Causality and self-signaling in economic games

Causality and Self-Signaling in Economic Games

Matthew Cashman

A.B. Chemistry & Philosophy Hamilton College, 2008

SUBMITTED TO THE SLOAN SCHOOL OF MANAGEMENT IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MANAGEMENT RESEARCH

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

FEBRUARY 2020

©2020 Massachusetts Institute of Technology. All rights reserved.

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Causality and Self-Signaling in Economic Games

by

Matthew Cashman

Submitted to the Sloan School of Management on January 17, 2020 in Partial Fulfillment of the Requirements for the Degree of Master of Science in

Management Research

ABSTRACT

Our ability to cooperate is one of the cornerstones of our success as a species, and the story of how humans have been able to put aside immediate personal gain in favor of a longer view is widely studied. We add to this literature by exploring certain seemingly irrational behaviors observed in economic games. This is a set of behaviors that look like misunderstandings of causality, and which are irrational or must be the result of a misunderstanding on a classical economic reading. However, modes of cognition such as those reflected in self-signaling theory may serve to explain how the seemingly irrational might sometimes be quite sensible. We elicit these behaviors using real-time multiplayer economic games and suggest mechanisms whereby players may incorporate the value of receiving certain signals themselves into their utility calculations, thus making for rational behavior-and rational inference-in cases where it is not obviously so. In particular, we demonstrate order effects in economic games with no information flow and we show systematic biases in estimates of what one's partners have done versus the population. Both of these phenomena are consistent with a combination of self-signaling and a limit on the direction of inference in time.

Thesis Supervisor: Drazen Prelec

Title: Digital Equipment Corp. Leaders for Global Operations Professor of Management Professor of Management Science

Professor of Economics

1 Introduction

Years of scholarship devoted to understanding the roots of our species' remarkable ability to cooperate have resulted in an extensive literature offering compelling accounts of how that behavior arose (e.g. Rand & Nowak 2013, Nowak 2006, Axelrod & Hamilton 1981). These

explanations are ultimate explanations; they are forces deep in our evolutionary past that motivate what manifests much closer to us as features of modern human psychology. Precisely how these motive forces leading to cooperation have shaped mechanisms found in modern human psychology, and exactly what those mechanisms are, is a field still ripe for exploration. One potential mechanism in our present-day psychological apparatus that has yet to be fully explored is the idea that we might guide our behavior not just based on the direct benefits of that behavior but also based on a sort of psychological utility feedback loop. We may, in fact, do certain things in part because we wish to know that we have done them. We might also use this phenomenon to make valid inferences about what other people have done. These sorts of strange feedback loops could help explain many puzzling phenomena we observe, from the domain of financial decision-making to that of morality. The underlying evolutionary game theory may be the same in aggregate, but the proximate psychology and therefore the expected behavior may shift noticeably should these mechanisms be at play. What if we choose healthy food in part because we want to believe in our own resolve, rather than just because we believe

it is good for us? What if we avoid taking advantage of others in part because we don't want to learn that we're opportunists, instead of a simple fear of retribution? In either case we might predict that people are interested in what a certain act allows them to learn about themselves (and, via that knowledge, about others) in a way not captured by more conventional theories.

2 Background

There has been an appreciation of these sorts of feedback loops and their ability to influence behavior for some time. Perhaps most notably, Quattrone and Tversky (1984) describe an experiment in which people believe one of two things: either that being able to hold their arm in cold water for a long time (a painful task) signifies good health, or that it signifies poor health. In this experiment, subjects tended to modify their behavior such that they received the pleasurable message, the message that they were in good health, whether that message was dependent on more or less pain tolerance. That is, subjects were motivated to keep their arms in the water either shorter or (more indicatively) longer because doing so allowed them to receive a pleasurable message, a message that had positive utility.

In the same paper Quattrone and Tversky describe an experiment that analogizes to voting. They examine the diagnosticity of one's own actions (vote for a candidate or not) with respect to the behavior of others. The theory they examine depends on meaningful correlation of behavior among a certain group of voters that is not present in a competing faction, which they refer to as being "like-minded". That is, if A-voters tend to behave alike in comparison to B-voters, and I am an A-voter, and I observed that I actually went through with voting for candidate A, then I can infer that other A-voters are also likely to do so. They investigate this

with a clever manipulation that hinges on voters' expectations of who will swing the vote: it will either be the turnout of registered "A" and "B" voters, or it will be turnout in non-aligned

voters (A and B-voter turnout expected to be the same). When taking the viewpoint of A-voters, subjects in this vignette study suggested higher turnout among A-voters when winning the election depended on A-voter turnout relative to the condition where non-aligned voters were the swing vote. Quattrone and Tversky found that, in the condition where A- vs. B-voters

were expected to be the swing vote, subjects when voting A estimated a significantly higher chance of A winning than when abstaining, relative to the non-aligned swing vote condition. Voting, in this case, seems diagnostic of outcomes. The authors also note a correlation: in the A/B swing vote condition, subject willingness to vote and diagnosticity of voting correlate (when considering both whose party has higher turnout and the likelihood of one party winning).

A 1992 Shafir and Tversky paper reports a disjunction effect among subjects playing a series of

single-shot Prisoners' Dilemmas (PDs). The authors manipulated whether the subjects had information about others' moves: in the case where subjects knew others' moves, cooperation was considerably lower (16% when Cooperate is played by the other, 3% when Defect) than in the case where the moves were unspecified (37% overall). This demonstrates a violation of Savage's sure thing principle (Savage 1954): players know that their counterpart's move is either Cooperate or Defect, and in the case of Cooperate they cooperate at a 16% rate, and in the case of Defect 3%. Given knowledge about their behavior in either state we know will ultimately obtain, the Sure Thing Principle states that they should prefer to cooperate at some rate in between 3% and 16%. But they do not, raising the specter of some sort of violation of causality in thinking or behavior. Among many explanations, the subject could, for instance, be looking to his or her own behavior for clues about the behavior of others. However, this

manipulation does not reliably specify to the subject whether the other player's move has been made already or not. The subject only knows that the other player's move is in the past when he or she is in the "bonus" condition, and receives information about the other player's move.

In the regular condition the subject does not know whether their counterpart has made a decision or not.

Self-signaling theory as we will treat it was formalized by Bodner and Prelec (2001). In brief, Bodner and Prelec suggest that an agent's utility can be described as being made up of two parts, outcome utility and diagnostic utility. Outcome utility is a measurement of the expected value of the causal consequences of an action. It is equivalent to utility in classical economic theory. Diagnostic utility is the utility the agent receives from learning about a change some poorly-known underlying disposition, such as generosity, propensity to alcoholism, or persistence:

Utility of choosing x from C= Outcome utility of x + Diagnostic utility of choosing x from C Bodner and Prelec formalize this as so:

Total utility of choosing action x from set C, U(x, C, 0), is equal to u(x, 0), outcome utility (classic economic utility), plus diagnostic utility, where x is the chosen action, 6 the latent parameter representing a poorly-known underlying disposition (such as persistence), f(6Ix, C) is an interpretation function which serves to update self-image given a particular action choice x, and V(6) is a meta-utility function which assigns utilities to values of 0. Diagnostic utility is then, the utility gained from a change in one's estimate of theta. One's estimate of 0 is updated in light of information gleaned about 0 from observed action choices.

Applying to Quattrone and Tversky's cold water experiment, 0 is cold-water tolerance, x is the act of withdrawing one's arm at a particular time (thus defining a length of time the arm was in the water), and V(0) is what the experiment manipulates: in the case where subjects are led to believe that higher cold-water tolerance is indicative of good health, we have modified V(6) to assign positive utility to higher 0 and vice-versa.

We can extend this model to doing valid inference about populations in the following way: represent an underlying disposition common to a population, such as propensity to go to the trouble of voting for candidate A, as 0, and assume some correlation in 6 among would-be A-voters. Given this correlation, it is possible for our agent to observe his own vote and make a valid inference about what the rest of the population might do (Bodner, Prelec 2001). Having

observed that you successfully voted for candidate A, and believing there to be some commonality among supporters, you are inclined to believe the likelihood that a given A-supporter will vote for A is higher after having observed that you yourself have actually done it relative to before. Our agent can do valid inference on the basis of observing his own behavior and raise his estimate of turnout for A after having voted.

In the subsequent portion of this paper we will share empirical evidence for some puzzling behaviors to which self-signaling theory might be applied and offer some theoretical explanations.

3 Empirical Investigation

All choices participants make in the following three experiments are incentivized. They are

paid more for correct answers in the case of comprehension checks, and they are paid more for more accurate answers in the case of predictions. The economic games work as standard. Each of the following three experiments was pre-registered on osf.io. For each experiment we have pooled all data collected together (pilots and full-scale experiments). Without pooling all available data, we lack the necessary power to draw conclusions.

3.1 Experiment 1, Sequential Prisoner's Dilemma 3.1.1 Motivation

How might we explore behaviors that seem to indicate a misunderstanding of causality? Driven

by a hint of evidence (Morris et al. 1998) that self-signaling might be profitably explored in the

context of economic games that involve a dilemma, we subsequently chose to develop an experimental platform built on the oTree framework (Chen et al. 2016), starting with a replication of Morris et al. with a prisoner's dilemma.

There has already been some empirical investigation of order effects in sequential games with no information flow. Morris et al. report more cooperation among first-movers (66%) than among second-movers (53%) in an experiment run with 89 German university students, but do not replicate the effect in a subsequent experiment with 221 subjects, showing 43%

cooperation in both cases. Abele & Erhart (2005) report two Public Goods Game (PGG) experiments with German undergaduates (N=86 and N=192). In both experiments, there was no significant difference between the sequential conditions, though their data do suggest that subjects of any order in a sequential game contribute less than subjects in a simultaneous game

(p< 0.001). Given our observed effect sizes, it seems likely that all of these experiments were

under-powered with respect to differences in cooperation rate by order. Morris et al. and Abele et al. report a difference in cooperation rates depending only on whether a person is a first-mover or a subsequent-first-mover in a PD or PGG, respectively, something that should not matter given the formal expression of the game. Players do not observe what others' moves are; the only difference in information among the players is the knowledge of where they are in the sequence (e.g., in a PGG one could go first, second, or third). That there is a difference in play given sequence suggests a misunderstanding of causality that could potentially be modeled with self-signaling.

But self-signaling theory itself isn't sensitive to the direction of time. On a first reading, players in a sequential PD or PGG should not care about their order-at least not for self-signaling reasons. In order to explain an effect with a direction in time, we need to add complexity to our model. We have chosen to do this by assuming a hyper-prior about causality only moving forward in time. This means that what's in the past isn't something one might influence, and so considering it isn't worth the effort. Our agent doesn't do inference about the past via self-signaling. What's happened already is immutable and, if not known to the agent, it is "known" to the universe in a broader sense and therefore the agent cannot self-signal about it.

Consequently, we now assume that self-signaling in the case of a sequential PD would only apply going forward. That is, one can make inferences about the behavior of others based on one's own behavior only if the others have not yet behaved. With a sequential PD, we are answering the question: What will other people do? What they have done is fixed.'

1 See Bums et al. 2012 for a discussion of how future behavior is seen as more intentional than past behavior. There may be some sense where, insofar as a behavior is not intentional, it cannot be the reflection of an underlying 0

We also considered whether a cognitive shortcut related to theory of mind (ToM) might be at play. It could be that order effects are due to a resource-rational shortcut: doing ToM

calculations is resource intensive, so one might want to do this in situations where a significant reward is possible. And it is possible that those situations tend to be situations where one can actually influence future actions of the people targeted for mentalizing. So, one might

mentalize about players who have not yet moved, but not bother spending resources thinking about the motivations of those who have already made a decision. Your own actions can't influence actions already taken, so perhaps it isn't worth a query about the whys and

wherefores. If we also believe those ToM calculations lead to different behavior in a sequential game depending on order, we then see how mentalizing might provide an explanation for, e.g., Morris et al.'s results. We can test this by manipulating ToM directly: we can instruct second-movers in a sequential PD to consider what the first-mover was thinking and see if this removes the difference seen in likelihood of cooperation by order in comparison to a manipulation matched for length but containing no ToM content.

Finally, we considered that we might also profitably investigate self-signaling via a manipulation that swamps the self-signal with real information. A group given actual information about aggregate behavior would not be expected to self-signal because there is nothing to self-signal about: using your own update of 0 to make an inference about the population is not informative when you believe you already have the ground truth about the population. We implement this in the sequential PD by varying, between-subjects, whether or not they are supplied with information about population-level behavior. Subjects in the population info condition are told that about half of the players in this game choose to

cooperate, and the other half choose to defect, while subjects in the no-info condition receive no information about the population.

3.1.2 Methods

The experiment is a one-shot sequential prisoner's dilemma, one in which players make their moves one after another. In our version of the game there is no information flow between players. Players know which position in the sequence they are, but do not know anything about their opponent's move. Only that he or she will make it (or has made it). In this experiment, players were grouped together to play in real time on an online platform.

The experiment is a 2 x 2 x 32 design, order crossed with population information and Irrelevant / Theory of Mind / Regular Asynchronous conditions (Figure 1).

2 Earlier pilots included additional conditions, namely a "distractor" theory of mind manipulation that involved theorizing about a person who is not involved. We ignore this condition in the analysis presented here.

Order: First Second

Population Info: Yes No Yes No

Regular _{Cell 1 2 3}

4 Async.

ToM 5 6 7 8

Irrelevant 9 10 11 12

Figure 1: 2 x 2 x 3 design matrix for Sequential Prisoner's Dilemma experiment

The subjects were 3072 U.S.-based Amazon Mechanical Turk workers. Players arrive at the experiment web page, are consented, and then engage in a real effort task as an attention and activity verification (transcribing nonsense sentences) 3.

After this, participants are placed into a chat room with their counterpart. They are given 30 seconds to chat with each other for the purposes of verifying that they are, indeed, playing against a real person in real time. The benefit of doing this is two-fold in the context of Mechanical Turk: First, Turk participants often believe experimenters to be lying or stretching the truth-particularly in the case of a claim they are playing against another person. While it may often be the case that one's counterpart is a person and selected a move, the other player's move may be randomly sampled from data collected weeks ago. Having this brief chat

interaction works to assure the participant that the experimenter is on the level. Second, having a brief but genuine interaction with a counterpart adds a psychological sense of reality that may otherwise be absent with an abstract, possibly computer-generated opponent.

Participants then arrive at an explanation of the rules of the game (see Appendix 1). In the rules explanation, the game is described not as a standard PD with "cooperators" and "defectors", but as an allocation task where players choose to "keep" an initial endowment or "transfer" the endowment to the other player, in which case it is multiplied before it reaches that other player (Bear, Rand personal communication). Instructions also include if-then statements about the consequences of certain moves to aid understanding. We hoped this would make the prisoner's dilemma, which subjects often find difficult to understand, more intuitive.

Players are asked to read the instructions (which appear on every page of the experiment, for reference), and then they progress to a series of comprehension questions:

1. Does the other player know what your move is?

2. If the other person TRANSFERS their money, what earns you the most money?

3. If the other person KEEPS their money, what earns you the most money?

3 Since subjects are grouped together for this real-time experiment, we must ensure that those who are being grouped are active directly before they are put into groups. If they are not, we may end up with responsive players grouped with non-responsive partners. Since the game is real-time and requires interaction, this means the group could not progress.

4. If you choose to TRANSFER your money, do you make more money if the other person TRANSFERS or KEEPS?

5. What year is it?

Of 3072 subjects, about half (1446) passed all of the comprehension check questions. Data

analysis is limited to these 1446 responses. Since subjects are grouped together playing a real-time game we are obliged to collect data from all subjects, including those who have failed comprehension checks.

After the comprehension questions in asynchronous conditions, Player 1 is invited to make a move. The screen will also contain any additional manipulation, such as Theory of Mind, Irrelevant, or nothing. On this screen players are also given either no information about population-level behavior or the knowledge that about 50% of players cooperate4: "About half

(50%) of other players choose to TRANSFER, and half choose to KEEP".

Player 1 then experiences a wait screen while Player 2 makes a move on a screen analogous to the one Player 1 saw. The payoff matrix is as follows (Figure 2):

Player 2 Transfer Keep (cooperate) (defect) Transfer _{$0.33, $0.33 $0.00, $0.50} (cooperate) Keep $0.50, $0.00 $0.16, $0.16 (defect)

Figure 2: Experiment 1 Sequential Prisoner's Dilemma payoff matrix

Both players then predict what the other player's move was, answering the question, "How likely is it that the other person in this game TRANSFERRED?" on a scale from 0-100. Players also answer the question about the population, "How likely is it that an average person who plays this game would TRANSFER?".

Players then exit the experiment and are paid. Median total pay per subject (including bonuses for accurate predictions) is $0.68, yielding an hourly rate of $8.25 per hour at 5 minutes' duration.

3.1.3 Results

Our preregistered predictions for the sequential PD were as follows:

1. First-movers will cooperate more than second-movers

We see good evidence that first-movers cooperate more than second-movers when pooling data from both pilots and the main experiment (Figure 3).

All Subjects who passed comp. checks, N=1361

* 0.5 0.4 00 0.3 'U 0.2 G. 0.1 0.0 first second Order

Figure 3: Percent cooperation among first- versus second-movers in a sequential PD, pooled data, 1361 of 1446 non-simultaneous condition respondents, standard

The difference between first- and second-movers' rates of cooperation is significant in a logistic regression, p < 0.05, with the first-movers showing 54.3% cooperation vs. second-movers at 48.7%.

When pooling data from all conditions and pilots plus the main experiment, we do produce an order effect. The effect is not present, p > 0.05 using

just

2. The increase in cooperation seen among first-movers is due to their performing a theory-of-mind calculation, while second-movers do not

Actual effect size in this experiment was smaller than estimated, so unfortunately we did not have enough power to resolve differences between individual cells and this question remains unanswered. We do note that both Theory of Mind (ToM) and Regular Asynchronous groups do seem to show the decrease in cooperation with increasing order to a similar degree, suggesting that the ToM manipulation does not eliminate the order effect (Figure 4).

All Subjects who passed comp. checks, no % info, N=761

0.6 0.5 0 0 0.4 8 CL 0 0 U 0.3 C d. 0.2

-r

"T

Figure 4: Percent cooperation split by treatment, 761 of1446 non-simultaneous condition respondents with no % info, standard error bars. reatments are Theory of Mind (ToM), a regular asynchronous condition (Regular_ Async), and an irrelevant distractor condition matched for ToM in time (Irrelevant). Differences not

significant.

The population information manipulation was under-powered, but results were consistent with theory.

I

3. The increase in cooperation among first-movers is due to self-signaling; people wish to

believe the other player will cooperate, and so produce evidence that is consistent with that hypothesis. Therefore, giving them information about the population will reduce the order effect.

We again lack the statistical power to examine the interaction between information about the percentage cooperators in the population and order. The pattern is somewhat suggestive of the predicted effect, however, with a larger difference in means evident in the no-information

condition vs. the information condition, where we expect the true information about the population's behavior to have wiped out self-signaling (Figure 5).

All Subjects who passed comprehension checks, N=1361 0.6 - 0.5 0.4 0.3 'U W 0.1 order CL o fi rst m second 0.0

NO info YES info

Pop. percent cooperators info

Figure 5: Percent cooperation by order and information treatment, pooled, 1361 of 1446 non-simultaneous condition respondents, standard error bars. Onder split by whether respondents received information about population-level behavior before making their decision. Differences not significant.

Predictions of the other player's move, for which we did not pre-register a hypothesis, show a striking pattern (Figure 6).

All Subjects who passed comp. checks, N=1361 1 0 -1 U 3 V CL .0 w -2 U C ~-3 Keep Transfer Decision

Figure 6: Difference in estimates ofpartner t move vs. population average, 1361 of 1446 non-simultaneous condition respondents, standard error bars.

This data suggests that people tend to believe that their partner is different from the population average in some way relevant to the game (p < 0.001). People who choose to keep their endowment (defectors) think that their partners are more likely to defect than the population at large, and people who choose to transfer their endowment believe their partners are more likely to cooperate than the population. This pattern persists in all cells when split by order and treatment.

The difference between partner and population predictions was not predicted, but turned out to be notable. This may suggest some sort of magical thinking ("If I move a certain way my partner does too"), or perhaps self-signaling if there is the belief that the relevant underlying 0

is strongly correlated between the subject and his counterpart in a group, but not between the subject and the rest of the population. In this case we have to speculate about why the subject may act as if that correlations is present. Being two Mechanical Turk workers on the platform working on the same HIT at the same time may be enough to suggest this correlation. In addition, recall that the two subjects had a very brief chance to write to each other in a chat

room before they knew the nature of the task. Though they did not know what the task would be and could not have colluded, nor did they have enough time for communication of any substance, we believe the chance to interact in real time may have heightened this sense of similarity.

3.2 Experiment 2, Sequential Public Goods Game 3.2.1 Motivation

After investigating Morris et al.'s findings via a sequential prisoner's dilemma, we turned to a public goods game as a next step. The PGG, an n-player prisoner's dilemma, provides a number of advantages over the PD: it is easier to understand since it maps readily to real-life experiences (compare, for instance, to raising money for a town swimming pool), it is easily expanded to more than two players, and it allows for partial cooperation (any contribution between zero and the full endowment is generally allowed). This allows us to increase the quality of the data we collect and investigate the shape of the curve describing the decay in cooperation with increasing order.

Given our modifications to self-signaling theory to account for the fact that inference only goes forward in time, we would predict a decline in cooperation going forward. But we would predict this only among conditional cooperatorss-players who want to cooperate only if the other player is expected to cooperate. In a three-person sequential PGG, people who move first will cooperate most, followed by those who move second and then those who move third. This is because conditional cooperators wish to estimate the likelihood of other players cooperating and adjust their own behavior on the basis of that estimate. If the likelihood of others choosing to cooperate is high, it is likely that a conditional cooperator will choose to cooperate. Agents can inform their estimates of whether or not others will cooperate by observing their own behavior. Accordingly, we predict that the variation in cooperation rates with order will be

driven by players SVO-classified as individualistic (Murphy et al. 2011). Given this model, we would also expect that subjects would believe their own move is more diagnostic of others'

moves when going forward in time relative to going backwards. In particular, we can predict correlations between one's own move and the bets one makes on the moves of partners are stronger going forward in time vs. going backwards.

3.2.2 Methods

Experiment 2 is a one-shot sequential public goods game similar in setup to Experiment 1. The

PGG setup is standard, with an endowment of $1 and a multiplier of 2x. Groups are made up of

three players, and we manipulate order: the players go in sequence with no information flow. As with the PD, the only difference in information among the players is knowledge of where

5 Conditional cooperators (Fischbacher et al. 2001) are players who want to cooperate, but only if they expect their counterparts will also cooperate. However, see Burton-Chellew et al. (2016) for a discussion of the role of misunderstanding.

they are in the sequence. The design is straightforward, with each participant assigned to one of four conditions: orders 1-3 and a simultaneous-move condition.

The experiment flow is similar to the PD as well. Subjects were 1668 U.S.-based Amazon Mechanical Turk workers. Players arrive at the experiment web page, are consented, and then engage in a real effort task transcribing nonsense sentences. After this, they are placed in a chat room for 30 seconds after all players in their group have arrived to ensure participants believe the experiment is, in fact, a real game in real time with real people.

After the chat, subjects are then given an explanation of the rules of the game (which appear on every subsequent page for reference). The PGG is framed as a question of how much to

contribute to a "community fund". Instructions include if-then statements about the consequences of certain moves to aid understanding. Subjects are then asked a series of comprehension questions:

1. Do any of the other players know how much you decide to contribute?

2. No matter what the other players do, what earns you the most money? TRANSFERRING to the community fund or KEEPING your endowment?

3. What year is it?

As with the PD, responses to the comprehension questions are only relevant to data analysis: players continue on whether or not they have answered correctly.

Of 1668 subjects, 69% (1151) passed all of the comprehension check questions. Data from

batches 1, 7, and 8 were excluded due to technical problems resulting in server crashes during the experiment. Analysis is limited to 902 responses which passed comprehension checks and were not in batches 1, 7, or 8. Since subjects are grouped together playing a real-time game we are obliged to collect data from all subjects, including those who have failed comprehension checks.

After having completed the comprehension questions, players make their move. The contribution page includes a graphic at the top highlighting their place in the sequence of moves in red' (Figure 7).

6 Players in the simultaneous condition do not see any indication of sequence since they are moving simultaneously.

Contribute

Time left to complete this page:1:52

Player 2

_Player

₃

Youaraymoutoflplayers togo You are the first player to make a decsion. There are 3 players in this game, an endowment of $1.00,

and amultiplier of 2.

Enter cents as fractions of a dollar,like "0.50"Is 50 cents. How much will you contribute?

m+

Instructions

in this study you have been randomly and anonymously formed Into groups with other players.

Farh nf wm will h nlven mnnvi start called ww arnwmnmt A fn

Figure 7: Decision screen for a first-mover in the PGG task highlighting order in red

They are asked how much they wish to contribute to the community fund, and enter their chosen amount. If there are players in the sequence after our example subject, the subject sees a wait screen.

Once all players have moved, subjects move on to a series of prediction questions:

1. How much do you think [the first other player] chose to contribute to the community

fund?

2. How much do you think [the second other player] chose to contribute to the community fund?

3. How much do you think the average person chooses to contribute to the community

fund?

4. What do you think [the first other player] predicted you'd contribute to the community fund?

5. What do you think [the second other player] predicted you'd contribute to the

community fund?

After the prediction questions, subjects complete an SVO slider battery (Murphy et al. 2011, code based on Bakker 2019). Players then exit the experiment and are paid. Median total pay per subject (including bonuses for accurate predictions) is $3.16, yielding an hourly rate of

$18.48 per hour at 10 minutes' duration.

3.2.3 Results

Our preregistered predictions for the sequential PGG were as follows:

1. Players in a sequential PGG contribute less the closer to the end of a sequence of

contributors they are, and that they do this even when the only information they have is their position in the sequence (they have no information about the actions of others). We see good evidence for this prediction (Figure 8).

"r

I

_I,

I

comp checks, N=902, All batches excluding those with errors (1,7,8) Passed 0.6- 0.5- 0.4-0 0.3 C 0 U 0.2-0.1 0.0 -a-1 Treatments

Figure 8: Rates of cooperation in the PGG task split by order (1, 2, 3) vs. simultaneous movers (sim). 902 of1668 respondents who passed comprehension checks and were not affected by errors. Difference between 1 and 2 + 3 or 1 and 2 or 3 are all significant at p <

0.05, standard error bars

The difference between first-movers' mean contribution and that of second- or third-movers' (or second and third movers' combined) is significant, with p < 0.05. We do not see a significant difference between second- and third-movers, which could be an artifact of low power or a problem for our theory. If this is a sort of magical thinking leadership effect (e.g. Levati et al. 2007), and we do not see a progression from position 1 to position n, our theory will have been falsified.

2. The correlation between one's own contribution and prediction of others' contributions is stronger going forward (if one is order = 1, predicting 2 and 3) than going backwards. The subject views his or her action as more diagnostic with respect to the future than to the past.

There is significant support for this hypothesis (Figure 9).

2 3 sim

-

Prediction

Prediction

Prediction

of Player 1

of Player 2 of Player 3

Order=1

0.21***

0.42***

Order=2

0.22***

0.43***

Order=3

0.10*

0.09

*=p<.05, **=p<.001, ***=p<.0001

Figure 9: Partial correlations of one's own move with predictions of other players'moves. Players in earlier positions (e.g., Order=l) show higher

correlations with their own move as that move is further forward in time, controlling for other correlations. The

first-mover

If we consider the partial correlations7 between one's own move and that of one's bets on the moves of others in his or her group, we see a significant increase in correlation when looking forward vs. back. For example, a first-mover's decision (Order=1) correlates with that player's prediction of his group's second-mover at 0.21 and the third-mover at 0.42. This fits with our

suggestion that people are acting as if they can influence or learn about the future but not the past. If this a self-signaling mechanism at work, we could suggest that people believe they are learning more from observing their own actions about those who have yet to more rather than those who have already moved, and that they are updating their estimate, as self-signaling

suggests, in the direction of their own move.

3. The majority of the effect over order will be driven by participants classified via SVO

slider task as individualistic. Individualistic participants will tend to decrease their contribution with order, whereas prosocial participants will tend to keep their contribution the same regardless of order.

Again, we see some evidence for this in the data (Figure 10).

7 We did not specify the precise analysis in the preregistration, but partial correlations seems like the correct tool. The pattern is the same if we consider simply Pearson's R correlations.

Passed comp checks, N=902, All batches excluding those with errors (1,7,8) 0.7- 0.6- 0.5-0 0.4-U 0.31 0.2

-SVO category

Figure 10: Change in contribution with order is driven by subjects SVO-classifed as individualistic. 902 of 1668 respondents who passed comprehension checks and were not affected by errors. For Individualistic-classified respondents, the difference between 1 and 2 + 3 or 1 and 2 or 3 are all significant at p < 0.05, and the interaction between SVO and order is significant p < 0.05 when grouping 2 and 3. Standard error bars

There is no significant difference among prosocial contribution levels, while we do see a difference between the first-mover data and the combination of second- and third-mover contributions ( p < 0.05). The interaction between SVO category and order is significant when grouping positions 2 and 3 as well (p <0.05). As with the aggregated data, we do not see the hypothesized difference between positions two and three.

3.3 Experiment 3, Population-level Inference via Self-Signaling in a Voting Task

3.3.1 Motivation

Experiment 3 is a task designed to directly test inference about the group via observation of change in one's own 0. We operationalize this facet of self-signaling theory in a voting context. Groups of seven people must decide together whether they would like to take a risky bet or a sure-thing payoff, thus voting on the basis of a characteristic they have brought to the

experiment: risk tolerance. The entire group must take the same bet, and the bet taken is determined by majority vote. The only interaction players have after having received the instructions is via voting; they do not discuss. Before and after voting, between subjects, players bet on the outcome of the vote. Players'bets on the outcome of the vote are the dependent variable of interest; the difference between these measures should give us insight into the extent to which they are updating their expectation of the vote's outcome on the basis of observed change in their own 0.

3.3.2 Methods

The task is presented with a cover story: players are told they are part of a farming village, and they have all just harvested crops. Each player has harvested differing proportions of wheat, beans, and rice. They must all take their crops to market in order to earn money, but the village has only one cart with which to move the goods-so they must all go to the same market. There are two markets, one of which they must choose together: a nearby market with lower prices ("Alton Market"), and a market that has higher prices but which requires travel on a dangerous road ("Bedford Market"). There is a 50% chance that all of their goods will be lost on the road to Bedford Market due to a dangerous ford. While each player has harvested different quantities of wheat, beans, and rice, the proportions are generated such that for any given combination of prices every player has the same expected value in their harvest. Subjects were 276 U.S.-based Amazon Mechanical Turk workers. Players arrive at the

experiment web page, are consented, and then engage in a real effort task transcribing nonsense sentences. After this, they are placed in a chat room for 30 seconds after all players have

arrived in their group to ensure participants believe the experiment is, in fact, a real-time game with real people.

Payers then arrive at an explanation of the game. This is available on every subsequent page of the experiment. Players are then informed of their harvest (i.e., endowment of crops), and are shown the mix of crops that other players in their group have been given. After this, their choice (between Alton Market and Bedford Market) is explained to them. In particular, payoffs for selling their crops at each market are calculated for them and displayed.

Players in the "before" condition are given the opportunity to bet on the outcome of the vote. They are given an endowment of $0.10 and asked to predict what proportion of the vote will be for Alton Market, on a scale from 0-100. They are paid more the closer their answer is to the truth, with a perfect response resulting in a 10x reward ($1.00).

"Before" players then choose whether or not to pay a small cost to vote,' and vote. Players in the "after" condition are first presented with the opportunity to vote, and are then subsequently asked to bet on the outcome.

Players then exit the experiment and are paid. Median total pay per subject (including bonuses for accurate predictions) is $2.28, yielding an hourly rate of $13.03 per hour at 10.5 minutes' duration.

3.3.3 Results

Our preregistered predictions for the self-signaling voting task were as follows:

1. Voters are theorized to increase their expectation of their preferred outcome after voting

for it. If a subject s votes for candidate A, we expect s's expectation of the outcome of the vote, Proportion voting for A, to be larger after s has voted relative to before. We do not see any evidence for the hypothesis in the data (Figure 11).

Paid to vote, N=276 70- _N.S. U 50 -0 0-2~0 C 0 40 W4-C 0 after before Treatment

Figure 11: Predictions of the outcome of the Alton-Bedford vote before vs. after voting. All

respondents, no significant difference between measures, standard error bars

However, further investigation revealed a problem with the design of the experiment. All subjects were asked to predict the percentage of the group voting for Alton with a slider on a scale from 0-100. Since the percentage of voters for Bedford is just 1 less the Alton voter percentage, this theoretically should be adequate. But when we examine the distribution of predictions split by vote, we see an interesting pattern (Figure 12).

Distribution of prediction by vote

0

I I I1

20 40 60 80 100

Prediction, percent who will vote for own candidate

Figure 12: A strongly bimodal distribution in Bedford (risk-seeking) voters'prediction of the outcome of the vote indicates Bedford voters are confused by the slider setup, sometimes treating it as indicating A-vote percentage (correct), sometimes B-vote percentage (incorrect).

There is a strong bimodality in Bedford voters' distribution of predictions, suggesting that they have not correctly understood the input slider. It appears as if about half of Bedford voters misinterpret the input slider to be about their own vote, Bedford, rather than Alton.

A-voters, N-121

4 Discussion

We have succeeded in producing laboratory models of phenomena that allow further

investigation of apparent misunderstandings of causality. However, given under-estimation of effect size in all cases, these experiments would ideally be replicated with pre-registration at sufficient scale. The work presented here is very much a first step towards investigation of order effects, self-signaling, and other misunderstandings of causality. There is much work to be done to refine the experimental paradigms, further develop the theory, and hone

manipulations to rule out competing explanations. Magical thinking, wishful thinking, hyper-priors about information leakage, and many other theoretical explanations have yet to be definitively ruled out.

We believe developing the PGG paradigm is a good way forward given its ease of

understanding and lack of a limit to two players relative to the PD. Natural extensions to the present PGG paradigm include increasing group size such that we might resolve the shape of the curve with which cooperation decays given increasing order. An effect that is limited to the first-mover would falsify our present thinking, and some sort of steady decay would fit. We might also investigate self-signaling further by introducing players who are (probabilistically) a computer rather than a human. It is impossible to self-signal about something so different to oneself, so we would predict that any effects based on self-signaling would decay with the increasing probability that other players are bots. We might also vary the probability that one's own decision is made by a computer to the same effect. We could also introduce trembling-hand type manipulations to introduce noise into the signal for the same purpose. Another avenue for development might be varying the amount of true information that players have about other players' (or the population's) moves. As the amount of direct information increases, the relevance of self-signaling as a means of learning about others declines.

In addition to moving forward with the PGG, the the voting task remains a very direct test of certain self-signaling predictions and might profitably be re-worked to increase ease and speed of understanding, remove problems with DV input mechanisms, and decrease the confounding effects of our measurements. An outcome prediction slider that is anchored at either end by Alton and Bedford, respectively, would serve to remind respondents to consider which destination they intend to (or have) voted for.

5 Works Cited

Abele, S., Bless, H., & Ehrhart, K.-M. (2004). Social information processing in strategic decision-making: Why timing matters. Organizational Behavior and Human Decision

Processes, 93(1), 28-46. https://doi.org/10/csmkp9

Abele, S., & Ehrhart, K.-M. (2005). The timing effect in public good games. Journal of

Experimental Social Psychology, 41(5), 470-481. https://doi.org/10/bkgg9d

Axelrod, R., & Hamilton, W. (1981). The evolution of cooperation. Science, 211(4489),

1390-1396. https://doi.org/10/fhfps5

Bakker, D. (2019). svotree: An implementation of SVO tasks in oTree [HTML]. https://github.com/dimaba/svotree (Original work published 2016)

Bodner, R., & Prelec, D. (2001). Self-signaling and diagnostic utility in everyday decision

making. 22.

Bodner, R., & Prelec, Drazen. (1997). The diagnostic value of actions in a self-signaling

model.

Burns, Z. C., Caruso, E. M., & Bartels, D. M. (2012). Predicting premeditation: Future behavior is seen as more intentional than past behavior. Journal of Experimental Psychology:

General, 141(2), 227-232. https://doi.org/10/bz6ngt

Burton-Chellew, M. N., El Mouden, C., & West, S. A. (2016). Conditional cooperation and confusion in public-goods experiments. Proceedings of the National Academy of Sciences,

113(5), 1291-1296. https://doi.org/10/f79v8z

Chen, D. L., Schonger, M., & Wickens, C. (2016). oTree-An open-source platform for laboratory, online, and field experiments. Journal of Behavioral and Experimental Finance, 9,

88-97. https://doi.org/10/b42

Chin, J. M. (2010). Moral uncertainty promotes prosocial behavior: Exploring the

self-signaling motivation for prosocial behavior [University of British Columbia].

https://doi.org/10.14288/1.0071257

Dhar, R., & Wertenbroch, K. (2012). Self-Signaling and the Costs and Benefits of Temptation in Consumer Choice. Journal of Marketing Research, 49(1), 15-25. https://doi.org/10/bbng3z Fischbacher, U., GAchter, S., & Fehr, E. (2001). Are people conditionally cooperative? Evidence from a public goods experiment. Economics Letters, 71(3), 397-404. https://doi.org/10/cxg259

Grossman, Z. (2015). Self-signaling and social-signaling in giving. Journal of Economic

Behavior & Organization, 117, 26-39. https://doi.org/10/f7smvz

Levati, M. V., Sutter, M., & van der Heijden, E. (2007). Leading by Example in a Public Goods Experiment with Heterogeneity and Incomplete Information. Journal of Conflict Resolution,

51(5), 793-818. https://doi.org/10/bcgzmc

Mijovic-Prelec, D., & Prelec, D. (2010). Self-deception as self-signalling: A model and

experimental evidence. Philosophical Transactions of the Royal Society B: Biological Sciences,

365(1538), 227-240. https://doi.org/10/c2tqv2

Morris, M. W., Sim, D. L. H., & Girotto, V. (1998). Distinguishing Sources of Cooperation in the One-Round Prisoner's Dilemma: Evidence for Cooperative Decisions Based on the Illusion of Control. Journal of Experimental Social Psychology, 34(5), 494-512.

https://doi.org/10/d4cs3w

Murphy, R. 0., Ackermann, K. A., & Handgraaf, M. (2011). Measuring Social Value

Orientation (SSRN Scholarly Paper ID 1804189). Social Science Research Network.

https://papers.ssrn.com/abstract=1804189

Nowak, M. A. (2006). Five Rules for the Evolution of Cooperation. Science, 314(5805),

1560-1563. https://doi.org/10/cz2pc6

Prelec, D., & Bodner, R. (2003). Self-signaling and self-control. 32.

Quattrone, G. A., & Tversky, A. (1984). Causal versus diagnostic contingencies: On self-deception and on the voter's illusion. Journal of Personality and Social Psychology, 46(2),

237-248. https://doi.org/10/dr4gxj

Rand, D. G., & Nowak, M. A. (2013). Human cooperation. Trends in Cognitive Sciences,

17(8), 413-425. https://ldoi.org/10/f47vv8

Robinson, A. E. (2006). The Impact of Causality, Strategies, and Temporal Cues on Games of

Decision.

https://smartech.gatech.edu/bitstream/handle/1853/11488/robinson alfred e 200608 phd.pdf? sequence=1&isAllowed=y

Savage, L. J. (1954). The sure-thing principle. Leonard J. Savage, The Foundations of

Statistics. New York: John Wiley, 21-26.

Shafir, E., & Tversky, A. (1992). Thinking through uncertainty: Nonconsequential reasoning and choice. Cognitive Psychology, 24(4), 449-474. https://doi.org/10/d6thrg

Tversky, A., & Shafir, E. (1992). The Disjunction Effect in Choice under Uncertainty.

6 Appendix

6.1 Decline in PGG Experiment Respondent Quality

We noticed a decline in the quality of PGG respondents over time as measured by percent that are passing the comprehension checks (Figure 13).

N=1668, All batches

exptime-continuous

Figure 13: Linear fit with a 95% bootstrapped confidence interval and a Lowess line on data describing pass vs. fail of comprehension checks, all subjects. The horizontal axis is a timeline of subjects completing the experiment with blocks of time between experiment batches

removed to make it continuous. We observe a marked decrease in subject quality over time.

Subject comprehension appears to fall from about 80% pass rate to about 65% over the course of the experiment. We are not sure about what is causing this change. We speculate that it may be depletion of the ranks of higher-quality respondents (who are also better and faster to get

1.0- 0.8-V -

0.6-a'

E

desirable, well-paying HITs such as ours). We do not, however, have any evidence for this theory.

6.2 Voting Experiment Confound

The manipulation in the voting experiment, Experiment 3, itself may be a confound since it appears that it may affect which market subjects vote for (Figure 14).

Paid to vote, N=276 0.7- 0.6-m 0.5-1l c50.4 -0 c0.2- 0.1-0.0 after before Treatment

Figure 14: Percentage voting

for

While the difference is not significant at p > 0.05, there may be some influence from the "before" measurement on subsequent voting behavior. In future iterations of this experiment this might be ameliorated by making the measurement more subtle, moving to a within-subjects design, or having within-subjects estimate the outcome of the vote both before and after their own vote, but incentivising only one of those votes.