Abstract
Making the best choice when faced with a chain of decisions requires a person to judge both anticipated outcomes and future actions. Although economic decisionmaking models account for both risk and reward in singlechoice contexts, there is a dearth of similar knowledge about sequential choice. Classical utilitybased models assume that decisionmakers select and follow an optimal predetermined strategy, regardless of the particular order in which options are presented. An alternative model involves continuously reevaluating decision utilities, without prescribing a specific future set of choices. Here, using behavioral and functional magnetic resonance imaging (fMRI) data, we studied human subjects in a sequential choice task and use these data to compare alternative decision models of valuation and strategy selection. We provide evidence that subjects adopt a model of reevaluating decision utilities, in which available strategies are continuously updated and combined in assessing action values. We validate this model by using simultaneously acquired fMRI data to show that sequential choice evokes a pattern of neural response consistent with a tracking of anticipated distribution of future reward, as expected in such a model. Thus, brain activity evoked at each decision point reflects the expected mean, variance, and skewness of possible payoffs, consistent with the idea that sequential choice evokes a prospective evaluation of both available strategies and possible outcomes.
Introduction
Evaluating alternative actions is central to decisionmaking. Many everyday situations require agents to generate a chain of actions (a path through a decisiontree), leading to a distribution of outcomes, which engenders uncertainty. This is a focus in ecology, in which animals forage to ensure intake exceeds minimal need constraints (Stephens et al., 2007), and in finance, in which traders reap bonuses by exceeding a target return from sequential transactions (Panageas and Westerfield, 2009). Common to these examples is that the distribution of possible outcomes (energy, money) differs for each available series of choices.
Decisionmaking models in finance, psychology, and ecology account for uncertainty (risk) and reward when valuing actions (Markowitz, 1952; Kahneman and Tversky, 1979; Stephens and Charnov, 1982). Growing neural evidence supports the idea that key components of an outcome distribution, such as mean and variance, are explicitly encoded in the brain. However, this literature focuses on immediate returns from single choices (Knutson et al., 2005; Abler et al., 2006; Yacubian et al., 2006; Plassmann et al., 2007; Elliott et al., 2008; De Martino et al., 2009), leaving a relative dearth of knowledge about sequential choice.
In game theory and classical dynamic programming, decisionmakers' strategies under every contingency are described by a set of actions that maximize subjective value (“utility”). In sequential choice, these utilities are called “continuation values” because action values are contingent on following a future strategy. Thus, in assigning continuation values, decisionmakers must make assumptions about what type of future choices they will make. Standard dynamic programming constrains decisionmakers to invoke only optimal choices in the future [optimal continuation value (OCV)]. Critically, an optimal strategy can be planned in advance, implying that “online” updating is irrelevant or irrational (dynamically inconsistent) in the absence of new information (Dekel et al., 1998; Epstein and Schneider, 2003). An OCV decisionmaker, when presented with decision 1 followed by decision 2, makes the same set of choices even if the order is reversed (as long as no new information is presented until after choices are made).
However, decisionmakers can assign utilities to options assuming that they might not take the optimal choice in the future. This might occur if choices became unexpectedly constrained, when planned strategies would no longer be available. All available strategies (rather than just the preplanned “optimal” strategy) are taken into account before each choice, for example, by assuming that future choices are distributed randomly [average continuation value (ACV)]. This entails that strategies are dynamically reevaluated and action values recalculated depending on which strategies are available. This scenario allows for dynamic inconsistency, in which future choices can depend on the order in which options are presented (Machina, 1989).
Thus, sequential decisionmaking poses two clear problems. First, do humans evaluate the distribution of outcomes when planning choice? Second, do individuals assume optimal future choices when making sequential decisions? Here, we tested different models of strategy valuation and planning, simultaneously acquiring neural data [using functional magnetic resonance imaging (fMRI)]. We hypothesized that neural activity evoked in singleshot decision paradigms also supports decision variables mediating sequential choice.
Materials and Methods
Behavioral experiment
The study was approved by the Institute of Neurology (University College London, London, UK) Ethics Committee. Seventeen subjects (age range, 22–36 years; seven male) participated; one dropped out from scanning because of claustrophobia and was excluded from analysis. Monetary earnings were between £18 and £28, including a fixed £10 participation fee. Stimuli were presented on a standard personal computer using Cogent presentation software (Wellcome Trust Centre for Neuroimaging, London, UK) run in MATLAB (version 6.5; MathWorks). Choices were made by keypress selections on the computer keyboard of a standard personal computer.
We provided instructions with a 15 min verbal tutorial, to ensure that subjects understood the paradigm. In each block of the task, subjects were required to make five sequential choices between a sure or risky alternative. For each trial, a lottery was represented on screen using a picture of four cards (Fig. 1), and subjects indicated by a button press their choice to gamble or selection of a fixed sure amount of £2. Different numbers on the cards (0, 1, 2, 3, 4) indicated monetary value in pounds. We used five different card combinations, generating five lotteries with matched expected value of £2 but with different variance. All lotteries had symmetrical outcome distributions (i.e., were unskewed), with a mean return of £2. On each block, the same five lotteries were presented once, using a randomized order of presentation, necessary if one is to detect dynamically inconsistent choices. Critically, no feedback about outcomes was given during the task. This constraint allowed us to focus on a situation in which individuals do not need to adjust their strategies in response to feedback, enabling us to distinguish whether individuals adhere to a predetermined strategy, regardless of the particular sequence in which the options are presented (consistent with classical dynamic programming models in which choices are made based on the optimal continuation value), or whether they continuously reevaluate, taking into account a range of available strategies on each trial.
We altered the distribution of possible outcomes and the relative utility of both the gamble and sure option by imposing financial targets (on each block). At the beginning of each block, a target appeared on the screen (four levels used outside scanner: 5, 9, 11, 13; two levels used inside scanner: 7, 12). Subjects were told that 10 trials would be randomly selected at the end of the experiment from all sessions (inside and outside scanner) in which all trials had an equal chance of being picked. For selected trials, if the required target had been reached in that block, the outcome of that trial would be paid out (i.e., 10 trials are chosen to determine pay, contingent on whether the target was reached at the end of each block). This would be £2 if they had picked the certain fixed amount or whatever the outcome of the lottery (determined by a random selection of one of the four cards), if they had chosen to gamble. If the target had not been reached, then no money would be won from the trial regardless of choice.
Subjects were instructed to try to win as much money as possible, remembering that the total amount won would depend on reaching the target and whether they chose to gamble or not. Subjects were informed that outcomes would be recorded for all their choices in the experiment but that these would not be shown at the time. Subjects were not explicitly told that only five types of gamble would be shown. However, to acquaint them with the task and to engender the idea that actual outcomes were recorded from each choice made, five practice blocks using identical gambles were run with full feedback.
Analysis.
We initially categorized trials by two factors, current target level and the variance (risk) of lottery presented. These data were assessed by ANOVA and multiple regression implemented in SPSS (SPSS for Windows, release 12.0.1, 2001; SPSS Inc.). We then analyzed choices by block. There are 2^{5} = 32 possible combinations of choices in each block, and we refer to each of these combinations or trajectories of choices as a strategy, denoting strategy n as s_{n} (n = 1, …, N). The frequency with which each strategy was chosen was compared with simulated strategy choice frequencies by χ^{2} test. We simulated blockwise choice frequencies using mechanistic binomial choice models (described below) and estimated the bestfitting parameters of these models using an application of the method of simulated moments (McFadden, 1989). This estimation is based on comparing observed frequencies of choices with simulated frequencies, derived from an underlying structural model. Free parameters were optimized with a nonlinear simplex search algorithm in MATLAB 7.0. We selected the bestfitting model and assessed relative model performances by a comparison of criterion values on an individual subject basis.
Behavioral modeling.
Our behavioral models have two components. The first component provides a model of valuation, calculating an expected utility per strategy (V_{n}, n = 1, …, N) given the distribution of outcomes a particular strategy generates. In other words, we apply a utility function to each distribution of outcomes, to generate a single number representing the subjective value of each strategy. The second component models how future choice is incorporated, in which we implement three separate continuation value models. These models specify how strategies are compared and which strategies influence the value of the current choice (to gamble or not to gamble). On the first trial in a block, there are 16 possible strategies given a choice to gamble and 16 possible strategies given a choice of the sure amount. This reflects the fact that there are five ordered binary choices between a lottery and a certain payout, so 32 (2^{5}) possible sets of choices are available in any given block. Each of these strategies has its own outcome distribution. Note that the set of available strategies reduces as sequential choices are made such that, by the fifth trial in a block, only two possible alternative strategies are available (to gamble or opt for the sure outcome). In other words, the set of possible strategies at any given trial is contingent on previous choices in the block and the order in which the gamble options have been presented.
Valuation model.
We used a mean–variance–skewness (MVS) model in which the distribution of outcomes is evaluated by a weighted linear sum of its statistical moments. We assign values (utilities), V_{s,t}^{h}, to available strategies s, on each trial t, for every target level h, and calculate utility given the predicted distribution of outcomes resultant from each strategy s. The set of strategies evaluated are contingent on previous choices in each block. For example, on trial t = 4, there will be four possible available strategies to evaluate, given a certain sequence of simulated or actual choices for trials t = 1,2,3. Each of these strategies will generate their own distribution of possible numerical outcomes. The probability distribution of outcomes for each strategy will alter depending on target level h.
Let B_{n,t}^{h} comprise the set of discrete outcomes given strategy s_{n} on trial t, where B_{n,t}^{h}(j) indexes the jth outcome from this set, and P_{j}(B_{n,t}^{h}(j)) indicates the probability of the jth outcome.
In this formulation, strategy value on trial t is specified as follows: E(X) denotes the expected, (mean) value of the distribution of outcomes from strategy s_{n}: Var(X) denotes variance of outcomes: and Skw(X) denotes skewness of outcomes: ρ is a coefficient reflecting aversion to variance in outcomes, and λ reflects the degree of positive skew seeking behavior. Using a free parameter for skewness allows modeling of a wide range of preferences (skewness is a secondorder approximation of risk), because this model accounts for preferences for relative losses and gains independently of the spread of outcomes.
We present the results using one valuation model (mean–variance–skewness) in the main text. We also tested alternative utility models, although these models are not directly statistically comparable because they are nonnested with a different number of parameters (see supplemental data, available at www.jneurosci.org as supplemental material). We did not implement more complex variants of prospect theory with differential weighting of relative losses and gains because an MVS model incorporates some aspects of this behavior while being much easier to fit (because it is linear in its arguments).
Continuation value models
To simulate choice, our continuation value models perform a tree search of all possible choice (action) and outcome (state) combinations from current trial t to the end of each simulated block. This search is contingent on (i.e., constrained by) previous choices. We recalculate the value of available strategies on each trial, and, as the block proceeds, the number of possible strategies available reduces such that by trial t there will be 2^{6−t} possible strategies remaining.
Optimal continuation value.
This model assumes that agents pick the choice combination (trajectory) of all possible alternatives that maximizes utility by the end of each block. In other words, the decisionmaker compares the current options on each trial (select the sure amount or to gamble), and only the trajectories giving a predicted outcome distribution that maximizes utility determines choice. Thus, OCV prescribes the choices for a prospective decisionmaker who acts in accordance with classical dynamic programming principles. Thus, in accordance with standard dynamic programming models, OCV is oblivious to the order in which options are presented.
We assign action values (Q) to the binary options gamble (specified as Q^{1}) or sure (specified as Q^{0}), calculated on every trial, as the decisionmaker progresses through the decision tree. These two action values are then compared with determine choice: n indexes the possible continuation trajectories (available strategies) or branches of the decision tree.
Note that the OCV will remain the same on every trial within a block when a subject adheres to the strategy selected at block outset. In the case of a deviation from an optimal trajectory, the next best (utility maximizing) strategy is taken from the remaining options available. Importantly, in the case of these deviations, the OCV model prescribes that appropriate correction is taken based on always trying to follow the utilitymaximizing strategy.
Average continuation value.
This model entails that agents calculate the average value or utility of each of the two alternative choices on each trial (i.e., choosing to gamble or take the sure amount) rather than forecasting with respect to optimal continuation trajectories. This model does not require that agents have an explicit plan of future choices and is akin to a model in which the current choice is made under an assumption that choices are made randomly for the rest of the block. Every possible strategy influences current continuation values. As such, the decisionmaker can be thought of as myopic:
Sure continuation value.
This model assumes that agents weigh up the current choice against a benchmark of taking the sure option for the remainder of the block. This implements a simple heuristic in which the choice to gamble or take the sure amount is made given a fixed benchmark: where t indexes the current trial (T) in the block (t = 1, …, 5).
For a given target, several strategies can lead to similar distributions of outcomes. However, strategies will differ in their outcomes depending on the target level. Moreover, a critical feature for all these decision rules is that subjects' previous choices within a block determine the remaining available strategies to be evaluated. These models assume that the full space of possible actions and outcomes is known. We make this simplifying assumption to render model estimation tractable (i.e., specifically we do not incorporate uncertainty about future options). This is not unreasonable given that the task has a simple repeating structure with the same five lotteries being presented on each block throughout practice, behavioral, and scanning sessions.
Numerical example.
We provide a simple numerical example of how these models work in practice (supplemental Fig. S1, available at www.jneurosci.org as supplemental material). Imagine that you are faced with a twostage sequential decision between a gamble (g) and a sure amount (s) of money, which is fixed at £2. The first decision (decision X) is whether to accept a 50:50 gamble giving either £4 or £1. The second decision (decision Y) is whether to accept a gamble giving a 75:25% chance of winning £3 or £0, or again opting for a sure amount of £2. There are four possible strategies to consider (ss, sg, gs, gg combinations, which we refer to as strategies A, B, C, D), each giving a different distribution of outcomes. In our models, these distributions are evaluated according to a utility function (U), to give four separate numbers, or utilities, one per strategy.
For example, we now assign numbers to these utilities for illustrative purposes: U(ss) = 10; U(sg) = 8; U(gs) = 4; U(gg) = 7. Imagine now having to choose a current action. In our model, if you were an optimal decisionmaker, you would compare the highest utilities given each choice [in this case, U(ss) for a sure choice, U(gg) for a gamble choice]. Because U(ss) > U(sg) > U(gg) > U(gs), you prefer to make a sure choice on the current trial. For the next decision, you again make a sure choice, now comparing U(ss) = 10 versus U(sg) = 8. Thus, you have selected strategy A. What if you are a decisionmaker conforming to an average continuation value model? In this case, you weigh up the average utility of outcomes from each current choice [i.e., you compare (U(ss) + U(sg))/2 = 9 with (U(gs) + U(gg))/2 = 5.5]. In this example, you also prefer to make a sure choice [as U(s, ·) > U(g, ·)]. For the subsequent decision, you compare U(ss) = 10 and U(sg) = 8 and again make a sure choice, following strategy A.
Now let us consider a situation in which the order of decisions is reversed (supplemental Fig. S2, available at www.jneurosci.org as supplemental material). There remain four strategies, but their order has changed such that we have the following: U(ss) = 10; U(sg) = 4; U(gs) = 8; U(gg) = 7. If you are an optimal continuation value decisionmaker, you choose the sure option as U(ss) > U(sg) > U(gg) > U(gs), followed by another sure option [as U(ss) > U(sg)]. The order has no effect on the ranking of the strategies, and you make a dynamically consistent choice by following strategy A again. If you are an average continuation value decisionmaker, you will compare (U(ss) + U(sg))/2 = 7 with (U(gs) + U(gg))/2 = 7.5 and pick the gamble choice initially. On the next decision, you make a sure choice, as U(gs) > U(gg), such that now you follow strategy B and have made a dynamically inconsistent choice as the order in which the options are presented has affected choices. A sure continuation value (SCV) decisionmaker would make a consistent choice in this example, in which the “sure, sure” strategy A has the highest utility but also can make dynamically inconsistent choices if this is not the case. The actions actually selected (and whether they will be dynamically consistent) will depend in practice on the specific utilities assigned to the available choices by the decisionmaker. Note that, in these models, order independence (dynamic consistency) only holds if there is no new information arriving, which is the case in this experiment in which no trialbytrial feedback is given, also that these models reflect different methods of valuation and planning (i.e., an anticipated selection of choices) rather than testing execution of a preformed plan (i.e., selfconsistency).
Action selection.
We account for randomness in choice by the addition of noise at action selection (modeled by a logistic choice function). Thus, the predicted probability of choosing to gamble on a given trial is given by the following: where σ is a free parameter.
Model estimation.
We based our model estimation on a comparison of the observed frequencies of blockbyblock choices (i.e., strategies) with simulated frequencies, derived from each of the underlying structural models outlined above. The models generate a choice per trial per simulated block (using the probabilistic action selection rule), from which we calculate the simulated frequency (ϕ) with which each strategy is chosen. We ran 1200 simulated blocks per model, across all six target levels, with a randomized trial order per block. These simulated frequencies are then compared with actual observed choices (z), using the method of simulated moments (McFadden, 1989). z(i) is a vector of choices over available strategies on block i, with its elements taking the value 1 for the chosen strategy and 0 otherwise: where y_{i} is a vector of observations from one block i (observed − simulated frequencies), Ω is a weighting matrix, and D, the criterion function, is the weighted sumofsquares difference between the observed and simulated frequencies across all blocks (i = 1, …, N). Optimization of D (which finds the bestfitting set of parameters θ) is performed in a twostep procedure. Initial unweighted estimates are derived with Ω = I (identity matrix). A weighted optimization is then performed. To estimate the precision of the observations, we calculate the covariance matrix (Ω) of the differences between simulated and observed frequencies that come out of an unweighted optimization: To make Ω invertible, it is necessary to aggregate unchosen strategies, otherwise the weighting matrix is rank deficient. The estimated precision is then the inverse of this covariance matrix (Ω^{−1}). We weight observations according to this precision in performing the weighted optimization to calculate an unbiased estimator. This method of moments criterion function D is not differentiable in the parameters (there are step changes in the value of the function as the parameters vary); hence, we use a simplex search method to optimize the parameters with respect to the criterion function D (by using the Nelder–Mead simplex algorithm implemented in MATLAB). We use the method of simulated moments to optimize the models because the problem of multinomial sequential choice is high dimensional and computationally difficult to integrate. This means that we cannot use Bayesian methods to get a measure such as the Bayesian information criterion. In these circumstances, the method of simulated moments provides a robust way of optimizing models, and the criterion function acts as a likelihood estimate that allows comparison of our model space.
The optimized criterion value D is a direct measure of the residual sum of squared error of each model. D (multiplied by the number of observations) is χ^{2} distributed (Hansen's J statistic) (Hansen, 1982). Relative model likelihoods calculated from χ^{2} statistics are not comparable for nonnested models. However, to the extent that the number of parameters are equal (for a given utility and noisy choice model), criterion values can be directly compared. Hence, inverse criterion values (D^{−1}), reflecting relative goodnessoffit, were directly compared for bestfitting models on an individual subject basis. All expected utility and prospect theory models have two free parameters (σ and ρ), whereas MVS models have three free parameters (ρ, σ, and λ).
Functional MRI
All subjects had previously completed the behavioral experiment and understood that the task structure and presented lotteries were identical. We used two target levels during scanning (7 and 12). Visual cues were projected onto a screen, visible via an angled mirror mounted on the MRI head coil. Choices were indicated by pressing a button box with the right index finger, and responses were recorded using Cogent presentation software.
Scanning parameters.
We acquired gradient echo T2*weighted echoplanar images (EPI) with blood oxygen leveldependent (BOLD) contrast, on a 3 T head scanner (Magnetom Allegra; Siemens Medical). Imaging parameters were as follows: 48 oblique transverse slices; slice thickness, 2 mm; gap between slices, 1 mm; repetition time, 3.1 s; echo time (TE), 30 ms; field of view, 192 × 192 mm^{2}. We used an EPI sequence that optimized for BOLD sensitivity in the orbitofrontal cortex (OFC) using a combination of an increased spatial resolution in the readout direction and a reduced echo time (Weiskopf et al., 2007). Together with the oblique orientation of the slice acquisition, this can compensate and recover for potential signal loss in OFC, one of our regions of interest. During the same experimental session, a T1weighted image was obtained for anatomical reference. To correct for geometric distortions induced in the EPIs at high field strength, we collected field maps based on dual echotime images (TE1, 29 ms; TE2, 19 ms) and processed these using the statistical parametric mapping SPM5 fieldmap toolbox (Hutton et al., 2002) to produce a voxel displacement map indicating the field distortions.
Images were realigned with the first volume, normalized to a standard EPI template, and smoothed using an 8 mm fullwidth at halfmaximum Gaussian kernel. Unwarping was performed using the routine in SPM5, correcting for distortions in each acquired image by combining the measured field maps with estimated susceptibilityinduced changes attributable motion. Realignment parameters were inspected visually to identify any potential subjects with excessive head movement. Data were analyzed in an eventrelated manner using a general linear model, with the onsets of each stimulus modeled as a δ function. To capture all variance of interest (i.e., the modulation of neural response preceding each choice), δ functions were placed halfway between the onset of the presentation screen and the subsequent keypress response. Regressors of interest (see Results) were generated by convolving the stimulus functions with a hemodynamic response function. Firstorder temporal derivatives of each of these convolved functions were included to ensure that any neural activity related to cognitive processes of interest within an approximately ±2s window period should be captured by the convolved δ functions placed at the halfway point (Friston et al., 1998). This also avoids the need to constrain the model by making predictions concerning the timing of the neural responses to the different regressors.
Our contrasts of interest purely concern responses parametrically modulated by specific stimulus dimensions, reflecting activity independent of the regressors modeling nonspecific responses to stimulus presentation. Covariates of no interest comprised the onsets of the target screens and subjectspecific realignment parameters from the image preprocessing to account for motionrelated artifacts in the images that were not eliminated in rigidbody motion correction. BOLD data from blocks in which a response had been missed were factored out by explicitly including a regressor for these error trials. All data were analyzed using statistical parametric mapping software (SPM5; Wellcome Trust Centre for Neuroimaging). Trialtypespecific β values of linear contrasts were estimated, and these were entered into t tests using randomeffects analysis to provide group statistics.
Presentation of data and images.
Figures are constructed by thresholding secondlevel SPM t images at p < 0.005, and superimposing data on a mean image across all participants. Stereotactic coordinates are reported in Montreal Neurological Institute (MNI) space (Mazziotta, 2001). For the contrasts of interest, results are reported at a threshold of p ≤ 0.001 uncorrected. We also report results with smallvolume correction for regions of interest dictated by previous studies at p < 0.05 (a 6 mm radius sphere centered on a priori coordinates) (for details, see supplemental Tables S6, S7, available at www.jneurosci.org as supplemental material).
Results
Behavioral
Trialbytrial choices
We first analyzed subjects' choices in terms of a decision to gamble or opt for the sure amount, on a trialbytrial basis, across all sessions (inside and outside scanner). We observed a linear relationship between risk (variance) of an individual gamble and the percentage of time that subjects chose the gamble over the sure alternative (Fig. 2). A repeatedmeasures ANOVA demonstrated a significant main effect of both riskiness of each gamble [F_{(2.86,42.95)} = 2.88, p = 0.049 (Greenhouse–Geisser corrected degrees of freedom, ε = 0.72); Mauchly's test for sphericity: χ_{(9)}^{2} = 19.49, p < 0.05] and target level [F_{(5,75)} = 16.32, p < 0.001 (withinsubjects contrasts; negative linear effect of risk: F_{(1,15)} = 9.64, p = 0.007, r = 0.63; linear effect of target: F_{(1,15)} = 42.37, p < 0.001, r = 0.86)] (Fig. 2). There was also a significant interaction between risk and target level (F_{(20,300)} = 2.72, p < 0.001] such that, at higher target levels, the slope of the linear relationship was reduced. There was no tendency for subjects to be more risk seeking at the beginning or end of the blocks, with neither a linear nor quadratic effect of time point within a block on the probability of choosing to gamble (risk × target level × time ANOVA; linear contrast: β = −0.023, r^{2} = 0.001, p = 0.679; quadratic contrast: p = 0.42).
Analysis of choices by block
Descriptively, subjects switched strategy in a systematic manner as the target level changed (Fig. 3). For low target, subjects tended to choose strategies involving fewer lottery gambles. However, for higher targets, as the chance of getting nothing increased, subjects chose strategies involving more lottery gambles, thereby increasing expected return. There was considerable heterogeneity in strategy selection, particularly for medium target levels. Analyzing group data across all subjects demonstrated that choices were significantly different from random (χ^{2} test against random choice, df = 155, p < 0.001).
A comparison of each of the decisionmaking models is illustrated in Figure 4A (for details of modeling analysis and alternative utility models, see Materials and Methods and supplemental text available at www.jneurosci.org as supplemental material). In absolute terms, a model of average continuation value obtained the lowest optimized weighted criterion value, or difference between predicted and actual choice frequencies (ACV: mean ± SEM, D = 6.3 ± 1.5; OCV: mean ± SEM, D = 11.9 ± 3.0; SCV: mean ± SEM, D = 29.5 ± 7.4). We also compared models with random choice (in which all strategies are selected with equal frequency) to give an absolute measure of accuracy. A random model obtains a mean criterion value of 78.3. The average distance (i.e., summed leastsquares error) between the array of observed frequencies and the array of modelsimulated frequencies is 6.3 for ACV and 78.3 for the random model. Therefore, the summed leastsquares error for ACV is 92% less than is the case for the random model. According to the ACV model, all 16 subjects were averse to variance (variance coefficient ± SD, 0.21 ± 0.03) and were positive skew seeking (skewness coefficient ± SD, 1.2 × 10^{−3} ± 0.6 × 10^{−3}). The value of the σ parameter (temperature parameter of the softmax/logistic function used to account for noisy choice) were low (average ± SD σ, 0.31 ± 0.62). This indicates that the valuation model performs well at explaining choice without modeling a large degree of additional randomness in action selection. It is important to note that, although on average the ACV model was superior, there was heterogeneity in the bestfitting model on a subjectbysubject basis (Fig. 4B). The ACV model was superior to the SCV model in 13 of 16 subjects (Fig. 4C). Both parametric and nonparametric tests of the criterion value statistics at the group level revealed that ACV obtained a significantly better fit than SCV (paired t test, p = 0.001; binomial test, p = 0.002) but was indistinguishable from OCV on these behavioral data alone (paired t test, p = 0.294; binomial test, p = 0.402).
Functional imaging
We analyzed fMRI data, initially using the average continuation value model. A linear utility model is akin to the general linear model used in fMRI analysis, enabling us to decompose neural activity according to the effect of three statistical moments of the outcome distribution (mean, variance, and skewness). For the design matrix, we parametrically modulated the magnitude of the neural response on every trial with four regressors indicating target level (high or low), expected value, variance, and skewness of the outcome distribution, respectively. The use of parametric modulators to model the neural response to complex stimuli with several dimensions is well established (Büchel et al., 1998; Wood et al., 2008), and the correlation of neural data with dynamically changing internal variables of a computational model has been implemented in several studies of the neural valuation system (O'Doherty et al., 2004; Samejima et al., 2005) (for review, see Corrado and Doya, 2007). We first include the target level to account for evoked activity differences solely attributable to changes in effort or concentration evoked by a difference between a low and high target and activity attributable to explicit tracking of the target or context. In addition, this removes correlations between the regressors induced by the fact that, at high targets, the expected value is naturally always low and the skewness is always positive (high chance of failing to reach target and receiving nothing).
We then performed a directed stepwise linear regression to analyze the contribution of each variable in turn to the BOLD signal, by sequentially orthogonalizing regressors. Because we had strong a priori regions of interest, derived from previous studies of singleshot decisionmaking for expected value and to a lesser extent variance and no a priori knowledge about skewness, we orthogonalized the regressors in this specific order. Thus, we first account for as much neural activity as possible with the expected value regressor, then explain residual activity with the variance regressor, and finally explain activity with the skewness regressor. Any residual activity correlating with skewness is therefore independent of expected value and variance. In addition, orthogonalization is necessary because correlations remain between the statistical moments even having accounted for gross differences attributable to the target level (correlation coefficients: expected value vs variance, 0.57; expected value vs skewness, 0.50; variance vs skewness, 0.01). It is important to note that including the target regressor changes our inference about activity tracking the predicted outcome statistics: we analyze activity tracking the conditional moments (i.e., expected value, variance, and skewness changes with respect to the current target level) rather than the raw unconditional statistics. This analysis conditional on current target is similar to previous studies investigating the tracking of value in different frames or conditions (De Martino et al., 2006, 2009; Elliott et al., 2008; Plassmann et al., 2008) and asks whether expected outcomes are encoded relative to context.
A key idea in this neural analysis is the principle of using neurophysiological data to arbitrate between models of decisionmaking that are difficult to distinguish using choice data alone. If future trials were not considered at all (i.e., participants were oblivious to the task structure and the need to attain a target) and instead if each lottery is compared with the sure amount in isolation, we would not expect to observe neural signals correlating with expected value or skewness (because all gambles were symmetric and had the same expected value) i.e., the null hypothesis. This hypothesis in itself is fairly trivial to refute using just behavioral data, because clearly our participants' choices are sensitive to the target level. Conversely, if individuals anticipate future outcomes in a strategic manner (using either of the two most likely strategies according to choice data, of OCV in which a specific set of choices are weighted, or ACV in which all possible future choices are weighted), the presence of such correlated signals can be interpreted as providing evidence that such strategies are taken into account and that the observed pattern of neural response tracks the anticipated distribution of outcomes (specifically in brain regions previously implicated in representing statistical moments of choice in single choice paradigms). We aim to discriminate specifically between OCV and ACV models using neural data, given that these models were indistinguishable solely from our behavioral analysis of sequential choice. We use the fact that these models predict different anticipated outcome distributions on a trialbytrial basis, together with finegrained neural signal changes (as opposed to categorical choice data), to give additional power to arbitrate between models.
Average continuation value model
Fluctuations in expected value for each choice correlated with activity in right medial OFC (mOFC) (MNI coordinates: 6, 50, −14; t = 4.16, p = 0.032, smallvolume corrected for region of interest) and nucleus accumbens (MNI coordinates: right nucleus accumbens, 4, 10, −6; t = 3.74, p = 0.036, smallvolume corrected for regions of interest). In other words, activity in these regions tracked the orthogonalized component of expected value (meaning the error in a projection of expected value onto target level), according to our model of online tracking of outcome distributions. Note that this regressor is linearly independent of that tracking the target level (supplemental Table S1, available at www.jneurosci.org as supplemental material) (Fig. 5B). The target regressor itself correlated with activity in areas including right middle frontal gyrus (MNI coordinates: 46, −2, 54; t = 5.76, p < 0.001 uncorrected), anterior cingulate cortex (MNI coordinates: 6, 44, 14; t = 5.08, p < 0.001 uncorrected), and paracentral lobule/supplementary motor area (MNI coordinates; −6, −24, 56; t = 4.66, p < 0.001 uncorrected) (supplemental Table S2, available at www.jneurosci.org as supplemental material) (Fig. 5A). We also tested the alternative ACV model that did not explicitly model the target separately (i.e., we ask whether there is BOLD signal that correlates with the unconditional statistics of the outcome distribution, not adapted to target level). In this model, no brain activity positively correlated with the overall expected value of a choice, even at a liberal threshold of p < 0.005 uncorrected significance.
We next examined neural activity accountable by changes in the average variance of possible future outcomes given each choice, having accounted for activity attributable to target and expected value. The orthogonalized component of variancerelated activity correlated with BOLD in anterior insula [MNI coordinates: right anterior insula, 40, 20, −6; t = 3.64, p = 0.028 smallvolume corrected for regions of interest (supplemental data, available at www.jneurosci.org as supplemental material); left anterior insula, −38, 20, 4; t = 3.84, p = 0.001 uncorrected)], right putamen (MNI coordinates: 26, 28, −8; t = 6.42, p < 0.001 uncorrected), and right anterior cingulate cortex (MNI coordinates: 8, 44, 16; p < 0.001 uncorrected) (supplemental Table S3, available at www.jneurosci.org as supplemental material) (Fig. 5C).
Having accounted for neural activity attributable to target, expected value, and variance of anticipated outcomes, we next sought to explain residual activity in terms of the (orthogonalized component of) skewness of the expected outcome distribution (calculated as the expected cubed deviations from expected outcomes). We observed activity correlating with skewness in medial frontal pole, left superior parietal cortex and postcentral gyrus, and left inferior frontal gyrus (p < 0.001 uncorrected) (supplemental Table S4, available at www.jneurosci.org as supplemental material) (Fig. 5D).
As an additional analysis, we estimated a separate general linear model in which we modeled neural responses covarying with subjectspecific expected utility on a trialbytrial basis, calculated according to the behavioral parameters estimated from subjects' parameters from the ACV model. As might be expected, the largest cluster of significant activity correlating with expected utility was found in medial prefrontal cortex (peak voxel MNI coordinates: −8, 56, −2; t = 3.87; extent = 66 voxels) (supplemental Table S5, available at www.jneurosci.org as supplemental material).
Optimal continuation value model
Given that the OCV model was statistically indistinguishable from ACV on the behavioral data alone, we implemented an analysis based on predictions from the alternative OCV model for each subject, to investigate whether neural activity correlated with the internal parameters of this model. In essence, we are asking whether brain activity can adjudicate between models. We formulated the fMRI design matrix in an identical manner, modeling activity as a series of spike events, with their heights modulated by parametric regressors corresponding to target level, expected value, variance, and skewness of the outcome distribution under a set optimal strategy but contingent on previous decisions in a block. As above, inferences were made at the group level. There was no significant activity in a priori regions of interest correlating with the regressors tracking the outcome distribution, at a threshold of p < 0.001 uncorrected significance. For completeness, we tested the alternative OCV model not explicitly modeling the target separately and also found that no brain activity positively correlated with the overall expected value of a choice (at p < 0.005 uncorrected). This suggests that neural activity in brain regions previously associated with economic decisionmaking is better captured by an ACV model, with online trialbytrial updating rather than the OCV model with a fixed preset strategy.
Discussion
We first asked how humans evaluate outcome distributions from different strategies in a sequential choice task. Using a mean–variance–skewness model, we find that our subjects are variance averse and positiveskew seeking. Positiveskewseeking manifests when participants excessively opt for the sure rather than risky option even with low targets (in which the chance of failing to reach the target is small). This implies an attraction of small chances of aboveaverage outcomes and dislike of smallprobability belowaverage outcomes. In effect, we observe a preference for relative gains over losses, similar to prospect theory.
fMRI data revealed brain activity correlating with the statistical moments of a distribution of outcomes in prototypical valuation and risksensitive areas. Previous studies of risky decisions have segregated risk and valuerelated activity in regions such as cingulate and insula cortices (risk) and ventral striatal and medial orbitofrontal areas (valuation) (Kuhnen and Knutson, 2005; Lee, 2005; Huettel et al., 2006; Rangel et al., 2008). Finding separable areas of brain activity parametrically varying with mean, variance, and skewness supports predictions from an MVS model, but we cannot rule out alternative neural implementations of subjective utility. It is possible that the variancerelated activity we see may reflect downstream emotional or physiological responses consequential on detecting increased risk rather than direct riskassessment itself. For example, the insula supports introceptive processing, and activity we observe could be explained either as risk representation or as an arousal response consequent on the perception of risk (Critchley et al., 2004). However, a parametric response to risk in these regions is less easily explained as an arousal response unless one invokes a monotonic relationship between risk and arousal.
Our finding that expected value correlates with activity in mOFC supports an hypothesis that mOFC integrates overall value given a predicted distribution of outcomes. In our task these variables related to a distribution of outcomes not from a single choice but from a set of serial choices, forecast to the end of each block. This corroborates a suggestion that neural response to value invokes an integrated, goaldirected, representation of choice (Quintana and Fuster, 1999; Fincham et al., 2002).
Our analysis includes a regressor accounting for the target (high/low), controlling for targetinduced changes in attention, concentration, or effort, and means that OFC responses track the moments of the outcome distribution conditional on target level. In effect, the mOFC BOLD signal tracks relative rather than absolute changes in expected value (because we did not see similar activity without controlling for target level). This suggests adaptive value encoding, similar to findings from direct neuronal recordings in monkeys in which a proportion of OFC neurons adapt to condition, manifesting similar ranges of response under different scales of outcomes (PadoaSchioppa and Assad, 2008; PadoaSchioppa, 2009; Kobayashi et al., 2010). This adaptation potentially can overcome the limited dynamic range of neuronal signaling, implying that responses to expected value are integrated with information about the target in generating action values.
We also identify a skewness response, comprising medial prefrontal and superior parietal cortex, also shown to reflect subjective value in tasks with stochastic outcomes (Peters and Büchel, 2009). It is unlikely that this simply reflects cognitive demand or planning, because we see a parametric response even having separately accounted for the target level. We detect a signal reflecting expected utility of each choice (incorporating the target, relative expected value, and risk), calculated as the subjectspecific combination of these statistical moments, in medial prefrontal cortex. This locus of activity overlaps with areas in which activity correlates with subjective utility (Daw et al., 2006). However, computations for sequential decisionmaking, in which outcomes are stochastic and forecast several trials into the future, are represented in a more anterior location to that found when decision utilities represent deterministic outcomes from singleshot choices (Plassmann et al., 2007).
Our second question related to how individuals account for their possible future choices when selecting actions. Behaviorally, we found individuals reevaluate on each trial (ACV) rather than comparing choices with the riskfree alternative (SCV). Furthermore, neural activity distinguished between behaviorally equivalent OCV and ACV models, with correlations of key variables from the latter rather than the former. However, classical dynamic programming models are based on OCV, in which decisionmakers assume a specific optimal series of future planned choices. These models insist on dynamic consistency (choices are independent of the order in which options are presented) and have been used to describe choice in computational (Sutton and Barto, 1998), ecological (Houston et al., 1988), and economic (Samuelson, 1969) settings. ACV implies that potential outcomes from a number of strategies influence current choice in the decisionmaking process (because ACV decisionmakers assume future choices are made randomly).
The likelihood that decisionmakers represent or weight outcomes of alternative strategies relates to the possibility that future actions may deviate from an optimal trajectory. This can be either intentional (as a result of exploration or future constraints on available choices) or by accident (lapses or mistakes). In reality, we are unlikely to follow a predetermined path in our strategic decisions. If we ignored alternative outcomes altogether, then deviations from an optimal strategy would lead to unpredicted and possibly far worse outcomes than originally envisaged. Indeed, there is good evidence that weighting of even potentially irrelevant alternative outcomes plays a role in paradoxes of choice (Allais, 1953; Loomes and Sugden, 1982; Birnbaum, 2008), with counterfactual outcomes being represented in prefrontal cortex (Ursu and Carter, 2005) and striatum (Lohrenz et al., 2007). An alternative reason why an “average plan” rather than an optimal strategy might be used is because of additional mental effort required in planning future actions. Predicted outcomes could instead be sampled from the whole range of possible alternatives to build up an average picture of what might transpire given current choice. ACV far outperforms random choice, which would be an alternative best heuristic if decisionmakers were completely ignorant of future options. Instead, ACV explicitly models the assumption that all future options are known, but that despite this the decisionmaker does not have a specific plan of their future choices.
Although using a fixed strategy model is possible in our task, it is likely to dramatically fail in situations in which an individual errs or if some planned alternatives are no longer available. An ACV decisionmaker considering all possible future outcomes is myopic (i.e., does not deterministically plan choices in advance). However, such a decisionmaker can mitigate future errors or constraints by weighting all possible action–outcome combinations, enabling recovery from error by selecting the best set of remaining choices without needing to assume a fixed strategy. In other words, it makes sense that dynamic programming should account for a decisionmaker's awareness that deviations from a specific policy may occur in the future. One method of implementing this is to optimize the average continuation value. ACV can also partly capture decision processes in which a proportion of (but not necessarily all) possible outcomes in a decision tree are considered. A more informed version of the ACV model might weight strategies according to the proportion of time that they are expected to be chosen either according to previous experience or based on a rational expectations model similar to quantal response equilibria models of choice (McKelvey and Palfrey, 1998). We necessarily test the joint hypothesis (that both MVS and ACV models are true), because our continuation value models are coupled to a valuation model. Thus, although the neural data support ACV over OCV contingent on MVS, it is possible that this could be bettered by a combination with an alternative utility model. We cannot draw direct statistical comparisons between utility models in the current framework (because the models are nonnested with different numbers of parameters), and these additional model variants remain to be tested in future work. It is possible that we do not see neural responses corresponding to an OCV model even in individuals who actually do follow this strategy purely because there is no requirement for a trialbytrial tracking of the outcome distribution if you have preplanned a trajectory of choices. However, the fact that we see responses corresponding to the ACV model suggests that, at least in some subjects, these average continuation values are being continuously tracked and reevaluated.
In conclusion, we provide behavioral and neural data showing how humans make sequential decisions, a central element in decisionmaking scenarios ranging from foraging to financial investment. Our design assumes a fixed bestfitting strategy across subjects and cannot rule out variation in strategies both between and within subjects (i.e., switching strategies through the experimental session). However, such heterogeneity would have the effect of obscuring our ability to differentiate between models. Rather than conforming to standard models of sequential decisionmaking, our data suggest that a set of possible strategies are neurally represented and drive choices. More generally, our findings indicate that strategic outcomes are evaluated by similar neural metrics as in singleshot choice, in which a behavioral preference for higherorder features of outcome distributions is mirrored by neural sensitivity to expected value, variance, and skewness. Thus, it seems that phylogenetically ancient circuitry subserving valuation and reward also enables the sophisticated representation of the future and its alternatives.
Footnotes

This work is supported by a Wellcome Trust Programme Grant (R.J.D.) and by the Swiss Finance Institute. We thank Nicholas Wright, Rosalyn Moran, Dominic Bach, Deborah Talmi, and Steven Fleming for many helpful discussions and Nikolaus Weiskopf for imaging advice.
 Correspondence should be addressed to Dr. Mkael Symmonds, Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, 12 Queen Square, London WC1N 3BG, UK. m.symmonds{at}fil.ion.ucl.ac.uk