Kelly Hennigan & Grace Tang
Back to Psych 204b Projects 2010
Background
(van den Bos et al. 2008; McClure, van den Bos, in press)
The winner's curse describes a phenomenon where winners in a common value auction (with an item of fixed but unknown value) tend to pay more than the item is worth. Assuming each bidder has an independent estimate of the value of the item, xi, and these estimates are distributed about the true value with error ε (figure 1), the most optimistic estimate will likely be an overestimate. Therefore, if bidders bid at their estimated values, the winner will generally pay more than the true value of the item and incur a net loss.
The optimal bidding strategy to avoid the winner's curse is to adopt the risk-neutral Nash equilibrium (RNNE) strategy, which states that the optimal bid is determined by this equation:
(Under the conditions of the experiment, the Y term is close to zero and is subsequently ignored)
Basically, bidders should adjust their estimates down by the error so that they do not end up paying more than the true value of the item.
However, even when informed of this optimum bidding strategy, bidders continue to bid above the RNNE amount, and end up losing money over many trials because they pay more than the true value of won items. This suggests that there might be some social value to winning (and social cost of losing).
To account for these social factors in the bidding process, the utility Ui of the outcome can be given by:
where bi is the bid, xo is the value of the item under auction, rwin is the social value associated with winning, and rlose is the social value associated with losing.
In this analysis, we examined the neural activity associated with the social value of winning or losing auctions.
Methods
Subjects
Data from 22 individuals was used. Subjects underwent a mathematics quiz given after the experiment to ensure they had the quantitative skills necessary for the experiment.
Auction task
Each subject was endowed with $30 at the beginning of the session. Subjects participated in auctions in groups of 5 or 6, bidding against each other.
The session consisted on 40 auction trials, during which subjects received a personal estimate of the item's value, xi, the error ε, and their current revenue (figure 2). Pictures of other participants were also displayed on the bottom of the screen.
Subjects entered their bids simultaneously. Individual bids were never revealed to other participants. After all the bids were submitted, the winning bidder was revealed. The winner was shown how much they won or lost, while no information about the true value or money won or lost by the winner was given to the subjects who lost the auction. The winner of each auction round won xo-b, where b was the winning bid for that round, while the other participants won $0.
MR acquisition
T2* echo planar images (EPI) and T1 structural images were acquired on 3T Siemens scanners at Baylor College of Medicine in Texas (TR = 2s). Bidding groups, comprised of five or six subjects, were scanned simultaneously.
MR Analysis
Pre-processing
Pre-processing and subsequent analyses were performed using SPM5 (Wellcome Department of Imaging Neuroscience, Institute of Neurology, London, United Kingdom). Images were realigned, normalized to an MNI template, and smoothed with a Gaussian kernel of 4mm full width half maximum.
Model
We estimated a GLM with four regressors of interest. The first regressor was included for the entry time period, i.e. the time during which participants were shown the estimate and error information and made their bids. Two separate regressors were created for the time periods during which the subject received the outcome of the trial, one for trials when the subject won, and another for when the subject lost. For trials in which the subject won, monetary outcome was used as a parametric modulator, giving rise to the final regressor. Six regressors of no interest were included for motion. Regressors were convolved with a canonical hemodynamic response function (SPM5).
We examined the brain activation associated with the social value of winning/losing the auction (Win>Lose) by contrasting the regressors for the outcome period for win vs loss trials. We also ran one-sample t-tests to find brain regions which activated significantly during the outcome periods for win/lose trials to baseline (mean activation across all trials).
A threshold of p<0.05 after FWE correction was applied (extent threshold = 4).
(Data at a more liberal threshold (p<0.001, uncorrected, extent threshold=4) is also presented.)
Results
Behavioral results
Participants consistently bid above the Nash equilibrium. Every participant lost money, and 8 participants lost all of the original $30 endowment (subjects who lost more than $30 were not required to pay for their losses above $30). The amount lost ranged from $6 to $66.4 (mean = $27.82).
Social value of winning/losing in the brain
Activation maps were overlaid on a representative subject's T1 image.
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Fig. 1: Contrast shows significantly greater BOLD signal change when subjects win vs. lose during the outcome onset (i.e., when subjects receive feedback about the outcome of the auction).
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Fig. 2: Opposite contrast, lose > win during outcome onset.
win and lose vs. baseline
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Fig. 3: Win > Mean activity
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Fig. 4: Lose > mean BOLD activity
Conclusions
Activity in the bilateral caudate (and medial prefrontal cortex) was found when 'win' trials were contrasted with 'lose' trials (p<0.05, FWE).
This suggests that the social value of winning is correlated with ... Caudate and reward-based behavioral learning: Caudate was previously found to correlate with short term reward, as well as associated with reward-based learning (Haruno et al. 2004) The Medial Prefrontal has been observed to be active in when inferring the mental states of others (Mitchell, Banaji & McCrae 2005).
At the less stringent threshold (p<0.001 uncorrected), activity was observed in the nucleus accumbens, anterior insula and DLPFC. Nucleus accumbens activation has been found in numerous studies to be associated with reward (Ernst et al. 2005) The insula has been observed to activate to unfair offers in ultimatum game (Sanfey et al. 2003) and predictions of loss (Knutson et al. 2007)
With the lose>win contrast,
mysterious region in the medial parietal lobe, visual areas...
win>baseline strict: MPFC, DLPFC, caudate, (Intraparietal sulcus?), visual areas lenient: FFA, anterior insula, NAcc, thalamus
lose>baseline strict: visual, mysterious transverse parietal area lenient: anterior insula, DLPFC, ACC, caudate, thalamus
References
Ernst, M., Nelson, E., Jazbec, S., McClure, E., Monk, C., Leibenluft, E., et al. (2005). Amygdala and nucleus accumbens in responses to receipt and omission of gains in adults and adolescents. Neuroimage, 25(4), 1279-1291.
Haruno, M., Kuroda, T., Doya, K., Toyama, K., Kimura, M., Samejima, K., et al. (2004). A neural correlate of reward-based behavioral learning in caudate nucleus: a functional magnetic resonance imaging study of a stochastic decision task. J Neurosci, 24(7), 1660-1665.
Knutson, B., Rick, S., Wimmer, G., Prelec, D., & Loewenstein, G. (2007). Neural predictors of purchases. Neuron, 53(1), 147-156.
McClure, S.M., Van den Bos, W. (in press) The psychology of common value auctions. In Attention and Performance XXIII: Decision Making
Mitchell JP, Banaji MR, Macrae CN. 2005. The link between social cognition and self-referential thought in the medial prefrontal cortex. J. Cogn. Neurosci. 17:1306–15
Sanfey, A., Rilling, J., Aronson, J., Nystrom, L., & Cohen, J. (2003). The neural basis of economic decision-making in the Ultimatum Game. Science, 300(5626), 1755-1758.
van den Bos, W., Li, J., Lau, T., Maskin, E., Cohen, J., Montague, R., et al. (2008). The value of victory: social origins of the winner's curse in common value auctions. Judgment and Decision Making, 3(7), 483-492.