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. 2011 Jan 24:5:1.
doi: 10.3389/fnhum.2011.00001. eCollection 2011.

Risk-sensitivity in sensorimotor control

Daniel A Braun  1 , Arne J Nagengast, Daniel M Wolpert
Affiliations

Risk-sensitivity in sensorimotor control

Daniel A Braun et al. Front Hum Neurosci. .

Abstract

Recent advances in theoretical neuroscience suggest that motor control can be considered as a continuous decision-making process in which uncertainty plays a key role. Decision-makers can be risk-sensitive with respect to this uncertainty in that they may not only consider the average payoff of an outcome, but also consider the variability of the payoffs. Although such risk-sensitivity is a well-established phenomenon in psychology and economics, it has been much less studied in motor control. In fact, leading theories of motor control, such as optimal feedback control, assume that motor behaviors can be explained as the optimization of a given expected payoff or cost. Here we review evidence that humans exhibit risk-sensitivity in their motor behaviors, thereby demonstrating sensitivity to the variability of "motor costs." Furthermore, we discuss how risk-sensitivity can be incorporated into optimal feedback control models of motor control. We conclude that risk-sensitivity is an important concept in understanding individual motor behavior under uncertainty.

Keywords: risk; risk-sensitivity; sensorimotor control; uncertainty.

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Figures

Figure 1
Figure 1
Representative subjective value and probability weighting functions from Prospect Theory. (A)The subjective value of money as perceived by an individual against its nominal value. The value function illustrates concavity for gains and convexity for losses. Note that the value function is steeper for losses than for gains leading to loss aversion. (B) The subjective probability as perceived by an individual against the actual numerical probability. The dashed line indicates no distortion of probabilities. The probability weighting function in red illustrates overweighting of small probabilities and underweighting of large probabilities as found when using explicit probabilities, for example, in questionnaire studies. The probability weighting function in green illustrates underweighting of small probabilities and overweighting of large probabilities as found in two recent motor control studies (Wu et al., ; Nagengast et al., 2010a,b).
Figure 2
Figure 2
Mean–variance trade-off. Six representative subjects of the mean–variance trade-off experiment (Nagengast et al., 2010b) ordered from the most risk-seeking to the most risk-averse. The five indifference points ± SD obtained using psychometric curve fits are shown in black. The best lines of fit were obtained using weighted linear regression and are shown in blue. The risk-attitude parameter θ is the line's slope and is shown in the right-hand corners of the subplots. In the experiment the null-hypothesis of risk-neutrality could be rejected with p < 0.05 for 11 out of 14 subjects (significance is marked with an asterisk in the plot).
Figure 3
Figure 3
Predictions of risk-sensitive optimal feedback control. A risk-neutral optimal control model (θ = 0) attempts to minimize the mean of the cost function. As a result, its policy (that is the motor command applied for a given state of the world) is independent of the noise variance N. In contrast, a risk-sensitive optimal control model minimizes a weighted combination of the mean and variance of the cost. Additional variance is an added cost for a risk-averse controller (θ < 0), whereas it makes a movement strategy more desirable for a risk-seeking controller (θ > 0). As a consequence, the policy of the controller changes with the noise level N depending on its risk-attitude θ. (A–C) Changes in motor command with the state for a low noise level (green) and for a high noise level (red) for the risk-neutral (A), risk-averse (B) and risk-seeking (C) controllers. In our experiments the state is given by the positional deviation of a cursor from the center of a target line (the cursor can deviate to the left or to the right leading to positive or negative State) and the control task is to reduce this deviation to zero. Consequently, all lines have a negative slope, as the control command needs to point into the opposite direction of the deviation. The slope of the lines is equivalent to the control gain of the controller.
Figure 4
Figure 4
Subjects’ control gains in experiment for different cost and noise conditions. (A) Experimental Session with low control costs. Results of a multi-linear regression analysis of the control gains for a representative subject. The lines show the average motor command that the subject produces for a given position (blue – low noise level, yellow – high noise level). The slope of the line is a measure for the positional control gain of the subject. (B) same as in (A) but for a condition with high control costs (green – low noise level, red – high noise level). (C) Positional control gain for the high noise condition plotted against the control gains of the low noise condition for all six subjects under low control costs (ellipses show the standard deviation). The dashed line represents equality between the gains. (D) as (C) but for high control costs. In both cost conditions, the gains have changed significantly as most ellipses do not intersect with the dashed diagonal. This change in gains is consistent with a risk-sensitive optimal feedback control model, but not with a risk-neutral model.
Figure A1
Figure A1
Violation of independence axiom. If we assume two lotteries A and B with the preference B s A and we create two new composite lotteries A′ and B′ by adding a “common consequence” lottery C to both A and B such that lottery A′ gives again lottery A with probability p and lottery C with probability 1 − p, and lottery B′ gives lottery B with probability p and lottery C with probability 1 − p, then the independence axiom of expected utility theory requires that we have the preference B′ s A′. We can see this immediately by comparing the expectation values of the composite lotteries and by subtracting the expected utility of the common consequence C. However, if we represent our preferences using the mean–variance approach, the preference between A′ and B′ also depends on the variance terms arising from the distance to C. The preferences are therefore not independent from the third consequence. For details see (Bar-Shira and Finkelshtain, 1999).
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