Nicolas Vuillerme
email: nicolas.vuillerme@imag.fr

Olivier Chenu

Alexandre Moreau-Gaudry

Jacques Demongeot

Yohan Payan

Artificial Tongue-Placed Tactile Biofeedback for perceptual supplementation: application to human disability and biomedical engineering

Keywords: Handicap, Biofeedback, Tactile display, Tongue Proprioception, Postural control, Human disability, Biomedical engineering

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The concept of "sensory substitution" was introduced and extensively studied by Paul Bach-y-Rita and colleagues in the context of tactile visual substitution systems [START_REF] Bach-Y-Rita | Vision substitution by tactile image projection[END_REF][START_REF] Bach-Y-Rita | Brain mechanisms in sensory substitution[END_REF]. These researchers evidenced that stimulus characteristics of one sensory modality (e.g., a visual stimulus) could be transformed into stimulations of another sensory modality (e.g., a tactual stimulation). Claiming that "we see with the brain, not the eyes" [START_REF] Bach-Y-Rita | Vision substitution by tactile image projection[END_REF][START_REF] Bach-Y-Rita | Brain mechanisms in sensory substitution[END_REF], Bach-y-Rita gives the example of a blind individual, without any pulse train coming form the retina, who can compensate and substitute with another modality: this person is able to navigate with a long cane and has a 3D perception of a room and/or a step. Here, whereas the interaction between the cane and the body is a tactile sensation in the hand, the blind person perceives a clear mental image of his 3D environment. Following the same idea, Bach-y-Rita has proposed tactile vision substitution systems to provide visual information to the brain through arrays of mechanical or electrical stimulators in contact with the abdomen, the chest, the brow, the back, the finger or the tongue. Optical images are recorded by a TV camera and transduced (through subsampling of the image) into vibratory or electrical stimulations that are mediated by the skin receptors. After sufficient training, subjects perceive an image in space rather than a tactile stimulation [START_REF] Bach-Y-Rita | Late post-acute neurological rehabilitation: Neuroscience, engineering and clinical programs[END_REF][START_REF] Sampaio | Brain plasticity: Visual acuity of blind persons via the tongue[END_REF]. Moreover, subjects fulfilled a shape-recognition task and even experienced a "projection" of the objects they tactually perceived in the external world [START_REF] Bach-Y-Rita | Brain mechanisms in sensory substitution[END_REF]. For instance, a sudden change in the zoom of the camera caused subjects to act as if they were approaching an obstacle. While the first tactile visual substitution systems were composed of 400 stimulators (20×20 matrix, Ø 1mm each) placed on different skin region of the human body, Bach-y-Rita recently converged to the electro-stimulation of tongue surface [START_REF] Bach-Y-Rita | Form perception with a 49-point electrotactile stimulus array on the tongue[END_REF], which overcomes the practical problems posed by the mechanical stimulation of the skin. Indeed, the human tongue is a highly dense [START_REF] Trulsson | Low-threshold mechanoreceptive afferents in the human lingual nerve[END_REF], sensitive and discriminative (spatial threshold = 2 mm) array of tactile receptors that are similar to the ones of the skin. Moreover, the high conductivity offered by the saliva insures a highly efficient electrical contact between the electrodes and the tongue surface and therefore does not require high voltage and current [START_REF] Bach-Y-Rita | Form perception with a 49-point electrotactile stimulus array on the tongue[END_REF]. Along these lines, a practical human-machine interface, the Tongue Display Unit (TDU) [START_REF] Bach-Y-Rita | Form perception with a 49-point electrotactile stimulus array on the tongue[END_REF], was developed and recently evaluated for blind persons [START_REF] Sampaio | Brain plasticity: Visual acuity of blind persons via the tongue[END_REF]. It consists in a 2D array of miniature electrodes (12×12 matrix) held between the lips and positioned in close contact with the anteriorsuperior surface of the tongue. A flexible cable connects the matrix to an external electronic device delivering the electrical signals that individually activate the electrodes and therefore the tactile receptors of the tongue.

The present paper aims at introducing the innovative technologies, based on the concept of "sensory substitution" [START_REF] Bach-Y-Rita | Vision substitution by tactile image projection[END_REF][START_REF] Bach-Y-Rita | Brain mechanisms in sensory substitution[END_REF] or "perceptual supplementation" [START_REF] Lenay | Sensory substitution: Limits and perspectives[END_REF], we are developing in the fields of human disability and biomedical engineering. Precisely, our goal is to design, develop and validate practical assistive biomedical and/or technical devices and/or rehabilitating procedures for persons with disabilities, using artificial tongue-placed tactile biofeedback systems.

Proposed applications are dealing with: (1) pressure sores prevention in case of spinal cord injuries (persons with paraplegia, or tetraplegia); (2) ankle proprioceptive acuity improvement for driving assistance in older and/or disabled adults; and (3) balance control improvement to prevent fall in older and/or disabled adults.

For each of these applications, a home-made TDU, consisting of a 6×6 matrix of electrodes (Figure 1A), thus reducing by a factor of 4 the overall dimension of the Bach-y-Rita's device [START_REF] Bach-Y-Rita | Form perception with a 49-point electrotactile stimulus array on the tongue[END_REF], was used. On the one hand, indeed, the addressed biomedical applications do not require the dense 12×12 TDU resolution of the tactile visual substitution systems. On the other hand, the size reduction allowed us to develop a wireless tongue placed tactile biofeedback device that will make its daily use comfortable from an ergonomic point of view (see Discussion section).

The following sections present three feasibility experiments performed on young healthy adults.

II. PRESSURE SORES PREVENTION

Objectives

A pressure sore is defined as an area of localized damage to the skin and underlying tissue caused by overpressure, shearing, friction or a combination of these factors. Its prevalence ranges from 23% to 39% in adults with spinal cord injuries [START_REF] Raghavan | Prevalence of pressure sores in a community sample of spinal injury patients[END_REF][START_REF] Nair | Nontraumatic spinal cord lesions: a prospective study of medical complications during in-patient rehabilitation[END_REF] since this population do not get the "signal" arising from the buttock area that allows healthy subjects to prevent pressure sores by moving their body in a conscious or subconscious manner. Located near bony prominences such as the ischium, sacrum and trochanter, pressure sores are recognized as the main cause of rehospitalization for patients with paraplegia [START_REF] Cardenas | Etiology and incidence of rehospitalization after traumatic spinal cord injury: a multicenter analysis[END_REF]. Their treatment, which can be medical or surgical, is always long, difficult and expensive.

Within this context, we developed an original system for preventing the formation of pressure sores in individuals with paraplegia. Its underlying principle consists in (1) putting onto the wheelchair seat area a pressure mapping system that allows real-time acquisition of the pressure applied on the seat/skin interface ; and (2) sending a "signal" to the individual through the TDU matrix each time an overpressure zone is detected by the pressure mapping system. This signal was voluntary chosen as a simple low-level message: if the overpressure is supposed to disappear with a postural change of the patient body in the front (respectively back, left and right) direction, the corresponding six electrodes of the front (respectively back, left and right) row of the matrix are activated (Figure2).

The purpose of the present experiment was to assess the performance of this system in young healthy adults.

Methods

Subjects

Ten young healthy university students (mean age: 26.2 years) were included in this study. They gave their informed consent to the experimental procedure as required by the Helsinki declaration (1964) and the local Ethics Committee, and were naive as to the purpose of the experiment.

Task and procedures

Subjects were seated comfortably in a chair. Electrostimulation of one of the four rows of the matrix (figure 2) were sent to the TDU and subjects were asked to move their chest according to the felt electro-stimulated direction. After each movement following the electrostimulated direction, a new record of the pressure map was provided. When the six electrodes placed in front of the electrode array were activated, subjects were asked to move their chest forward (a). In the same way, they were asked to move their chest backward, to the left or to the right if electrodes situated behind (b), on the left (c) and on the right (d) were activated.

Data analysis

The pressure map applied at the seat/skin interface was recorded at a frequency of 10Hz. By computing the differences between the two pressure maps (before versus after electro-stimulation), we determined whether or not the movement was adapted to the electro-stimulated information. In the first case, the result was marked as "one", and otherwise, as "zero". For each subject, the experiment was carried out 10 times, thus obtaining a total score out of 10.

Results

No difficulty was reported during the calibration stage. The procedure was completed by each seated and healthy subject. The mean score was 9.2, with a standard deviation at 0.8. This result demonstrates three points: (1) young healthy subjects have a strong and accurate perception of the electrostimulated information; (2) this information is both meaningful and correctly interpreted; and (3) the action resulting from the interpreted information is relevant since the corresponding postural changes decrease the overpressure area.

III. CAR DRIVING ASSISTANCE

Accurate proprioception at the ankle joint is critical for body orientation and balance control and represents a prerequisite for different functional activities such as walking, running or driving. Indeed, driving a car requires, among other things, accurate foot movements since the drivers control the stopping and the speed of the vehicle with podal actions on brake and accelerator pedals [START_REF] Egol | Lower-extremity function for driving an automobile after operative[END_REF]. Normal aging, disease or trauma can cause a loss of sensation in the feet and ankle (e.g., [START_REF] Corriveau | Evaluation of postural stability in elderly with diabetic neuropathy[END_REF][START_REF] Kenshalo | Somesthetic sensitivity in young and elderly humans[END_REF][START_REF] Perry | Evaluation of age-related plantar-surface insensitivity and onset age of advanced insensitivity in older adults using vibratory and touch sensation tests[END_REF][START_REF] Skinner | Age-related decline in proprioception[END_REF][START_REF] Wells | Regional variation and changes with ageing in vibrotactile sensitivity in the human footsole[END_REF]), which could prevent drivers from gauging the amount of pressure they're applying to the brake and gas pedals. Impaired ankle proprioception also may be a predisposing factor for chronic ankle instability, balance difficulties, reduced mobility functions, fall, injury and reinjury (e.g. [START_REF] Fu | Ankle joint proprioception and postural control in basketball players with bilateral ankle sprains[END_REF][START_REF] Halasi | Changes in joint position sense after surgically treated chronic lateral ankle instability[END_REF][START_REF] Madhavan | Influence of age on dynamic position sense: evidence suing a sequential movement task[END_REF][START_REF] Payne | Ankle injuries and ankle strength, flexibility, and proprioception in college basketball players[END_REF]). Therefore, it is legitimate to propose that a therapeutic intervention and/or a technical assistance designed to increase proprioceptive acuity at the ankle could be of great interest for performing functional activities such as walking, running or driving a car in the elderly and/or disabled persons.

Along these lines, we developed an original biofeedback system for improving proprioceptive acuity at the ankle joint whose underlying principle consists in supplying the user with supplementary sensory information related to the position of the matching ankle relative to the reference ankle position through the TDU.

The purpose of the present experiment was to assess the performance of this system in young healthy adults.

Methods

Subjects

Eight young male healthy university students (mean age: 24.8 ± 1.5 years) were included in this study. They gave their informed consent to the experimental procedure as required by the Helsinki declaration (1964) and the local Ethics Committee, and were naive as to the purpose of the experiment. None of the subjects presented any history of injury, surgery or pathology to either lower extremity that could affect their ability to perform the ankle joint position test.

Task and procedures

Subjects were seated comfortably in a chair with their right and left foot secured to two rotating footplates. The knee joints were flexed at about 110°. Movement was restricted to the ankle in the sagittal plane, with no movement occurring at the hip or knee. The axes of rotation of the footplates were aligned with the axes of rotation of the ankles. Precision linear potentiometers attached on both footplates provided analog voltage signals proportional to the ankles' angles. A handheld press-button allowed recording the matching. Signals from the potentiometers and the press-button were sampled at 100 Hz (12 bit A/D conversion), then processed and stored within the Labview 5.1 data acquisition system.

Subjects were barefoot for all testing, and care was taken to ensure that there were no discernible cues from the sole of the foot before testing. In addition, a panel was placed above the subject's legs to eliminate visual feedback about both ankles position.

The experimenter placed the left reference ankle at a predetermined angle where the position of the foot was maintained by means of a support (e.g., [START_REF] Walsh | Human forearm position sense after fatigue elbow flexor muscle[END_REF]). Subjects therefore did not exert any effort to maintain the position of the left reference ankle, preventing the contribution of effort cues coming from the reference ankle to the sense of position during the test (e.g., [START_REF] Walsh | Human forearm position sense after fatigue elbow flexor muscle[END_REF]). Two matching angular target positions were used: (1) 10° of plantarflexion (P10°) and (2) 10° of dorsiflexion (D10°). These positions were selected to avoid the extremes of the ankle range of motion to minimize additional sensory input from joint and cutaneous receptors (e.g., [START_REF] Burke | Responses to passive movement of receptors in joint, skin and muscle of the human hand[END_REF]). Once the left foot had been positioned at the test angle (P10° vs. D10°), subject's task was to match its position by voluntary placement of their right leg. When they felt that they had reached the target angular position (i.e., when the right foot was presumably aligned with the left foot), they were asked to press the button held in their right hand, thereby registering the matched position.

This active matching task was performed under two No-TDU and TDU experimental conditions. The No-TDU condition served as a control condition. In the TDU condition, subjects performed the task using a TDU-biofeedback system.

The underlying principle consisted of supplying subjects with supplementary biofeedback about the position of the matching right ankle relative to the reference left ankle position through the TDU. The following coding scheme for the TDU was used (Figure 3):

(1) when both ankles were in a similar angular position within a range of 0.5°, no electrical stimulation was provided in any of the electrodes of the matrix (Figure 3A);

(2) when the position of the matching ankle was determined to be outside the Dead Zone (DZ), electrical stimulation of either the anterior or posterior zone of the matrix (2×6 electrodes) (i.e. stimulation of front and rear portions of the tongue) was provided, depending on whether the matching right ankle was in a too plantarflexed (Figure 3B) or dorsiflexed (Figure 3C) position relative to the reference left ankle, respectively.

Several practice runs were performed prior to the test to ensure that subjects had mastered the relationship between ankle angular positions and lingual stimulations and to gain confidence with the TDU.

Five trials for each target angular position and each experimental condition were performed. The order of presentation of the two targets angular positions (P10° vs. D10°) and the two experimental conditions (No-TDU vs. TDU) was randomized. Subjects were not given feedback about their performance and errors in the position of the right ankle were not corrected.

Data analysis

Matching performance was quantified using two dependent variables [START_REF] Schmidt | Motor Control and Learning, 2nd Edition[END_REF].

(1) The absolute error (AE in degree), the absolute value of the difference between the position of the right matching ankle and the position of the left reference ankle, is a measure of the overall accuracy of positioning.

(2) The variable error (VE in degree), the variance around the mean constant error score, is a measure of the variability of the positioning.

Decreased values in AE and VE indicate increased accuracy and consistency of the positioning, respectively [START_REF] Schmidt | Motor Control and Learning, 2nd Edition[END_REF].

The means of the five trials performed in each of the two experimental conditions were used for statistical analyses. Two Conditions (No-TDU vs. TDU) × 2 Targets angular positions (P10° vs. D10°) analyses of variances (ANOVAs) with repeated measures of both factors were applied to the AE and VE data. Level of significance was set at 0.05.

Results

Analysis of the AE showed a main effect of Condition, yielding smaller values in the TDU than No-TDU condition (F(1,7) = 24.91, P < 0.01, Figure 4A). The ANOVAs showed no main effect of Target angular position, nor any interaction of Condition × Target angular position (Ps > 0.05).

Analysis of the VE also showed a main effect of Condition yielding smaller values in the TDU than No-TDU condition (F(1,7) = 31.37, P < 0.001, Figure 4B). The ANOVAs showed no main effect of Target angular position, nor any interaction of Condition × Target angular position (Ps > 0.05). These results suggested more accurate and more consistent matching performances, whatever the target angular position, when biofeedback was in use than when it was not.

These results provide evidence that electrotactile stimulation of the tongue can be used to improve ankle proprioceptive acuity.

IV. FALL PREVENTION

Postural control is a particularly complex system involving various sensory and motor components. Among the sensory inputs relevant to the regulation of postural sway, the importance of somatosensory information from the foot sole is now well established (e.g., [START_REF] Kavounoudias | The plantar sole is a 'dynamometric map' for human balance control[END_REF][START_REF] Meyer | The role of plantar cutaneous sensation in unperturbed stance[END_REF]). Clinically, alteration or loss of somatosensory information from the lower limbs resulting from normal aging or disease (e.g., diabetic peripheral neuropathy) [START_REF] Wells | Regional variation and changes with ageing in vibrotactile sensitivity in the human footsole[END_REF][START_REF] Meyer | The role of plantar cutaneous sensation in unperturbed stance[END_REF] is known to impair postural control (e.g., [START_REF] Corriveau | Evaluation of postural stability in elderly with diabetic neuropathy[END_REF][START_REF] Kenshalo | Somesthetic sensitivity in young and elderly humans[END_REF][START_REF] Simoneau | Postural instability in patients with diabetic sensory neuropathy[END_REF]). Progressive degeneration of sensory inputs from the lower extremities also represents a common clinical finding associated with aging (e.g., [START_REF] Kenshalo | Somesthetic sensitivity in young and elderly humans[END_REF][START_REF] Perry | Evaluation of age-related plantar-surface insensitivity and onset age of advanced insensitivity in older adults using vibratory and touch sensation tests[END_REF][START_REF] Skinner | Age-related decline in proprioception[END_REF][START_REF] Wells | Regional variation and changes with ageing in vibrotactile sensitivity in the human footsole[END_REF]) and has even been identified as important contributing factor to the occurrence of falls in elderly (e.g., [START_REF] Lord | Physiological factors associated with falls in an elderly population[END_REF][START_REF] Menz | Foot and ankle risk factors for falls in older people: a prospective study[END_REF]). Therefore, it is legitimate to propose that a therapeutic intervention and/or a technical assistance designed to increase somatosensory function of the plantar sole could be of great interest for controlling balance and preventing falls in the elderly and/or disabled persons.

Along these lines, we developed an original biofeedback system for improving postural control whose underlying principle consists in supplying the user with supplementary sensory information related to foot sole pressure distribution through the TDU. The purpose of the present experiment was to assess the performance of this system in young healthy adults.

Methods

Subjects

Eight young male healthy university students (mean age: 25.3 ± 3.3 years) were included in this study. They gave their informed consent to the experimental procedure as required by the Helsinki declaration (1964) and the local Ethics Committee, and were naive as to the purpose of the experiment. None of the subjects presented any history of motor problem, neurological disease or vestibular impairment that could affect their ability to perform the postural task.

Task and procedures

Subjects stood barefoot, foot together, their hands hanging at the sides, with their eyes closed. They were asked to sway as little as possible in two No-TDU and TDU experimental conditions. The No-TDU condition served as a control condition. In the TDU condition, subjects performed the postural task using a plantar pressure-based, tongue-placed tactile biofeedback system. A plantar pressure data acquisition system (FSA Inshoe Foot pressure mapping system, Vista Medical Ltd.), consisting of a pair of insoles instrumented with an array of 8×16 pressure sensors per insole (1cm² per sensor), was used. The pressure sensors transduced the magnitude of pressure exerted on each left and right foot sole at each sensor location into the calculation of the positions of the resultant ground reaction force exerted on each left and right foot, referred to as the left and right foot centre of foot pressure, respectively (CoP lf and CoP rf ). The positions of the resultant CoP were then computed from the left and right foot CoP trajectories through the following relation [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]:

CoP = CoP lf × R lf / (R lf + R rf ) + CoP rf × R rf / (R rf + R lf ),
where R lf , R rf ,CoP lf , CoP rf are the vertical reaction forces under the left and the right feet, the positions of the CoP of the left and the right feet, respectively.

CoP data were then fed back in real time to the TDU. Note that the TDU was inserted in the oral cavity all over the duration of the experiment, ruling out the possibility the postural improvement observed in the TDU relative to the No-TDU condition to be due to mechanical stabilization of the head in space.

The underlying principle of our biofeedback system was to supply subjects with supplementary information about the position of the CoP relative to a predetermined adjustable "dead zone" (DZ) through the TDU. In the present experiment, antero-posterior and medio-lateral bounds of the DZ were set as the standard deviation of subject's CoP displacements recorded for 10 s preceding each experimental trial.

The following coding scheme for the TDU was used (Figure 5):

(1) when the position of the CoP was determined to be within the DZ, no electrical stimulation was provided in any of the electrodes of the matrix (Figure 5, central panel);

(2) when the position of the CoP was determined to be outside the DZ, electrical stimulation was provided in distinct zones of the matrix, depending on the position of the CoP relative to the DZ (Figure 5, peripheral panels). Specifically, eight different zones located in the front, rear, left, right, front-left, front-right, rear-left, rear-right of the matrix were defined ; the activated zone of the matrix corresponded to the position of the CoP relative to the DZ. For instance, in the case that the CoP was located towards the front of the DZ, a stimulation of the anterior zone of the matrix (i.e. stimulation of the front portion of the tongue) was provided (Figure 5, upper panel). Black triangles, dashed rectangles and black dots represent the positions of the CoP, the predetermined dead zones and activated electrodes, respectively. There were 9 possible stimulation patterns of the TDU.

Central panel: no electrodes were activated when the CoP position was determined to be within the DZ.

Peripheral panels: 6 electrodes located in the front, rear, left, right, frontleft, front-right, rear-left, rear-right zones of the matrix were activated when the CoP positions were determined to be outside the DZ, located towards the front, rear, left, right, front-left, front-right, rear-left, rearright of the DZ, respectively. These 8 stimulation patterns correspond to the stimulations of the front, rear, left, right, front-left, front-right, rearleft, rear-right portions of the tongue dorsum, respectively.

Several practice runs were performed prior to the test to ensure that subjects had mastered the relationship between the position of the CoP relative to the DZ and lingual stimulations.

A force platform (AMTI model OR6-5-1), which was not a component of the biofeedback system, was used to measure the displacements of the centre of foot pressure (CoP), as a gold-standard system for assessment of balance during quiet standing. Signals from the force platform were sampled at 100 Hz (12 bit A/D conversion) and filtered with a second-order Butterworth filter (10 Hz low-pass cut-off frequency).

Three 30s trials for each experimental condition (No-TDU vs. TDU) were performed. The order of presentation of the two experimental conditions was randomized. Subjects were not given feedback about their performance.

Data analysis

Postural control was quantified using two dependent variables.

(1) The surface area (in mm²) covered by the trajectory of the CoP with a 90% confidence interval [START_REF] Tagaki | A new method of statokinesigram area measurement. Application of a statistically calculated ellipse[END_REF] (2) The density histograms of the sway path [START_REF] Harris | A method for the display of balance platform center of pressure data[END_REF] give a qualitative description of the distribution of the CoP displacements. Distributions were quantified by calculating an average CoP position and the percentage of time the subjects spent in 6 arbitrarily defined concentric circles around the average CoP in radial increments of 2.5 mm. Each of these areas was labelled from area 0-2. The means of the three trials performed in each of the two experimental conditions were used for statistical analyses. A one-way ANOVA 2 Conditions (No-TDU vs. TDU) was applied to the CoP surface area data. A 2 Conditions (No-TDU vs. TDU) × 6 Areas (0-2.5 vs. 2.5-5 vs. 5-7.5 vs. 7.5-10 vs. 10-12.5 vs. 12.5-15 mm) ANOVA with repeated measures of both factors was applied to the density histograms of the sway path data. Level of significance was set at 0.05.

Results

Analysis of the surface area covered by the trajectory of the CoP showed a main effect of Condition, yielding a narrower surface area in the TDU than No-TDU condition (F(1,7) = 25.24, P < 0.01, Figure 6). Analysis of the density histograms of the sway path showed a significant interaction of Condition × Area (F(5,35) = 7.81, P < 0.001, Fig. 7). The ANOVA also showed main effects of Condition (F(1,7) = 10.05, P < 0.05) and Area (F(5,35) = 19.36, P < 0.001). The decomposition of the interaction into its simple main effects showed that subjects spent more time in areas 0-2.5 and 2.5-5 mm in the TDU than No-TDU condition (P < 0.001 and P < 0.05, respectively), whereas they spent less time in 10-12.5 and 12.5-15 mm in the TDU than No-TDU condition (Ps < 0.05). In figure 7, condition differences in the dispersion of sway are illustrated by subtracting the percentages of time spent by the subjects in the TDU condition in a given area from those obtained in the No-TDU condition in that same area. Positive values indicate than subjects spent more time in an area in the TDU than No-TDU condition, whereas negative indicate than subjects spent less time in an area in the TDU than No-TDU condition. Clearly, subjects exhibited narrower sway dispersion in the TDU than No-TDU condition.

On the whole, these results suggested that young healthy adults were able to take advantage of an artificial tongueplaced tactile biofeedback to improve their postural control during quiet standing.

V. DISCUSSION & CONCLUSION

This paper presented three experiments evaluating the feasibility of an artificial tongue-placed tactile biofeedback for perceptual supplementation in the fields of human disability and biomedical engineering.

Proposed applications are: (1) pressure sores prevention in case of spinal cord injuries (persons with paraplegia, or tetraplegia); (2) ankle proprioceptive acuity improvement for driving assistance in older and/or disabled adults; and (3) balance control improvement to prevent fall in older and/or disabled adults.

Overall, the present findings evidence that electrotactile stimulation of the tongue can be used (1) as a part of a device designed to prevent the formation of pressure sores (experiment 1), (2) to improve proprioceptive acuity at the ankle (experiment 2) and (3) to improve postural control during quiet standing (experiment 3). Although these feasibility studies have been conducted in young healthy individuals, we strongly believe that our results could have significant implications in rehabilitative and ergonomical areas, for enhancing/restoring/preserving balance and mobility in individuals with reduced capacities (resulting either from normal aging, trauma or disease) with the aim at ensuring autonomy and safety in occupations of daily living and maximizing quality of life. Along these lines, the effectiveness of our biofeedback system in the three applications presented in this paper is currently being evaluated in paraplegic patients, individuals with somatosensory loss in the feet from diabetic peripheral neuropathy and persons with lower limb amputation, respectively.

Finally, although the current version of our human machine interface (Figure 1A) offers a realistic possibility of practical and cosmetically accepted device for users, the encouraging results evidenced in the present experiments already have led us to improve our system by making it wireless to increase its portability to provide a perspective for the application of this device/technology outside the laboratory framework and to permit its use over long-time period in real-life environment. Indeed, to be acceptable as part of a viable system, this device had to be lightweight, portable, and capable of several hours of continuous operations. The current ribbon TDU system does not meet these requirements yet. Within this context, in addition to the use of a portable, battery-operated ambulatory foot pressure device, we have developed, with the help of Coronis-Systems Company, a wireless radio-controlled version of the 6×6 TDU matrix. This consists in a matrix glued onto the inferior part of an orthodontic retainer including microelectronics, antenna and battery, which can be worn inside the mouth like a dental retainer (Figure 1B). The effectiveness of this wearable device will soon be tested in a daily life situation.

Figure 1A :

 1A Figure 1A: Wire TDU: 6×6 matrix and flexible cable Figure 1B: Wireless TDU: 6×6 matrix and embedded radio-frequency circuit

Figure 2 .

 2 Figure 2. Different patterns for electro-stimulation.

Figure 3 .

 3 Figure3. Sensory coding schemes for the TDU (lower panels) as a function of the position of the matching right ankle relative to the reference left ankle (upper panels). Black dots represent activated electrodes. There were 3 possible stimulation patterns of the TDU. A: no electrodes were activated when both ankles are in a similar angular position within a range of 0.5°. B: 12 electrodes (2 × 6) of the anterior zone of the matrix were activated (corresponding to the stimulation of the front portion of the tongue dorsum) when the matching right ankle was in a too plantarflexed position relative to the reference left ankle. C: 12 electrodes (2 × 6) of the posterior zone of the matrix were activated (corresponding to the stimulation of the rear portion of the tongue dorsum) when the matching right ankle was in a too dorsiflexed position relative to the reference left ankle.

Figure 4 .

 4 Figure 4. Mean and standard deviation for the absolute error (A) and the variable error (B) for the two No-TDU and TDU conditions. The two experimental conditions are presented with different symbols: No-TDU (white bars) and TDU (black bars). The significant P-values for comparison between No-TDU and TDU conditions also are reported (**: P<0.01, ***: P<0.001).

Figure 5 .

 5 Figure 5. Sensory coding schemes for the Tongue Display Unit (TDU) as a function of the position of the centre of foot pressure (CoP) relative to a predetermined dead zone (DZ).Black triangles, dashed rectangles and black dots represent the positions of the CoP, the predetermined dead zones and activated electrodes, respectively. There were 9 possible stimulation patterns of the TDU.

  is a measure of the CoP spatial variability. During quiet standing, increased values in CoP surface area indicate a decreased postural control, whereas decreased values express an increased postural control.

  5 mm (closest to mean CoP) to area 12.5-15 mm (farthest from mean CoP). During quiet standing, increased values of the percentage of time spent away from the average CoP indicate a narrower sway behaviour, whereas increased values of the percentage of time spent close to the average CoP indicate a wider sway behaviour.

Figure 6 .

 6 Figure 6. Mean and standard deviation of the surface area covered by the trajectory of the centre of foot pressure (CoP) obtained in the two No-TDU and TDU conditions. These experimental conditions are presented with different symbols: No-TDU (white bars) and TDU (black bars). The significant P-values for comparison between No-TDU and TDU conditions also are reported (**: P < 0.01).

Figure 7 .

 7 Figure 7. Condition differences in the dispersion of sway. For each area, the percentages of time spent by the subjects in the TDU condition were subtracted from those spent in the No-TDU condition. Thus, positive values (black bars) indicate that subjects spent more time in an area in the TDU than No-TDU condition, whereas negative values (white bars) indicate than subjects spent less time in an area in the TDU than No-TDU condition. The significant P-values for comparison between No-TDU and TDU conditions also are reported (*: P < 0.05; ***: P < 0.001).


Fabio Michelucci
email: fabio.michelucci@cerge-ei.cz.

Levent C ¸elik

Françoise Forges

Philippe Jehiel

Marco Pagnozzi

Margaret Meyer

Jakub Steiner

1 See the literature on information aggregation, information disclosure, efficient auctions.

2 In an interdependent value setting, this information affects bidders' expected valuations for the object and thus determines their bidding behaviour.

Hiding Information in Open Auctions with Jump Bids *

David Ettinger and Fabio Michelucci

We analyse a rationale for hiding information in open ascending auction formats. We focus on the incentives for a bidder to call a price higher than the highest standing one in order to prevent the remaining active bidders from aggregating more accurate information by observing the exact drop out values of the opponents who exit the auction. We show that the decision whether to allow jump bids or not can have a drastic impact on revenue and efficiency.

Economists have focused extensive attention on market environments where the aggregation of new information is important. 1 The possibility of aggregating new information is also the key feature of open auction formats, which often leads auction theorists and designers to advocate the use of open auction formats as opposed to sealed bid auction formats.

The existing literature ignores the possibility that bidders in an auction might have an incentive to manipulate the quality (precision) of the new information that can be aggregated. We show that jump bids can be used to achieve this objective. In this paper, the information that bidders can manipulate is who is active and who is not at any given price. 2 Under the standard modeling of the open ascending auction, when a bidder drops out his/her exact signal is pinned down by the remaining active bidders in equilibrium. However, when a jump bid is not matched by some of the bidders, the remaining active bidders can aggregate only coarser information of the private information of the bidders who did not match the jump bid. That is, a bidder may call a price in order to alter the information revelation process.

1 This way of manipulating information aggregation is relevant, as in practice most open auction formats allow bidders (or the auctioneer) to call a price higher than the highest standing price and bidders do make use of this option. It is a well documented fact that jump bids are quite prevalent in several auction contexts, including, among others, many FCC auctions (see, for instance, [START_REF] Cramton | The FCC spectrum auctions: An early assessment[END_REF], [START_REF] Plott | The simultaneous, ascending auction: dynamics of price adjustment in experiments and in the {UK3G} spectrum auction[END_REF], [START_REF] Börgers | Strange Bids: Bidding Behaviour in the United Kingdom's Third Generation Spectrum Auction[END_REF], [START_REF] Mark | A theory of jump bidding in ascending auctions[END_REF]), and online auctions (see, for instance, [START_REF] Easley | Jump Bidding Strategies in Internet Auctions[END_REF], [START_REF] Grether | A Field Experiment of Jump Bidding in Online Automobile Auctions[END_REF], [START_REF] He | The impact of jump bidding in online auctions[END_REF]). Furthermore, jump bidding is prevalent in important markets, which are perhaps not always explicitly regulated by auction rules such as corporate takeovers. In finance, a voluminous literature has tried to explain the existence of high initial offers (jump bids) leading to a takeover premium (see [START_REF] Burkart | Takeovers[END_REF] for a review).

The main available theoretical justifications for the observed use of jump bids are based on a signaling motivation. 3 Conceptually, the key difference between any explanation based on signaling and the one we propose is the following. In the former, the jump bid is meant to convey, early on in the auction, finer information about the private information of the bidder calling a price. In the latter, a jump bid is used to preclude opponents from aggregating finer information. We wish to provide an alternative theoretical explanation for jump bidding that could complement the existing ones, and might further guide the empirical investigations that test for jump bidding in auction markets, and corporate takeovers. Under our explanation, preventing more precise information is profitable, because the information would have a larger impact on the expected value of the strongest of the opponents rather than on the value of the bidder who jump bids. In practice, this could be the case if: 1) The object for sale is hard to evaluate; 2) The bidder who jump bids is an experienced bidder (thus less influenceable by the information held by other bidders); and 3) Some of the remaining bidders are more novice and strongly react to the information conveyed by the presence of one or more experienced bidders being active at high prices.

In the paper we provide possibility results via the use of different examples in specific auction frameworks. The examples differ in the exact nature of the asymmetry between players and the interdependence of payoffs. Our first example appears in Section 4, and it illustrates that a direct consequence of the foreclosure of finer information through a jump bid may be a decrease in the expected price paid by the bidder who jump bids, and consequently a decrease in revenues for the seller. It also shows that jump bidding may distort the equilibrium allocation in an inefficient way. However, this is not the end of the story as the indirect effects of allowing jump bidding cannot be neglected. Such effects are present in the more strategically rich environments that arise when more than one bidder has an incentive to foreclose information. We analyse this possibility in Section play in the complex dynamic game that is generated when one allows for jump bids. In particular, we are able to provide the following additional insights: 1) When a jump bid is placed to prevent another bidder from placing another jump bid later in the auction, the ultimate effect of allowing for jump bids might be that all bidders are at least weakly worse off than in an open ascending auction where jump bids are banned (set-up 2); 2) It is possible that although jump bids are not observed in equilibrium, the fact they are allowed drastically changes the equilibrium outcome (set-up 3); 3) Allowing for jumps bids can increase both revenue and efficiency in environments where an instance of the free-rider problem prevents the aggregation of information during the auction (set-up 4). This last point is particularly interesting as jump bidding is found to dampen competition and lower revenue by the existing theoretical literature. Considering the widespread observation of jump bids in open auctions, this is believed to open to a puzzle as to why jump bidding is allowed. Our observation that jump bidding may rise expected revenue could provide an answer.

Related Literature

Within the works that use signaling models to explain jump bidding, we can distinguish some that follow a preemptive motivation, and others that follow a pure signaling motivation.

The first contribution suggesting the preemptive motivation for jump bidding is [START_REF] Fishman | A Theory of Preemptive Takeover Bidding[END_REF].

Other related works include [START_REF] Hirshleifer | Facilitation of Competing Bids and the Price of a Takeover Target[END_REF], [START_REF] Bhattacharyya | The Analytics of Takeover Bidding: Initial Bids and their Premia[END_REF], [START_REF] Bernhardt | Promotion, Turnover and Preemptive Wage Offers[END_REF]. [START_REF] Fishman | A Theory of Preemptive Takeover Bidding[END_REF] presents a two-bidders independent private value model in which one of the two bidders has an informational advantage: she is able to discover her valuation prior to the start of the auction while the other bidder cannot. 4 If the first bidder's value is above some critical threshold, a jump bid that pre-empts the second bidder from investing and competing is placed. In this setting, a jump bid has an anti-competitive effect and reduces the seller's revenue. [START_REF] Bulow | Why Do Sellers (Usually) Prefer Auctions?[END_REF] also use an independent private value model with entry costs but focus on the comparison between a simultaneous auction and a sequential sale mechanism in which a new entrant can place a jump bid to deter further participation. Generally, in their framework, sellers prefer the simultaneous auction (without jump bids) even though it is less efficient because of the higher entry costs involved.

Interestingly, [START_REF] Roberts | When Should Sellers Use Auctions?[END_REF] show that if potential entrants receive an informative signal prior to the entry decision and if they may be ex-ante asymmetric, the jump bidding equilibrium of the sequential sale mechanism may induce higher revenues and efficiency than the simultaneous auction. This is because their equilibrium is separating and potential entrants with higher expected values are less deterred by jump bids.

The other leading justification for jump bidding was first proposed by [START_REF] Avery | Strategic Jump Bidding in English Auctions[END_REF]. Using a symmetric model with affiliated valuations, he shows that jump bidding can be employed to select the strongest bidder during the first stage. During this stage, strong bidders signal that their type is high by placing a jump bid. The signaling induces asymmetric bidding behaviour in the second stage of the game with a strong bidder committing to a more aggressive strategy than a weak bidder.

Such equilibrium behaviour can be viewed, as Avery points out, as a form of implicit collusion (it requires that the weaker bidder quits strictly below his expected value conditional on the informational content of the jump bid). More recently, [START_REF] Hörner | Costly Signalling in Auctions[END_REF] follow a similar approach, but they consider the use of a sealed bid format in the second stage (a First Price All Pay auction), and a pure private value structure. They show that an equilibrium exists in which signaling is not monotonic in the type.

Our setting differs in several dimensions from the ones mentioned above. For instance, we do not assume the existence of bidding costs or the successful implementation of an implicitly collusive strategy. The effect of a jump bid that we stress is the foreclosure of access to finer information. 5 There is also some information foreclosure in [START_REF] Fishman | A Theory of Preemptive Takeover Bidding[END_REF], but in his model the bidder who observes a jump bid can still acquire the finer information, even if, in equilibrium, he will not. In our model, the finer information is simply no longer available. Even though there is no cost of acquiring information in our setting, in an open ascending auction, the winner might experience a loss when winning. A bidder stays active at a price at which he would make a loss if he were to win because such loss is more than compensated by the potential profits of winning at a higher price, later in the auction. In other words, the bidder is active at lower prices in the hope of aggregating favorable information later on.

Thus, one can view the mentioned expected losses as the implicit cost of aggregating information. 6 It is in this environment that we show that allowing jump bids can increase both revenue and efficiency.

An efficiency improvement effect of limiting the amount of information disclosed in open ascending auctions is also found in Hernando-Veciana and Michelucci (2013). However, it is not due to the free riding phenomenon, that we illustrate here, and the improvement is achieved by switching to a different mechanism rather than by allowing jump bids. [START_REF] Compte | The Wait and See Option in Ascending Price Auctions[END_REF] and Compte and Jehiel (2007) also point out that bidders might be active in an open auction in order to aggregate new information. The difference is that in those two papers the acquired information is exogenous.

Empirical evidence on the effect of jump bids is not vast, and only a few studies have tried to test the signaling explanation for jump bids. That said, the support for a signaling explanation to jump 5 We are aware of only one other paper, [START_REF] Kirkegaard | The Phantom Made Me Jump! Preemptive Jump Bidding in (Takeover) Auctions[END_REF], that proposes that jump bids might be used to conceal information. However, there, bidders do so in response to the auctioneer using phantom bids.

6 See Hernando-Veciana and Michelucci (2011) for relevant economic applications where this applies.

4 bidding seems inconclusive. For instance, within the literature on takeovers [START_REF] Dimopoulos | Preemptive Bidding, Target Resistance, and Takeover Premiums[END_REF] compare the signaling/preempting explanation against a target resistance explanation, and they find support for the latter. [START_REF] Betton | Toeholds, bid jumps, and expected payoffs in takeovers[END_REF] find that jump bids are not limited to initial bids, and a similar observation can be made for FCC auctions [START_REF] Mark | A theory of jump bidding in ascending auctions[END_REF]) and online auctions [START_REF] He | The impact of jump bidding in online auctions[END_REF]). Finally, [START_REF] Grether | A Field Experiment of Jump Bidding in Online Automobile Auctions[END_REF]'s evidence on auctions of used cars points out at an ambiguous effect on revenue. Our first set-up illustrates that jump bids can happen at different points in the auction. Our last set-up shows that jump bids can also raise revenue. Overall, our alternative explanation might provide an input for additional empirical testing.

The remainder of the paper is organized as follows. Section 2 introduces the model. Section 3 presents a natural environment where a jump bid emerges in equilibrium. Section 4 studies environments that are strategically more complex, illustrates some properties of jump bids, and shows that there is no clear-cut effect of jump bids on revenue and efficiency. Section 5 concludes.

Auction setting

We analyse a slightly modified version of the Japanese Auction (JA), which aims to capture an element of the dynamic features of the English Auction (EA) that cannot be represented when adopting the standard JA format, the opportunity to call a price.

Environment

A set N of i : 1, ..., n bidders is present at the start of the auction. No further entry takes place after the auction has started, and a decision to exit the auction is irreversible. Bidder i's private information is represented by a unidimensional signal t i ∈ T i , while the vector t -i ∈ T -i contains the n -1 signals of i's opponents. Bidders' valuations are interdependent, i.e., v i (t i , t -i ), with v i weakly increasing in t j for all arguments. We also assume quasi-linear utility so that

u i (t i , t -i ) = v i (t i , t -i )-p
if the bidder i gets the object and pays price p, and u i (t i , t -i ) = 0, if bidder i does not get the object and no payment is required.

While t i is private to bidder i, the value functions v i as well as the cumulative distribution functions, F i , from which the signals t i are independently drawn7 are common knowledge among bidders. In some of the following analysis we assume a discrete type space. We find this more convenient to illustrate our point, but it should be apparent that an environment with a continuous type space can always be constructed to derive the same insights.

Auction Rules

We consider two versions of the Japanese auction. The standard Japanese auction without jump bids, which we call the C game 8 and the J game, a Japanese auction in which jump bids are allowed. The latter is defined as follows.

The price starts from a very low value, which we normalize to zero, and it is increased at a constant pace by an exogenous device such as a clock. Bidders are considered active only if they are currently pressing a button. At any point in time, i.e., at any price p ≥ 0 indicated by the clock at a specific instant of time, each bidder faces a decision with three alternatives: exit at p by releasing the button, remain active by keeping their hands on the button or, finally, call a price. The identity of the bidders who quit is publicly revealed so that a bidder knows exactly against whom he is competing at any time during the auction.

Using the third option, a bidder can interrupt the exogenous price increase. We assume that any jump bid offer must exceed the current price by a minimum increment ε > 0 arbitrarily small. 9 The clock is then stopped at the price indicated at that time and the price that has been called is publicly revealed. In case that more than one bidder simultaneously stops the clock, the right to call the price is assigned randomly by the auctioneer to one of the bidders who proposed the highest called price. 10

At this point any bidder who was active when the clock was stopped can react and decide to call a higher price (exceeding the current price called by a minimum increment ε > 0 arbitrarily small).

The process is iterated until no bidder wants to raise the price called further. 11 Then, the identity of the bidder who called the highest jump bid is publicly revealed, and all the bidders who were active independently decide whether they want to be active at the price being called. Then, the identities of the bidders who do not match the jump bid are publicly revealed.

The auction ends either when a price is called and no other bidder matches it or when, in the continuous price increase phase, the penultimate bidder quits. In the first case, the winning bid is given by the price that was called, in the second, by the price at which the penultimate bidder exited.

We use the following tie-breaking rule. If the k last active bidders (with k ≥ 2) leave the auction at the same price, p, the good is sold at price p with a probability 1/k to each of the k last active bidders.

The solution concept adopted is Perfect Bayesian Equilibrium (PBE). In order to rule out less interesting equilibria, we also assume that bidders do not play weakly dominated strategies.

8 The price continuously increases. 9 With this ε, we avoid the possibility of infinite (but converging) sequences of jump bids. 10 The fact that other bidders also had stopped the clock is not revealed nor are the bidders' identities. 11 An alternative rule could prohibit bidders from further raising the price before the identity of those who do not want to match the first jump bid is observed. Such alternative formulation would, in some part, make the analysis more cumbersome by adding an extra channel for bidders to interfere in each others optimal jump bidding strategy.

Information Aggregation and Jump Bids

This section presents a natural set-up where a jump bid emerges in equilibrium and for which the rationale behind jump bidding is simple to understand from a strategic viewpoint.

We model an auction framework in which one bidder might benefit more than others from aggregating new information during the auction. A jump bid can be used to prevent this bidder from acquiring the finest information.

A piece of land is for sale. Bidder 1 and Bidder 2 are local actors, they know the characteristics of this piece of land for the type of production they are respectively interested in. Let us say, for instance, that Bidder 1 (call it Robinson) would like to produce coconuts and Bidder 2 (call it Friday) would like to produce whisky. Bidder 3 is also interested in buying the piece of land in order to produce whisky and export it. He does not know the value of the piece of land and he has access to a wider market that is further away so that he would incur a transportation cost. We represent this situation with the following framework.

SET-UP 1.

• For i : 1, 2, v i = t i with F i uniform on [0, 1] and F 1 and F 2 independent. • v 3 = βt 2 -α with α > 0 and 6α ≥ β ≥ 2α + 1.
Before analyzing this set-up, we define p * with the following equation: p * = βp * -α. p * ∈ (0, 1) and it is unique since α > 0 and

β ≥ 2α + 1. If t 2 < p * , v 2 (t 2 ) > v 3 (t 2 ), if t 2 > p * , v 2 (t 2 ) < v 3 (t 2 ) and if t 2 = p * , v 2 (t 2 ) = v 3 (t 2
). p * is the value of t 2 for which bidder 2 and bidder 3's valuations cross. This is coherent with our story. The foreigner cares more about the value of the piece of land than Friday but he faces a fixed cost. The linearity of the value function, and the use of uniform distributions only help the presentation but are not crucial for the arguments. For the same reason, we do not endow Bidder 3 with any private information. The slopes of Bidder 1 and Bidder 2's values being set equal to 1 and the intercepts to 0 are only a normalization. What is crucial is that the values of Bidder 2 and Bidder 3 cross, and that the two slopes differ enough, which is guaranteed by the above conditions on α and β. 12 The fact that the slopes need to differ enough is natural for the type of environment we want to model. The key element is that one (or more) bidder(s) can become substantially more competitive relative to the others if they aggregate some specific information during the auction. For more examples where this might be relevant, see section 7.1 in [START_REF] Ettinger | Hiding Information in Open Auctions[END_REF]. Now let us begin with the C game, where jump bids are not allowed. Since bidders do not play weekly dominated strategies, in any considered equilibrium of the C game, Bidder 1 and Bidder 2 leave the auction when the price reaches their respective valuations for the good, which is their unique weakly dominant strategy. In contrast, Bidder 3 does not know his valuation for the good at the beginning of the auction. As long as Bidder 1 and Bidder 2 are still active, it is not costly for Bidder 3 to wait and see, since the probability that both bidders will leave exactly at the same price is equal to zero. If Bidder 2 leaves first, Bidder 3 can compute his expected valuation conditional on this observation and stay active up to this price or leave immediately. The real issue for Bidder 3 is what to do when Bidder 1 leaves first for a price strictly lower than p * . In that case, it is not obvious that he should decide to stay active and learn finer information about the value of the good by observing Bidder 2's bidding behaviour. If he does so, Bidder 3 incurs a loss equal to α + (1 -β)p when Bidder 2 leaves for a price p strictly lower than p * . Thus, he stays active only if his potential gain compensates for this type of loss. This is the case. The condition β > 2α + 1 guarantees that even if Bidder 1 leaves the auction immediately, the expected profit of Bidder 3, if he stays active up to 1, is positive.

These elements are summarized in the following result.

RESULT 1. In any equilibrium of the C game : Bidder 1 and Bidder 2 leave the auction when the price reaches their respective valuations for the good. With probability 1, Bidder 3 stays active until Bidder 2 leaves the auction. If Bidder 2 leaves the auction at price p ≤ p * , Bidder 3 leaves immediately after Bidder 2, and if Bidder 2 leaves the auction at price p > p * , Bidder 3 stays active up to βp -α. Now, let us consider the J game. There is only one bidder, Bidder 1, who has an incentive to jump bid, and we assume that he places at most one jump bid (see the proof in the Appendix). Therefore, we can refer to a jump bid simply as (p 1 , p 2 ), where p 1 is the price at which the clock is stopped and p 2 the price called.

The key element of the environment we present is that Bidder 1 would prefer Bidder 3 not to discover the value of t 2 by observing at which price Bidder 2 leaves the auction, since Bidder 3 may become a strong opponent. By placing a jump bid (p 1 , p 2 ), Bidder 1 may manage to alter this revelation process. If Bidder 2 is active before the jump bid and does not choose to follow the jump bid, Bidder 3 can only assume that t 2 ∈ [p 1 , p 2 ]. Therefore, he will stay active up to max(p 2 , β p1+p2 2 -α).

Such a jump bid can be profitable for Bidder

1 if E[max(v 2 , v 3 )|t 2 ∈ [p 1 , p 2 ]] > max(p 2 , β p1+p2 2 -α).
We will see that values of (p 1 , p 2 ) exist such that this is the case. This observation is formalized in the following result.

RESULT 2. An equilibrium of the J game exists with the following properties.

• Bidder 1. If t 1 ≤ p * , she stays active until the auction reaches t 1 , leaves at that price and never calls a price. If t 1 > p * , she stays active until the auction reaches

p 1 = -4α+6αβ+2βt1-β 2 t1 4-8β+5β 2 < p * , calls p 2 = 4α(β-1)+β 2 t1 4-8β+5β 2
> p * at p 1 and then stays active up to t 1 without calling a price.

• Bidder 2 stays active until the auction reaches t 2 , leaves at that price or when a price higher than t 2 is called and never calls a price.

• Bidder 3. As long as no price is called, he stays active as long as Bidder 2 does also. If Bidder 2 leaves when the price is lower than p * , Bidder 3 immediately leaves. If Bidder 2 leaves at a price p > p * , he stays active up to βp -α. If Bidder 1 places a jump bid (p 1 , p 2 ), when Bidder 2 is still active at p 1 , Bidder 3 follows the jump bid. If Bidder 2 also does, Bidder 3 stays active as long as Bidder 2 does so and then stays active up to βp -α, p being the price at which Bidder 2 leaves the auction. If Bidder 2 does not follow the jump bid, Bidder 3 leaves when the auction reaches max(p 2 , β p1+p2 2 -α).

Proof. See the Appendix

The intuition of this equilibrium is as follows. Bidder 2 cannot use a jump bid in order to profitably hide information from Bidder 3, since he would always call a price lower than his valuation and Bidder 3 would always stay active after the jump bid. Therefore, he sticks to a simple strategy: always to stay active when the price is lower than his valuation for the good. Bidder 3's behaviour also has the flavour of a standard best response to the behaviours of the other two bidders. He has no information to hide or to signal. Notice that, in the proposed equilibrium, Bidder 3 always matches a jump bid regardless of his characteristics. This is not costly since, in equilibrium, Bidder 1 has a strictly higher value than p 2 and remains active after the jump bid.

Figure 1: Example with β = 3, α = 1, (p 1 , p 2 ) = (0.4, 0.8), v 3 ((p 1 + p 2 )/2) = v 2 (p 2 ) = p 2 information from Bidder 3 with a jump bid (p 1 , p 2 ). The jump bid is directly profitable if and only

if E[max(v 2 , v 3 )|t 2 ∈ [p 1 , p 2 ]] > max(p 2 , β( p1+p2 2 ) -α).
For this condition to be satisfied, we must have

p 2 > p * otherwise E[max(v 2 , v 3 )|t 2 ∈ [p 1 , p 2 ]] = E[v 2 |t 2 ∈ [p 1 , p 2 ]] ≤ max(p 2 , β( p1+p2 2 ) -α) and p 1 < p * otherwise E[max(v 2 , v 3 )|t 2 ∈ [p 1 , p 2 ]] = E[βt 2 -α|t 2 ∈ [p 1 , p 2 ]] ≤ max(p 2 , β( p1+p2
2 ) -α). With her jump bid, Bidder 1 should mix cases in which Bidder 2 has the highest valuation and cases in which Bidder 3 has the highest valuation.

In order to better understand the extra profit that Bidder 1 derives from a jump bid, first let us focus on a jump bid that does not alter the allocation that one would have in the equilibrium of the C game. This requires t 1 > βp 2 -α, that is a high enough valuation of Bidder 1 relatively to the price called. Consider then a jump bid (p 1 , p 2 ) with

p 1 < p * < p 2 such that E[v 3 |t 2 ∈ [p 1 , p 2 ]] = p 2 .
The jump bid affects the outcome of the auction only when t 2 ∈ [p 1 , p 2 ] (considering Bidder 2 and Bidder 3's strategies). In that case, without the use of a jump bid, Bidder 1 would win the auction and her expected payment would be E

[max(v 2 , v 3 )|t 2 ∈ [p 1 , p 2 ]] = p * -p1 p2-p1 ( p1+p * 2 ) + p2-p * p2-p1 (β( p * +p2
2 ) -α), the average value of the bold segment of figure 1. For the same values of t 2 , suppose that Bidder 1 placed a jump bid (p 1 , p 2 ); she would then win the auction and pay

p 2 = E[v 3 |t 2 ∈ [p 1 , p 2 ]] (by construction),
which is the average of segment CF on figure 1. Hence, on this graph the extra profit due to the jump bid is represented by the triangle ABC. COROLLARY 1. In the equilibrium of the J game stated in result 2, the equilibrium jump bid placed by Bidder 1, (p 1 , p 2 ), is such that p 1 is decreasing in Bidder 1's type, while p 2 is increasing in Bidder 1's type.

The corollary above points out that jump bids can happen at different times during the auction process, and one should expect later jump bids to come from lower types bidders.

It might be intuitive that the higher a bidder's type the higher the jump bid placed, but perhaps less so why the higher a bidder's type, the earlier the jump. To see why note that with the jump bid, Bidder 1 mixes states of the world in which Bidder 2 holds a higher value than Bidder 3 with other states of the world in which the reverse holds. Placing a higher jump bid means to hide (potentially) more states of the world for which Bidder 3 would have the highest value, and this needs to be counterbalanced by adding states of the world for which Bidder 2 would have the highest value.

Otherwise, the expected value of Bidder 3 (conditional on Bidder 2 not matching Bidder 1's jump bid) would increase too much.

Let us also remark that when Bidder 1 calls a price, she does not intend to send a message about her valuation for the good. Even though her jump bid perfectly reveals her private information, neither of the two other bidders care about that information. 13 The following result shows that allowing for jump bids in set-up 1 has a negative impact on the efficiency attainable by the open ascending auction. 14 RESULT 3. In the equilibrium of the J game that we considered, the allocation is less efficient and the expected revenue is strictly lower than with the equilibrium of the C game.

Proof. Regarding expected revenue, we obtain the result with simple computations. The jump bid only affects the outcome of the auction when t 1 > p * and t 2 ∈ [p 1 , p 2 ]. In that case, in the C game, the expected revenue is equal to :

p * -p 1 p 2 -p 1 p * + p 1 2 + t1+α β -p * p 2 -p 1 t 1 + p * 2 + p 2 -( t1+α β ) p 2 -p 1 t 1
and in the J game, it is equal to p 2 . The difference between these two terms is equal to

(β-2) 2 (βt1-t1-α) 4(β-1)β 2
, which is strictly positive when t 1 > p * with our assumptions. Hence, the result.

The allocations in the C game and in the J game only differ in cases when Bidder 1 calls a price. In the J game, Bidder 1 calls a price when t 1 > p * . In this case, in the C game, at the equilibrium, the auction is always efficient. In the J game, if t 1 > p * and t 2 ∈ ( t1+α β , p 2 ], Bidder 2 does not stay active after the jump bid and Bidder 3 also immediately leaves the auction after observing that Bidder 2 does not match the jump bid. Bidder 1 obtains the good although

v 3 > v 1 since v 3 > β t1+α β -α = t 1 = v 1 .
Therefore, the jump bid may reduce the efficiency of the auction.

We also need to check that the interval ( t1+α β , p 2 ] is not empty for some values of t 1 . This is the case

if p 2 -t1+α β ≥ 0. p 2 -t1+α β = (β-2) 2 (βt1-t1-α) β(5β 2 -8β+4)
. This formula is increasing in t 1 and equal to 0 when t 1 = p * . Hence, the condition is satisfied.

In order to get an intuition of this negative effect of jump bids on efficiency, let us go back to figure 1. A measure of the profit due to the jump bid is the area of the triangle ABC. It is decreasing in p 1 . Therefore, the lower p 1 (and the higher p 2 since we assume that p 2 = 2α-βp1 β-2 ), the higher the profit. This is true since t 1 ≥ βp 2 -α is equivalent to p 2 ≤ t1+α β . In these cases, the jump bid does not modify the allocation. If bidder 1 calls a higher price p 2 , she also pools together with her jump bid values of t 2 for which she would have lost the auction. This comes with a loss. However, when 13 In fact, she can choose a jump bid that perfectly reveals her private information precisely because it is not costly to reveal that information.

14 Other Perfect Bayesian Equilibria such that only jump bids that do not modify the equilibrium allocation of the C game (as of the type used to illustrate the revenue result) might be obtained. However, that requires a less plausible choice of out of equilibrium beliefs, to rule out the use of the more profitable distortionary jump bids proposed in result 2. the price is slightly higher than t1+α β , this loss is much lower than the increase in profits (as shown in figure 2 in which the loss is represented by the area of the triangle F GH and the extra profit the area of the trapeze BCDE) so that Bidder 1 calls a price strictly higher than t1+α β . Therefore, the allocation is inefficient when a price is called and t 2 ∈ ( t1+α β , p 2 ]. We will see in the remainder paper that the negative effect on efficiency and revenue is not a general property of jump bids.

Strategically More Complex Environments

In this section, we consider strategically more complex environments in order to illustrate properties of jump bids. In particular, the set-ups that follow capture some dynamic features that were absent in the setting considered earlier, as there was only one bidder with the incentive to jump bid. We have selected specific set-ups to illustrate in the simplest possible way the most interesting effects that this extra complexity brings.

In the first subsection (set-up 2), we show that a bidder may be induced to jump bid by the anticipation of someone else hiding some information later on. Eventually, everybody may be strictly worse off in the J game than in C game. This is interesting because it is generally thought that jump bids are anti-competitive and thus should be banned by the seller, but that bidders who place them are strictly better off when jump bids are allowed. Here, instead, the bidder who calls a price would be better off in the C game. 15

In the first example of the second subsection (set-up 3), we show that a bidder may be induced to 15 Proposition 4 below illustrates a stronger result that even if one bidder was the only one allowed the option to jump bid, it is possible that she would be willing to pay to avoid having such an option. are observed in equilibrium, the equilibrium outcome is drastically affected by the fact that bidders have such an option. Finally, the second example of the second subsection (set-up 4) illustrates that there are instances in which the C game fails to aggregate new information, while, surprisingly, allowing for jump bids raises both revenue and efficiency.

A Jump Bid to Prevent Another Jump Bid

We consider a setting in which a bidder might be induced to jump bid by the anticipation that another bidder may strategically hide some relevant information (via a jump bid) later on. This, in turn, induces one of these two bidders to further anticipate her jump bid. The setting is therefore suggestive of the fact that the dynamic environment we study rapidly becomes extremely strategically complex once one departs from the set-ups of section 3.

An interesting feature illustrated by this subsection is that all the bidders as well as the seller are worse off with the equilibrium of the J game than with the equilibrium of the C game. We consider the following setting: SET-UP 2.

• t 1 ∈ {8, 9, 10}, p(t 1 = 8) = p(t 1 = 9) = p(t 1 = 10).

• v 1 (t 1 ) = t 1 • v 2 (t 1 = 8) = 8.5; v 2 (t 1 = 9) = 14; v 2 (t 1 = 10) = 16 • v 3 (t 1 = 8) = 0; v 3 (t 1 = 9) = 0; v 3 (t 1 = 10) = 20
We begin with the analysis of the C game. Any equilibrium of the C game has the following properties. Bidder 1 knows the value of v 1 and therefore stays active till v 1 is reached. Bidder 2 exits at 8.5 if Bidder 1 exits at 8, she exits at 14 if Bidder 1 exits at 9 and she exits at 16 if Bidder 1 exits at 10. Bidder 3 exits immediately if Bidder 1 exits at 8 or at 9 and he exits at 20 if Bidder 1 exits at 10. The expected revenue in the C game is 11.

Consider the effect of allowing jump bids in this setting. In order to choose their jump bidding strategy, bidders need to take into account that at any current price, if they let the price increase without calling a price, the other bidders might have an incentive to call a price later in the auction and modify the way the information is aggregated in their favor. In this case, the key element is that Bidder 2, once she discovers that t 1 = 8, would like to prevent Bidder 3 from discovering whether t 1 is equal to 9 or to 10. She can do so by calling a price 10 after having observed that Bidder 1 is still active at price 8. In that case, Bidder 1 immediately leaves; Bidder 3's expected value for the good is also 10 (since he cannot distinguish between the two states t 1 = 9 and t 1 = 10), and therefore he also immediately leaves the auction. In this case, Bidder 3 always makes a zero profit. However, anticipating the unfolding of the game, Bidder 3 can do better by placing a jump bid from price 0 to 9.

In this case, if Bidder 1 immediately leaves, Bidder 2 has an expected value of (8.5+14)/2 = 45/4 > 9.

Thus, Bidder 2 stays active and Bidder 3 immediately leaves after having observed that Bidder 1 has left. Bidder 2 wins the auction at a price 9 and obtains 9/4. Instead, if Bidder 1 stays active after the jump bid (and up to 10), Bidder 3 stays active up to 20 and Bidder 2 up to 16. This yields Bidder 3 an expected profit of 4/3, making the jump bid profitable. However, this is not yet the equilibrium.

In fact, anticipating Bidder 3 jump bid from 0 to 9, Bidder 2 is better off placing a jump bid from 0 to 10 (recall that if the two jump bids are called at the same time, the highest is selected). In fact, this yields Bidder 2 expected profits of 17/6 > 9/4. RESULT 4. An equilibrium of the J game exists in which Bidder 2 calls a price 10 at the beginning of the auction and no other bidder stays active after the jump bid.

In this equilibrium of the J game, the expected revenue is 10, which is less than 11 under the C game. Bidder 3 never wins under the J game and therefore is strictly worse off. Bidder 1 never wins in either cases. Bidder 2 is also strictly worse off (in expectations) as her expected profits are 11/3 in the C game and 17/6 in the J game.

All the bidders and the seller prefer the equilibrium outcome of the C game. Thus, this equilibrium of the J game is Pareto dominated by the equilibrium of the C game.

Information Aggregation and Ex-Post Regret

In the next two subsections, we focus on an environment where, in the C game, some bidders might experience ex-post regret in equilibrium (see [START_REF] Hernando-Veciana | Second Best Efficiency and the English Auction[END_REF] for other relevant applications). 16 Bidders accept to suffer from ex-post regret in equilibrium because the expected losses are more than compensated by the expected gains that follow from aggregating further information during the auction. Hernando-Veciana and Michelucci (2013) have shown that in such environments, the open ascending auction is prone to rushes, which are detrimental to efficiency. They also show that alternative mechanisms that limit the information that bidders can aggregate can increase efficiency.

Here, we provide a novel, unrelated insight, that arises when bidders benefit in an asymmetric way from the aggregation of new information.

In particular, in the first subsection, in the C game the private information that bidders hold is aggregated in a very desirable way thanks to the possibility of the wait and see strategy described in the introduction. In this case, allowing bidders to call a price causes both efficiency and revenue to drop. Conversely, in the next subsection, in the C game the information fails to aggregate because the cost of staying active when other competitors are also active may lead to a free rider issue. This results in no bidder being willing to acquire finer information by staying active in the auction. In this latter case, the possibility of jump bidding allows the bidder with the ex-ante higher valuation to hide the piece of information causing such a free-rider issue. She may then profitably win the auction. This boosts both efficiency and revenue.

The hidden impact of allowing jump bids

We start with the scenario where the aggregation of information is very smooth in the C game. This setting also illustrates that in the J game, the anticipation of a future jump bid may induce a bidder to quit earlier than he would in the C game and that, even though no jump bid is observed in equilibrium, the equilibrium outcome in the J game substantially differs from the one in the C game.

SET-UP 3.

• t 1 ∈ {5, 6, 7} with P (t 1 = 5) = P (t 1 = 6) = P (t 1 = 7) = 1 3 .

• v 1 (t 1 ) = t 1 .

• v 2 (t 1 = 5) = 0, v 2 (t 1 = 6) = v 2 (t 1 = 7) = 9.

• v 3 (t 1 = 5) = v 3 (t 1 = 6) = 0, v 3 (t 1 = 7) = 12.
For both uninformed bidders, winning if t 1 = 5 entails a significant loss as they learn that t 1 = 5 when p = 5 and both value the object at a price zero.

In the C game, the information is aggregated in a desirable way during the auction.

RESULT 5. In any equilibrium of the C game, Bidder 1 stays active until his private value is reached.

Bidder 2 quits as soon as Bidder 1 quits if that happens at a price lower than 5, and otherwise stays active until the price reaches 9. Bidder 3 quits as soon as Bidder 1 quits if that happens at a price lower than 6, and otherwise stays active until the price reaches 12.

The C game allows Bidder 2 and Bidder 3 to share the risk of winning when t 1 = 5, (the expected loss being ( 13 )( 1 2 )5 = 5 6 for each). Furthermore, the two bidders can split the benefits of being active at higher prices in a way that allows both bidders to recover the expected losses. In the case t 1 = 6, expected revenue is equal to R C = ( 13 )5 + ( 1 3 )6 + ( 13 )9 = 20 3 . The expected value of the winner is equal to E C = ( 13 )9 + ( 1 3 )12 = 21 3 = 7. Now, if we allow jump bids, the smooth sharing of costs and benefits becomes unattainable and given that Bidders 2 and Bidder 3 can be active at low prices only if they do so jointly, they both quit early. RESULT 6. In any equilibrium of the J game, Bidder 1 stays active until the price reaches his private value, Bidder 2 and Bidder 3 leave the auction at a price lower than 5.

To understand why such behaviours arise at the equilibrium, note that as soon as the price rises just above 5, Bidder 2 learns that t 1 = 5 and thus that v 2 = 9. Conversely, at that price Bidder 3 is still uncertain regarding his exact value. Bidder 2 can hide such information from Bidder 3 by calling a price equal to 7 when the current price is still in (5, 6). The jump bid pulls together the two cases, t 1 = 6 and t 1 = 7, for Bidder 3, who consequently bids up to E(v 3 |t 1 = 5) = 6. With the jump bid, Bidder 2 makes a sure profit of 2 as opposed to winning only if t 1 = 6 if he lets the price increase continuously. The latter strategy yields 1 2 (9 -6) = 3 2 < 2; therefore, Bidder 2 cannot commit not to call such a price. Then, Bidder 3 anticipating that Bidder's 2 jump bid will pre-empt him from winning in the only profitable case, he is no longer willing to stay active over the price p = 5. Since Bidder 3's presence is necessary for Bidder 2 (his expected gain with the jump bid strategy is 4 3 but his expected loss if he does not share the risk is 5

3 ), the equilibrium outcome is that they both quit the auction at a price lower than 5. 17 This brings a revenue lower than 5 for any value of t 1 , and it inefficiently always allocates the object to Bidder 1.

In such a context, if the seller is not aware of the implications of allowing bidders to call a price, he/she may be wrongly induced to believe that the bidders' valuations were low.

We report the main results of this subsection in the two propositions below. PROPOSITION 1. An equilibrium of the J game may exist whose allocation and revenue differ from the allocation and the revenue of any equilibrium of the C game even though in this equilibrium of the J game no price is ever called.

PROPOSITION 2. A bidder may accept to pay a strictly positive amount to the auctioneer for prohibiting him to call a price. 17 Any other jump bid by Bidder 2 or Bidder 3, it is also not profitable.

say that, in this framework, Bidder 3 is a free rider whose presence is detrimental to both revenue and efficiency. Now, let us consider the J game. RESULT 8. An equilibrium of the J game exists in which Bidder 2 calls a price 10 at the beginning of the auction and no other bidder stays active at that price.

Bidder 2, by calling a price 10, prevents all the possible information aggregation. That way, she also prevents the free-rider problem. Eventually, Bidder 2 wins with probability 1 at price 10, which yields her an expected profit of 1 2 . That is better than in the C game where she never wins. Expected revenue goes up from R C ∈ [8; 9) in the C game to R J = 10 in the type of equilibrium of the J game that we mention. Similarly, the expected value of the winner increases from 19 2 in the C game to 21 2 in the J game.

The different environments that we have introduced allow us to conclude the section with the following result.

COROLLARY 2. Allowing bidders to call a price can decrease or increase revenue and efficiency depending on the considered setting.

Proof. We only need to provide examples where all these possibilities are covered. The settings in the previous subsections prove that revenue and efficiency can drop. The setting above proves that they can increase.

Conclusion

We have analysed a version of the Japanese auction that allows bidders to stop the continuous price increase and call a price at any point during the auction. We have looked at how the possibility to call a price affects the way information is aggregated and have shown that bidders may have an incentive to alter the aggregation of information by placing jump bids to hide the drop-out value of some of their opponents. This is a novel explanation to jump bidding that contrasts with the traditional one based on signaling, for which more rather than less information is available after a jump bid. The general wisdom that comes with the traditional approach is that jump bids are anticompetitive. We show instead that the strategic environment is so rich that this is not always the case. Our analysis shows that the possibility of placing jump bids strongly affects both revenue and efficiency. Therefore, a jump bid may be profitable, but what is the most profitable jump bid?

Université

To find the optimal jump bid, first consider a jump bid (p 1 , p 2 ) with p 1 < p * < p 2 < t 1 and such that β p1+p2 2 -α < p 2 . Bidder 1 can obtain a higher revenue by placing a jump bid (p 1 , p 2 ) with p 1 > p 1 such that β p 1 +p2 2 -α = p 2 . As a matter of fact, the outcome is the same with both jump bids except when t 2 ∈ [p 1 , p 1 ). In that case, with a jump bid (p 1 , p 2 ), Bidder 1 obtains the good and pays p 2 while with a jump bid (p 1 , p 2 ), she obtains the good and pays a price strictly lower than p 1 and therefore strictly lower than p 2 . Therefore a jump bid (p 1 , p 2 ) with β 2 (p 1 + p 2 ) -α < p 2 cannot be part of a best response for Bidder 1.

Consider now a jump bid (p 1 , p 2 ) with p 1 < p * < p 2 < t 1 and such that β p1+p2 2 -α > p 2 . Bidder 1 can obtain a higher revenue by placing a jump bid (p 1 , p 2 -ε) with ε > 0 and arbitrarily small. If t 2 ∈ [p 1 , p 2 -ε], Bidder 1 obtains the good at a price β p1+p2 2 -α with a jump bid (p 1 , p 2 ) while she obtains it at a price β p1+p2 2 -α -β ε 2 with a jump bid (p 1 , p 2 -ε). If t 2 ∈ [p 2 -ε, p 2 ], Bidder 1 obtains the good at a price β p1+p2 2 -α with a jump bid (p 1 , p 2 ), while she obtains it at an expected price βp 2 -βε 2 -α or does not even obtain the good (if βp 2 -α > t 1 ) with a jump bid (p 1 , p 2 -ε).

20

The expected revenue is therefore equal to (p 2 -p 1 )(t 1 -(β p1+p2

Figure 2 :

 2 Figure 2: Example with β = 3, α = 1, (p 1 , p 2 ) = (0.4 -, 0.8 + 3 )

  Paris Dauphine, LEDa, CEREMADE, and CIRANOCERGE-EINow, let us consider Bidder 1. If t 1 ≤ p * , she can only derive a profit if t 2 < t 1 ≤ p * . In that case, when Bidder 2 leaves the auction, Bidder 3 also leaves immediately. Then, calling a price is costly and cannot be profitable. Staying active up to her valuation is a best response to Bidder 2 and Bidder 3's strategies. If t 1 > p * , leaving the auction at a price strictly lower than her valuation is a dominated strategy. Is it possible to place a profitable jump bid? Consider a jump bid (p 1 , p 2 ) with p 2 ≤ p * . If Bidder 2 is still active after the jump bid, calling the price is costless but useless. If Bidder 2 does not stay active after the jump bid, Bidder 3 also immediately leaves. With the jump bid, Bidder 1 obtains the good and pays p 2 ; without the bid, she would also have won but she would have paid a strictly lower expected price for the good. Therefore a jump bid (p 1 , p 2 ) with p 2 ≤ p * cannot be part of a Bidder 1's best response.Consider a jump bid (p 1 , p 2 ) with p 1 ≥ p * . If Bidder 2 is still active after the jump bid, calling the price is costless but useless. If Bidder 2 does not stay active after the jump bid, Bidder 3 stays active up toβ p1+p2 2 -α (> p 2 ). If t 1 > β p1+p22 -α, with the jump bid, Bidder 1 obtains the good and pays β p1+p2 2 -α, without the jump bid, she would also have won the good and she would have paid the same expected price. If t 1 ≤ β p1+p2 2 -α, she does not derive any profit after the jump bid, although she may have obtained a strictly positive profit without jump bid if βp 1 -α < t 1 . Therefore a jump bid (p 1 , p 2 ) with p 1 ≥ p * cannot be part of a Bidder 1's best response.Suppose now that Bidder 1 places a jump bid (p 1 , p 2 ) with p 1 < p * < p 2 < t 1 . Again, if Bidder 2 stays active after the jump bid, it is useless and costless. Suppose that Bidder 2 does not stay active after the jump bid, so that Bidder 1 stays active up to max(p 2 , β p1+p2

2

-α) and this may be profitable

if max(p 2 , β p1+p2 2 -α) < E[max(t 2 ; βt 2 -α)|t 2 ∈ [p 1 , p 2 ]].

We comment with more details in the next section. Other advanced explanations include bidders' impatience, bidding costs, and irrationality.

Throughout the paper we use the convention to refer to the bidder who places a jump bid along the equilibrium path (in the game that allows jump bids) as "she", and to all the other bidders as "he".

Except in some specific cases that we will describe.

We could make assumptions less restrictive than these and derive equilibrium jump bids but these ease our exposition.

Also in section 5.1 Bidder 3 is active at prices at which, conditional on winning, he would make a loss. However, such event does not arise in equilibrium.

Proof. Take the setting above. Suppose that Bidder 2 is the only bidder allowed to jump bid. Since in the J game he never wins, he would be willing to pay up to his expected profits in the C game to restrict his strategies space to the choice of quitting or staying active. 4.2.2 A free-rider problem and the existence of efficiency and revenue enhancing jump bids

In this setting, we show that in the C game, perverse incentives may impede the aggregation of information and that the enlarged strategy set of the J game may alleviate such a problem and bring higher revenue and efficiency.

SET-UP 4.

The setting is similar to the previous one in so far as both Bidder 2 and Bidder 3 may have an incentive to wait and see. However, here one of them, Bidder 2, has an ex-ante value strictly higher than Bidder 1's. This means that if Bidder 2 were the only bidder competing with Bidder 1, she would profitably be active over the price of 9 to be able to discover the value of t 1 . Bidder 3 could potentially benefit from the active presence of Bidder 2 over the price of 9. However, if both bidders are active at that price, they share the expected losses but not the expected gains. In fact, if Bidder 3 infers that t 1 = 10, he always wins against Bidder 2. However then Bidder 2 prefers to stay active only until the price is 8 to avoid incurring a loss. In turn, if that is the case, Bidder 3 also must quit before the price reaches 9, as his expected value is lower than Bidder 1's. Hence, no aggregation of information is possible.

RESULT 7. In any equilibrium of the C game, Bidder 1 leaves the auction when the price reaches t 1 .

Bidders 2 leaves at a price strictly lower than 9 and higher than 8, and Bidder 3 leaves as soon as Bidder 2 leaves.

The auction performs very poorly as Bidder 1 always wins at a price in [8, 9), which implies that both revenue and efficiency would be higher if Bidder 3 were excluded from the competition. We can

Proof of Result 2

Consider the following strategies:

• Bidder 1. If t 1 ≤ p * , stay active until the auction reaches t 1 , leave at that price and never calls a price. If t 1 > p * , stay active until the auction reaches

, call

at that price and then stay active up to t 1 without calling a price. If a jump bid ( p 1 , p 2 ) is called such that p 1 < p 1 and p 2 ∈ [p 1 , 4α(β-1)+β 2 t1 4-8β+5β 2 ], follow the jump bid and immediately call 4α(β-1)+β 2 t1 4-8β+5β 2 . If a jump bid ( p 1 , p 2 ) is called such that p 1 < p 1 and p 2 > 4α(β-1)+β 2 t1 4-8β+5β 2 , follow the jump bid if and only if p 2 < t 1 and stay active up to t 1 without calling a price.

• Bidder 2. Stay active until the auction reaches t 2 , leave at that price or when a price higher than t 2 is called. Never call a price.

• Bidder 3. Never call a price. As long as no price is called, follow behaviour A : Stay active as long as Bidder 2 does also; if Bidder 2 leaves when the price is lower than p * , immediately leave; if Bidder 2 leaves at a price p > p * , stay active up to max{β -α, βp -α}. Considering Bidder 3's strategy, it is obvious that Bidder 2 cannot derive any profit whatever his strategy. Besides, the proposed strategy for Bidder 2 is the only non dominated strategy without jump bids. Bidder 3 at any price p stays active at least up to his expected value for the good conditional on the equilibrium information available at p. Leaving at a lower price could not raise his profit. As we mentioned when we considered the C game, even though the wait and see strategy for value lower than p * may appear costly, it is eventually always profitable. Bidder 3 may also consider calling a price. This could only be profitable if it would deter Bidder 1 from calling a price, so that Bidder 3 could have access to finer information about Bidder 2's type. However, considering Bidder 1's strategy, no jump bid can deter Bidder 1 from calling price 4α(β-1)+β 2 t1 4-8β+5β 2 (or this would require calling a price higher than 4α(β-1)+β 2 t1 4-8β+5β 2 ).

-α)) with a jump bid (p 1 , p 2 ) and

either obtains the same expected revenue with the two jump bids or a strictly lower expected revenue with (p 1 , p 2 ) than with (p 1 , p 2 -ε) (when βp 2 -α > t 1 ). Hence, we can restrict to jump bids (p 1 , p 2 ) with p 1 < p * < p 2 < t 1 and such that β p1+p2 2 -α = p 2 . In such a jump bid, p 2 = 2α-βp1 β-2 . Considering Bidder 2 and Bidder 3's strategies, the expected revenue of Bidder 1 calling a price (p 1 ; 2α-βp1 β-2 ) with p 1 < p * < t 1 is

With simple computations, we show that it is maximized in p 1 = -4α+6αβ+2βt1-β 2 t1 4-8β+5β 2

, which implies p 2 = 4α(β-1)+β 2 t1 4-8β+5β 2 . Q.E.D.