued to more than 90 periods. The problem was caused by a bug in the ending script of the loop and was fixed after detection. Their first 90 periods’ decisions were still valid for the purpose of this experiment. After finishing 90 periods, participants were invited to fill in a short questionnaire (see A2), in which their age, gender and strategies used for the game were asked. They could also fill in their email address if they wanted to enter the draw for payment.
After the experiment, two participants were randomly drawn by random number generator and they were contacted and paid in private.
3.3 Hypotheses
Whether participants played randomly or not are indicative of choice bias, either towards status quo or over-alternation. So, the first hypothesis relates to the randomness of participants’ game-playing behavior.
Hypothesis 1:
H0: The choice sequences of participants were generated in a random manner.
H1: The choice sequences of participants exhibited dependencies upon other factors.
Rational players with consistent preferences should play the game with same
1 In control group, 5 participants missed periods and completed 85, 88, 83, 87, 87 periods respectively.
12 strategies regardless of the presence of default options, which draws forth the null hypothesis of Hypothesis 2. The alternative hypothesis assumes irrational game-playing behavior dependent upon default options.
Hypothesis 2:
H0: There was no difference in measurements of choice dependency between control
and treatment groups.
H1: Measurements of choice dependency were different between control and treatment
groups.
The measurements for the two hypotheses include conditional probabilities, number of runs, earnings and coefficients of logit equations.
4. Results
According to 37 questionnaire responses2 after the game, the average age of the
participants is 23.5. 17 participants are male and 20 are female. Nationalities of the participants range from Chinese, Dutch, British, Australian, etc. Regressing treatment against age, gender and nationality, the F-test was not rejected (p > 0.05), which means there was no statistical indication that the assignment of treatment was conditional on any of these background characteristics. This supports the assumption that the assignment was random.
4.1 Conditional Probabilities
Across all participants, option A was played with probability of 40.33%, more than predicted by game theory. However, the standard deviation was rather high, being 13.91%, as participants’ strategies differed a lot and there were two extreme responses where option B was chosen in every period. Participants in control group on average chose A with probability of 42.85%, and those in treatment group chose with 38.27%. If participants made decisions independently for each period, conditional probabilities should be the same regardless of their order in this Bernoulli process, where option A
13 was represented as 0 and option B as 1. Conditional probabilities should also not differ between treatment and control for rational participants. Denote the conditional probability that participants chose A after an A by P(0|0) and that after two consecutive As by P(0|0,0), etc. Since a sequence of four As was already rare, probabilities conditional on more than three same choices were not calculated. Table 1 summarizes the mean conditional probabilities of selecting A(B) on condition that previous decisions were all A or all B over control and treatment groups, compared to the mean probabilities of A(B), which shed light on when participants stayed with previous decisions and when they changed their options.
Table 1
Mean Probabilities and Conditional Probabilities over Control and Treatment Groups
Control Treatment Control Treatment
P(0|0) 50.60% 61.01%a P(1|1) 62.22% 73.09% P(0|1) 37.78% 26.91% P(1|0) 49.40% 38.99% a P(0|0,0) 49.89% 57.59%a P(1|1,1) 57.78% 69.32% P(0|1,1) 42.22% 30.68% P(1|0,0) 50.11% 42.41% a P(0|0,0,0) 46.13% 47.55%a P(1|1,1,1) 52.24% 65.51% P(0|1,1,1) 47.76% 34.49% P(1|0,0,0) 53.87% 52.45% a P(0) 42.85% 38.27% P(1) 57.15% 61.73%
a Two participants in treatment group chose B in all periods, so probabilities conditional on having
chosen A were undefined. These mean probabilities were calculated over 20 participants.
After choosing one option for the last period(s), the conditional probabilities of choosing it again was generally higher than choosing the other option in the current period, the differences being more distinct for treatment group. The conditional probabilities of choosing A became lower in both control and treatment groups as the consecutive number of latest As increased. The same was also true for option B. On the other hand, the conditional probabilities of choosing A became higher as the consecutive number of latest Bs increased. And vice versa. Though participants were
14 presented with all their previous choices, they tended to focus more on the last period. It was very likely that participants would choose A(B) if their last period’s choice was A(B), while after a sequence of three A(B)s, participants were more likely to switch to the other option. In general, the conditional probabilities of staying with the same option were higher than the mean probability of that option throughout the whole game in both groups, except for P(1|1,1,1) in control group. On the contrary, the conditional probabilities of switching to the other option after choosing A(B)s were lower than the mean probability of A(B), except for P(0|1,1,1) in control group. Participants were preliminarily proved to be status quo biased.
As can be directly seen from conditional probabilities, participants in treatment group were more inclined to stick to their previous choices than those in control group. To study the effect of default options, tests were performed to compare P(0|0), P(0|0,0), P(0|0,0,0), P(1|1), P(1|1,1), P(1|1,1,1) between control and treatment groups. Due to small sample and non-normal distribution, nonparametric test - Wilcoxon rank-sum test was applied. Table 2 shows all the results of rank-sum tests.
Table 2
Results of Wilcoxon Rank-sum Tests for Conditional Probabilities between Control and Treatment Groups
Prob > |z| Prob > |z|
P(0|0) 0.0555 P(1|1) 0.0375
P(0|0,0) 0.2479 P(1|1,1) 0.0552
P(0|0,0,0) 0.8261 P(1|1,1,1) 0.0726
Note: The number of observation for P(0|0), P(0|0,0), P(0|0,0,0) in treatment group was 20, exclusive of two participants that did not choose any A in all periods.
The second null hypothesis that there was no difference between treatment and control groups regarding conditional probabilities of same choices was rejected at significance level of 10% in four cases, three of which were for option B. The differences in conditional probabilities of option B were more significant than those of
15 option A. And the differences in first-order conditional probabilities of both options were more significant than second- and third-order conditional probabilities. Thus, the presence of default option that preselected their last period’s choice made participants more status quo biased, especially for option B which guaranteed them at least one point. And participants were more inclined to make choices conditional on the last period’s decision.
4.2 Runs Analysis
A run in a Bernoulli sequence is a series of consecutive 0 or 1 and the number of runs can be used to determine whether the elements of the binomial sequence are mutually independent. In the case of this experiment, if participants made their choices independently between two options in every period, their sequence of choices should stand the runs test. The expected number of runs for a Nash-playing participant should be 41, with the expected number of playing A being 30 and that of playing B being 60 (2×30×60
90 + 1 = 41 ). The total number of runs was counted for each participant
individually, the frequencies of which were then categorized into groups to form frequency distributions. Table 3 exhibits the frequency distributions of number of runs for control and treatment groups separately, followed by Figure 3 showing the cumulative frequency distributions.
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Table 3
Frequency Distributions of Number of Runs over Control and Treatment Groups
Number of Runs
Control Treatment
Frequency Percentage Frequency Percentage
1-8 0 0.00% 3 17.65% 9-16 1 7.69% 2 11.76% 17-24 0 0.00% 3 17.65% 25-32 4 30.77% 3 17.65% 33-40 3 23.08% 3 17.65% 41-48 3 23.08% 3 17.65% 49-90 2 15.38% 0 0.00% Total 13 100.00% 17 100.00%
Note: Participants who had missing periods (5 in control group and 5 in treatment group) were omitted.
Figure 3
Cumulative Frequency Distributions for Control and Treatment Groups
The number of runs for each individual discriminates between control and treatment groups. The frequency distribution for control group skewed more towards the expected number of runs than that for treatment group. By using two-sample
0.00% 10.00% 20.00% 30.00% 40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 8 16 24 32 40 48 90 Cum ula tiv e F re qu ency Number of Runs control treatment
17 Kolmogorov-Smirnov test, the differences in cumulative frequency distributions between control and treatment groups can be detected. According to the results (see A3), the null hypothesis that two cumulative frequency distributions were the same could not be rejected (p > 0.05), which might be attributed to small sample.
In order to gain more insight into individual behavior regarding number of runs, a two-sided Wald–Wolfowitz runs test was conducted upon each participant. Denote the number of 0 (option A) as 𝑛0 and the number of 1 (option B) as 𝑛1, the expected
number of runs for each participant is
𝑅̅ = 2𝑛0𝑛1 𝑛0+ 𝑛1
+ 1 and the expected variance is
𝑠𝑅2 =
2𝑛0𝑛1(2𝑛0𝑛1− 𝑛0− 𝑛1) (𝑛0+ 𝑛1)2(𝑛0+ 𝑛1− 1)
Then construct test statistics
Z =𝑅 − 𝑅̅ 𝑠𝑅
where R is the actual number of runs for each participant.
In control group, the first null hypothesis that the choice sequence was produced in a random manner was rejected at 5% significance level in 6 out of 18 (33.33%) cases; in treatment group, the first null hypothesis was rejected in 12 out of 20 (60%) cases. Participants with default options seemed to have made choices more dependently and those without default more randomly. However, by means of two-sided z-test, the second null hypothesis that there were differences in proportions of rejections between two groups could not be rejected at 5% significance level (see A4).
4.3 Earnings Analysis
If participants played the game with Nash equilibrium mixed strategy, their expected payoff in each period should be 2. In response to that, the computer would also play Nash approximately, earning expected payoff of 2. Therefore, in equilibrium total earnings of the participant and the computer should be the same. Since the sum of earnings in each period was the same, participants would be able to gain more than the
18 computer in total if they played better against the computer. In this experiment, however, only 11 out of 40 (27.5%) participants gained no less the computer, with 1 participant earning exactly the same as the computer. Out of the 11 participants, 7 were from control group. In general, participants were not able to beat the computer and the reason could be that participants changed their choices too often or too little. In order to verify which reason was valid, measurement of either bias was constructed by using actual earnings and hypothetical earnings. Hypothetical earnings were the earnings that could have been gained by the participant if he/she had chosen the other option. Each participant’s earnings in all periods were divided into two categories: one for periods where participant changed options, the other for periods where participant stayed with last period’s choices. For each category, the sum of actual earnings was compared to that of hypothetical earnings. In the changing category, hypothetical earnings were those that could have been gained if participants had not changed their choices. If participants alternated too often, they would have less actual earnings than hypothetical earnings. The over-alternation bias was calculated as difference between actual and hypothetical earnings divided by the number of periods where participants changed. In the staying category, hypothetical earnings were those that could have been gained if participants had not stuck to the status quo. If participants were status quo biased, they would have less actual earnings than hypothetical earnings. The status quo bias was calculated as difference between actual and hypothetical earnings divided by the number of periods where participants stayed. Were there not any bias towards changing too often or too little, the bias measurement should be 0. Table 4 summarizes the mean over-alternation bias and status quo bias for control and treatment groups.
Table 4
Mean Values of Bias Measurement over Control and Treatment Groups
Control Treatment
Over-alternation Bias 0.3092 0.6520a
Status Quo Bias -0.3445 -0.5259
a Two participants that chose B in all periods and made no changes were omitted, so the mean was
19 Only 5 out of 18 (27.78%) participants in control group and 2 out of 20 (10.00%) in treatment group had less actual earnings than hypothetical earnings in changing periods. The positive mean value of over-alternation bias infers that on average participants actually gained more when they did change their options. Therefore, there was little tendency to alternate too often. 13 out of 18 (72.22%) participants in control group and 16 out of 22 (72.73%) in treatment group had less actual earnings than hypothetical earnings in staying periods, meaning that had participants not stayed with their previous choices, they could have earned more. The negative values of status quo bias measurement indicated that on average participants showed tendency to stick to status quo and therefore incurred high opportunity costs. In comparison, treatment group had higher absolute values of both bias than control group, which implies that status quo bias generated more severe consequences, namely less earnings in treatment group. In order to see whether default options imposed significant effects on both bias, rank-sum tests were applied, the results of which are shown in Table 5.
Table 5
Results of Wilcoxon Rank-sum Tests for Bias Measurement between Control and Treatment Groups
Prob > |z|
Over-alternation Biasa 0.2726
Status Quo Bias 0.7857
a Two participants that chose B in all periods and made no changes were omitted.
The second null hypothesis that there was no difference in either bias between control and treatment groups could not be rejected. So, the presence of default options that preselected participants’ previous choices did not significantly make participants stay with the status quo more often compared to those without default options. And the absence of default options did not significantly make participants alternate their choices more often than those with default options. In general, participants behaved quite
20 similarly regardless of the existence of default choices, that is, more biased towards status quo and less towards too much alternation.
4.4 Logit Equations
From the above analysis, participants did not make choices independently for each period. In order to identify the sources of failure, this part applies logit regression to each individual’s choice sequence. The estimating equation was inspired by Brown and Rosenthal (1990) who tested choice dependencies of O’Neill’s experimental data (1987). Since choices were binary in this experiment, estimating log odds of one choice makes no essential difference to the other. To be consistent with option A being 0, B being 1 for the participant and option C being 0, D being 1 for the computer, the logit equation goes as follows:
logit(B) = 𝑎0+ 𝑎1𝑙𝑎𝑔(𝐵) + 𝑎2𝑙𝑎𝑔2(𝐵) + 𝑏1𝑙𝑎𝑔(𝐷) + 𝑏2𝑙𝑎𝑔2(𝐷)
+ 𝑐1𝑙𝑎𝑔(𝐵)𝑙𝑎𝑔(𝐷) + 𝑐2𝑙𝑎𝑔2(𝐵)𝑙𝑎𝑔2(𝐷)
where the dependent variable logit (B) is the logarithm of odds ratio of B. The independent variables are the first and second lags of both the participant and computer’s choices, the interaction between first lags of the participant and the computer’s choices as well as the interaction between second lags, which account for the potential condition that previous successes predicted current choice.
The logit equation was run for each individual, and likelihood-ratio tests were performed to isolate the prediction power of separate indicators. If the participant made each choice independently, all the variables should not have explanatory power on the probabilities of choosing B. The results are displayed in Table 6.
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Table 6
Results of Significance Tests for Logit Equations of Option B
Null Hypothesis:
Number of Rejections of the Null Hypothesis at 0.05 Significance Level
Control Group Treatment Groupa
1) 𝑎1, 𝑎2, 𝑏1, 𝑏2, 𝑐1, 𝑐2 = 0 10 17
2) 𝑎1, 𝑎2 = 0 5 9
3) 𝑏1, 𝑏2, 𝑐1, 𝑐2 = 0 9 8
4) 𝑐1, 𝑐2 = 0 1 3
5) 𝑏1, 𝑏2 = 0 8 8
a Two participants that chose B in every period were not included. There were three cases where some
of the variables perfectly predicted success. Those variables were deleted from the model to maintain equivalent number of observations, so some coefficients of the null hypotheses did not exist and rejections were based on the rest of the variables.
As can be inferred from the first test, the logit equation in its whole fit the choice sequences of numerous participants, especially those from treatment group, suggesting that there existed internal or external factors that influenced their decisions, which is in support of alternative hypothesis 1. The second test implied that more participants in treatment group relied on their own previous choices, which could be seen as internal factors, so there were more rejections of the null hypothesis that the first two lags of own choices were extraneous to the current choice. The third test was run to see if choices of the opponent (i.e. the computer) and results in the last two periods jointly had a significant effect, in other words, the effect of external factors. Nearly half of both groups were dependent upon the external factors. Test four and test five distinguished between the impact of previous results and opponent’s previous choices. Both groups had equal number of rejections against the null hypothesis that the opponent’s previous choices were irrelevant. Meanwhile, a few more participants in the treatment group depended on previous results. To conclude, participants with default treatment were more dependent when making decisions, especially upon their own previous choices, which in another way confirms the effectiveness of default options.
22 Returning to the logit regression model, the significance and value of individual variables also illustrate choice dependency. Focusing on coefficients that were significant at 5% level, those of the first lagged term of the participant’s own choices have positive values in general, which means that if the participant had chosen option B in the last period, it would increase the likelihood that B would be chosen again in the current period. The negative coefficients of the second lagged indicators for own choices including one that perfectly predicted not choosing B in treatment group suggested that these participants tended to change every two periods. As to the opponent’s choices, if the computer chose option D in the last period, most participants were inclined to choose B as if the computer would stick to option D in the following period, and this behavior was more consistent in treatment group. The same behavior also happened in accordance with the computer’s choice in the second last period for treatment group, as indicated by the positive value of 𝑏2 . The results of previous periods also predicted choices for some participants. Including the one that completely determined choosing B in control group, the positive results in the last period encouraged participants to choose B again, while fearing that wining would not last for long, after two periods of winning participants were more likely to switch options, as indicated by the negative value of 𝑐2.
4.5 Strategies Analysis
In the treatment group where 15 participants described their strategies used for the game, four of them played “random” or “guessed”. Indeed, these four participants all stood the runs test, indicating that their actual number of runs was approximately the same as expected for a random manner. Besides, three of them showed no significance in the logit regression model and achieved statistical randomness. However, as inferred from conditional probabilities, two of them were more likely to switch after same choices, while the other two were more likely to stay. Still, one of them could be qualified as playing unconditionally since the differences of conditional probabilities were quite small. Except this participant who also obtained 0 in status quo bias measurement, all
23 other three showed negative values, meaning that they could have earned more if they had not stuck to previous choices. Six participants tried to figure out the computer’s strategy, of whom three scored higher earnings than the computer. One successful participant believed that option B had a higher chance of giving 3 points and seemed to be more consistent, while the opposite was true for option A. He played status-quo-biasedly with high conditional probabilities on same choices and too few runs. But meanwhile, he played A one third of the times and B two thirds, which was exactly the Nash mixed strategy, and obtained high positive value of over-alternation bias. Each time of changing options yielded good results for this participant, serving as one reason for his higher earnings than the computer. In addition, from his logit equation, one could only tell some prediction power of first-lag interaction, meaning that he played quite random. The other two reflected on the computer’s reaction through previous results and behaved in a way similar to backward induction. Both of them played approximately Nash, with high conditional probabilities on same choices, too few runs, significant dependencies upon self or computer’s choices, but higher hypothetical earnings in both changing and staying periods. Proper timing of changing options or staying with previous options and some luck yielded high earnings for these two participants. The strategies described by some participants also cast light on why they stayed with previous choices. Participants stuck to option B for the reason that option B was “safer and less emotional” or by choosing B one “got something always”; in the meantime, one would stick to option A because the game was thought to be “random”. Also, it seemed reasonable to stick to the other option if one option gave “less points” several times. Switching from option B to option A happened out of boredom or curiosity, or simply because one “felt like it”, while the other way of switching was because “the result was disastrous” or “playing A seemed quite bad”.
The strategies explained by 18 participants in treatment group looked quite different. Only two participants claimed to “have chosen randomly”. One of them in fact was more status-quo-biased while the other played almost random. There were two extreme cases where the participants chose option B in all periods, displaying absolute
24 status quo bias. One of them regarded the computer as payoff maximizer and refused to choose any A because he thought the computer would in response choose D and earn 4 points “all the time”; in comparison, option B would guarantee 1 point “all the time” as the computer would maximize payoff by choosing C. This was naïve because the computer could never know the participant’s choice before it was made. Moreover, to think one step further for backward induction, if the computer chose one option all the time, it would also be a good reason to exploit the computer. Refusing to switch options could only result in being exploited by the computer and avoiding any chance of getting the highest payoff of 4. The other participant wanted to “get something every time” so she chose B in every period, eliminating any probability of getting nothing. Five participants tried to find the pattern of computer’s choice. One of them first calculated the expected payoff of each option and decided to choose B all the time as she thought the computer’s choice was “completely random”, but later she realized that the computer tried to “understand her choice pattern” and so she alternated choices according to her points gained. Then she found out that focusing on her earnings affected her randomness and started “clicking randomly” in later periods. Another participant stated that at the beginning he often changed strategies because he thought the computer was “learning from his choices”. He claimed to have figured out the computer strategy in the end, which could be true as his actual earnings in last 20 periods were twice as many as hypothetical earnings. The other three participants incorrectly considered the computer to choose without pattern or to choose D more often overall. Unlike participants in control group, those with default treatment tended to explain more in strategies about when they switched option. Interestingly, six participants pointed out that they changed options based on whether they were winning or losing. Keeping one option until one was losing seemed to be a common strategy. Switching also took place when participants wanted to see if one would “get a different result”, or when the computer changed options, or at a strategically regular interval of “every 2/3 turns”.
25 Quite a few participants in both groups considered option B as the safer one and chose B or stayed with B more often. Several participants in both groups found out that the computer reacted to their choices and playing random was the optimal strategy. Though the fact that the computer would respond to participants’ behavior was clearly stated in the instruction, still some participants believed that the computer played purely random, which was unexpected. In line with the data, participants in control group played more randomly and also more strategically, suggesting that they paid more attention to the choices that they made. This can also be shown by the difference between average total time spent on the experiment: the mean time spent for control group was more than four times the mean time spent for treatment group. Participants in treatment group tended to continuously click on “submit” with the same option and only occasionally think about when they should change options. The default option did indeed make a difference on how participants played the game.
5. Conclusion and Discussion 5.1 Main Findings
The general existence of status quo bias in strictly competitive games can be proved through this experiment of iterated constant-sum game, though the distinction between players with and without default options was not statistically conspicuous. From the aspect of conditional probabilities, strategically random players should make each decision unconditionally upon previous choices, within the framework of the mixed Nash equilibrium strategy. Whereas, results showed that participants were more likely to choose the same option after having chosen that option in previous period(s). The differences between participants in control and default treatment groups were not well significant, with treatment group being a little more status quo biased than control group. The runs analysis further explored the randomness of participants’ game-playing behavior, using number of runs as measurement. The cumulative frequency distribution of number of runs for control and treatment group did not differ significantly, but larger number of rejections in runs tests for treatment group in a way verified the fact that
26 participants with default options changed too little. Observed from earnings in periods where participants changed or stayed, participants in this experiment on average gained more than they could in changing periods but less than they could in staying periods. Adhering to the status quo caused great opportunity costs for both control and treatment groups. To gain more insight into the sources of choice dependencies, logit equations with internal and external factors as dependent variables were regressed for each participant. The logit regression model fitted quite a few participants overall, implying that there were universal dependencies for decision-making in the game. Internal factors of own previous choices exerted more influence on participants with default treatment, while both groups had some participants that relied on external factors such as the computer’s choices and results of previous periods. Specifically, one’s own previous choice, computer’s previous choice and results in the last period positively predicted current decision. Lastly, what the participants recalled in the strategy part of the questionnaire was basically reflected in the data. Participants would also rely on intuition, emotion and belief when making decisions.
5.2 Methodological Issues and Limitations
The constant-sum game in question has the disadvantage of weak sense of loss, since the participants could not lose points throughout the game. Their loss incurred in this game should be referenced to the earnings of the computer. However, the strategy part of the questionnaire indicated that some participants played the game with reference point of their own earnings, since they went for option B much more often, which guaranteed at least one point. One point for them generated the same feeling of winning something. This could result in less attention paid to the game and extreme behavior of sticking to option B for more-than-normal periods. Furthermore, as reasoned by some participants, option B was safer, so risk preferences could also have played a role in their decision-making behavior. This sets forth the question of whether the design of the game matrix is proper for studying behavior under competitive circumstances. But in real economic world, there are situations where the success of one party does not
27 impose loss on the other party and risk attitudes in some way also drive inertia behavior under competition. Taking grabbing market share as an example, one party chooses to implement a new aggressive market strategy hoping to gain more share, while the other adheres to the conservative strategy that at least ensures a certain share. Their changing or staying behavior resembles that observed in this constant-sum game. Another issue concerning the game is about belief, that is, what the participants believed the computer was doing. Quite a few participants tried to learn the computer strategy from previous results, but wrong beliefs like computer played purely random or that it systematically played one option more led to suboptimal behavior. The instructing statement that the computer tries to play best against the participants might have been ignored by some participants.
The form of playing against computer online also poses some restrictions on the experiment. Firstly, O’Neill (1987) pointed out that preprogrammed computers could not represent how natural opponents would play in a game, so that choices of the participants might not be representative. Computer in this experiment had fixed memory of last 6 periods’ results throughout the whole game, which was indeed not natural as human. But the computer strategy was made to react with proper randomness in order to resemble natural opponent. In addition, on the page of the game, participants were given all the results of previous periods. A rational and strategic decision-maker could learn from computer’s choices as from a natural opponent in the real world with transparent information. Secondly, not seeing a real opponent in front would lessen the feeling of involvement in competition (O’Neill, 1987), thus leading to loss of attention during the experiment. To deal with that, earnings of the participants and those of the computer were placed on top in the results table. As recalled in the questionnaire, participants did care about “lost” or “won” when they made decisions. This is a sign showing that they had the feeling of competing with the computer. Thirdly, as explained above in experimental procedure, some participants had problems loading the page and missed periods due to page refreshing, which could also lead to distraction. Last but not least, in treatment group, participants only needed to click “submit” if they stayed with
28 previous choice, while in control group they needed to actively select and then confirm, so click twice. This has made the treatment group more status quo biased than they would be if they had to click their choices before submitting. The powerfulness of the default option can be utilized in real life to nudge people into better decision-making.
Limited number of participants, most of whom were university students is another problem compromising the robustness of results. And there lacks a question about academic background in the questionnaire, which might be explanatory for different choice patterns.
With regard to incentives, this experiment used random incentive system to determine monetary payoff for the participants. Random incentive system was proved to elicit true preferences (Starmer and Sugden, 1991) and have no contamination effects (Cubitt et al., 1998) in within-subject decision-making tasks. As for between-subject design, Baltussen et al. (2012) found that between-subject random incentive treatment had a negative impact on risk aversion but the authors elucidated that the lower risk aversion was caused by larger winnings in preceding lottery games. Outcomes of previous tasks dominated the reasons for risk aversion bias, hence similarly in this experiment of iterated games where previous results mattered, the random incentive had little influence on how subjects perceived the riskiness of each option.
5.3 Possible Further Research
As it could make a difference in game-playing behavior when absolute loss instead of relative loss would occur, the payoffs in the game matrix could be modified so that participants might incur real loss. For instance, subtract 2 from each payoff in the matrix to make the game zero-sum. In this case, to avoid negative earnings on participants’ side, the initial endowment should be set no less than 168, since the most extreme negative outcome would be that participants chose A in every period, earning 0 point in the first 6 periods and -2 in all following 84 periods. This modification will enhance the feeling of competition and also reduce the effect of risk preferences. If each option has the chance of loss, then less opportunistic behavior like “getting something always”
29 will come about. In this setup, one could expect less status quo bias or even more over-alternation bias. More complicated game matrix with more options is also a good way of further exploring inertia behavior under competition. With more options at hand, participants might be less inclined to think thoroughly and find optimal strategy, so more status quo bias would be expected.
In this experiment, participants had to make choices in the first period whether they were in control or treatment group. It could also be interesting to apply default options in the very first period. With different results of the first period, winning participants might be more status quo biased than the losing participants if they do not change the default. This, however, is based on the fact that the first 6 periods’ choices of the computer are fixed, so defaulting A or B gives disparate results. Another modification could be to randomize the computer’s first choices. On this condition, individual behavior might diverge depending on their results in the first periods.
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References
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https://www.researchgate.net/publication/2387128_Strategic_Interaction_In_It erated_Zero-Sum_Games.
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Cubitt, R. P., Starmer, C., & Sugden, R. (1998). On the validity of the random lottery incentive system. Experimental Economics, 1 (2), 115-131.
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Johnson, E. J. & Goldstein, D. (2003). Do defaults save life? Science, 302 (5649), 1338-1339.
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Madrian, B. C. & Shea, D. F. (2001). The Power of suggestion: inertia in 401(k) participation and savings behavior. The Quarterly Journal of Economics, 116 (4), 1149-1187.
O’Neill, B. (1987). Nonmetric test of the minimax theory of two-person zerosum games. Economic Sciences, 84, 2106-2109.
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31 Samuelson, W. & Zeckhauser, R. (1988). Status quo bias in decision making. Journal
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Appendix
A1. Instructions Introduction:
This experiment is designed for a desktop or laptop computer: please don't use a phone!
Welcome to this online experiment about decision-making. Thank you for your participation and please read the instruction carefully. You can earn money according to your performance in the experiment. After the experiment, two of you will be randomly selected and contacted in private to get paid.
In this experiment, you will make a series of decisions. For each decision round, you will play against the computer that tries to compete with you. The computer's payoff will not be paid. Your monetary payoff will depend on your total points earned throughout the experiment. After the experiment, your points will be converted to euros. Specifically, 1 point equals 10 cents.
Game:
In each decision-making round, you and the computer will play a simple game. You choose between A and B, while the computer simultaneously chooses between C and D. After you and the computer take the move, you will immediately see the choices and earnings of you and those of the computer for that round. And the choices and earnings of you and those of the computer in all previous rounds will be shown on the right side of the webpage. The table below summarizes the points to you and the computer for each combination of decisions you and the computer make in a single round. Your choices and corresponding points are shown in green, and those of the computer are shown in blue.
C D
A 4, 0 0, 4
B 1, 3 3, 1
33 depends on the move that you and the computer take. For example, if you choose A and the computer chooses C in one round, you will get 4 points and the computer will get 0 point. Note that this example does not have any implication for the choices you should make in the experiment. In all four situations, the sum of points you and the computer get is 4, so either you win 4 or the computer wins 4, or the 4 points are divided into 1-3 or 1-3-1. The computer will try to play best against your choice and win as many points as possible. You can see your total earnings and those of the computer during the experiment. If you are one of the two randomly selected to get paid, your total earnings will be paid out at the rate that 1 point equals 10 cents.
Before we start the game, we will ask you a few questions to check your understanding.
Question:
1. How many points do you get if you have chosen B and the computer has chosen C? (Answer: 1)
2. After 10 rounds, the computer has earned 18 points, how many points have you earned? (Answer: 22)
A2. Questionnaire
Thank you for your participation in the experiment. You are kindly invited to fill in a short questionnaire and your information will only be used for this research. If you want to enter the draw for payment, please fill in your email address below.
Age: _____ Gender: _____ Nationality: _____
What is your strategy when playing the game? _____ E-mail address: _____
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A3. Kolmogorov-Smirnov Test Results
Two-sample Kolmogorov-Smirnov Test for Equality of Cumulative Frequency Distributions
Smaller Group Differences P-value Exact
Control: 0.4286 0.276
Treatment: 0.0000 1.000
Combined K-S: 0.4286 0.541 0.575
Note: Ties exist in combined dataset; there are 9 unique values out of 14 observations.
A4. Test for Proportions of Runs Test Rejections between Control and Treatment Groups
Control group Treatment group
𝑛1 = 18 𝑛2 = 20
Number of rejections 𝑦1 = 6 Number of rejections 𝑦2 = 12
𝑝1 =1 3 𝑝2 = 0.6 H0: 𝑝1 = 𝑝2 H1: 𝑝1 ≠ 𝑝2 𝑝̂ =𝑦1+ 𝑦2 𝑛1+ 𝑛2 = 6 + 12 18 + 20= 9 19 Z = (𝑝1− 𝑝2) − 0 √𝑝̂(1 − 𝑝̂)(𝑛1 1+ 1 𝑛2) = 1 3 − 0.6 √ 9 19 (1 − 9 19)( 1 18 + 1 20) = −1.64 > −1.96
