Strategic allocation of working memory resource

Visual working memory (VWM), the brief retention of past visual information, supports a range of cognitive functions. One of the defining, and largely studied, characteristics of VWM is how resource-limited it is, raising questions about how this resource is shared or split across memoranda. Since objects are rarely equally important in the real world, we ask how people split this resource in settings where objects have different levels of importance. In a psychophysical experiment, participants remembered the location of four targets with different probabilities of being tested after a delay. We then measured their memory accuracy of one of the targets. We found that participants allocated more resource to memoranda with higher priority, but underallocated resource to high- and overallocated to low-priority targets relative to the true probability of being tested. These results are well explained by a computational model in which resource is allocated to minimize expected estimation error. We replicated this finding in a second experiment in which participants bet on their memory fidelity after making the location estimate. The results of this experiment show that people have access to and utilize the quality of their memory when making decisions. Furthermore, people again allocate resource in a way that minimizes memory errors, even in a context in which an alternative strategy was incentivized. Our study not only shows that people are allocating resource according to behavioral relevance, but suggests that they are doing so with the aim of maximizing memory accuracy.

, where we assumed that γ < 2k. 21 So far, we have considered a trial with a givenJ. Now, we ask how, for a givenJ total , τ, and γ, the observer should set p high , p med , and p low to minimize the expected cost across the entire experiment. We refer to this expected cost as the "overall expected cost" (OEC); it is equal to OEC(p high , p med , p low ) = 0.6C(p high ·J total ) + 0.3C(p med ·J total ) + 0.1C(p low ·J total ).
We denote the resulting cost-minimizing proportions by p * high , p * med , and p * low . Each of these is a function ofJ total , τ, 22 and γ.

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Estimation of p * high , p * med , and p * low . We assume the observer calculates and uses these cost-minimizing propor-24 tions. While the brain may be able to do this optimization in a way we do not even begin to try to answer, we find 25 the values of p * high , p * med , and p * low with fmincon in MATLAB's Optimization Toolbox (MathWorks). We begin the 26 optimization from ten different starting points, to lower the probability of finding a local minimum, and choose the 27 proportions corresponding to the lowest OEC. Note that this optimization is different from the optimization completed 28 to estimate the ML parameters (explained below): the former is necessary to calculate the log-likelihood of a single 29 parameter combination, and is thus completed thousands of times within one ML parameter estimation.

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For each participant and each model, we estimated the parameters using maximum-likelihood estimation. The likelihood of the parameter combination θ for a given trial is defined as p(data|model, θ). In Experiment 1, the only data is the saccade landing,ŝ.
integrate over J with 500 equally spaced bins. To calculate the log-likelihood, we take the logarithm of this value. To 33 calculate the log-likelihood of a particular parameter combination θ, we sum the logs of the likelihoods across trials. In Experiment 2, we model the memory estimation as described in Experiment 1. The goal of this section is to derive 42 model predictions for the additional behavioral data: the circle wager. 43 We assume that on every trial, the observer chooses a circle radius r noisily around the value that maximizes the 44 expected utility (EU) of that trial. The EU is calculated as the product between the utility of setting a circle with radius 45 r and the probability that the true stimulus lies within the circle bounds (i.e., a hit). The observer calculates the utility 46 as the number of points awarded for circle radius r raised to a power α that accounts for risk preferences, 120e −rα . An α > 1 corresponds to risk-seeking behavior (corresponding to smaller circles on average), while an α < 1 corresponds 48 to risk-averse behavior (corresponding to larger circles on average).

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The probability of a hit is equivalent to the bounded integral of the posterior p(s|x) over the region described by the circle. For a two-dimensional Gaussian distribution, this is equivalent to a cumulative Rayleigh distribution evaluated at r: We assume that the observer's decision noise follows a softmax rule, such that the probability of choosing r is Here, β, the inverse temperature parameter, controls the decision noise level: a lower β corresponds to more decision 50 noise. parameters. Its six free parameters are thenJ total , τ, α, β, p high , p med . In the Minimizing Error model, observers allocate 57 resource in order to minimize expected behavioral loss across the experiment exactly as described in Experiment 1.

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In Experiment 2, this strategy is a myopic: the observer does not take into account the subsequent decision they must 59 make, but first maximizes performance in terms of estimation error, then maximizes EU. Its five free parameters are 60J total , τ, γ, α, and β. The optimal resource allocations p * high , p * med , p * low depend on parametersJ total , τ, and γ.

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While observers in all models maximize the EU on every trial for a given J, the Maximizing Points model observer 62 additionally allocates resources in order to maximize the expected utility across the entire experiment. We define the 63 cost of a single trial as the negative EU on that trial: The expected cost on that trial, for a given J, is an average of the costs for all possible radii r reported on that trial.
However, J itself is a random variable, drawn from a distribution determined by a priority-specificJ. Thus, we must also marginalize over J to calculate the expected cost of a trial in each priority condition: C wager (J) ≡ E(C wager |r, J) = C wager (r|J)p(r|J)dr = −EU(r, J)p(r|J)p (J |J, τ) drdJ We numerically integrated over r and J to obtain theC wager for a givenJ. The OEC for this experiment is thus: OEC(p high , p med , p low ) = 0.6C(p high ·J total ) + 0.3C(p med ·J total ) + 0.1C(p low ·J total ).
In the Maximizing Points model, the cost-minimizing proportions p * high , p * med , and p * low are a function of all parameters 65J total , τ, α, and β. We obtain these values through the optimization methods described in Experiment 1.
We used the same optimization method as described in Experiment 1.

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In Experiment 2, we found a positive correlation between error magnitude and the radius of the circle wager. We 75 believe this correlation was driven by a knowledge of internal noise, but it is possible that it is driven by a knowledge 76 of stimulus-dependent noise. For example, in orientation perception, targets with orientations closer to the cardinal axis 77 are perceived more accurately than obliquely-oriented objects (Girshick, Landy, & Simoncelli, 2011). We considered 78 that there was a similar effect for memories of locations; perhaps objects closer to the cardinal axes are remembered 79 with a different fidelity than those farther away. We did a regression to see if there was a trend between distance from 80 cardinal axis (up to 45 • ) and estimation error. The oblique effect in memory of locations of objects was inconclusive.

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For seven of eleven participants, stimulus location did not significantly predict error (p > 0.05), but the remaining 82 four participants had greater error when moving farther from the cardinal axes (M ± SEM regression weights: 1.40 ± 83 0.84, p < 0.05).

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relationship between the two. However, there may still be some stimulus-dependent relationship that can still be 86 driving the correlation between error and circle size. We decided to conduct a permutation test, which allows us to 87 investigate this question without needing to describe or parameterize the relationship between the stimulus location