Abstract
The sunk cost effect refers to the fact that human decisions are consistently influenced by previous irrecoverable and irrelevant costs. Recent neuroimaging experiments suggest that the dorsolateral prefrontal cortex (dlPFC) plays a pivotal role in the sunk cost effect yet the causal and neurocomputational role of the dlPFC remains elusive. In this study, two cohorts of healthy human male and female adults were recruited to complete a novel two-step decision-making task during the anodal-sham or cathodal-sham high-definition transcranial direct current stimulation (HD-tDCS) over the dlPFC, respectively. Consistent with previous studies, we showed that the sunk cost deterred participants from making further investment and therefore engendered a de-escalation effect. Such behavior can be captured by a weighted mental accounting model with a recalibrated reference point in which the direction and magnitude of the sunk cost effects hinge on the decision weights apportioned to the option values. Interestingly, transcranial stimulation did not influence participants' initial willingness to incur sunk costs but only altered sunk costs' downstream effects. Specifically, anodal stimulation over the right dlPFC amplified the de-escalation effect of sunk costs whereas cathodal stimulation yielded the opposite result. HD-tDCS also changed the decision weights of the mental accounting model, providing a causal and computational link between PFC and sunk cost effects.
SIGNIFICANCE STATEMENT Traditional economic theory assumes that decisions only concern the marginal costs and benefits yet human choices are notoriously susceptible to previously-incurred costs (termed the sunk cost effect). In the current study, we showed that direct current stimulation (DCS) of the right dorsolateral prefrontal cortex (dlPFC) altered sunk cost effects in participants' subsequent choices. Such effects can be captured by a mental accounting model where transcranial stimulation modulates the decision weights assigned to different options in the value integration process. These findings help elucidate the computational and causal role of the dlPFC in the context of sunk costs.
Introduction
According to classic economic theory, rational decision-making only concerns marginal costs and benefits (Samuelson, 1952; Edwards, 1954), yet such an assumption is often violated in real-life observations. Specifically, it is well-established that the current choice can be influenced by previously-received, and thus nominally irrelevant, outcomes (termed the “sunk cost effect”; Staw, 1981; Arkes and Ayton, 1999; Soman and Cheema, 2001; Sweis et al., 2018). However, the direction and the magnitude of sunk cost effects remain controversial. Extant literature suggests that the sunk cost effect can manifest as the tendency to continue or escalate an endeavor after the occurrence of a substantial amount of cost (“throwing good money after bad”; Staw, 1976, 1981). For example, people may continue to play tennis despite developing a tennis elbow just because they have already paid a hefty membership fee to join the tennis club (Thaler, 1980). On the other hand, other studies indicate that previous costs can also cause de-escalated commitments (a reverse sunk cost effect) to pursue an unfinished goal (McCain, 1986; Garland et al., 1990; Heath, 1995; Heath and Soll, 1996; Zeelenberg and Dijk, 1997; Friedman et al., 2007; Macaskill and Hackenberg, 2012; Roth et al., 2015; Nash et al., 2019). For instance, a movie-goer may be more reluctant to purchase another ticket after accidentally losing the original one (as opposed to losing an equivalent amount of money), by arguing that the movie is not worth the price of two tickets (Thaler, 1999). The latter scenario is especially true when the sunk and further costs are of the same physical modality (time, effort, or money) or when the prospects are well-specified (Garland et al., 1990; Heath, 1995; Roth et al., 2015), possibly because of easier calculation of the “total costs” in the cost-benefits comparison process (Thaler, 1999).
The empirical investigation of the neural basis of the sunk cost effect mainly focused on the brain processes accompanying commitment escalation associated with the sunk cost (the “standard” sunk cost effect; Zeng et al., 2013; Haller and Schwabe, 2014; Bogdanov et al., 2015; Fujino et al., 2016, 2018; Wang et al., 2022). In line with earlier findings that stressed the pivotal role of the dorsal lateral prefrontal cortex (dlPFC) in exerting cognitive control in goal-directed tasks (McClure et al., 2004; Kable and Glimcher, 2007; Hare et al., 2009; Weber and Johnson, 2009; Figner et al., 2010), these studies also highlighted the involvement of the dlPFC in implementing certain social rules leading to commitment escalation. For example, heightened dlPFC activity has been associated with escalated commitment in the sunk cost condition, possibly reflecting the enforcement of the “do not waste” rule (Haller and Schwabe, 2014; Fujino et al., 2016). Furthermore, the dlPFC can project its influence into the value comparison process and alter subjects' behavior accordingly (Haller and Schwabe, 2014; Fujino et al., 2016; Wang et al., 2022), via its functional connectivity with regions heavily involved in valuation and moral judgment such as the ventromedial PFC (vmPFC) and the insula (Greene et al., 2001; Sanfey et al., 2003; Rangel et al., 2008). Finally, early causal evidence has linked the dlPFC's activity to the escalation of commitment. For example, anodal transcranial direct current stimulation (tDCS) over the right dlPFC can elicit a pronounced commitment escalation effect associated with the sunk cost in human subjects (Bogdanov et al., 2015). Nevertheless, existing causal evidence regarding the role of the dlPFC is still scant and these results are further complicated by the fact that sunk costs can lead to either commitment escalation or de-escalation, depending on the specificity of decision contexts.
To this end, recent research aimed to develop a general framework to reconcile the seemingly opposite behavioral patterns of decision-making following sunk costs (Roth et al., 2015; Wang et al., 2022) using mental accounting theory (Thaler, 1985, 1999). This approach suggests that people effectively concatenate sunk and prospective costs together, leading to the natural emergence of the commitment escalation effect because of the convex shape of the value function in the loss domain (as opposed to the concave value function in the gain domain). This line of research additionally hypothesizes that people assign different decision weights to the available option values in the value comparison process. The decision weight, therefore, can be viewed as the individual parameter capturing the direction and the magnitude of the decision bias associated with sunk costs. Interestingly, dlPFC activity has been shown to encode the decision weight parameters, and the degree of its functional connectivity with the vmPFC is similarly associated with variation in decision weights across subjects (Wang et al., 2022). Taken together, these results highlight the critical role of the dlPFC in the generation of the behavioral effects of sunk costs, regardless of the effects' directions.
Recent work combining computational modeling and noninvasive brain stimulation methods to investigate the role of the dlPFC in value-based decision-making have made promising inroads in quantifying the computational and causal roles specifically implemented by the dlPFC, stressing its effect in exerting cognitive control in tasks such as temporal discounting, norm enforcement and reinforcement learning (Ott et al., 2011; Buckholtz, 2015; Buckholtz et al., 2015; Shen et al., 2016). Yet, despite pervasive evidence suggesting dlPFC's involvement in sunk cost experiments, the relationship between the dlPFC activity and the direction of the sunk cost decision bias (commitment escalation or de-escalation) remains controversial. Indeed, although earlier literature suggested heightened dlPFC activity was related to commitment escalation (Zeng et al., 2013; Haller and Schwabe, 2014; Bogdanov et al., 2015; Fujino et al., 2016, 2018), other studies attributed stronger dlPFC response to commitment de-escalation (Wang et al., 2022). It therefore begs the question of the exact computational role played by the dlPFC in a sunk cost environment.
To address this question, we designed a two-step decision task where human subjects were asked whether to incur sunk costs and then to make a choice about purchasing a lottery ticket. We recruited two cohorts of subjects and, while subjects performed the task, we applied anodal-sham and cathodal-sham high-definition tDCS (HD-tDCS) over subjects' right dlPFC to manipulate brain activity. Anodal tDCS has been shown to increase neuronal resting potentials and therefore has an excitation effect in the targeted brain areas, whereas cathodal tDCS has the opposite effect (Jacobson et al., 2012; Filmer et al., 2014). We chose the right dlPFC as the target brain area for two reasons. First, previous fMRI studies implicated the role of the right dlPFC in the sunk cost effect (Haller and Schwabe, 2014; Fujino et al., 2016). And second, specific computational role for the dlPFC has been hypothesized in our previous work (Wang et al., 2022). We systematically tested how the decision bias of sunk costs was manipulated by the HD-tDCS over the right dlPFC, and found that the anodal stimulation increased commitment de-escalation effect and the cathodal stimulation yielded the opposite effect, in both model-free and model-based analyses.
Materials and Methods
Participants
Eighty-four healthy right-handed volunteers (42 females; age: 18–27 years; mean = 21.11, SD = 1.98) were recruited into two cohorts (anodal and cathodal) for the study. A total of five participants were removed from the final analysis for either not following the task instructions (n = 3), or the equipment failures (n = 2), leaving 39 participants (19 females; age: 18–25 years; mean = 21.08, SD = 1.88) in the anodal group and 40 (21 females; age: 18–25 years; mean = 21.03, SD = 1.94) in the cathodal group. All participants were free of neurologic and psychiatric history and provided with written informed consents before the experiment. This experiment was approved by the Ethics Committee of the School of Psychological and Cognitive Sciences, Peking University.
Task and design
This study used a sham-controlled single-blind design, with the type of HD-tDCS stimulation (anodal or cathodal) as a between-group variable. Subjects in the anodal (cathodal) group received anodal (cathodal) and sham stimulation in a counterbalanced order over the right dlPFC in two experiment sessions, which were separated by ∼24 h to avoid carry-over effects of tDCS (Shen et al., 2016).
In each session, participants completed a two-step decision task (Fig. 1A). In this task, at the beginning of each trial, the face value of a lottery ticket (50% chance of winning ¥20) and the amount of the entrance fee were presented on the screen and subjects had up to 4 s to decide whether to pay the entrance fee to earn the privilege to purchase the lottery ticket later in the trial. The trial ended if subjects refused to pay the entrance fee or no choice was made within 4 s. Otherwise, the trial proceeded and the lottery ticket price was revealed where subjects had to make their second choice as to whether to purchase the ticket at the listed price or quit the trial (without paying for the ticket). In either case, the already paid entrance fee was irrecoverable and therefore “sunk.” Critically, in our task, subjects incurred sunk costs on their own will, thus circumventing the confounding issue of sunk cost's agency effect (Thaler, 1980). All the information (entrance fee, ticket price, and ticket face value) related to the ticket purchase remained on the screen during the trial and a 2-s intertrial interval (ITI) was introduced between trials (Fig. 1A).
The face value and the winning probability of the lottery ticket were fixed to ¥20 and 50% in the task, respectively. We independently manipulated the amounts of the entrance fee and the lottery ticket price. The entrance fee had five different levels: [0, 1, 2, 3, 4], and the lottery ticket price had six levels: [2, 4, 6, 8, 10, 12]. Each combination of the entrance fee and the lottery ticket price was repeated three times, rendering a total of 90 trials per session. To discourage participants from detailed mental calculation, a random and independent jitter between −0.2 and 0.2 (with the precision of 0.01) was added to the entrance fee and the lottery ticket price in each trial. Subjects also performed a brief practice session (six trials) before the beginning of their first session, and the screen locations of “Pay” and “Quit” (corresponding to the left and right keyboard buttons) were randomized across subjects.
Procedures
Subjects went through the safety screening to ensure their eligibility for the HD-tDCS. They were then given detailed task instructions and an ¥18 endowment before each experiment session. Participants were told that one randomly selected trial would be honored as their final payoff for each session to ensure incentive compatibility. In addition, subjects also received ¥140 for their participation, making their average total payments close to ¥194 (SD = 13).
The right dlPFC was localized with the 10/20 EEG system at F4. In each session, participants performed the decision task while receiving either active or sham HD-tDCS. The anodal group received right dlPFC anodal stimulation (F4+), and sham stimulation in a counterbalanced order and so did the cathodal group (F4– and sham). The experiment was implemented in Psychtoolbox 3 (PTB-3; http://www.psychtoolbox.org/) in MATLAB R2010b.
HD-tDCS
The HD-tDCS stimulation was delivered using a multi-channel stimulation adapter (SoterixMedical, 4 × 1-C3) which was connected to a battery-driven constant current stimulator (SoterixMedical, Model 1300-A). Five Ag-AgCl sintered ring electrodes were held in plastic casings filled with conductive gel, embedded in an EEG cap, and attached to the adaptor device. We arranged those five electrodes on the skull in a 4 × 1 ring configuration (Edwards et al., 2013; Villamar et al., 2013a; Shen et al., 2016). The return electrodes were spaced ∼7.5 cm radially around the central electrode and at the corners of a square (Villamar et al., 2013a, b). Specifically, for our right dlPFC stimulation, the central electrode was placed above F4 in the 10/20 EEG system, with the other four return electrodes' locations corresponding roughly to the C4, F8, Fp2, and Fz (Fig. 1B). As suggested by previous HD-tDCS studies, we used central anodal stimulation for the excitatory modulation (F4+) and central cathodal stimulation for inhibitory modulation (F4– ; Filmer et al., 2014). For both the anodal and cathodal stimulation sessions, the current intensity was set to 2.0 mA, which created ∼0.5 mA/cm2 current density at the central electrode, and ∼0.125 mA/cm2 peak current density at the return electrodes. Stimulation started 5 min before the task, and was delivered during the entire course of the task (∼15 min), with an additional 30-s ramp-up period at the beginning of the stimulation and 30-s ramp-down period at the end. Previous studies have shown that such intensity and total charge were safe for humans and sufficient to influence dlPFC activities (Minhas et al., 2010; Borckardt et al., 2012; Kuo et al., 2013; Shen et al., 2016). The placement of electrodes was the same for the sham condition with only the 30-s ramp-up period and a 30-s ramp-down period at the beginning and the end of the session. Our participants' reports suggested they could not distinguish whether a session was a sham or an active tDCS treatment (χ2(1) = 1.253, p = 0.263), and no adverse effects were reported.
Behavioral statistical analysis
Choice behavior were analyzed using R v3.5.1 (http://www.r-project.org) and Rstan. We used both model-free and model-based (see below) analyses to test for the influence of HD-tDCS on the sunk cost effects. Behavioral analyses focused on the trials where subjects paid the entrance fee (valid trials; Fig. 1C,D). All p values reported were two-tailed.
In the model-free analysis on the behavioral effects, we estimated the indifference point (IP) of subjects' lottery purchase choice at each sunk cost level. Specifically, we fitted a logistic function to the lottery purchase choice with the lottery price as the independent variable:
At the IP, subjects would choose the “Purchase” or “Quit” option with equal probability and the IP was calculated as following:
Lottery prices and the corresponding lottery purchase choices from each condition and each entrance fee (sunk cost) level were submitted into separate linear mixed models to fit β0 and β1. The package 'lme4' in R was used (Bates et al., 2015), and both the intercept and the independent variable (lottery price) had a fixed effect across all subjects and a random effect for each subject. The IPs were then calculated at each stimulation condition and each entrance fee (sunk cost) level separately with larger IP indicating people are willing to pay more for the lottery. Those IPs were then used as the dependent variable in the following ANOVA. Finally, to directly evaluate the effects of sunk costs on lottery purchase decisions, for each subject and each condition, we ran another regression with IP as the dependent variable and sunk costs as the independent variable to obtain the slope of sunk costs, βIP-SK.
Ideally, the effect of HD-tDCS on the sunk cost associated behavioral bias can be assessed by specifying a more comprehensive linear mixed model that includes active and sham conditions for each group (anodal/cathodal) and the interaction term between sunk costs and conditions (active vs sham) reflects the modulatory effect of tDCS on the sunk cost bias. Unfortunately, such estimation did not converge in the model fitting process in R and we therefore reported the linear mixed-effect regression results from separate conditions here.
Mental accounting model
The classic prospect theory typically defined the reference point as the status quo (Kahneman and Tversky, 1979), whereas previous research in the mental accounting literature proposed that people would incorporate prior outcomes into prospect valuation, effectively shifting their internal reference points (Thaler, 1980, 1999). We adopted this approach and augmented the model with an additional parameter to capture subjects' decision weights assigned to the values of lottery purchase and opt-out options (weighted mental accounting model, M1). More formally, the difference in subjective values between purchase and opt-out options was defined as (assuming no probability distortion):
In our previous work, we verified the validity of this model in explaining individual differences of the sunk cost effects, as well as its adequacy and stability (Wang et al., 2022). In the current study, we again tested it against three alternative models. First, we tested the classic prospect theory model where the reference point was defined as the status quo (prospect theory model, M2). In such a baseline model, the option value difference was calculated as the following:
Second, we compared our candidate model (M1) with the one introduced by the original mental accounting paper (original mental accounting model, M3; Thaler, 1980). The only difference between these two models was the introduction of the parameter κ (decision weight) in our model. The option value difference of model 3 (M3) was defined as:
Such a model shifts subjects' reference points and effectively places them in the loss domain and therefore could account for the commitment escalation effect (Thaler, 1980).
Lastly, we tested a candidate model where the opt-out option value was 0 and no weighting parameter was included, and the subjects treated the sum of the entrance fee and the lottery price as a full compound price of the lottery. In such a model (compound price model, M4), the option value difference was calculated as the following:
Again, the option value of opting out under such model is 0. Such a model can be viewed as a special case of M1 where the decision weight parameter κ is set to 0, and therefore should predict a predominant de-escalation effect across subjects. The same choice function and value function were applied to all the models tested.
Estimation procedure
We estimated all the candidate models (M1–M4) with the same hierarchical Bayesian analysis (HBA) approach, which has been shown to improve the precision of subject-level parameter estimates and provide more reliable parameter recovery (Ahn et al., 2011; Hill et al., 2017). Under the HBA framework, individual-level parameters were sampled from group-level distributions. Since in the current design each subject completed the task twice (active vs sham condition), we assumed that at the individual level, each subject had different parameters for each condition (taking M1 for example, one subject from the anodal group would have six parameters κanodal, βanodal, μanodal for the anodal condition and κsham, βsham, μsham for the sham condition), and each pair of parameters (for example, κanodal and κsham) was sampled from a two-dimensional joint normal distributions with correlation coefficient ρ: N(μ1, μ2, σ12, σ22, ρ). In order to avoid the bias to the posterior distribution within a small sample size, we used normal and half-Cauchy distributions to specify the priors of the group-level means [μ1 and μ2 ∼ N(0, 1)] and SDs [σ1, σ2 ∼ Cauchy (0, 2.5)], respectively. The Lewandowski-Kurowicka-Joe (LKJ) covariance matrix was used to specify the priors of the group-level covariance between conditions (Lewandowski et al., 2009). In the model fitting, all parameters were restricted to the following ranges (κ ∈ [0, 1] in M1, β ∈ [0, 2], μ ∈ [0, +∞) in M1–M4) by proper mathematical transformations. We performed posterior inference with standard Markov chain Monte Carlo (MCMC) sampling methods implemented in Rstan (v2.15.1; Carpenter et al., 2017). A total of 40,000 samples were drawn after a burn-in period of 20,000 samples with four chains (5000 samples after 5000 burn-in samples for each chain). R-hat values of all model parameters were <1.01, indicating satisfactory convergence of the MCMC chains (Ahn et al., 2017). The Watanabe–Akaike information criterion (WAIC) was used to assess the performance of all the models (Watanabe, 2010). To validate the model fit while controlling for overfitting, we further performed a cross-validation analysis. Specifically, for each participant in each group (anodal and cathodal), we randomly selected half of the valid trials (where subjects paid the entrance fee) in both the active stimulation and sham conditions to estimate parameters (estimation trials) by the HBA method described above for each candidate model. We then used these parameters to predict choices in the remaining other half valid trials in each condition (test trials). This process was repeated 20 times for each subject and we measured the predictive accuracy by calculating the prediction accuracy index (proportion of correctly predicted choices in the test trials) for all four models (M1–M4).
Data availability
All the data and codes that support the findings in this study will be available upon request.
Results
HD-tDCS and the entrance fee decisions
Since the lottery purchase decisions were contingent on subjects paying the entrance fee (valid trials), we first analyzed subjects' entrance choices. We conducted an entrance fee × stimulation (active vs sham) × group (anodal vs cathodal) ANOVA with gender and stimulation sequence (stimulation first vs sham first) as between-subject control variables on how many trials subjects paid the entrance fee. This analysis only revealed a significant effect of the entrance fee (F(4,284) = 113.096, p < 0.001, all other ps > 0.05). Indeed, subjects' first choices were sensitive to the magnitude of the entrance fee across all conditions: their willingness to participate in a certain trial decreased as the amount of the entrance fee increased in both the anodal and cathodal groups, regardless of the stimulation content (active stimulation or sham; Fig. 1C,D). On average subjects paid entrance fees in 83.9% and 86.1% of the trials in the anodal and cathodal groups, respectively. Importantly, there was no significant main effect of stimulation (F(1,71) = 1.151, p = 0.287), stimulation × group interaction (F(1,71) = 1.956 p = 0.166), or stimulation × entrance fee level interaction (F(4,284) = 0.449, p = 0.703). The three-way interaction effect of stimulation × group × entrance fee was not significant (F(4,284) = 0.369, p = 0.760) either (Fig. 1C,D). These results therefore ruled out the possibility that the HD-tDCS stimulation effect on the lottery purchase behavior was because of the difference of valid trials.
Experimental design and simulated HD-tDCS electric field intensity map. A, In each trial, subjects first chose whether to pay the entrance fee to enter the lottery game. Once the entrance fee was paid, the lottery price was revealed and subjects had to decide whether to purchase the lottery at the listed price. B, The current flow produced by the anodal HD-tDCS was restricted within the ring of reference electrodes. C, D, There was no difference between the numbers of trials subjects paid entrance fee as a function of the sunk cost in the anodal (F(4,140) = 0.181, p = 0.889) and cathodal groups (F(4,144) = 0.656, p = 0.570). Error bars: mean ± SEM.
Sunk cost-sensitive lottery purchase decision
In contrast to the prediction of the prospect theory, our data showed subjects' lottery purchase decisions were affected by the entrance fee paid earlier (sunk cost) in the trial. As shown in Figure 2, regardless of the stimulation content (active vs sham) or type (anodal vs cathodal), the proportion of lottery purchase choices varied both as a function of the lottery price and the sunk cost level. To formally test whether subjects were sensitive to sunk costs when they made lottery purchase decisions, we conducted a two-way ANOVA with two factors: entrance fee (sunk cost) and lottery price, and the proportion of lottery purchase choice as the dependent variable by controlling gender and stimulation sequence as nuance variables. These analyses revealed significant main effect of sunk costs in all conditions (anodal group: anodal condition F(4,52) = 38.612, p < 0.001, sham condition F(4,48) = 9.330, p < 0.001; cathodal group: cathodal condition F(4,64) = 22.882, p < 0.001, sham condition F(4,44) = 18.868, p < 0.001), as well as significant main effect of lottery prices (anodal group: anodal condition F(5,65) = 169.348, p < 0.001, sham condition F(5,60) = 75.806, p < 0.001; cathodal group: cathodal condition F(5,80) = 176.408, p < 0.001, sham condition F(5,55) = 134.776, p < 0.001).
Proportion of lottery purchase choices. Color maps represent the proportions that subjects chose to purchase the lottery at corresponding lottery price and sunk cost levels. Upper panels, Results from the anodal group (left, anodal; right, sham). Lower panels, Results from the cathodal group (left, cathodal; right, sham).
The above results indicated that in general, sunk costs led to a commitment de-escalation pattern, regardless of the stimulation type (anodal or cathodal) or content (active stimulation or sham). Next, to quantitatively illustrate the effects of the brain stimulation, we conducted the following model-free and model-based analyses.
Model-free analysis of the dlPFC stimulation on the behavioral effects of sunk costs
We then assessed the impact of the sunk cost (entrance fee) on lottery ticket purchase choices and the impact difference between stimulation conditions. To this end, we first calculated the lottery purchase IP for each sunk cost level in each session. Since the IP reflects how much subjects are willing to purchase the lottery, the escalation effect of sunk cost should correspond to the positive correlation between the IP and the entrance fee level. Conversely, the de-escalation of commitment effect would lead to a negative correlation between the IP and the entrance fee. An individual example of analysis was given in Figure 3A, where the IPs for both the anodal (Fig. 3A, top panel orange) and sham (Fig. 3A, bottom panel gray) conditions were plotted against their corresponding sunk cost levels (Fig. 3A, middle panel). The effect of HD-tDCS stimulation can be inferred from the change of the regression coefficients between IPs and sunk costs (βIP-SK) in the active stimulation as well as in sham sessions in each group.
Model-free analysis of HD-tDCS over the dlPFC. A, Behavioral data from an example subject for illustration. IPs of the lottery price were estimated for different sunk cost levels for both the anodal (top panel, orange) and sham (bottom panel, gray) conditions using logistic regressions. The IPs were then regressed to the sunk costs to obtain βIP-SK in each condition (middle panel). B, As the sunk cost level increased, subjects' IPs decreased differently between the anodal (orange) and sham (gray) conditions in the anodal group (F(4,108) = 2.936, p = 0.024), suggesting a more prominent de-escalation effect of the anodal stimulation. C, In the cathodal group, the sunk cost × stimulation (cathodal, blue vs sham, gray) interaction effect was only marginally significant (F(4,112) = 2.279, p = 0.082). Error bars represent SEM.
To test the HD-tDCS impact on the effects of sunk costs at the group level, we conducted a two-way ANOVA with sunk cost levels and stimulation contents (active vs sham) as within subject variables on the IPs in both the anodal and cathodal groups, respectively, again controlling for the gender and stimulation sequence. In the anodal group, this analysis revealed a significant main effect of sunk cost (F(4,108) = 50.023, p < 0.001) but a nonsignificant main effect of stimulation content (anodal vs sham; F(1,27) = 0.028, p = 0.869). Critically, the interaction effect between the stimulation and sunk costs was significant (F(4,108) = 2.936, p = 0.024; Fig. 3B), indicating that the anodal HD-tDCS over the right dlPFC altered the effects of sunk cost on IPs. Similar results were obtained for the cathodal group, with a significant main effect of the sunk cost (F(4,112) = 29.680, p < 0.001) but no effect of the stimulation (cathodal vs sham; F(1,28) = 0.071, p = 0.791). However, the sunk cost × stimulation interaction effect was only marginally significant (F(4,112) = 2.279, p = 0.082; Fig. 3C) in the cathodal group. Furthermore, to test whether the anodal and cathodal stimulation yielded different effects, we added group (anodal vs cathodal) as a between-subject variable and ran a three-way ANOVA (group × stimulation × sunk cost). Consistent with previous results, this analysis revealed a significant three-way interaction among sunk cost, stimulation and group (F(4,220) = 2.493, p = 0.047).
Taken together, these results suggested that subjects in general showed a de-escalation effect of sunk costs, and the right dlPFC anodal stimulation increased such a tendency, whereas the cathodal HD-tDCS stimulation tended to have the opposite effect.
Model-based analysis of the dlPFC stimulation
We further developed a quantitative mental accounting model in which subjects integrated the entrance fee (sunk cost) into the valuation and comparison processes of prospects, as suggested previously (Kahneman and Tversky, 1979; Thaler, 1980). The key idea in this model is a subject specific decision weight κ that captures the relative weights subjects assign to the choice options when they compare option values. In the classic prospect theory, the reference point is assumed to be the status quo, the decision weight has no influence since the value for the opt-out option (reference option) is always 0. However, under the mental accounting framework, the subjective value of the reference option is recalibrated by the sunk cost (Thaler, 1980, 1985, 1999). Intuitively, when κ is close to 0, subjects mainly focus on the prospect of purchasing the lottery and the sunk costs are simply added to the ticket price to augment the overall costs, thus rendering subjects less likely to purchase lotteries (Heath, 1995; Thaler, 1999). As κ increases toward 1, subjects are mainly concerned with the prospect of the opt-out option and therefore tend to purchase tickets more often under higher sunk costs to avoid appearing wasteful (Thaler, 1980). Therefore, the decision weight κ can be viewed as an idiosyncratic measure of the de-escalation effect, with the smaller κ indicating more prominent de-escalation effects (less escalation effect). Similar ideas about assigning decision weights to option values have also been applied in recent eye-tracking research on value-based decisions (Krajbich et al., 2010; Crockett et al., 2017) and neuroimaging studies of sunk cost effects (Haller and Schwabe, 2014; Fujino et al., 2016; Wang et al., 2022). Our simulation results showed that when κ was small, IPs were negatively correlated with sunk costs (de-escalation effect), with this relationship switching to the positive correlation (escalation effect) when κ was larger (Fig. 4A).
Model-based analysis. A, The IP function of the sunk cost varies depending on the specific decision weight parameter κ in the weighted mental accounting model. When κ is small, IPs decrease as the amount of sunk cost increases (commitment de-escalation). When κ is large, the relationship between IPs and sunk costs reverses (commitment escalation). B, In both the anodal (orange) and cathodal (blue) groups, our weighted mental accounting model (M1) performed the best among all the candidate models. C, Predictive accuracy for all the models. Weighted mental accounting model's performance exceeded all the other models in both the anodal (orange) and cathodal (blue) groups (all ps < 0.001). D, Compared with the sham (gray) condition, the anodal stimulation (orange) decreased the decision weight κ in the anodal group (paired t test, t(38) = −3.468, p = 0.001), whereas the cathodal stimulation (blue) had the opposite effect in the cathodal group (paired t test, t(39) = 7.942, p < 0.001). The group × stimulation interaction effect was also significant (F(1,71) = 37.401, p < 0.001). Solid dots represent the group mean; **p < 0.01, ***p < 0.001. Error bars represent SEM.
We fitted our weighted mental accounting model to subjects' behavioral data with a hierarchical Bayesian algorithm (HBA) and summarized the model fitting results in Table 1. We also tested the performance of our model (weighted mental accounting model, M1) against the traditional prospect theory model (prospect theory model, M2; Kahneman and Tversky, 1979), Thaler's mental accounting model (original mental accounting model, M3; Thaler, 1980), and a compound price model (M4; for details, see Materials and Methods). Simply put, the prospect theory model did not include sunk costs to recalibrate the reference point; Thaler's mental accounting model did not contain the individual decision weight parameter κ, and the compound price model was a special case of model M1 (κ = 0). Model comparison results confirmed the weighted mental accounting model's (M1) better performance in explaining choice behavior relative to other candidate models (Fig. 4B). Moreover, the cross-validation analysis also demonstrated that M1 performed the best among all the candidate models in terms of predictive accuracy in both the anodal and cathodal groups (all ps < 0.001; Fig. 4C).
Model fitting results
We went further and tested the effects of HD-tDCS under this theoretical framework. In line with the model-free results that the right dlPFC anodal stimulation increased subjects' de-escalation tendency, we also found that in the anodal group, compared with the sham condition, the sunk cost de-escalation effect was significantly more prominent in the anodal condition (paired t test, t(38) = −3.468, p = 0.001; Fig. 4D). On the contrary, the decision weight was significantly larger in the cathodal condition relative to the sham condition in the cathodal group (paired t test, t(39) = 7.942, p < 0.001; Fig. 4D), indicating a lower de-escalation effect. A two-way ANOVA on the decision weights κ with the stimulation (active vs sham) and group (anodal vs cathodal) as independent variables again revealed a significant interaction effect between the group and the stimulation (F(1,71) = 37.401, p < 0.001). In summary, we found that anodal and cathodal HD-tDCS over the right dlPFC exerted opposite effects, with the anodal stimulation increasing the de-escalation effect and the cathodal stimulation alleviating the de-escalation effect under the mental accounting framework.
The correlation between model-free and model-based results
Lastly, if both the model-free and model-based approaches depicted the same behavioral effects caused by the HD-tDCS, we should expect tight correlations between these approaches. To test this, we examined the correlation between the model-free IP-sunk cost regression coefficients βIP-SK (Fig. 3A) and the model-based decision weight κ in each session, as well as the differences in βIP-SK and κ induced by HD-tDCS across subjects. As expected, we observed significantly positive correlation between variables βIP-SK and κ in both the anodal (r = 0.569, p < 0.001) and sham (r = 0.761, p < 0.001) conditions in the anodal group (Fig. 5A,B), and in the cathodal (r = 0.767, p < 0.001) and sham (r = 0.771, p < 0.001) conditions in the cathodal group (Fig. 5D,E), after controlling for the nuance variables including the risk attitude β, inverse temperature μ from the model-based analysis of each individual (Pearson correlations between κ and μ: r = 0.063, p = 0.435; κ and β: r = 0.447, p < 0.001, across all conditions).
Correlations between model-free and model-based results. The model free index of the sunk cost effect (βIP-SK) positively correlated with the decision weight parameter κ from the weighted mental accounting model in both (A) the anodal condition and (B) sham condition in the anodal group; and in both (D) the cathodal condition and (E) sham condition in the cathodal group. The change of βIP-SK (Δ βIP-SK) also positively correlated with the change of κ (Δ κ) between the active and sham conditions in the anodal group (C) and the cathodal group (F). Gray shades represent the 95% confidence interval (CI).
Similarly, we further tested whether the βIP-SK difference (ΔβIP-SK) correlated with the decision weight κ difference (Δκ) between the active stimulation and sham conditions. Indeed, for both groups, we found significant positive correlations [anodal group: r = 0.445, p = 0.006 (Fig. 5C); cathodal group: r = 0.676, p < 0.001 (Fig. 5F)], again after controlling for the potential confounds of Δβ and Δμ. Overall, our model-free and model-based results corroborated with each other and supported the hypothesis that the anodal and cathodal stimulation over the right dlPFC led to opposite effects.
Discussion
The sunk cost effect is one of the most recognized decision biases and is believed to underlie a variety of “irrational” behaviors, such as continuing an endeavor that is doomed to fail or quitting too early when a project is in fact profitable (Staw and Ross, 1989; Heath, 1995). It has been proposed that the setup of inflexible mental accounts might be the culprit for such a bias (Thaler, 1980). Interestingly, however, most of the early studies on the sunk cost effect mainly focused on the commitment escalation phenomenon, ignoring significant heterogeneity among people, some of whom demonstrated the commitment de-escalation effect. Importantly, although previous research has recognized the critical role of the dlPFC in mediating sunk cost effects, the exact computation that the dlPFC implements remained elusive (Zeng et al., 2013; Haller and Schwabe, 2014; Bogdanov et al., 2015; Fujino et al., 2016, 2018). Built on a modified mental account framework (Wang et al., 2022), we investigated the causal effects of HD-tDCS over the dlPFC via a two-step decision task. Consistent with our previous research, we found an overall de-escalation effect associated with sunk costs, regardless of the stimulation type (anodal or cathodal) or stimulation content (active stimulation or sham). Specific to the current study, we also showed that HD-tDCS over the right dlPFC indeed causally altered subjects' attitude toward sunk costs: relative to the sham condition, anodal stimulation led to further commitment de-escalation whereas cathodal stimulation tended to have the opposite effect. These results could be interpreted as the change of decision weights κ in our mental accounting model: anodal stimulation over the right dlPFC prompted subjects to focus more on whether the profit prospect was worth the “total cost” (sum of the sunk and prospective costs) and therefore led to further commitment de-escalation. In contrast, cathodal stimulation over the dlPFC directed subjects to ruminate more on the costs already “sunk” and thus to be more likely to stick to the “do not waste” rule compared with the sham condition.
The dlPFC has long been implicated as a crucial structure in arbitrating between different decision systems in the brain, ranging from contemplating the trade-off between monetary amount and delay in the intertemporal choice tasks (McClure et al., 2004; Kable and Glimcher, 2007; Figner et al., 2010), to food options competing along the tastiness and healthiness axes (Hare et al., 2009, 2011). It is believed that the dlPFC might represent certain abstract rules or context information that influence valuation and decision-making processes via functional connections to brain areas such as the ventral striatum and vmPFC (Aron et al., 2004; Hare et al., 2009; Kahnt et al., 2011; Buckholtz, 2015). For example, previous studies on social cognition using tDCS and transcranial magnetic stimulation (TMS) provided causal evidence that altering the activities of the dlPFC intervened in the valuation processes where social norm integration was necessary and thus biased participants' action selection (Baumgartner et al., 2011; Hayashi et al., 2013; Ruff et al., 2013; Buckholtz et al., 2015). Similarly, studies focusing on the sunk cost effect also ascribed the representation of social rules such as “do not waste” to the realm of the dlPFC functions (Haller and Schwabe, 2014; Bogdanov et al., 2015; Fujino et al., 2016, 2018; Wang et al., 2022). Recent work investigating the sunk cost effect and mental disorders also found that people with gambling disorders experienced difficulty stopping “the loss-chase,” paralleled by changes in the cognitive control brain network function (Worhunsky et al., 2017; Fujino et al., 2018). Inspired by these studies, we also applied tDCS over the dlPFC and showed that such modulation altered subjects' behavioral sensitivities toward sunk costs, from both the model-free and model-based perspectives.
It is also worth mentioning that we used F3/F4 in the 10–20 EEG system to localize the dlPFC, an approach adopted by a number of previous conventional and HD-tDCS studies focusing on cognitive control in tasks including social cognition, language, working memory, delay discounting and risky decisions (Fecteau et al., 2007a, b; Kekic et al., 2014; Nikolin et al., 2015, 2018, 2019; Shen et al., 2016; Chua et al., 2017; Huang et al., 2017; Guo et al., 2018; for review, see Dedoncker et al., 2016; Parlikar et al., 2021). However, HD-tDCS's effects may not be focal enough (Kuo et al., 2013) and the electric current might flow into a relatively large brain area around the central electrode (Kuo et al., 2013; Shen et al., 2016). Also, we adopted a single-blind rather than a double-blind design. Although this is consistent with many previous tDCS studies (Nikolin et al., 2015, 2018, 2019; Guo et al., 2018; Verveer et al., 2021), participants might be aware of the stimulation difference according to experimenters' operations. However, this is unlikely since most subjects did not report differences in their subjective feelings in our postexperiment survey.
In our weighted mental accounting model, the decision weight parameter κ acted on option values that were recalibrated by the sunk cost. This extension to the classic mental accounting framework is able to capture the individual differences of sunk cost effects across subjects and more importantly, accounts for the less mentioned de-escalation effect or reverse sunk cost effect (Garland et al., 1990; Heath, 1995; Roth et al., 2015; Wang et al., 2022). Similar ideas have been proposed in previous studies on decision-making and reinforcement learning to describe the asymmetric cognitive processes of different decision related option attributes (Krajbich et al., 2010; Li and Daw, 2011; Crockett et al., 2017). Our model was thus able to reconcile the discrepancies between recent neuroimaging research that proposed the sunk cost effect was predicated on the degree to which subjects followed top-down principles such as “do not waste” (Staw and Ross, 1989; Haller and Schwabe, 2014; Fujino et al., 2016; Wang et al., 2022), and other studies that suggested subjects might fall prey to the de-escalation effect if they held a relatively rigorous mental budget in a sequential investment decision task (Heath, 1995). In addition, the neural encoding of decision weights results resonate with the well-established hypothesis that the sustained activity of the dlPFC instantiates working memory representations (Miller et al., 2018; Romo and Rossi-Pool, 2020). In our task, all the necessary information including the sunk cost, the lottery price and the lottery face value remained on the screen while subjects made their choices to minimize the working memory load. However, it is likely that the activity of the dlPFC and thus the decision weight reflect the degree of emphasis on certain abstract rules, which prioritize either the escalation or de-escalation tendency, consistent with the dlPFC's general role in the flexible use of working memory.
It is worth noting that the impact of the cathodal stimulation was only marginally significant in the model-free analysis (Fig. 3C). One possibility is that the cathodal stimulation effect may be inherently less reliable than that of the anodal stimulation (Jacobson et al., 2012; Bogdanov et al., 2015; Dedoncker et al., 2016). Alternatively, the model-free measurement of the tDCS effect might have been susceptible to the subtle influence of other variables in the tasks. Indeed, by teasing apart the contribution of other factors such as subjects' risk attitudes and choice consistency, our model-based approach revealed a significant effect of cathodal stimulation (Fig. 4D). More importantly, the strong correlations of the anodal and cathodal stimulation effects between the model-free and model-based measurements further confirmed the validity of our results and highlighted the unique contribution of the model-based approach (Fig. 5).
We are aware of only one recent study providing evidence of a causal relationship between the dlPFC and sunk cost effects. This study showed that anodal tDCS over the dlPFC made subjects more likely to commit the escalation effect, while cathodal stimulation had no effect (Bogdanov et al., 2015). Although this pattern appears to contradict our findings, one of possible explanation for such discrepancies may be the difference of the decision environments between tasks. Previous studies have emphasized how the framing of decision contexts might influence the direction and magnitude of the sunk cost effect across subjects (Friedman et al., 2007; Sleesman et al., 2012; Roth et al., 2015). It therefore is possible that the effect of tDCS depends on subjects' behavioral bias in the baseline (sham) condition. In fact, the results from both the previous (Bogdanov et al., 2015) and our studies can be viewed as the anodal stimulation over the dlPFC exacerbating the behavioral bias induced by the sunk cost in the sham condition: more commitment escalation under anodal stimulation if commitment escalation was present in the sham condition (Bogdanov et al., 2015), and more commitment de-escalation under anodal stimulation if the sham condition already yielded commitment de-escalation. Taken together, these results might suggest a context-dependent effect of the tDCS on the sunk cost bias. Another possibility may be related to how the sunk cost scores were defined and calculated. The sunk cost score in Bogdanov's study was calculated as the percentage of investment decisions between “no prior investment trials” and “low/high prior investment trials.” However, the sunk cost investment decisions in the “no prior investment trials” could also be interpreted as subjects' initial willingness to incur sunk costs (but not decisions following zero sunk cost; i.e., with no prior investment) in a two-step investment task. Indeed, the tDCS stimulation did not change their subjects' initial investment decision (Bogdanov et al., 2015), which was consistent with our findings with respect to subjects' entrance fee payment decisions (Fig. 1C,D). It further highlighted the specificity of the stimulation: the tDCS did not have a ubiquitous effect on subjects' choices. Instead, its effect was only evident when the sunk cost was involved. A final possibility was that we used the HD-tDCS for both the anodal and cathodal stimulation, which had been shown to have a more constrained effect on the target brain area than the conventional tDCS (Shen et al., 2016). This difference, together with different electric current intensities use in the studies, may help explain the null results of cathodal stimulation in the aforementioned research (Bogdanov et al., 2015).
In summary, by combining computational modeling with causal brain activity modulation, the current work showed that anodal (cathodal) HD-tDCS over the right dlPFC increased (decreased) subjects' commitment de-escalation effect toward sunk costs in a two-step decision task. These results were robust to both model-free and model-based perspectives, and they helped to illustrate the computational role played by the dlPFC in a sunk cost environment: flexibly assigning decision weights to options whose valuation was recalibrated by the sunk cost. Our results help reconcile long-standing debates regarding the direction and the magnitude of sunk cost effects and their underlying neural mechanisms. These findings may also speak to the development of cognitive interventions designed to change maladaptive decisions in professional and daily life events in the healthy population (Staw, 1976; Thaler, 1980; Arkes and Blumer, 1985; Heath, 1995), as well as clinical applications in populations with anxiety and impulsive and compulsive disorders (Robbins et al., 2012; Hunter et al., 2022).
Footnotes
J.L. was supported by Chinese National Science Foundation Grants 31871140 and 32071090 and the National Science and Technology Innovation 2030 Major Program No. 2021ZD0203700/2021ZD0203704. We thank Dr. Peter Sokol-Hessner for constructive feedbacks of the manuscript.
The authors declare no competing financial interests.
- Correspondence should be addressed to Jian Li at leekin{at}gmail.com