Michal Krol

On the Existence and Social Optimality of Equilibria in a Hotelling Game with Uncertain Demand and Linear-Quadratic Costs

This paper examines a variant of the Hotelling two-stage mill-pricing duopoly game with “linear-quadratic” transport costs and the uniform customer distribution subject to a random shock. The demand is equally likely to be found anywhere in a fixed interval of feasible product characteristics, with the ex-post differentiation of tastes parametrized to reflect the degree of uncertainty. It turns out that, for uncertainty big enough, the presence of a linear component in the cost function no longer rules out an analytical solution to the game, which is a common problem in spatial competition models. In particular, a subgame-perfect equilibrium is shown to exist in which the firms’ locations approach the socially efficient ones as uncertainty further increases, regardless of the curvature of the cost function. When the demand uncertainty reaches maximum, mill-pricing is equivalent to spatial price discrimination under the most general conditions.

The right look for the job: decoding cognitive processes involved in the task from spatial eye-movement patterns

The aim of the study was not only to demonstrate whether eye-movement-based task decoding was possible but also to investigate whether eye-movement patterns can be used to identify cognitive processes behind the tasks. We compared eye-movement patterns elicited under different task conditions, with tasks differing systematically with regard to the types of cognitive processes involved in solving them. We used four tasks, differing along two dimensions: spatial (global vs. local) processing (Navon, Cognit Psychol, 9(3):353–383 1977 ) and semantic (deep vs. shallow) processing (Craik and Lockhart, J Verbal Learn Verbal Behav, 11(6):671–684 1972 ). We used eye-movement patterns obtained from two time periods: fixation cross preceding the target stimulus and the target stimulus. We found significant effects of both spatial and semantic processing, but in case of the latter, the effect might be an artefact of insufficient task control. We found above chance task classification accuracy for both time periods: 51.4% for the period of stimulus presentation and 34.8% for the period of fixation cross presentation. Therefore, we show that task can be to some extent decoded from the preparatory eye-movements before the stimulus is displayed. This suggests that anticipatory eye-movements reflect the visual scanning strategy employed for the task at hand. Finally, this study also demonstrates that decoding is possible even from very scant eye-movement data similar to Coco and Keller, J Vis 14(3):11–11 ( 2014 ). This means that task decoding is not limited to tasks that naturally take longer to perform and yield multi-second eye-movement recordings.

“Economies of Experience”—Disambiguation of Degraded Stimuli Leads to a Decreased Dispersion of Eye-Movement Patterns

We demonstrate economies of experience in eye-movement patterns-that is, optimization of eye-movement patterns aimed at more efficient and less costly visual processing, similar to the priming-induced formation of sparser cortical representations or reduced reaction times. Participants looked at Mooney-type, degraded stimuli that were difficult to recognize without prior experience, but easily recognizable after exposure to their undegraded versions. As predicted, eye-movement dispersion, velocity, and the number of fixations decreased with each stimulus presentation. Further analyses showed that this effect was contingent on recognition, and the selection of information from the stimulus could be informed by the identity of the presented object. Finally, our study demonstrates that after exposure to the undegraded version of the stimulus, eye-movement patterns associated with its degraded and undegraded versions become more similar. This suggests that eye-movement patterns can evolve to facilitate the optimal processing of a given stimulus via experience-driven perceptual learning.

The trickle-down effect of predictability: Secondary task performance benefits from predictability in the primary task

Predictions optimize processing by reducing attentional resources allocation to expected or predictable sensory data. Our study demonstrates that these saved processing resources can be then used on concurrent stimuli, and in consequence improve their processing and encoding. We illustrate this “trickle-down” effect with a dual task, where the primary task varied in terms of predictability. The primary task involved detection of a pre-specified symbol that appeared at some point of a short video of a dot moving along a random, semi-predictable or predictable trajectory. The concurrent secondary task involved memorization of photographs representing either emotionally neutral or non-neutral (social or threatening) content. Performance in the secondary task was measured by a memory test. We found that participants allocated more attention to unpredictable (random and semi-predictable) stimuli than to predictable stimuli. Additionally, when the stimuli in the primary task were more predictable, participants performed better in the secondary task, as evidenced by higher sensitivity in the memory test. Finally, social or threatening stimuli were allocated more “looking time” and a larger number of saccades than neutral stimuli. This effect was stronger for the threatening stimuli than social stimuli. Thus, predictability of environmental input is used in optimizing the allocation of attentional resources, which trickles-down and benefits the processing of concurrent stimuli.

On the equivalence of quantity competition and supply function competition with sunk costs

This paper considers competition in supply functions in a homogeneous goods market in the absence of cost or demand uncertainty. In order to commit to a supply schedule, firms are required to build sufficient capacity to produce any quantity that may be prescribed by that schedule. When the cost of extra capacity (given the level of sales) is strictly positive, any Nash equilibrium outcome of supply function competition is also a Nash equilibrium outcome of the corresponding Cournot game, and vice-versa. Conversely, when the cost-savings from reducing output (given the capacity level) are sufficiently small, any outcome of iterated elimination of weakly dominated strategies in the supply function game is also an outcome of the same process in Cournot, and vice-versa.