www.sciencemag.org/cgi/content/full/321/5897/1849/DC1
Supporting Online Material for
Understanding Overbidding: Using the Neural Circuitry of Reward to
Design Economic Auctions
Mauricio R. Delgado, Andrew Schotter, Erkut Y. Ozbay, Elizabeth A. Phelps*
*To whom correspondence should be addressed. E-mail: [email protected]
Published 26 September 2008, Science 320, 1849 (2008)
DOI: 10.1126/science.1158860
This PDF file includes
Materials and Methods
SOM Text
Figs. S1 to S3
Tables S1 to S7
References
Supporting Online Material
METHODS
Participants
Twenty-two right-handed volunteers answered a posted advertisement and were recruited
to participate in this study. Five participants were excluded; two for excessive motion, two due
to scanner malfunction, and one who reported misunderstanding the task instructions. The
remaining 17 participants (8M, 9F; age: M = 23.77, SD = 3.38) were included in the final
analysis. All participants gave informed consent according to the Institutional Review Board at
New York University.
Procedure
Participants were instructed they would be playing two types of games, a two-person
auction and a lottery. In the auction game, at the start of each trial participants were given a
randomly drawn value for the fictitious good being sold. These random values were drawn from
a finite set with equal probability. Participants were told that they were bidding against a male
confederate whom they knew was a confederate and using an unknown, but predetermined
strategy (the Nash equilibrium strategy). They met the confederate before the imaging session
and both the confederate and participant listened to the instructions together before being
separated to play the game. In the lottery, after receiving a value for their good, the participant
was assigned a bid and was asked whether or not he or she wanted to play a game of chance
against a computer. In the lottery game the probability that the participant won, and hence
received a payoff, was exactly equal to the probability that they would win with the same bid in
the auction given the bidding strategy of the confederate.
Both auction and lottery games used two treatments. In one, participants were paid in
money, while in the other they received points. Participants were told that they would keep the
monetary gains accumulated in a randomly selected monetary reward block (1 out of 4 possible
blocks). They would also get the points gained in randomly selected points block (1 out of 4
possible blocks), which could earn them a spot in an anonymous “top 10 players list” to be
disseminated at the conclusion of the study. This design aimed to investigate the utility of
winning or losing as a function of type of social competition (human vs. computer) and type of
incentive (money vs. points).
In the auction game, participants were faced with four values for the fictitious good sold
(6, 8, 10, 12) and could place only one of four different bids (2, 5, 7, 8) during a decision phase.
The auction used the first price rule in which the participant who submitted the highest bid wins
and has to pay a price equal to his or her winning bid. As previously stated, they were also told
that the opponent would play with an unknown fixed-strategy throughout, which was bidding
according to the following equilibrium bid function: (value:equil bid – 6:2, 8:5, 10:7, 12:8). In
other words, the equilibrium bid strategy prescribes that when a bidder received a value of 12, he
or she bid 8, when the value of 10 is received the bid should be 7, and so on. A pair of bidding
strategies is an equilibrium if, given the strategy used by one’s opponent, no subject wishes to
deviate from his or her prescribed bidding strategy. We determined the equilibrium bid strategy
for the auction by demonstrating that if both participants use this strategy, and posit that their
opponent is also using it, then neither will wish to deviate since they could not make more
money by doing so. This was done by calculating for each possible value the expected payoff
for each potential bid, and verifying that the equilibrium bid yields the highest expected payoff.
Given this, the strategy, is, by definition, a Nash equilibrium.
Importantly, in each trial the value generated for the participant and the opponent (or
lottery) was randomly generated. Participants knew only their own value in each round and, in
terms of feedback, participants found out if they won the auction (payoff = “value” – “bid”) or
lost the auction (did not have the high bid, payoff = 0) in the outcome phase of each trial. They
were not told what their opponent’s bid was. In case of identical bids, ties were broken at
random, thus a tie was never indicated as a potential outcome. Importantly, a loss did not signify
a monetary or points loss per se, but merely not winning the auction.
In the lottery game, participants were presented with the same values (6, 8, 10, 12) and
were assigned a bid (2, 5, 7, 8). They were given the option to play or not play the lottery. On
most trials, the assigned bid was lower than the value, but at times the bid could exceed the
good’s value, thereby causing a loss if the subjects won the lottery. Such catch trials occurred
once during a scanning session to ensure that participants paid attention to the assigned values
and bids. During this game, participants assigned bids were matched against independent goods
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and bids generated by the computer. As in the auction game, the opponent (in this case a
computer) played a fixed equilibrium strategy. During the outcome phase, the participant found
out if they won or lost the lottery (based on the value of their assigned bid and the computer’s
bid). The actual lottery trial distribution and how it matched with subjects actual bids are
presented in Table S1.
There were four total types of trials (money auction and lottery, points auction
and lottery) presented 26 times per experimental session and blocked into 8 fMRI runs of 13
trials each (4 alternating rounds of money and points incentive each). The values 6,8,10 and 12
were presented 6 or 7 times each per type of trial. A single trial lasted 30 seconds, with 4
seconds for each decision, followed by a 12 second inter-stimulus interval, and each outcome
displayed for 2 seconds, followed by a 12 second inter-trial interval. Behavioral measures such
as reaction time and choice (i.e., bid) were collected, along with post-experimental ratings. The
post-experimental questions probed participants’ interest in the auction and lottery games as well
as their overall interest in social competition (i.e., beating a human vs. a computer). For
example, on a Likert scale (1-7, 7 being very important), the following question was posed:
“With respect to your opponent, how important to you was it that you were "beating" a human
being in the auctions versus "beating" a computer in the lotteries? An indirect measure of risk
aversion was also acquired via the Holt-Laury procedure, a series of gambles assessing
participants’ risk preferences (1). Finally, neuroimaging data was collected throughout the
experiment, with analysis focusing on the outcome phase, when participants received positive
(“win”) or negative (“loss”) feedback about their auction or lottery decision.
fMRI Acquisition and Analysis
A 3T Siemens Allegra head-only scanner and a Siemens standard head coil were used for
data acquisition. Anatomical images were acquired using a T1-weighted protocol (256 x 256
matrix, 176 1-mm sagittal slices). Functional images were acquired using a single-shot gradient
echo EPI sequence (TR = 2000 ms, TE = 20 ms, FOV = 192 cm, flip angle = 75°, bandwith =
4340 Hz/px, echo spacing = 0.29 ms). Thirty-five contiguous oblique-axial slices (3 x 3 x 3 mm
voxels) parallel to the AC-PC line were obtained. Analysis of imaging data was conducted using
Brain Voyageur software (Brain Innovation, Maastricht, The Netherlands). Functional imaging
data preprocessing included motion correction (using a threshold of 2.5 mm or less), slice scan
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time correction (using sinc interpolation), spatial smoothing using a three-dimensional Gaussian
filter (4-mm FWHM), and voxel-wise linear detrending and high-pass filtering of frequencies (3
cycles per time course). Structural and functional data of each participant were transformed to
standard Talairach stereotaxic space (2).
A random-effects general linear model (GLM) analysis was conducted during the
outcome phase including each condition (money auction, money lottery, point auction, point
lottery) and associated outcome (“win” and “loss” feedback) as predictors. The motion
parameters in the x, y, z direction were also included as predictors of no interest. The primary
contrast of interest was “win” x “loss” (p < 0.001, cluster threshold of 3 mm3 contiguous voxels)
which explored regions associated with feedback and outcome processing during both auction
and lottery games. For each region of interest (ROI) defined by this contrast, mean beta weights
for each predictor were then extracted for further analysis. Specifically, two 2-way repeated
measures ANOVA were conducted separately for win and loss outcomes respectively. The two
within-subjects factors were type of social competition (auction or lottery) and type of incentive
(money or points). Finally, pending a significant main effect of type of social competition, posthoc t-tests probed differences between auction and lottery trials for trials that resulted in a “win”
or a “loss” separately.
SOM TEXT: SUPPLEMENTARY RESULTS
Behavioral Results
Behavioral results in the neuroimaging experiment included subjective data, reaction time
and choice data, which measured bidding behavior by participants. The subjective ratings data
(post-questionnaires) suggested that participants were more interested in playing the auction than
the lottery (t(16) = 3.41, p < 0.005), and they were more interested in playing for money than for
points (t(16) = 3.27, p < 0.005). Surprisingly, they showed no differences when rating their
feelings upon winning an auction compared to winning a lottery (t(16) = 1.10, p = 0.29), despite
reporting that it was more important to beat a human versus a computer opponent (t(16) = 7.38, p
< 0.0001). This pattern of results with respect to competition was mirrored by the reaction time
data during the decision phase (see Table S2). Reaction time was influenced by the type of
social competition (F(1, 16) =29.51, p < 0.0001), as participants took longer during auction
trials, but not type of incentive (F(1, 16) =0.48, p = 0.51).
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During the auction decision phase, participants were asked to place a bid. Data from
individual’s choices demonstrated that participants overbid according to the prescribed
equilibrium, as suggested by a comparison of the number of trials (percentage of choices) where
participants overbid with the number of trials where their bid matched equilibrium (t(16) = 3.04,
p < 0.008). This was observed irrespective if the incentive was money (t(16) = 3.30, p < 0.005)
or points (t(16) = 2.40, p < 0.03), and exacerbated by considering a comparison of overbid trials
with all other trials (bids at or below equilibrium, t(16) = 4.03, p < 0.001). The breakdown of
choice behavior during auction trials, by specific values is given in Table S3. Finally, there were
no significant correlations between individuals’ scores during the Holt-Laury risk aversion scale
and their bidding strategy, in support of the idea that risk aversion is not a necessary condition
for overbidding.
An exploratory analysis was also conducted to examine any gender effects in the bidding
data (9F/8M participants). Using gender as a between-subjects variable and bidding strategy and
incentive as within-subjects variables, we found no effect of gender (F(1,15)=1.56, p=0.23) or
interaction with bidding strategy (F(1,15)=0.52, p=0.48) or incentive (F(1,15)=0.35, p=0.56).
These results are deemed exploratory, however, as a larger sample is necessary to truly access
gender differences in a competitive auction environment.
Neuroimaging Results & Discussion
A contrast of all wins vs. loss trials was conducted to identify functional ROI’s for further
analysis (see Table S4). Mean beta weights extracted from all ROIs were then input into two
separate ANOVAs for win and loss outcomes separately. Only the striatum ROIs displayed
significant results from these analyses, with voxels located in the ventral portion of the striatum
and extending into the caudate nucleus, thus referred to as the ventral caudate nucleus. Within
the right ventral caudate nucleus ROI, a main effect of incentive (F(1, 16) = 9.67, p < 0.01) was
observed during win trials, driven primarily by a larger response to monetary compared to points
rewards (t(16) = 3.11, p < 0.01), but no main effect of social competition (F(1, 16) = 0.36, p =
0.55) or interaction (F(1, 16) = 0.04, p = 0.85) was seen. Instead, differences between auction
and lottery trials were apparent only in the context of losses (F(1, 16) = 5.29, p < 0.05). Of
particular interest, post-hoc t-tests showed that mean beta weights for win trials during the
auction game (irrespective of incentive) were not significantly different from the lottery game
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(t(16) = -0.60, p = 0.55). In contrast, mean beta weights for losses led to a more pronounced
decrease from baseline during auction compared to lottery trials (t(16) = -2.30, p < 0.05). No
main effect of incentive (F(1, 16) = 0.22, p = 0.64) or interaction with social competition (F(1,
16) = 0.01, p = 0.91) were observed during loss trials. Interestingly, differential responses
between win and losses, previously reported in the ventral caudate nucleus (see 5 for review),
were observed during both auction (t(16) = 5.80, p < 0.0001) and lottery (t(16) = 2.30, p < 0.05)
trials when the incentive was monetary. When points served as the incentive, however, a trend
was observed during auction trials (t(16) = 2.01, p = 0.06), but no significant differences were
observed during lottery trials (t(16) = 0.50, p = 0.63), highlighting the main effects previously
discussed (Figure S1).
The left ventral caudate nucleus ROI were similar to the right striatum ROI as a main
effect of incentive (F(1, 16) = 5.05, p < 0.05) was observed during win trials, but no main effect
of social competition (F(1, 16) = 0.06, p = 0.80) or interaction (F(1, 16) = 0.08, p = 0.79) was
observed. Instead, differences between auction and lottery trials were apparent only in the
context of losses (F(1, 16) = 6.06, p < 0.05). Of particular interest, post-hoc t-tests showed that
mean beta weights for win trials during the auction game (irrespective of incentive) were not
significantly different from the lottery game (t(16) = -0.80, p = 0.25). In contrast, mean beta
weights for losses led to a more pronounced decrease from baseline during auction compared to
lottery trials (t(16) = -2.46, p < 0.05). No main effect of incentive (F(1, 16) = 0.02, p = 0.91) or
interaction with social competition (F(1, 16) = 0.39, p = 0.54) were observed during loss trials.
Similar to the right ventral caudate nucleus, responses between win and losses were observed
during both auction (t(16) = 5.20, p < 0.0001) and lottery (t(16) = 3.48, p < 0.005) trials when
the incentive was monetary. When points served as the incentive, however, a trend was observed
during auction trials (t(16) = 1.93, p = 0.07), but no significant differences were observed during
lottery trials (t(16) = 1.01, p = 0.33).
Thus, the main finding during the outcome phase was activation of the striatum
bilaterally, with manipulations of type of social competition solely with respect to losses. One
interpretation of these findings is that the striatum could be coding for reward and punishment
values of a decision, here symbolized by a win or a loss (3-5). An alternative interpretation is
that the dynamic nature of the auction game leads to learning and updating, and decisions (i.e.,
amount to bid) may change with respect to the previous experience or your beliefs regarding
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your opponent. This interpretation is further supported by the vast array of data that link the
striatum with prediction error signals during affective learning (for review see 3, 9). More likely,
activation in the striatum in this specific design could be integrating both these ideas and
representing reward-action values, or coding for the value of the selected action (either the
auction bid or decision to play the lottery), with the purpose of teaching the system to optimize
future decisions (3-9).
It is also notable that the right and left striatum were the only ROIs identified in the
outcome contrast that also showed a modulation with respect to competition. While prefrontal
cortical regions (observed only at lower thresholds in this experiment) have been involved in
outcome processing (10) and error monitoring (11), the striatum has been linked with modulation
of outcomes with respect to social (12-14) and incentive (15-19) factors.
The observed results in the striatum are consistent even when controlling for value. The
current design contains both low (6, 8) and high (10, 12) values, which could potentially
influence the overall result (e.g., correlation between losses and low values being responsible for
auction observations). Using low and high values as regressors of interest during the feedback
phase, we contrasted overall Win x Loss (High Win + Low Win > High Loss + Low Loss) to
identify regions of interest that show differential responses to positive and negative outcomes
during the experiment (p < 0.001, 3 contiguous voxels). The main ROIs identified by this
contrast were the right (x, y, z = 10, 10, 0) and left (x, y, z = -9, 10, 1) striatum, once again at the
level of the ventral caudate nucleus. We extracted beta weights for predictors as a function of
competition (auction, lottery) and value (high, low), as well as outcome (win, low). We then
input the mean beta weights for the “loss” trials separately into a repeated measures ANOVA
and probed for a main effect of competition, while hypothesizing that no main effect of value or
interaction with competition would occur during these loss trials, potentially suggesting a lack of
correlation between loss trials and expected value (high and low values). We performed the
same ANOVA with the “win” trials separately for comparison purposes.
Consistent with our previously reported results in the right striatum, we observed a main
effect of competition during losses (F(1,15)= 7.42, p < 0.05) that was not observed during wins
(F(1,16)=1.03, p=0.33). No significant results were found for main effect of value either during
losses (F(1,15)=3.84, p=0.07) or wins (F(1,16)=0.01, p=0.92). No interaction was observed
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during losses (F(1,15)=0.05, p=0.83) or win trials (F(1,16)=3.75, p=0.07). Similar, but slightly
weaker, results were observed in the left striatum, with a main effect of competition during
losses (F(1,15)=3.80, p=0.07), that was not observed during wins (F(1,16)=0.12, p=0.73). No
main effect of value or interaction with competition was observed. Thus, the main finding that
striatum signal during losses is more significant during a socially competitive game such as the
auction is replicated when extra predictors of value are included.
An exploratory analysis was also conducted to examine any gender effects in the bidding
data (9F/8M participants). Using gender as a between-subjects variable and mean beta weights
from the striatum ROIs, we observed no main effect of gender for either right (F(1,15)=0.92,
p=0.35) or left striatum (F(1,15)=0.35, p=0.56), and no interactions with incentive (money,
points), feedback (win, loss) or type of competition (auction, lottery) separately in either ROI.
These results are deemed exploratory as a larger sample is necessary to truly access gender
differences in a competitive auction environment.
Finally, another exploratory analysis was conducted during the decision phase using a
random-effects general linear model (GLM) analysis with each condition (money auction, money
lottery, point auction, point lottery) as predictors (Table S5). Statistical maps were generated
contrasting Auction versus Lottery predictors, irrespective of incentive (p < 0.001, 3 contiguous
voxels). Notably, the left lateral inferior parietal cortex (BA 40; x, y, z= -44, -41, 40) showed the
largest response during auction decisions (Figure S2). Mean beta weights from this region
showed a graded response based first on social competition (i.e., auction), then incentive (i.e.,
money), in accordance with primate studies that suggest this area is involved in processing the
subjective desirability of an action (20). These results are deemed exploratory because the task
was not designed to optimally compare auction and lottery trials during the decision phase when
different cognitive and emotional factors may be interacting.
Methods and Additional Results for Behavioral Study
Participants: A computerized recruitment program enlisted the participation of 120
volunteers for this study. All participants were undergraduate students at New York University.
They were randomly assigned to a condition. The number of participants in Loss-Frame, BonusFrame and Baseline treatments were 52, 46, 22 respectively. No subject participated in more than
one session. Participants were paid a $7 participation fee in addition to their profits in the
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experiment. Experimental dollars were used with the conversion rate: 1 USD = 60 experimental
dollars. All participants gave informed consent.
Procedure: The experiment was conducted at the New York University Center for
Experimental Social Science (C.E.S.S.). The experiment involved 3 sessions called the Baseline,
Bonus-Frame and Loss-Frame treatments. In each session one of the three treatments was
administered. Participants were seated in isolated terminals and the relevant instructions were
given. The experiment was administered by using MultiStage, a generalized program developed
for running economic experiments.
In each condition, participants engaged in a two bidder auction with a random opponent
for 30 rounds. The Baseline was a typical first price auction: A private valuation of the fictitious
good for each subject was drawn uniformly and independently between 0 and 100 experimental
dollars, rounded to the cents. Participants knew their own valuation and the distribution of his
opponent, and were asked to submit a bid. The participant who had submitted the highest bid in
his group won the fictitious good. The payoff was equal to the value minus bid for the winner
and zero for the loser.
The Loss-Frame auction was identical to the Baseline except that subjects were told that
"at the beginning of each round you will be given a sum of 15 experimental dollars which are
yours to keep if you win the auction. This will be your initial endowment. If you lose the auction,
you will have to give this initial endowment of experimental dollars back. Only the person who
wins will be able to keep them. In other words, your payoff in this auction will be equal to your
value minus your bid plus your initial endowment of 15 experimental dollars if you win the
auction and zero if you lose since by losing you must give back your initial endowment".
The Bonus-Frame auction was again identical to the Baseline except they were told that:
"... the payoffs for the auction you engage in will differ slightly from what is described above
since in addition to receiving your value minus your bid if you win, you will also be given an
additional sum of 15 experimental dollars. Only the winner will receive this sum so if you lose
your payoff is zero".
Note that in both of the non-baseline treatments, only the winners get additional 15
experimental dollars; the only difference between these treatments is the way it is framed. Hence,
independent of the form of the utility function, in equilibrium, bid functions should be the same.
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More precisely, in the risk neutral Nash equilibrium of the two non-baseline treatments, subjects
should bid what they bid in the baseline plus 15 experimental dollars.
Supplementary Behavioral Results & Discussion
Results for the behavioral study are presented in the main manuscript. Here, we present
additional analysis and statistics that corroborate the findings on an individual level.
Supplementary table 6 presents Kolmogorov-Smirnov statistics for the equality of the
distributions for individual bid/value ratio in the loss and bonus treatments (Table S6).
Estimations at the individual level (α and β) of the bid functions (b(v) = α + βv) are also
presented for each of the three treatments in the behavioral study: loss, bonus and baseline (Table
S7). The cumulative distribution function for individual-by-individual α and β values are
observed in Figure S3. Supp Figure 3 shows that the distribution of betas in the baseline
treatment is above the distributions of betas both in loss and bonus treatments. Moreover, for the
high betas the distribution of betas in the loss treatment is below the distributions of betas in the
bonus treatment. Such a difference is not observed in the distribution of alphas in the loss and
bonus treatments. These individual results are in line with the pooled data. By using one-sample
t-test, we found that individually estimated betas are not significantly different from the beta
estimates of the pooled data (in loss treatment t=0.2028 with p=0.8402 and in bonus treatment
t=-0.0353 with p=0.9720).
Finally, one question addressed by this study of interest to economics and with respect to
auction designs is how one can design an auction that maximizes the revenue of the auctioneer.
Such an auction design is called “optimal” by economists. Our behavioral study motivated by
our fMRI data shows that individuals increase their bids when the design of the auction
emphasizes losses. Hence, higher revenues are obtained in an auction when loss stakes are high.
From a mechanism design perspective, however, optimal auction must take into account the
behavioral differences of the bidders who may fear potential losses.
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Experimental Instructions for fMRI study
You are about to engage in an experiment in decision-making. Various research
foundations have provided money to fund this project and if you make good decisions you may
be able to make a significant amount of money that will be paid to you when the experiment is
over.
The experiment will be divided up into eight blocks of thirteen trials per block. Thus,
there are a total of 104 trials. During the experiment you will engage in two types of tasks – an
auction and a lottery. These auctions and lotteries will be mixed randomly throughout the blocks,
so that you will play both games in each block, but with no specific pattern of order. You will be
playing for two types of incentives: points – where we will tell you how you rank with other
participants that performed the task; and money – where accumulated winnings are yours to
keep.
As you arrive in the lab you will be paired with another person who will engage in the
experiment with you. He or she will interact with you via a computer while you are in the fMRI
machine and your payoffs will depend on the decision both you and your cohort make.
However, it is important to note that this person is not a naïve subject like yourself, but a
confederate who works for us and has helped design the experiment. He or she will be playing
against you throughout the experiment, and will also be paid in the same way as you, but also
will be playing with a strategy that he or she has been coached on. While we are not going to
divulge the nature of this strategy to you at this time, you should know that it will remain
consistent throughout the experiment.
In addition to the confederate, you will also meet the fMRI technician, who will be
helping us run through the experiment.
The Auction
When playing the auction game, you will be bidding against your experimental cohort to
buy one unit of a fictitious good. The good will have one of four values: $12, $10, $8 or $6.
Which value you get will be randomly determined with equal probability so that there is a ¼
chance that your value will be $12, a ¼ chance that it will be 10, etc etc. The same will be true
for your experimental cohort—he or she will have a good valued at $12, $10, $8, and $6 with
equal probability. Note that while you will be told the value of your good at the beginning of the
trial, you will not be told the value of your cohort’s good. The same is true for your cohort; they
will not know the value of your good. It is also important to note that your cohort will be
bidding with a pre-determined strategy that will be consistent throughout the auction trials. We
have instructed him/her about this strategy, but it will remain unknown to you. This strategy is
part of the experiment.
Once you are told your value and your cohort is told his or hers, each of you will have the
opportunity to bid for the good. When bidding, you will be restricted to choose one of four
possible bids: $2, $5, $7, or $8. The rules of the auction are simple: the person placing the
highest bid wins the good, and pays a price equal to his or her bid. The payoffs for winning are:
Payoff from winning = (Value – bid);
Payoff for losing is $0.
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So for example, say that in the first trial you receive a value for the good equal to $12 and
you make a bid of $7, while your cohort receives a value of $8 and makes a bid of $2. Because
you have placed the higher bid ($7 > $2) you will win the good and will have to pay $7. Your
payoff will therefore be $12-$7 = $5. Your cohort, on the other hand, will receive a payoff of $0.
So what happens if both of you place the same bid? There is a tie, and in this case the
computer will “flip a coin” and to determine who wins. Obviously, the probability of you
winning a tie is equal to that of the probability of your cohort winning the tie (1/2). If you win
the tie you will get the normal payoff (equal to your value minus your bid). If you lose, you will
get nothing.
Note that there is the possibility of overbidding. If you enter a bid greater than your
value, you run the risk of losing money. If you win that bid, you will lose the difference between
your bid and the value of the good. For example, if you overbid $8 for a good of value $6, you
will lose $2 ($8 - $6 = $2).
When trial 1 of an auction block is over, we will proceed to trial 2. In this trial, you will
again receive a good of random value and be asked to make a bid. The random value you get in
this trial is completely independent of the one you received in trial 1; and once again, each value
has an equal probability (1/4) of occurring. So, no matter what value you got last trial, there is an
equal probability that you will receive one of the four possible values this trial. The exact same
thing is true for your cohort.
As we have explained, it is important to remember that your opponent in the auctions is a
confederate who has participated in the designing of this experiment, and thus, is not naïve to it.
Further, he or she will be playing with a fixed consistent strategy throughout the experiment, but
will most definitely be getting paid for their efforts, and thus will be motivated to play the game.
Entering your bids:
At the beginning of any round your computer monitor will show you your value by
stating:
“The good is worth $__.
Bid $2, $5, $7, $8.”
You will be given a hand held instrument with four buttons on it. Each button will be associated
with one of the four possible bids so that if you want to bid $5 you will push the $5 button while
if you want to bid $7 you will push the $7 button, etc. After the value appears on the screen, you
will have four seconds to make your bid. Failure to make a bid in the allotted time will result in
an automatic loss. In between auction questions, there will be a relatively long delay of about 10
seconds before the next auction question appears.
After both you and your cohort have made your bids, the computer will compare them,
calculate your earnings, and show both of you the outcome. If you have won the computer will
display:
“High bid. Payoff = $__”
If you have lost, the computer will display:
“Low bid. Payoff = $0.”
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The Lottery Experiment
Only you, the subject in the fMRI machine, will participate in the lottery trials. Your
cohort who plays in the auction experiment will not be involved. Because of this, during the
lottery trials, you will not be bidding against a live human subject as you were in the auction
trials; instead, you will be playing against the computer.
Unlike the auction trials, in each lottery trial, instead of being given just a value you will
be given a value-bid pair. Both the value and the associated bid will be random. So, for example,
if the value you get is $12, then your associated bid could be $8, $7, $5 or $2 with equal
probability (1/4). Once you are given your value and associated bid, you will have the option of
whether to play or not.
The computer on the other hand, will be bidding against you with the same strategy as
your human opponent uses in the auctions. It will randomly generate a value for itself in the
same way that it generates your value, and then according to its pre-determined strategy, it will
bid. Since both your human and computer opponents use the same strategy to bid, the
probability that you will win should be the same in the lotteries and the auctions.
Your payoff in the lottery trials will be the same as it was in the auction trials. If you win,
then your payoff is Payoff = (Value – bid); if you lose, your payoff is $0. Note that there is no
monetary penalty for losing in the lottery.
For each lottery trial you will be given a value-bid pair. Your only decision is whether or
not you want to play. A lottery trial will be set up as follows:
“Lottery:
The good is worth $__.
Your bid = $__.
Press 1—play.
Press 2—not play.”
However, just like with the auction trials, there is always the possibility of an overbid.
Since the computer is randomly assigning you your bid, it is possible that the assigned bid will
be an overbid. As such, it is very important that you read the bid and the value closely. While
you are welcome to choose to play with an overbid, you should realize that if you win the lottery,
you will lose the difference between the bid and the value.
Overall setup of experiment
The experiment will be divided up into eight blocks (sections), each of which will consist
of thirteen trials (questions). Thus, there will be a total of 104 questions throughout the
experiment. There will be no particular ordering of the questions; all blocks will contain a
randomized mixture of both auctions and lotteries, with the numbers presented also being
randomly determined.
However, there will be a division with respect to the eight blocks: that of incentive. In
this experiment, besides playing for monetary gain (as we have thus far discussed), you will also
be playing for points. These trials will be run in exactly the same manner as the dollar games
described above, but there will be no dollar payoffs to you as a result of your bids. Instead, we
13
will add up your points and tell you how you rank with respect to other subjects who have
participated in the study. Thus, if you do well in these point blocks, you will rank highly in
comparison to the other subjects.
Specifically, four of the eight blocks will be played for points, and four will be played for
money. Incentives will alternate from block to block. Before each block begins (there are eight
blocks), you will be told whether the block will be a point block or a monetary block. Within
that block of thirteen trials, all trials will be for either money or points; there is no mixing of
incentives within blocks.
When you are finished with the experiment, we will choose one of your dollar blocks
(including both auction and lottery rounds) at random and pay you the amount of money you
earned in that block. At the culmination of the entire experiment—i.e., when all of our subjects
have completed it—we will also compile a high score list, which will be sent out to you via
email. No names will be used in the rankings for confidentiality purposes, but by using your
subject number, you will be able to see how you rank compared to the other participants.
14
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15
Table S1. Distribution of bids for both Lottery and Auction trials. Lottery bids were assigned,
while Auction bids were averaged across participants. Bids are horizontal (2, 5, 7, 8) while
values are vertical (6, 8, 10, 12). Bids were matched as closely as possible between lottery and
auction trials as depicted by the total number of bids across values.
Lottery Bids (assigned)
6
8
10
12
Total
2
16.67%
14.29%
0%
8.33%
9.82%
5
83.33%
35.71%
35.71%
16.67%
42.86%
7
0%
50%
21.43%
41.67%
28.28%
8
0%
0%
42.86%
33.33%
19.05%
Auction Bids (averaged across participants)
6
8
10
12
Total
2
22.20%
13.07%
7.40%
2.55%
11.31%
5
75.21%
37.35%
15.76%
10.65%
34.74%
7
0.43%
46.72%
33.98%
25.96%
26.77%
16
8
2.13%
2.87%
42.86%
60.84%
27.18%
Table S2. Descriptive reaction time data from neuroimaging experiment. Times are in
milliseconds.
Reaction Time (milliseconds)
Auction
Lottery
Money
Points
Mean
2150
1834
1981
2002
Median
2130
1872
1977
1971
St.Dev.
279
332
293
284
Table S3. Equilibrium and overbid decisions during the fMRI auction game for each individual
value per incentive. Totals reflect the average for values 6, 8 and 10, as no overbidding is
possible when the value is 12.
MONEY
Value
Equilibrium
6
16%
8
32%
10
37%
12
63%
Total (6-10) 28%
POINTS
Overbid
84%
66%
44%
65%
Equilibrium
20%
44%
35%
57%
33%
Overbid
80%
49%
42%
57%
Table S4. Outcome Phase contrast of Win x Loss trials, p < 0.001.
Outcome Phase: Win vs. Loss Trials
Brain Region
Left superior occipital gyrus
Right ventral striatum: caudate nucleus
Left ventral striatum: caudate nucleus
Left fusiform gyrus
Left fusiform gyrus
Brodman's Area
(BA)
BA 19
BA 37/21
BA 37
17
Talairach Coordinates
(x,y,z )
(-26, -64, 30)
(10, 2, 1)
(-9, 5, -1)
(-44, -49, -7)
(-39, -57, -13)
Number of
Voxels
122
83
116
83
177
ROI p < value
0.0006
0.0005
0.0005
0.0006
0.0004
Table S5. Decision Phase contrast of Auction x Lottery trials, p < 0.001.
Decision Phase: Auction vs. Lottery
Auction > Lottery
Brain Region
Right precuneus
Left inferior parietal lobe
Right precuneus
Lottery > Auction
Right paracentral lobe
Right cuneus
Left inferior frontal gyrus
Left middle temporal gyrus
Right middle occipital gyrus
Right inferior frontal gyrus
Left cuneus
Left lingual gyrus
Right superior temporal gyrus
Right lingual gyrus
Left cuneus
Left middle temporal gyrus
Left medial frontal gyrus
Right lingual gyrus
Right fusiform gyrus
Right Hippocampus
Left middle temporal gyrus
Left parahippocampal gyrus
Right middle temporal gyrus
Brodman's Area
(BA)
BA 7
BA 40
BA 19
Talairach Coordinates
(x, y, z)
(6, -65, 44)
(-44, -41, 40)
(11, -73, 39)
Number of
Voxels
160
218
92
ROI p < value
(11, -32, 48)
(11, -89, 28)
(-54, 17, 19)
(-51, -59, 18)
(21, -90, 15)
(52, 28, 13)
(-16, -91, 12)
(-7, -60, 9)
(61, -39, 8)
(8, -71, 7)
(-8, -94, 6)
(-56, -33, 3)
(-9, 48, 2)
(2, -81, -7)
(27, -43, -8)
(32, -21, -8)
(-37, -24, -10)
(-26, -28, -12)
(48, -34, -12)
130
375
627
528
2072
207
2253
1065
451
1856
897
3160
125
1686
567
117
150
617
136
0.0005
0.0005
0.0005
0.0004
0.0004
0.0005
0.0003
0.0005
0.0005
0.0004
0.0005
0.0004
0.0005
0.0004
0.0004
0.0004
0.0005
0.0004
0.0005
BA 4
BA 19
BA 45
BA 39
BA 19
BA 45
BA 18
BA 30, 19
BA 22, 42
BA 17
BA 17
BA 21
BA 10
BA 18
BA 37
BA 20/21
BA 35
BA 20
0.0005
0.0006
0.0005
Table S6. Kolmogorov-Smirnov statistics for the equality of the distributions of individual-byindividual bid value ratio averages in the loss and bonus treatments in the behavioral study.
K-S Statistic
P value
Reject Equality of
Distributions?
Bid/Value
0.2918
0.019
YES
18
Table S7. Individual-by-individual estimations (α and β) of the bid functions
(b(v) = α + βv) for each of the three treatments in the behavioral study: loss, bonus and baseline.
Loss
α
β
9.919141 0.6479847
4.720401 0.6425769
14.95177 0.6473061
8.148945 0.9641106
17.63813 0.4087365
7.325954 0.7507925
14.72235 0.8849755
6.917519 0.7920502
18.83097 0.6927926
8.468124 0.5642198
18.32981 0.7693515
12.69093 0.9632519
7.595222 0.7155383
6.197822 0.9452072
21.23432 0.4812441
22.24826 0.656263
10.56555 1.005955
10.35742 0.8323147
18.48169 0.7480905
9.021985 0.6235372
15.82074 0.6446262
12.84547 0.4226389
10.3741 0.7347557
21.10602 0.5209144
6.136736 0.7966224
17.89721 0.5584491
15.51911 0.776598
9.581149 0.6908863
-1.247954 0.9111353
4.728478 0.9539073
11.4787 0.7855934
12.13583 0.6088027
-0.872467 0.8703968
16.75871 0.5643411
-4.406765 0.8812832
2.696669 0.7804496
18.37303 0.5550862
13.12433 0.7067662
21.17476 0.5167618
13.47376 0.7989588
1.699087 0.7628618
Bonus
Α
β
8.652137 0.4758416
7.756369
1.005251
10.64434 0.8079708
-19.33583
1.001413
20.93126 0.7132936
19.9902 0.6453473
12.76921 0.6008256
14.64525 0.4805113
4.658813
0.715584
16.64266 0.7201985
20.35244 0.4683794
16.05134 0.4665202
15.0093 0.7219506
12.05498 0.5537742
-2.301492 0.9736962
14.38764 0.7139722
16.8656700 0.6810258
12.21917 0.9728449
12.19666
0.537016
22.59074 0.3449986
5.812759 0.9077426
1.935698 0.8682723
14.88523 0.6940362
12.95929 0.4956494
16.50712 0.6911757
10.4353 0.8954579
17.15828 0.4451198
12.25263 0.6905911
13.63616
0.576942
0.5594087 0.6823685
1.856223 0.8733236
14.58249 0.5937087
13.04895 0.8741735
15.86708 0.4109571
0.7437788 0.6992694
12.69809
1.000863
19.68753 0.6536439
-4.206955 0.6404166
5.355051 0.8823126
0.0699811 0.6933685
3.279988 0.7634166
19
Baseline
Β
0.5083711
0.5631919
0.4366957
0.5288461
0.5736948
0.6551092
0.5504933
0.6920061
0.6911667
0.71785
0.6353496
0.592185
0.6428888
0.6742655
0.4825259
0.7822831
0.6805169
0.5003907
0.5136719
0.5756223
0.760932
0.7212702
19.52405
6.364241
9.942049
0.8217906
4.951505
8.221475
21.74656
9.997553
10.20036
15.72456
6.445052
0.8152677
0.8610829
1.017873
0.9195056
0.8521507
0.8606224
0.5287596
0.6498664
0.9578538
0.7382101
0.6364328
-0.0637984
5.899172
17.54107
1.903102
-2.762338
0.8487893
0.9323578
0.6277891
0.7372607
0.7503069
20
Figure S1
Right ventral caudate nucleus: Outcome Phase
Parameter Estimates (% signal change)
0.5
LOTTERY
AUCTION
0.4
0.3
0.2
Type of
Outcome
0.1
WIN
LOSS
0
-0.1
-0.2
-0.3
-0.4
Money
Points
Money
Points
Supplementary Figure 1: BOLD response in the right ventral caudate nucleus during the
outcome phase and broken down by type of social competition (Auction, Lottery), type of
incentive (Money, Points) and type of feedback (Win, Loss).
21
Figure S2
-8.00
Parameter Estimates (% signal change)
Left inferior parietal cortex: Decision Phase
-4.40
-4.01
8.00
4.40
4.01
t(16)
0.9
0.8
0.7
Type of
Incentive
0.6
0.5
MONEY
0.4
POINTS
0.3
0.2
0.1
0
AUCTION
p < 0.001
LOTTERY
Supplementary Figure 2: BOLD response in the inferior parietal cortex (BA 40) to Auction vs.
Lottery trials in the Decision phase.
22
Figure S3
Cumulative Distribution Functions for Estimated α for Each Subject
Percentage of Subject whose Alpha Estimate is Lower than the
Corresponding α
1.20
1.00
0.80
Loss
Bonus
0.60
0.40
0.20
0.00
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
α values
Cumulative Distribution Functions for Estimated β for Each Subject
Percentage of Subject whose Beta Estimate is Lower than the
Corresponding β
1.000
0.900
0.800
0.700
0.600
Loss
Bonus
0.500
Baseline
0.400
0.300
0.200
0.100
0.000
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
β values
Supplementary Figure 3: Cumulative distribution function for individual-by-individual
estimated α and β values (b(v) = α + βv) in each treatment in the behavioral study: loss, bonus,
baseline
23