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, x_i, 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.

Fig. 1: Distribution of individual estimates, x_i, around the true value x_o

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 assumed to have a negligible effect)

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 monetary value alone does not sufficiently explain bidding behavior. Given the competitive environment of auctions, social factors may also contribute to bidders' evaluation process. Including the social value of winning and losing, the utility U_i of the outcome can be given by:

where b_i is the bid, x_o is the value of the item under auction, r_win is the social value associated with winning, and r_lose 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. While being scanned, subjects participated in auctions in groups of five, bidding against each other.

The session consisted of 40 auction trials, during which subjects received a personal estimate of the item's value, x_i, the error ε (which was the same for all participants in any given trial), 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 x_o-b, where b was the winning bid for that round, while the other participants who lost the round had no change in revenue.

Fig. 2: Auction task

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

Figure 3: GLM design matrix

We estimated a GLM with four regressors of interest. Two separate regressors were created for the time periods during which the subject were presented with 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, which was the third predictor. A final regressor was included for the entry time period, i.e., the time during which participants made their bids based on private estimate and error information. This regressor was included to control for potential carryover effects of BOLD activity into the time window of interest, the outcome period. Six additional regressors of no interest were included for motion. Regressors were convolved with a canonical hemodynamic response function (SPM5). Figure 3 (right) shows a graphic representation of the GLM design matrix.

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 and losing in the brain

Activation maps were overlaid on a representative subject's T1 image.

Figure 4 (below) shows voxels in which BOLD signal activity was significantly greater for win trials than for loss trials during the outcome onset (i.e., when subjects receive feedback about the outcome of the auction). We found robust activation in the dorsal striatum (Figure 4a) as well as a region in the medial frontal cortex (Figure 4b). A region associated with reward-related processing, it has been proposed that the dorsal striatum maintains information about rewarding outcomes (O'Doherty et al., 2004). A more liberal threshold revealed other regions associated with reward representation such as the nucleus accumbens and the dorsolateral prefrontal cortex (Figure 4c).

Figure 4a. p<0.05 FWE corrected, extent threshold = 4 voxels : caudate activation

Figure 4b. p<0.05 FWE corrected, extent threshold = 4 voxels : medial frontal cortex activation

Figure 4c. p<0.001 uncorrected : Nucleus accumbens, anterior insula, DLPFC

The opposite contrast (lose > win) shown in Figure 5 revealed greater BOLD activity in part of the precuneus, located in the posteromedial parietal lobe. As part of the default network, the precuneus is a region engaged in self-referential mental representations, and decreases in activation are reported during engagement of non-self related activities (Cavanna & Trimble, 2006).

Figure 5a. p<0.05 FWE corrected, extent threshold = 4 voxels: precuneus

Figure 5b. p<0.05 FWE corrected, extent threshold = 4 voxels

Social value of winning and losing compared to baseline activity

When compared to the mean BOLD signal, both the beta values of win trials and lose trials showed patterns of activation similar to the win vs. lose contrast. Figure 6 below shows BOLD activity greater for win trials that baseline in the caudate, and at the more liberal threshold, regions including the FFA, anterior insula, and nucleus accumbens.

Figure 6a. p<0.05 FWE corrected, extent threshold = 4 voxels: caudate

Figure 6b. p<0.001 uncorrected - additional areas of activation include FFA, anterior insula, NAcc, thalamus

Activation greater for loss trials than baseline is shown in Figure 7. Noteably, we still see activation in the precuneus above baseline, suggesting that participants were engaged in social/self-referential processing during loss feedback that may not be attributed to default, non-task related processing.

Figure 7a. p<0.05 FWE corrected, extent threshold = 4 voxels: precuneus

Figure 7b. p<0.05 FWE corrected, extent threshold = 4 voxels

Figure 7c. p<0.001 uncorrected - additional areas of activation include anterior insula, DLPFC, ACC, caudate, thalamus

Conclusions

Our main finding of interest was that activity in the bilateral caudate was found when 'win' trials were contrasted with 'lose' trials (p<0.05, FWE). The caudate was previously found to correlate with short term reward, as well as associated with reward-based learning (Haruno et al. 2004). Noteably, because we included an additional regressor for revenue change in our model, the win beta values used in this contrast are independent of monetary gains/losses incurred when participants won a trial, and therefore this contrast reflects differential BOLD activity that is orthogonal to changes in revenue. To the extent that this activation reflects differences related to social evaluations, it is noteworthy that the caudate activation in this case could therefore be associated with the social value of winning.

Medial prefrontal cortex activity was also observed with the win>lose contrast. The medial prefrontal cortex 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). Activity in the insula has previously been found to activate in response to unfair offers in ultimatum game (Sanfey et al. 2003) and predictions of loss (Knutson et al. 2007)

With the lose>win contrast, activation was found in the precuneus. This is interesting because activity in the precuneus was previously found to correlate with reflective self awareness and rating one's personality traits against those of others (Kjaer et al. 2002, Lou et al. 2004). Possibly, the higher activity found in the precuneus during the lose trials might reflect social evaluation of other players in the session.

References

Cavanna, A.E., and Trimble, M.R. (2006). The precuneus: a review of its functional anatomy and behavioural correlates. Brain (129): 564-583.

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.

Kjaer, T., Nowak, M., Lou, H. (2002). Reflective self-awareness and conscious states: PET evidence for a common midline parietofrontal core. Neuroimage. 17(2):1080-6.

Knutson, B., Rick, S., Wimmer, G., Prelec, D., & Loewenstein, G. (2007). Neural predictors of purchases. Neuron, 53(1), 147-156.

Lou, H., Luber, B., Crupain, M., Keenan, J., Nowak, M., Kjaer, T., Sackeim, H., Lisanby, S. (2004). Parietal cortex and representation of the mental Self. Proc Natl Acad Sci U S A. 101(17):6827-32.

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

O'Doherty, J., Dayan, P., Schultz, J., Deichmann, R., Friston, K., and Dolan, R.J. (2004). Dissociable Roles of Ventral and Dorsal Striatum in Instrumental Conditioning. Science 304(5669), 452.

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.