Léa Huguet

Marine Le Dudal

Marine Livrozet

Dominique Bazin

Vincent Frochot

Joëlle Perez

Jean-Philippe Haymann

Isabelle Brocheriou

Michel Daudon

Emmanuel Letavernier

High frequency and wide range of human kidney papillary crystalline plugs

Keywords: Kidney stone, Calcium, Phosphate, Papilla, Crystal, Plug

Most of kidney stones are supposed to originate from Randall's plaque at the tip of the papilla or from papillary tubular plugs. Nevertheless, the frequency and the composition of crystalline plugs remain only partly described. The objective was to assess the frequency, the composition and the topography of papillary plugs in human kidneys. A total of 76 papillae from 25 kidneys removed for cancer and without stones were analysed by immunohistochemistry combined with Yasue staining, field emission-scanning electron microscopy and Fourier transformed infrared micro-spectroscopy. Papillary tubular plugs have been observed by Yasue staining in 23/25 patients (92%) and 52/76 papillae (68%). Most of these plugs were made of calcium phosphate, mainly carbonated apatite and amorphous calcium phosphate, and rarely octacalcium phosphate pentahydrate. Calcium and magnesium phosphate (whitlockite) have also been observed. Based upon immunostaining coupled to Yasue coloration, most of calcium phosphate plugs were located in the deepest part of the loop of Henle. Calcium oxalate monohydrate and dihydrate tubular plugs were less frequent and stood in collecting ducts. At last, we observed calcium phosphate plugs deforming and sometimes breaking adjacent collecting ducts. Papillary tubular plugging, which may be considered as a potential first step toward kidney stone formation, is a very frequent setting, even in kidneys of non-stone formers. The variety in their composition and the distal precipitation of calcium oxalate suggest that plugs may occur in various conditions of urine supersaturation. Plugs were sometimes associated with collecting duct deformation.

Introduction

The prevalence of renal lithiasis has increased over the last decades [START_REF] Trinchieri | Epidemiological trends in urolithiasis: impact on our health care systems[END_REF][START_REF] Denstedt | Epidemiology of stone disease in North America[END_REF][START_REF] Hesse | Study on the prevalence and incidence of urolithiasis in Germany comparing the years 1979 vs. 2000[END_REF][START_REF] Ogawa | Epidemiology of stone disease over a 40-year period in Japan[END_REF]. However, the exact origin of kidney stones is still a matter of debate. Three pathways for initial crystalline germ nucleation and growth have been proposed: [START_REF] Trinchieri | Epidemiological trends in urolithiasis: impact on our health care systems[END_REF] nucleation of crystals on a Randall's plaque at the tip of the papilla, [START_REF] Denstedt | Epidemiology of stone disease in North America[END_REF] growth in renal cavities and (3) formation of tubular plugs [START_REF] Coe | Three pathways for human kidney stone formation[END_REF]. Development of endoscopic procedures has recently allowed to visualize both Randall's plaque and tubular plugs at the tip of papillae. Randall's plaques were first described by Alexander Randall eight decades ago [START_REF] Randall | The origin and growth of renal calculi[END_REF]. Randall's plaques are discrete, flat layers of calcification arising in the interstitium of renal papillae. Development of these deposits and urothelium erosion would provide nuclei for calcium oxalate stone formation [START_REF] Evan | Randall's plaque of patients with nephrolithiasis begins in basement membranes of thin loop of Henle[END_REF]. These plaques are found in a large number of calcium oxalate stone formers but also in the absence of nephrolithiasis [START_REF] Low | Endoscopic mapping of renal papillae for Randall's plaques in patients with urinary stone disease[END_REF]. Tubular plugs are described as apatite deposits in renal tubules. They represent a common phenomenon in stone formers [START_REF] Linnes | Phenotypic characterization of kidney stone formers by endoscopic and histological quantification of intrarenal calcification[END_REF]. Tubular plugs have been associated with epithelial cell damaging, interstitial fibrosis, papillary scarring and nephron loss [START_REF] Coe | Plaque and deposits in nine human stone diseases[END_REF]. Both Randall's plaques and tubular plugs are thought to be precursors for idiopathic calcium oxalate stone formation [START_REF] Randall | The origin and growth of renal calculi[END_REF][START_REF] Evan | Randall's plaque of patients with nephrolithiasis begins in basement membranes of thin loop of Henle[END_REF][START_REF] Low | Endoscopic mapping of renal papillae for Randall's plaques in patients with urinary stone disease[END_REF][START_REF] Linnes | Phenotypic characterization of kidney stone formers by endoscopic and histological quantification of intrarenal calcification[END_REF][START_REF] Coe | Plaque and deposits in nine human stone diseases[END_REF].

Randall's plaque surface has been shown to be related to stone formation risk factors such as the decrease of diuresis volume and urinary pH and the increase of calciuria [START_REF] Kuo | Urine calcium and volume predict coverage of renal papilla by Randall's plaque[END_REF]. It has also been related to the amount of lithiasis clinical events. Tubular plugs seem to be related to high urinary pH, lower citraturia and increased calcium phosphate supersaturation [START_REF] Coe | Plaque and deposits in nine human stone diseases[END_REF][START_REF] Evan | Renal histopathology of stone-forming patients with distal renal tubular acidosis[END_REF]. They are thought to be more involved than Randall's plaque in case of stone formation post-bypass surgery, brushite stone, hyperparathyroidism or renal tubular acidosis [START_REF] Coe | Plaque and deposits in nine human stone diseases[END_REF][START_REF] Evan | Renal histopathology of stone-forming patients with distal renal tubular acidosis[END_REF].

We previously analysed human papillae from healthy parts of kidneys removed for cancer [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF]. As nephrolithiasis is a very frequent disease affecting more than 10% of the population, and because urine is a supersaturated milieu, we hypothesized that normal kidney would be the siege of incipient stone formation in tubules and actually, we have been surprised not only by the high frequency of incipient Randall's plaque but also papillary tubular plugs [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF]. Our purpose is to perform a descriptive analysis of papillary tubular plugs observed in papillae from non-stone formers. We describe herein that papillary plugs are extremely frequent and present an unexpected variety of crystalline phases. In addition, we observe collecting duct deformation at the contact of tubular plugs formed in the loop of Henle.

Materials and methods

Papillae collection

We studied kidneys removed for cancer from 25 patients between 2011 and 2014 as previously described [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF]. Patients gave a written consent in the Urology Unit relative to kidney removal and tissue conservation. Therefore, 76 papillae have been anonymously collected and coded after extemporaneous examination and no data relative to patients have been recorded according to French legislation and Helsinki declaration for patient's safety, except for the age and the sex of the patient. Papillae (with adjacent inner and outer medulla) have been embedded in paraffin after fixation by formaldehyde.

Yasue staining and immunohistochemical analyses

Tissue sections (4 µm) have been performed and stained by the Yasue procedure to reveal tissue calcifications. To assess the topography of incipient plugs, Yasue staining has been followed by classical immunostaining. Proximal convoluted tubules and descending loops of Henle were immunostained with a rabbit anti-aquaporin 1 (AQP1) antibody (1:600, SC-20810 clone H-55, Santa Cruz Biotechnology). Ascending loops of Henle were immunostained with a mouse monoclonal anti-Barttin antibody (1:100, SC-365161 clone A-3, Santa Cruz Biotechnology). Distal convoluted tubules and, to a lesser extent, collecting ducts were immunostained with rabbit anti-WNK 4 antibody (1:400, clone NB600-284, Novus Bio). Collecting ducts were immunostained with a goat anti-aquaporin 2 (AQP2) antibody (1:200, SC-9882 clone C-17, Santa Cruz Biotechnology). Samples were revealed with Single Stain MAX PO (mouse, rabbit or goat) Histofine (Nichirei Biosciences, Japan). For negative controls, the primary antibodies were replaced by an equal concentration of rabbit or mouse IgG (Dako).

Field emission-scanning electron microscopy (FE-SEM)

Tissue sections (4 µm) were investigated with a Zeiss SUPRA TM 55VP field emission-scanning electron microscope (FE-SEM). Measurements were performed at a low voltage (1.4 keV) and without the usual deposits of carbon at the surface of the sample.

FTIR micro-spectroscopy

Microcalcifications were characterized using Fourier transform infrared micro-spectroscopy (µ-FTIR). Tissue sections (4 µm) were deposited on low-emission microscope slides (MirrIR, Keveley Technologies, Tienta Sciences, Indianapolis). FTIR analysis was performed in serial sections adjacent to tissue sections stained with Yasue technique. FTIR hyperspectral images were recorded with a Spectrum spotlight 400 FT-IR imaging system (Perkin Elmer Life Sciences, Courtaboeuf, France), with a spatial resolution of 6.25 micrometre and a spectral resolution of 8 cm -1 . The spectra were recorded in the 4000-700 cm -1 mid-infrared range. Each spectral image covering a substantial part of the tissue consisted of about 30,000 spectra.

Results

High frequency of tubular plugs

Among 25 kidneys removed for cancer, we analysed 76 papillae in the healthy parts of the kidney for a median of three papillae per patient. For each patient, we analysed one 4-µm-thick section of the papilla, cut in the longitudinal axis. As previously described, we distinguished distal interstitial structures (incipient Randall's plaques) from intratubular plugs [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF]. Sixty-eight percent (52/76) of the papillae

High variety of crystalline phases forming tubular plugs

We analysed 110 tubular plugs from 24 papillae (14 patients) by µFTIR spectroscopy to characterize crystalline phases. In many cases, two different crystalline phases were associated components of a single plug (Table 2). The association between CA and ACCP was frequent and identified in 60.9% (67/110) of the plugs. Calcium oxalate was observed in 9.1% (10/110), either pure COM in 4.5% (5/110) of plugs or mixed COM + COD in 4.5% (5/110) of plugs, all these calcium oxalate plugs were observed in only one individual and did not contain calcium phosphate.

Plugs' topography

We performed Yasue staining coupled to immunohistochemistry to determine plug localisation along the nephron, in 25 papillae, with inner and outer medulla. Proximal convoluted tubules and descending loops of Henle were labelled with anti-aquaporin 1 (AQP1) antibody, ascending loops of Henle with anti-Barttin antibody, distal convoluted tubules with anti-WNK4 antibody and collecting ducts with anti-AQP2 antibody. In most cases, there was no colocalization of tubular plugs with AQP1, AQP2, Barttin or WNK4 staining, evidencing that most of the plugs were present in the deepest part of the loop of Henle (Fig. 5b-d, f). In rare cases, calcium phosphate plugs were found in WNK4-positive structures, i.e. distal tubule (Fig. 5d). A notable exception was the exclusive presence of calcium oxalate plugs in the collecting ducts expressing AQP2 (Fig. 5a).

Tubular remodelling in contact with plugs

In many patients, calcium phosphate tubular plugs probably present in the thin ascending loop of Henle (not expressing AQP1, Barttin, WNK4 or AQP2) were able to deform and sometimes to tear up the adjacent collecting duct epithelial cells (Fig. 6a-f).

Discussion

Although urolithiasis is a growing public heath challenge worldwide, the first step of kidney stone formation is still a matter of debate. It seems likely that crystal adhesion to epithelial tubular cells may initiate the process [START_REF] Khan | Role of renal epithelial cells in the initiation of calcium oxalate stones[END_REF]. Cellular and rodent models are of interest to characterize the early phases of stone formation and evidence the importance of calcium oxalate or calcium phosphate tubular plugging [START_REF] Khan | Retention of calcium oxalate crystals in renal tubules[END_REF]. Nevertheless, these models do not reproduce human kidney stone disease by many aspects (kidney stone composition, absence of Randall's plaque, etc.). The main purpose of this study was to analyse incipient papillae calcifications to draw up an inventory of the renal calcifications that may be observed in "normal" papillae, and to assess the early processes potentially leading to kidney stone formation.

We have been previously surprised by the frequency of tubular plugs in non-stone formers' kidneys [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF]. These plugs may co-exist with Randall's plaque in the same papilla but their spatial distribution is different: incipient plaques are interstitial deposits limited to the tip of the papilla and surrounding a large number of tubular or vascular lumens, whereas plugs are present inside the tubule, often isolated and sparse in the papilla. Of notice, we analysed a single 4-µm-thick section for each papilla, so that we obviously underestimated the frequency of tubular plugs. Even in these limiting conditions, we observed tubular plugs in 92% of patients. Since we collected papillae slices, we cannot estimate the number of plugs contained in one human papillae; this would require micro-or nano-computed tomography analyses of the whole papilla. There is no proof that these plugs may be the first step toward kidney stone formation but their high incidence is puzzling and evidences that tubular plugging is a common mechanism. It would be of interest to collect papillae from stone formers and to compare the topography, composition and number of tubular plugs to non-stone formers papillae, but this stands beyond the scope of this study. We ignore why there is a such high incidence of tubular plugs in "normal" papillae but it seems likely that calcium phosphate supersaturation at the tip of the papilla is the main driving force leading to plug formation. The high incidence of plug formation also suggests that other mechanisms are required before clinical stone formation and that tubular plugging is not sufficient. Conversely, the observation of calcium phosphate tubular plugs in human papillae should not necessarily be considered as a specific pathological process.

The second matter of interest was the various compositions of these plugs. As expected, CA and ACCP were frequently observed, and associated in many cases. We also observed an OCP plug in one patient. OCP is a rare stone component in common lithogenic conditions and has been described in pregnancy-related stones [START_REF] Meria | Stone formation and pregnancy: pathophysiological insights gained from morphoconstitutional stone analysis[END_REF]. Pregnancyrelated stones occur rapidly in calcium phosphate supersaturation conditions (high urine pH, hypercalciuria, hyperphosphaturia). OCP is an unstable calcium phosphate that may transform into CA over time and one may hypothesize that in some individuals, OCP plugs may be a transitory phase toward CA plug formation. The identification of calcium carbonate plugs (calcite) was also surprising since pure calcite stones are rare in humans. Calcite formation requires high pH (carbonate) and high calcium concentration. The proportion of carbonate is highly variable in calcium phosphate kidney stones and it would be of interest to understand in which part of the tubule calcium carbonate may be observed [START_REF] Levillain | Crystalline phase differentiation in urinary calcium phosphate and magnesium phosphate calculi[END_REF]. Unfortunately, colocalization experiments in this patient were not contributive. We also observed calcium and magnesium phosphate (whitlockite) in one individual. Whitlockite may be identified in infection stones and we previously identified by FTIR large amounts of whitlockite in incipient Randall's plaque [START_REF] Verrier | Topography, composition and structure of incipient Randall plaque at the nanoscale level[END_REF][START_REF] Carpentier | Relationships between carbonation rate of carbapatite and morphologic characteristics of calcium phosphate stones and etiology[END_REF][START_REF] Daudon | Vibrational spectroscopies to investigate concretions and ectopic calcifications for medical diagnosis[END_REF]. The patient affected by whitlockite plugs had also abundant interstitial calcifications (Randall's plaque). Interestingly, brushite signature has not been identified in any case, although its presence in tubular plugs has been mentioned previously [START_REF] Evan | Contrasting histopathology and crystal deposits in kidneys of idiopathic stone formers who produce hydroxy apatite, brushite, or calcium oxalate stones[END_REF]. One may hypothesize that brushite plugs may be observed in kidney stone formers only.

We did not find any colocalization of calcium phosphates with AQP1 or Barttin. In some exceptional cases, we identified calcium phosphates plugs in collecting ducts (colocalization with AQP2) and sometimes in distal tubule (colocalization with WNK4). It seems likely that most of calcium phosphate deposits stood in the deepest part of the loop of Henle. This is not unexpected. Actually, plugs are usually considered to result from excessive urinary supersaturation and calcium concentration is predicted to be maximal at the tip of the papilla [START_REF] Tournus | A model of calcium transport along the rat nephron[END_REF]. Accordingly, plugs were frequent in the papilla and the inner medulla but sparse in the outer medulla and cortex. Interestingly, in the papillae in which calcium oxalate was found, no association with phosphates was observed. Moreover, COM and COD crystalline plugs stood in moderately dilated collecting ducts expressing AQP2. This observation is in accordance with preliminary reports evidencing calcium oxalate plug formation in collecting ducts [START_REF] Coe | Plaque and deposits in nine human stone diseases[END_REF]. COM and COD plugs found in dilated collecting ducts suggest either that multiple plugs were responsible for upstream duct dilation, or that dilated collecting ducts disturb urine flow, promoting calcium oxalate crystals retention.

An unexpected finding was the observation in several patients that plugs could deform adjacent structures, especially collecting ducts and in some cases promote collecting duct epithelium rupture. Whether this process allows elimination of large plugs in urine through collecting ducts remains to be confirmed. Interestingly, animal models of kidneys stones and nephrocalcinosis were characterized by calcium phosphate plugs in kidney tubules and crystal translocation to the interstitium, highlighting that calcium phosphate deposits can, in some conditions, cross the epithelium [START_REF] Khan | Calcium oxalate crystal deposition in kidneys of hypercalciuric mice with disrupted type IIa sodium-phosphate cotransporter[END_REF][START_REF] Khan | Histological aspects of the "fixed-particle" model of stone formation in animal studies[END_REF]. Long-term experimental animal models of intratubular calcium phosphate precipitation could be of help to analyse whether calcium phosphate tubular plugs are able to promote collecting duct deformation and at last their rupture.

The major limit of our study is the absence of clinical data, due to study design. We were, therefore, unable to correlate urine biochemistry to the presence of plugs or to their composition. Further, we presumed that tubular plugs may be a first step toward kidney stone formation but we ignore whether plugs present in the ascending loop of Henle could migrate downstream, toward collecting duct and renal cavities. According to tubule diameter downstream of the thin part of the loop of Henle, this hypothesis seems likely but remains speculative. At last, we are extremely cautious with our results relative to plug localization in the tubular lumen. Actually, massive plugs may mask epithelial cells, this is the reason why we did not provide quantitative data regarding plug localization. The observations reported above deserve further studies. As nephrolithiasis is a frequent disease, the collection of healthy papillae from stone formers affected by cancer should be possible in the long term, and a comparison between papillae from stone formers and non-stone formers could provide meaningful outcomes.

Conclusion

The systematic analysis of renal papillae sections from nonstone formers highlights that tubular plug formation is a very frequent process probably affecting most of the people. The polymorphism and the unexpected variety of these plugs suggest that their composition may vary according to urine composition and to the part of the nephron where they form. In addition, tubular plasticity in response to these plugs is impressive. The observation of tubular plugs deforming and sometimes eroding collecting ducts deserves further studies to understand their potential role in kidney stone formation.

Fig. 1 Fig. 2 Fig. 3 Fig. 4

 1234 Fig. 1 Carbonated apatite plug. a Yasue staining (photonic microscopy, × 600) and b, c scanning electron microscopy (× 250 and × 1000). d Plug reflectance FTIR cartography. e Reflectance spectrum showing a peak at 1032 cm -1 , specific for apatite (theoretically around 1030 cm -1 )

Fig. 5

 5 Fig. 5 Immunohistochemistry and Yasue staining (photonic microscopy, × 600). a AQP2-positive colocalization (calcium oxalate), b AQP2-negative colocalization (calcium phosphate), c AQP1-negative

Fig. 6

 6 Fig. 6 Collecting duct deformation induced by crystalline plugs. Immunohistochemistry and Yasue staining (photonic microscopy, × 600). Crystalline plugs promote collecting duct deformation (a-d) and in some cases rupture of the epithelium (e-f) (red arrows). Immu-

Table 1 .

 1 Although CA and ACCP were observed in many papillae and patients, OCP was observed in only one individual, calcium oxalate was observed in only one patient (but in several papillae), as well as calcite and whitlockite (Figs.[START_REF] Trinchieri | Epidemiological trends in urolithiasis: impact on our health care systems[END_REF][START_REF] Denstedt | Epidemiology of stone disease in North America[END_REF][START_REF] Hesse | Study on the prevalence and incidence of urolithiasis in Germany comparing the years 1979 vs. 2000[END_REF][START_REF] Ogawa | Epidemiology of stone disease over a 40-year period in Japan[END_REF].

	Seven compounds have been identified: carbonated apatite
	(CA; 90/110), amorphous carbonated calcium phosphate
	(ACCP; 79/110), calcium oxalate monohydrate (COM;
	10/110), calcium oxalate dihydrate (COD; 5/110), calcium
	carbonate (calcite; 3/110), whitlockite (3/110) and octacal-
	cium phosphate pentahydrate (OCP; 1/110). The various
	crystalline phases present in plugs, papillae and patients
	are summarized in

Table 1

 1 Number of plugs, papillae and patients (%) for each identified crystalline phase ACCP amorphous carbonated calcium phosphate, CA carbonated apatite, COM calcium oxalate monohydrate, COD calcium oxalate dihydrate, OCP octacalcium phosphate pentahydrate

			CA	ACCP	COM	COD	Calcite	OCP	Whitlockite
		Plugs	90 (81.8%)	79 (71.8%)	10 (9.1%)	5 (4.5%)	3 (2.7%)	1 (0.9%)	3 (2.7%)
		Papillae	24 (100%)	20 (83.3%)	3 (12.5%) 2 (8.3%)	1 (4.2%)	1 (4.2%)	1 (4.2%)
		Patients	14 (58.3%)	12 (50%)	1 (7.1%)	1 (7.1%)	1 (7.1%)	1 (7.1%)	1 (7.1%)
	Table 2 Number of plugs containing pure or mixed		CA	ACCP	COM	COD	Calcite	OCP	Whitlockite
	crystalline phases	CA	17 (15.5%)	67 (60.9%)	0	0	3 (2.7%)	1 (0.9%)	2 (1.8%)
		ACCP		9 (8.2%)	0	0		0	0
		COM			5 (4.5%)	5 (4.5%)	0	0	0
		COD				0	0	0	0
		Calcite					0	0	0
		OCP						0	0
		Whitlockite							1 (0.9%)

ACCP amorphous carbonated calcium phosphate, CA carbonated apatite, COM calcium oxalate monohydrate, COD calcium oxalate dihydrate, OCP octacalcium phosphate pentahydrate

Physiology Unit, AP-HP, Hôpital Tenon, 75020 Paris, France

Pathology Unit, AP-HP, Hôpital Tenon,

Paris, France 7 Service des Explorations Fonctionnelles Multidisciplinaires, Hôpital TENON, 4 rue de la Chine, 75020 Paris, France

Funding This work has been supported by the Agence Nationale de la Recherche (ANR-13-JSV1-0010-01, ANR-12-BS08-0022), the Société de Néphrologie (Genzyme Grant), the Académie Nationale de Médecine (Nestlé-Waters award), Convergence-UPMC CVG1205 and CORDDIM-2013-COD130042.

Conflict of interest All authors declare to have no conflict of interest.

Human and animal rights statement All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent Informed consent was obtained from all individual participants included in the study.


Yukio Koriyama
email: yukio.koriyama@polytechnique.edu.

Ali Ozkes

Colin Camerer

Nobuyuki Hanaki

Guillaume Hollard

Keigo Inukai

Nagore Iriberri

Jean-François Laslier

Ugur Özdemir

Thomas Palfrey

Eduardo Perez-Richet

Phil Reny

Ariel Rubinstein

Inclusive Cognitive Hierarchy in Collective Decisions *

Keywords: JEL classification: C92, D72, D91 Collective decision-making, Bounded rationality, Cognitive hierarchy, Information aggregation

We study the implications of structural models of non-equilibrium thinking, in which players best respond while holding heterogeneous beliefs on the cognitive levels of others. We introduce an inclusive cognitive hierarchy model, in which players are capable of projecting the self to others in regard to their cognitive level. The model is tested in a laboratory experiment of collective decision-making, which supports inclusiveness. Our theoretical results show that inclusiveness is a key factor for asymptotic properties of deviations from equilibrium behavior. Asymptotic behavior can be categorized into three distinct types: naïve, Savage rational with inconsistent beliefs, and sophisticated.

Introduction

In her presidential speech to the American Political Science Society, [START_REF] Ostrom | A behavioral approach to the rational choice theory of collective action: Presidential address, American Political Science Association[END_REF] called for a behavioral theory of collective action based on models of boundedly rational individuals, pointing to the fact that behavior in social dilemmas is affected by many structural variables, such as group size. The objective is put forward with an understanding of the "thin" model of rationality as a limiting case of bounded or incomplete rationality, as in [START_REF] Selten | Reexamination of the perfectness concept for equilibrium points in extensive games[END_REF].

Collective decision-making processes entail most of the time, if not always, asymmetric information. And studying strategic thinking in private information games is crucial, as they are common tools for modeling interactions in bargaining, matching, contracts, financial markets, and political systems, inter alia. If strategic naïveté is prevalent, peculiarities due to private information can be more crucial than what is predicted by equilibrium analysis, and thus policy responses may be astray, as noted by [START_REF] Brocas | Imperfect choice or imperfect attention? Understanding strategic thinking in private information games[END_REF].

Models of non-equilibrium strategic thinking have been proposed to explain structural deviations from equilibrium in a variety of games. A sizable part of bounded rationality literature is devoted to the models of cognitive hierarchy, starting with the level-k model due to [START_REF] Nagel | Unraveling in guessing games: An experimental study[END_REF] and [START_REF] Stahl | On Players Models of Other Players: Theory and Experimental Evidence[END_REF], which allow for heterogeneity among individuals in levels of strategic thinking. In the level-k model, a foundational level (level-0) represents a strategically naïve initial approach to a game, and a level-k player (hereafter Lk ), where k ≥ 1, is assumed to best respond to others with level k -1. The construction of levels resonates with rationalizability, as in [START_REF] Bernheim | Rationalizable strategic behavior[END_REF], due to the fact that the decisions made by a level-k player survive k rounds of iterated elimination of strictly dominated strategies in two-person games.

Closely related, the cognitive hierarchy (CH) model introduced by Camerer et al. ( 2004) allows for heterogeneity in the beliefs on others' levels.1 For each k ≥ 1, a level-k player best responds to a mixture of strictly lower levels, induced by the truncation up to level k -1 from the underlying level distribution, which is either obtained from maximum likelihood estimations applied to data, or calibrated from previous estimates. The set of level-1 strategies in the CH model (hereafter CH1) is exactly the same as that of L1, and Lk and CHk differ for k ≥ 2. Experimental studies provide various sorts of evidence that the CH model delivers a better fit for explaining the actual behavior of players in certain games. Common to these models is the assumption that level-k players do not assign any probability to levels higher than or equal to k, and this emerges from the idea that the cognitive limits among players have in fact a hierarchical structure.

In both of the aforementioned models, self-awareness as described in [START_REF] Camerer | A cognitive hierarchy model of games[END_REF] is precluded,2 that is, the possibility for players to assign positive probability to the events in which other players have the same cognitive level as themselves is ruled out.

Here, we propose, instead, an inclusive cognitive hierarchy (ICH) model, in which players are allowed to project themselves to others in regard to their cognitive level. The ICH model thus allows for inclusiveness while maintaining the hierarchical structure of cognitive levels and the partial consistency induced by truncation of the underlying level distribution as in other cognitive hierarchy models. In this paper, we study the role of inclusiveness both empirically and theoretically. On the empirical side, we present the results of our laboratory experiment on binary collective decision making. On the theoretical side, we provide theorems which describe asymptotic properties of the deviation from rational behavior. Since Condorcet's classical work in 1785, mathematical support has been provided for the idea of increasing the accuracy of collective decisions by including more individuals in the process. In his seminal Essai, Condorcet considered non-strategic individuals voting to make a decision on a binary issue where each alternative is commonly preferred to the other in one of the two states of the world. Each individual independently receives an imperfectly informative private signal about the true state of the world and votes accordingly. Under majority rule, the probability of reaching a correct decision monotonically increases with the size of the electorate and converges to certainty in the limit.

Although enabling strategic behavior may imperil the validity of the basic assumptions in the original model, the asymptotic property survives in various circumstances of collective decision-making (e.g. Austen-Smith and [START_REF] Banks | Information aggregation, rationality, and the Condorcet jury theorem[END_REF] and [START_REF] Feddersen | Voting behavior and information aggregation in elections with private information[END_REF]). Studies have documented conditions under which the probability of making a right decision increases and converges to certainty as the group size increases, even when strategic players may vote against their signals.

However, assuming full rationality, especially complete mutual consistency of beliefs, may be too demanding when the strategy space is big and a large number of players are involved. In large games, the collective behavior of strategic players may differ qualitatively from that in small games. This intuition finds support from experimental evidence and information aggregation by a large group is not, to say the least, exempt from it, especially when a major presumption of the strategic models is that voters take into account their probability of being pivotal, as in [START_REF] Downs | An economic theory of democracy[END_REF]. As shown by [START_REF] Esponda | Hypothetical thinking and information extraction in the laboratory[END_REF], perfectly accurate hypothetical thinking to extract information from others' strategies that is required in most strategic voting models might be too strong as an assumption, even untenable, when it is assumed for all individuals. 3 Battaglini et al. (2008) document in another experimental study an increase in irrational, non-equilibrium play as the size of the electorate increases. 4Collective performances are correlated across challenges, as demonstrated in [START_REF] Woolley | Evidence for a collective intelligence factor in the performance of human groups[END_REF], hence a good knowledge about the behavioral basis in collective decision-making procedures is essential to understanding the more general phenomena of our societies. We argue in this paper that the cognitive hierarchy approach can be fruitful in that regard, once inclusiveness is incorporated.

There are at least three reasons why we believe that the consequences implied by inclusiveness should be studied. First, the observed limitations of existing models in the extant literature on strategic thinking call for studies with an explicit focus on inclusiveness, 5 for which our experimental analysis provides a clear support. Second, our post-experimental questionnaire also suggests inclusiveness. 6 Figure 1(a) shows that a fairly large proportion (96%) of our experimental subjects exhibit a positive degree of inclusiveness: 'sometimes', 'most of the time', or 'always'. They are further asked to provide a subjective estimation of the percentage among other members who used the 'same reasoning' (Figure 1(b)). The answers vary and a vast majority of them are far from 0%, which suggests that ruling out inclusiveness would be too extreme as an assumption. 7 Third, our analytical results imply that the presence of inclusiveness simply matters (Theorem 1) in a certain class of games (Theorem 2). A novel finding of this paper is that a lack of the inclusiveness condition may imply that the deviation from rational behavior asymptotically diverges away without a bound, which we argue is not coherent with the spirit of the cognitive hierarchy models. 8 One of the objectives of this paper is to shed light on the role of the behavioral assumptions that account for the asymptotic properties of the deviation from the equilibrium thinking. et al. (2005) provide an extensive survey on the studies of collective decision-making in committees and the Condorcet Jury Theorem. [START_REF] Palfrey | Experiments in Political Economy[END_REF] provides a comprehensive survey on experiments in political economy and particularly in strategic voting. [START_REF] Costinot | On Optimal Voting Rules under Homogeneous Preferences[END_REF] study voting rules and show that the optimal voting rule is the same when players are sincere, playing according to Nash equilibrium, to level-k, or a mixture of these. [START_REF] Bhattacharya | Voting with endogenous information acquisition: Experimental evidence[END_REF] test experimentally the theoretical predictions about individual behavior and group decisions under costly information acquisition. They find poor support for the comparative statics predictions which are delivered theoretically. [START_REF] Alaoui | Endogenous depth of reasoning[END_REF] introduce a model of strategic thinking that endogenizes individuals' cognitive bounds 5 Colman et al. (2014) point to the observed weak performance of the CH model in common interest games. [START_REF] Georganas | On the persistence of strategic sophistication[END_REF] conclude that level-k models have a high performance in some games, but not in others. "Equilibrium plus noise" models such as quantal response equilibrium (cf. [START_REF] Goeree | Quantal Response Equilibrium: A Stochastic Theory of Games[END_REF]) often miss systematic patterns in participants' deviations from equilibrium, which is a main feature observed in our experimental data.

Related Literature

Gerling

6 As the questionnaire is not incentivized, the interpretation should be performed with care.

7 Heterogeneity of the answers implies that the participants' degree of inclusiveness was also heterogeneous. This observation is coherent with the assumption of hierarchical belief structure that we describe in detail in Section 2.

8 Evidence from experimental studies on public good games suggest that as group size increases, individual behavior bears convergent and stabilizing tendencies. See [START_REF] Isaac | Group size and the voluntary provision of public goods: experimental evidence utilizing large groups[END_REF]. as a result of a cost-benefit analysis. Their framework allows individuals to reason about opponents whom they regard as more sophisticated as well. [START_REF] Hanaki | The strategic environment effect in beauty contest games[END_REF] study how the strategic environmental effects depend on the group size in beauty contest games, finding support for the presence of the effects in large groups.

The paper proceeds as follows. We introduce the inclusive cognitive hierarchy model formally in the following section. We furthermore provide a numerical comparison of the individual behaviors and the performance of collective decisions under different specifications of the cognitive hierarchy in a game of information aggregation. In Section 3 we introduce our experimental design that features novelties due to our modeling concerns and signal setup. Section 4 provides the results of the experiment, and the models are compared in terms of the fit to the data. Section 5 provides theoretical results focusing on linear quadratic games. We present our main theorems, which provide a characterization of games according to the asymptotic properties of the strategic thinking. We conclude by summarizing our findings and presenting further research questions in Section 6. The proofs of the theorems are relegated to Appendix.

The Model

Let (N, X, u) be a symmetric normal-form game where N = {1, . . . , n} is the set of players, X ⊂ R is a convex set of pure strategies, and u : X n → R n is the payoff function. Each player forms a belief on the cognitive levels of the other players. Let g k (h) denote the probability that a k th -level player assigns independently for each of the other players to belong to the h th -level.

In the standard level-k model, a naïve, non-strategic behavior is specified as the initial level (level-0, or L0). For k ≥ 1, a level-k (Lk ) player holds the belief that all of the other players belong to exactly one level below herself:

g k (h) = 1 if h = k -1, 0 otherwise. (L)
In the cognitive hierarchy model, each k th -level (CHk) player best responds to a mixture of lower levels. Let f = (f 0 , f 1 , . . .) be a distribution over N which represents the composition of levels. Each k th -level player holds a belief on the distribution of the other players' levels that is a truncation up to one level below herself:

g k (h) = f h k-1 m=0 f m , for 0 ≤ h ≤ k -1 and k ≥ 1. (CH)
Thus, these two models share the following assumption:

Assumption 1 (Strong overconfidence) g k (h) = 0 for all h ≥ k.

Assumption 1 enables us to say that what we call "levels" here indeed has a hierarchical structure. To see this, consider g k (h) in the format of a k-h matrix. Assumption 1 implies that the upper-diagonal entries are all zeros, and thus the remaining non-zero elements have a pyramid structure with strictly lower-diagonal entries. Each level-k player assigns non-zero probabilities only to the levels strictly lower than herself. In that sense, players are assumed to be overconfident.9 Alternatively, the following assumption can be considered:

Assumption 2 (Weak overconfidence) g k (h) = 0 for all h > k.

Assumption 2 is weaker than Assumption 1. As in Assumption 1, zero probability is assigned for all strictly upper-diagonal entries, and thus a hierarchical structure among levels is still preserved. However, the diagonal entries are not restricted to be zero. A level-k player is allowed to assign a non-zero possibility for the other players to have the same level as herself.

In what follows, we formally introduce the inclusive cognitive hierarchy (ICH) model, in which the strong overconfidence condition is weakened to allow for inclusiveness.

Inclusive Cognitive Hierarchy

Fix an integer K > 0 that prescribes the highest level considered in the model. 10 In the ICH model, we consider a sequence of mixed strategies σ = (σ 0 , . . . , σ K ), in which for each k ∈ {1, ..., K}, σ k ∈ ∆(X) is a best reply, assuming that the other players' levels are drawn from the truncation of the underlying distribution f up to level k. Note that a sequence of truncated distributions g = (g 1 , . . . , g K ) is uniquely defined from f . As in the other models, we focus on level-symmetric profiles in which all players of the same level use the same mixed strategy.

Definition 1 A sequence of level-symmetric strategies σ = (σ 0 , . . . , σ K ) is called inclusive cognitive hierarchy strategies when there exists a distribution f over N under which supp (σ k ) ⊂ arg max

x i ∈X E x -i [u (x i , x -i ) |g k , σ] , ∀k ∈ {1, • • • , K} ,
where g k is the truncated distribution induced by f such that

g k (h) = f h k m=0 f m for h ∈ {0, • • • , k} , (ICH)
and the expectation over x -i is drawn, for each player j = i, from a distribution

γ k (σ) := k m=0 g k (m) σ m .
We note that Definition 1 is analogous to the definitions used in the standard level-k model and the cognitive hierarchy model. It simply replaces the assumptions on beliefs, (L) and (CH), with (ICH). Building on previous studies (as developed by [START_REF] Camerer | A cognitive hierarchy model of games[END_REF], we maintain the assumption that the underlying distribution f of levels follows a Poisson distribution with coefficient τ :

f k = τ k k! e -τ .
Note that the expectation of the distribution is τ , which thus represents the overall expected level among the players.

Condorcet Jury Theorem

A group of n individuals makes a binary collective decision d ∈ {-1, 1}. The true state of the world is also binary, ω ∈ {-1, 1}, with a common prior of equal probabilities. The payoff is a function of the realized state and the collective decision as follows:

ũ(ω, d) =    0 if ω = d, q if ω = d = 1, 1 -q if ω = d = -1,
with q ∈ (0, 1) for all individuals.11 Each individual i ∈ {1, • • • , n} receives a private signal s i ∈ S, distributed independently conditional on the true state ω. A collective decision is made by the majority rule. Upon receiving signal s i , individual i casts a vote v i ∈ {0, 1}, and the collective decision is determined by the sign of ( i v i -n/2).

We do not restrict the signal space S to be binary. 12 We assume S ⊂ R so that S is an ordered set, and we assume that the commonly known distribution satisfies the monotone likelihood ratio property, that is, the posterior distribution Pr[ω = 1|s i ] is increasing in s i .

A strategic Condorcet Jury Theorem claims that asymptotic efficiency is obtained among the rational individuals with homogeneous preferences and costless information acquisition, as described above. More precisely, it claims that, under the Nash equilibrium behavior, the probability of making a right decision converges to one as n goes to infinity. In the following subsection we ask whether the asymptotic, collective efficiency would be obtained under the cognitive hierarchy models in which individuals may show systematic deviations from the Nash behavior.

Asymptotic efficiency: a numerical example

In order to underline the differences implied by different behavioral assumptions, we provide numerical computation results using the game described above. The model parameters are chosen so that the game coincides exactly with the one with asymmetric payoffs in our experiments (Section 3). 13 Our aim here is to highlight numerically the behavioral consequences of the inclusiveness condition, by comparing the probabilities of making a correct group decision and the individual strategies predicted by each model. Four different behavioral specifications are compared: Nash equilibrium (NE), the standard level-k model (L), the Poisson cognitive hierarchy model (CH), and the Poisson inclusive cognitive hierarchy model (ICH).

Figure 2(a) shows the probability of making a correct decision as a function of the group size. Convergence to one in Nash equilibrium reflects the strategic Condorcet Jury Theorem, as a benchmark for the cognitive hierarchy models under our consideration. The ICH model also exhibits asymptotic efficiency. The quality of group decisions shows a stark contrast both in the L model and in the CH model with the ICH model. The probability approaches 0.5 in large groups, which is as bad as pure noise, implying that the Condorcet Jury Theorem fails to the worst extent without considering the inclusiveness of the agents.

The key difference in asymptotic properties between CH and ICH stems from the inclusiveness condition. With inclusiveness, each player in the ICH model chooses the optimal behavior that maximizes the expected utility given that the other players may have the same level. As is shown analytically in Section 5, the distance of the ICH strategies from the Nash equilibrium does not diverge away even in a large group, since the strategy at each level is obtained as a best reply to the other players' strategies, which may match each other with a positive probability. A highest-level player in the ICH model is thus capable of choosing his strategy fully rationally, correcting the biases caused by the lower levels. Without inclusive players, the deviations from the rational behavior accumulate in the group and thus deteriorate the quality of the collective decision.

Figure 2(b) depicts level-1 and level-2 strategies under the L, CH, and ICH models, and the Nash equilibrium, as a function of the group size n. 14 In the L model, strategies hit the boundary for both level-1 (except for n = 3) and level-2 values. In the CH model, level-1 is the same as in the L model, while the level-2 strategy is decreasing, meaning a divergence from the Nash behavior. In the ICH model, strategies are increasing in both level-1 and level-2, in accordance with the Nash behavior. We observe that the convergence of the ICH2 strategy is remarkably quick, so that it visibly coincides with the Nash equilibrium.

These figures demonstrate a stark contrast among the behavioral assumptions under our consideration. In particular, the figures suggest that the inclusiveness condition plays a key 14 Strategy is defined as a cutoff value in the interval [0, 10]. See Section 4.1 for a precise definition.

role in describing the asymptotic behavior. In the following sections, we show the results from our stylized laboratory experiment which provides statistical evidence for our scrutiny of the models.

Experimental Design

All of our computerized experimental sessions were held at the Ecole Polytechnique Experimental Economics Laboratory. 15 In total we had 140 actual participants in 7 sessions, in addition to the pilot sessions with more than 60 participants. In each session, 20 participants took part in 4 phases (together with a short trial phase) which lasted about one hour in total. Earnings were expressed in experimental currency units (ECUs) and exchanged for cash, to be paid immediately following the session. Participants earned an average of about 21 Euros, including a default 5 Euros for participation. Complete instructions and details can be found in our online appendix. 16 The instructions pertaining to the entire experiment were read aloud at the beginning of each session. Before each phase, the changes from the previous phase were read aloud, and an information sheet providing the relevant details of the game was distributed. These sheets were exchanged with new ones before each phase.

We employed a within-subject design where each participant played all 4 phases consecutively in a session. Each phase contained 15 periods of play, and thus each participant played for a total of 60 periods under a direct-response method. Since the question of our research relates to the strategic aspects of group decisions, our experiment was presented to participants as an abstract group decision-making task where neutral language was used to avoid any reference to voting or elections of any sort.

In the beginning of each period, the computer randomly formed groups of participants, of a size that was commonly known and predetermined for each phase (either n = 5, 9, or 19).17 Then, a box was shown to each participant with one hundred squares (to be referred as cards from now on), all colorless (gray in z-Tree). At the same time, the unknown true color of the box for each group was determined randomly by the computer. The participants were informed that the color of the box would be either blue or yellow, with equal probability. It was announced that the blue box contained 60 blue and 40 yellow cards, whereas the yellow box contained 60 yellow and 40 blue cards.

After confirming to proceed to the next screen, 10 cards drawn by the computer with random locations in the box were shown to the participants, this time with the true colors. These draws were independent among all participants but were drawn from the same box in the same group. Having observed the 10 randomly-drawn cards, the participants were required to choose either blue or yellow by clicking on the corresponding button. Then, the decision for the group was reached by majority rule, which was resolute every time, since we only admitted odd numbers for group sizes and abstention was not allowed. Once all participants in a group had made their choices, the true color of the box, the number of members who chose blue, the number of members who chose yellow, and the earnings for that period were revealed on the following screen. A new period started after everyone confirmed.

In one of the four phases, the group size was set at n = 5 and the payoffs were symmetric. Each participant earned 500 ECUs for any correct group decision (i.e., a blue decision when true color of the box was blue, or a yellow decision when true color of the box was yellow). In the case of an incorrect decision, no award was earned. In the other three phases, each treatment differed only in the size of the groups (5, 9, or 19) where asymmetric payoffs were fixed. The correct group decision when the true color of the box was blue awarded each participant in the group with 900 ECUs, whereas the correct group decision when the true color of the box was yellow awarded them with 200 ECUs.18 Lastly, we implemented a random-lottery incentive system where the final payoffs at each phase were determined by the payoffs from a randomly-drawn period. 19Let us underline that asymmetry in remuneration is introduced in our experiment in order to observe the effect of a prior bias on the participants' behavior. It is not surprising that an informative strategy (i.e. voting for the choice favored by the signal), or one close to it, is employed by a large majority of the participants under symmetric awards. 20 When it is commonly known that one of the alternatives may provide a larger award, in addition to the change of the symmetric Nash equilibrium shifting towards the ex ante preferable alternative, each individual's behavior may shift, and furthermore, such shifts may be heterogeneous across individuals. Consequentially, each individual may hold heterogeneous beliefs over the strategies employed by the other individuals in the group. The accumulated effects of such heterogeneous belief formation may hamper the performance of group decision-making, which is one of our main concerns in this paper.

At the beginning of each session, as part of the instructions, participants played through two mandatory trial periods. Each session concluded after a short questionnaire. According to the anonymously-recorded questionnaire, 44% of the participants were female. The age distribution was as follows: 31% between the age of 19 and 22, 26% between 23 and 29, 14% between 30 and 39, and 29% between 40 and 67. Heterogeneity in their professions was relatively high: 46% administrative staff, 37% undergraduate students ("Polytechniciens"), 12% Ph.D. students, 1% master students, and 3% researchers. 6% of the participants had previously taken an advanced course in game theory, while 14% had taken an introductory course. 39% said that they had some notions about game theory, while 41% claimed to have no knowledge of game theory.

Experimental Results

In this section we present our experimental results at both the individual and group levels.

Cutoff Strategies

Under our experimental design, a pure strategy of an individual is a function from the realized signal to a binary vote. It is straightforward to show that the best reply of an individual, given any belief on the strategies of other group members, is a cutoff strategy. There exists a threshold for each individual such that she votes for blue if and only if her signal induces a higher posterior probability of a blue state than the threshold. Since the posterior belief over the two states varies monotonically as a function of the number of blue cards among the 10 revealed ones, a cutoff strategy in our experiment is that each individual votes for blue when the number of observed blue cards is higher than the cutoff value, and for yellow otherwise. Special cases include voting for one of the colors regardless of the signal. The cutoff value is considered as an extreme value (either 0 or 10) for such a behavior.

We have observed in our data that a majority of participants used a cutoff strategy with randomization. Randomization occurs with two or more realized values of the signal, with the degree of randomization varying monotonically in the right direction (i.e. a higher probability of voting for blue given more blue cards in the signal). We regard such a behavior as a consequence of decision-making with an error or other uncertainties which are not explicitly formalized in the model. 21More precisely, let x i ∈ [0, 10] be the (continuous) cutoff strategy of voter i, and let s t i ∈ {0, 1, • • • , 10} be the (discrete) signal realized in period t, i.e. the number of blue cards out of the ten revealed ones. We assume that the probability that i casts a vote v t i ∈ {0(yellow), 1(blue)} depends on the distance between the signal and the cutoff value, following the logistic distribution:

ϕ v t i |s t i , x i , ε i = (exp ε i (s t i -x i )) v t i 1 + exp ε i (s t i -x i ) (1)
where ε i is the error coefficient. As ϕ pins down a distribution over voting profiles (v i ) n i=1 , the social decision d made by the simple majority rule is a function of the realized signals s = (s i ) n i=1 , cutoff values x = (x i ) n i=1 , and the error coefficients ε = (ε i ) n i=1 . Hence, for fixed ε, we can write the expected payoff as a function of the cutoff strategy profile x ∈ [0, 10] n : 22

u (x) = E ω,s [ũ (ω, d (s, x, ε))] .
Let T be the set of periods under consideration. Provided the observation (s t i , v t i ) t∈T , our estimation of the cutoff strategies is obtained by maximizing the likelihood:

L s t i , v t i t∈T x i , ε i = t∈T ϕ v t i s t i , x i , ε i .
Some robustness checks are in order. During each period t, what we observe for each individual i is only one realized signal s t i and the casted vote v t i . Therefore, a cutoff estimation requires a pool of observations over periods. We checked estimations over different intervals of periods, 23 suggesting that no change in strategy occurred over time, including the possibility of learning. We thus concluded that the estimation is robust including observations over all 15 periods. 24 We also tested different assumptions on ε i : (i) common across phases for each individual, (ii) common across individuals for each phase and session, (iii) common across individuals and sessions for each phase, and (iv) common for individuals, sessions and phases. Comparing the estimated cutoff values between any pair of different assumptions on ε i , the R 2 values are all close to one, implying that the estimation is robust. 25 In the following, we use the estimated cutoff values under assumption (iii), as our main concern here is the change in subjects' behavior as a function of the group size.

In Figure 3, histograms of the estimated cutoff values are shown for each phase. Several remarks are in order. First, we see a clear shift in the distribution from the symmetric payoffs to the asymmetric ones. Most notably, for each of the group sizes of 5, 9 and 19 with asymmetric payoffs, a peak in the frequencies is clearly visible on the intervals [0, 1), representing 6%, 9% and 9% of all cutoff values, respectively. As the cutoff value 0 corresponds to the behavior of voting for blue regardless of the obtained signal, the presence 22 Technically, we can extend the strategy space beyond the interval [0, 10], given our specification of the errors in (1). However, we chose not to do so, as the main intuition of the model is unchanged, and we prefer to avoid any possible confusion caused by a cutoff value defined outside of the signal space.

23 Our trials include

T = {1, • • • , 15}, {1, • • • , 10}, {6, • • • , 15}, {1, • • • , 7}, {9, • • • , 15}, {1, • • • , 5}, {6, • • • , 10},
and {11, • • • , 15}. The t-tests did not reject the null hypothesis for any pairs of intervals at the p = 0.10 level, 24 We also counted the number of votes consistent with the cutoff estimation obtained from T = {1, • • • , 15}. Overall, 90.3% of actions are consistent. Inconsistent actions are spread across periods, and the t-statistics of the comparison between the first and last 7 periods are 1.06, 1.32 and 1.39 for the number of inconsistent blue actions, yellow actions and the sum, respectively, none statistically significant at the p = 0.10 level. 25 The R 2 values for the 6 pairs are respectively, 0.90, 0.89, 0.89, 0.99, 0.99 and 0.99.

of the peaks suggests that a certain amount of participants used the signal-independent voting strategy, or at least one close to it. Second, about half of the estimated cutoff values are included in the interval [4, 5) with asymmetric payoffs. The percentages in this interval for group sizes of 5, 9, and 19 are 54%, 51%, and 66%, respectively. Note that the unbiased strategy is represented by the cutoff value of 5. A cutoff value lower than 5 corresponds to a strategy biased in favor of voting for blue, the ex ante optimal choice. According to our estimation, roughly between one half and two thirds of participants used a cutoff strategy slightly biased towards the ex ante optimal choice. Third, no single player used a cutoff strategy higher than 8 in any phase with asymmetric payoffs. It is worth underlining that no signal-independent voting behavior to the other extreme direction (i.e. a cutoff value of 10, which corresponds to voting regardless of the signal for yellow, the ex ante suboptimal alternative) is observed with asymmetric payoffs. Fourth, a non-negligible amount of voting behaviors in favor of yellow are observed, even though they are rather a minority. The frequencies of cutoff values higher than 5 are 15%, 17% and 9%, respectively, in the three phases with asymmetric payoffs.

The model fit

In what follows, we evaluate three models: level-k (L), cognitive hierarchy (CH), and inclusive cognitive hierarchy (ICH), estimating the parameters which fit best to our experimental data.

Our aim is to find out the sequence of level strategies σ = (σ 1 , • • • , σ K ) and the level distribution f = (f 0 , • • • , f K ), which best fits the observed cutoff values (x i ) i∈N , provided an exogenously fixed level-0 strategy σ 0 .26 Throughout the paper, we maintain the assumption that f follows the truncated Poisson distribution with coefficient τ . Estimation of the level distribution f thus boils down to finding the best-fit τ . Level strategy σ k (k ≥ 1) is the best response of a player holding a belief g k . Remember that the distinction between the models stems uniquely from the way in which g k is defined (see (L), (CH) and (ICH) in Section 2).

We assume that the observed cutoff values (x i ) i∈N are drawn from the distribution K k=0 f k σ k with a logistic error. More precisely, the probability that the realized cutoff strategy is x i is:

φ (x i |f, σ, ρ) = K k=0 f k E y [ (x i |y, ρ)| σ k ]
where expectation over y is drawn from the distribution σ k ,27 and

(x i |y, ρ) = exp x i -y ρ ρ 1 + exp x i -y ρ 2
is the density function of the logistic distribution with mean y and the error coefficient ρ. Our estimation maximizes the likelihood, choosing the Poisson distribution f and the error coefficient ρ:

i∈N φ (x i |f, σ, ρ) .
Given σ 0 , the variables to be estimated are thus τ and ρ.

Level-0 Strategy

Before proceeding to the comparison of the models, we briefly discuss the choice of the level-0 strategy, which can be supported by the idea of saliency. As discussed in [START_REF] Crawford | Fatal attraction: Salience, naivete, and sophistication in experimental Hide-and-Seek games[END_REF], inter alia, some naturally occurring landscapes that are focal across the strategy space may constitute salient non-strategic features of a game and attract naïve assessments. For instance, a strategy space represented by a real interval, say [m, M ], may have its minimal point m, its maximal point M , and its midpoint m+M 2 as salient locations. In our game, two of these deserve close attention. First, if we expect that a non-strategic, level-0 player would evaluate her choices while disregarding others' strategic incentives, such a behavior corresponds to a strategy of the ex ante favored choice, i.e. always voting for blue. A salient location would then be 0. Second, if we expect that a non-strategic player would choose a strategy which maximizes the probability of making a right decision regardless of the winning point (and thus is not payoff-maximizing), then a salient strategy is the midpoint 5. Furthermore, a uniform randomization over all available pure strategies is often chosen as the level-0 strategy in the literature (see discussion in [START_REF] Camerer | A cognitive hierarchy model of games[END_REF].

Table 1 provides a comparison based on the maximum likelihood estimation described in the previous subsection, under the ICH model with the group size n = 5. We used a grid search, with both τ and ρ varied from 0.1 to 10 with an increment of 0.1. Among the four salient strategies of σ 0 we specified above, the highest log-likelihood is attained at σ 0 = 0, followed by the uniform distribution, with the lowest at σ 0 = 5 and 10. Note that the estimated τ * hits the upper bound grid value 10 for σ 0 = 5 and 10. Provided the Poisson assumption imposed on f , a high value of τ * means that the model can best fit the data by assigning the largest possible probability to the highest level-K (and the smallest to level-0). Therefore, one can expect at best the model to provide as a good fit as the symmetric Nash equilibrium. We thus conclude that setting such values of the level-0 strategy is of limited interest for describing the deviation from rational behavior, since these values of σ 0 can provide only a limited capacity of explaining our data beyond the Nash model. 28 Between the uniform σ 0 and σ 0 = 0, the latter fits better, which is coherent with the observations from our experiments with asymmetric payoffs that (i) there is a peak at 0 for all n, and (ii) no single player used a cutoff strategy higher than 8 in any phase. In what follows, we continue our analysis by setting the level-0 strategy to be σ 0 = 0.

τ * ρ * ICH1 (σ 1 ) ICH2 (σ 2 ) LL σ 0 = 0 3.

Results

Level-k Model

An Lk player (level-k player in the L model) maximizes her payoff holding a belief that all other individuals play the L(k -1) strategy. Since each individual has an incentive of correcting biases caused by all other players, the best reply of a Lk player is biased toward the opposite direction with respect to the Nash equilibrium as compared to the L(k -1) strategy, and the degree of amplification increases as n increases. 29 We see in Table 2 that the cutoff strategy of the L1 player hits σ 1 = 10 as a response to the L0 strategy σ 0 = 0 for all values of n = 5, 9, and 19. A similar argument applies to L2 in the opposite direction, implying σ 2 = 0. Such an oscillation continues in the L model, and a bang-bang solution is obtained perpetually as k increases. For all n, we observe that the most likely value τ * hits the lower bound 0.1. This is not surprising, given the bang-bang strategies σ 1 = 10 and σ 0 = σ 2 = 0. As the truncated Poisson density of level 1, g 2 (1) = τ 1+τ +τ 2 /2 , is minimized at τ = 0, the model fits best to the data observed in Figure 3, at the lowest value of τ . We underline that Battaglini et al. (2010) are the first to observe a limitation of the level-k model in binary-state, binary-action committee decision making in a laboratory experiment.

CH Model

The CH model stipulates that a CHk player (a level-k player in the CH model) maximizes her expected payoff holding a belief that other n -1 players have levels up to k -1. In particular, a CH1 player holds the belief that all other players have level 0, which is exactly the same as the belief of an L1 player. In our game, the CH1 strategy is σ 1 = 10 for all n = 5, 9 and 19.

In Table 2, we observe that the values of τ * are high for all n. This means that the best-fitting Poisson distribution assigns the highest density to level 2. Since the CH model attributes the corner values σ 0 = 0 and σ 1 = 10 for levels 0 and 1, the model fits to the data by assigning minimal densities to these corner strategies. We do not observe either a particular trend in τ * , or a convergence of the CH2 strategies to Nash equilibrium, as n increases. 30 A key observation is an increasing sensitivity of the CH2 strategy for large n. Not only is the best-reply function in our game decreasing, but the slope of the best-reply function becomes steeper as n increases. Thus, the sensitivity of the best reply to the belief over the other players' strategies also increases as n increases. As we show later in Theorem 1, such an increasing sensitivity leads to a divergence of the CH strategy in asymptotically expanding games, as n increases. We provide further discussions in Section 5.

ICH Model

We observe that the estimated strategies of ICH1 and ICH2 are both increasing in n. 31Comparing these values with the Nash equilibrium, the increase of both ICH1 and ICH2 is in line with the increase of the Nash equilibrium strategy with respect to n (Figure 2(b)). The intuition is that the Nash strategy monotonically converges to the unbiased strategy (i.e. 5), since all individuals equally share the prior bias caused by the asymmetric payoffs, and such an individual share converges to zero as n increases. As we discuss later in Section 5, ICH1 would converge to a value opposite to the prior bias with respect to the Nash equilibrium, and ICH2 would approach to the Nash equilibrium (Theorem 1). Our ICH estimations from the data are consistent with these theoretical predictions.

The estimated values of τ * are 3.9, 5.0 and 6.7, for n = 5, 9 and 19, respectively. For these values of τ , it is interesting to compute the probability assigned to their own level by an inclusive player. For a level-2 player, it is g 2 (2) = τ 2 /2 1+τ +τ 2 /2 , which is equal to 61%, 68% and 74%, respectively. For a level-1 player, it is g 1 (1) = τ 1+τ , which is equal to 80%, 83% and 87%, respectively. These values are consistent with the responses we observed in Figure 1(b).

Another observation is that the best-fitting τ values are increasing as the group size increases. Given that τ is the expectation of the level drawn from the Poisson distribution, a larger τ corresponds, ceteris paribus, to a higher value in the expected cognitive levels. Therefore, an increase in the estimated values of τ may be interpreted as evidence that the average cognitive level increases as the group size grows larger. This observation is not consistent with the findings of [START_REF] Guarnaschelli | An experimental study of jury decision rules[END_REF], in which evidence of lower expected levels with larger groups is reported, reflecting a larger cognitive load in large groups.

Comparison

Our maximum likelihood estimations show that the ICH model best fits the data for all n = 5, 9 and 19. The result is robust for the estimations done by each session separately: the log-likelihood is improved under the ICH model and the differences are significant at the p = 0.01 level, with the Wilcoxon test both against L and CH. From the observations above, we conclude that our laboratory experiments provide clear evidence that the inclusive cognitive hierarchy model (ICH) fits better to the data as compared with the standard level-k model (L) and the cognitive hierarchy model (CH).

Our data also provides us with statistics which allow a comparison of the fits between the above models and Nash equilibrium. The log-likelihood of the best fitting symmetric Nash equilibrium to our data is -222.9, -246.2 and -244.2 with ρ * = 1.6, 1.3 and 1.3, for n = 5, 9 and 19, respectively. The Bayesian information criterion thus suggests that the ICH model fits better than Nash equilibrium, followed by the L and CH models. Our observation that Nash equilibrium fits better than the L and CH models is consistent with the findings of Battaglini et al. (2010) in an experiment on the swing voter's curse.

Asymptotic Properties

The main findings from our experimental results in the previous section are that (i) the maximum likelihood estimation exhibits a better fit of the ICH model to the data than L and CH models, and (ii) clear deviation from Nash behavior is observed, in an information aggregation problem with asymmetric payoffs.

Our objective in this section is to provide theoretical explanations according to the asymptotic property of the games. We show below that the distance of the L and CH strategies from the Nash equilibrium diverges, while that of the ICH strategy is bounded (Theorem 1) for the games in which the best-reply functions are asymptotically expanding. On the other hand, the strategies in all of those models are bounded (Theorem 2) for the games in which the best-reply functions are not asymptotically expanding. These analytical results suggest that whether or not the inclusiveness condition matters in describing the asymptotic behavior depends on the asymptotic property of the best-reply functions.

For the sake of tractability, we focus on linear quadratic games [START_REF] Currarini | Information sharing networks in linear quadratic games[END_REF]) which have desirable features for our analysis. First, they are fully aggregative games [START_REF] Cornes | Fully aggregative games[END_REF]), in which the payoff of each player is affected by the action profile of the players through the aggregate of the strategies of all players and her own strategy. This fits well to our current objective, as our goal here is to understand analytically how the optimal strategy of a player would be affected by the belief over the type of the other players. The fact that the strategy of the other players appears explicitly in an aggregative form allows us to obtain straightforward insights on the relationship between the shape of the best-reply functions and the players' beliefs over the strategies of the other players. Second, in a more technical convenience, linear quadratic games have a property such that, when a player holds a stochastic belief over the strategies of the other players, the maximizer of her expected payoff coincides with the best reply against the pure strategy which takes the expected value of the aggregate. This is because the linearity of the derivative allows us to switch the order of the partial derivative and the expectation. Then, facing heterogeneous beliefs over the other players' strategies, our analysis can simply focus on the best reply against the expectation of the beliefs, which provides us with a high tractability of the models. 32Consider n individuals, each of whom takes an action x i ∈ R. The payoff of player i in a linear quadratic game is a function of her own action x i and the aggregate of the other players' actions X -i = j =i x j in the following form:

u i (x i , X -i ) = λ t x + x t Γx (2) where x = (x i X -i ) t , λ = (λ x λ X ) t and Γ = γ 11 γ 12 γ 12 γ 22 .
There are several games of interest which fall into the class of linear quadratic games.

Example 1 (A simple quadratic game) Suppose u i (x i , x -i ) =j x j

2

. Then, λ t = (0 0) and

Γ = -1 -1 -1 -1 .
Example 2 (Cournot competition) Consider a Cournot competition. Suppose that the inverse demand function is linear P (Q) = a -bQ, and each firm has a constant marginal cost c i . Let q i be the quantity produced by firm i and Q -i := j =i q j . The profit of firm i is:

Π i = q i (a -b (q i + Q -i ) -c i ) .
Then, λ t = (a -c i 0) and

Γ = -b -b 2 -b 2 0 .
Example 3 (Keynesian beauty contest games) Suppose that each of n players chooses a number x i simultaneously, and each player's payoff is quadratic with respect to the distance between her own choice and the average of all players' choices multiplied by a constant p ∈ (0, 1). Then,

u i (x i , X -i ) = -x i -p x i + X -i n 2 .
Then, λ t = (0 0) and

Γ = -1 -p n 2 1 -p n p n 1 -p n p n -p n 2 .
Example 4 (Public good provision game) Suppose that each agent contributes x i to a public good and the cost is quadratic:

u i (x i , X -i ) = θ i (x i + X -i ) -c i x 2 i .
Then, λ t = (θ i θ i ) and Γ = -c i 0 0 0 .

We impose some regularity conditions on the linear quadratic game in the form (2). First, we assume γ 11 < 0. This implies that u i has a unique maximizer for any X -i and thus the best-reply function is well-defined. It is straightforward to show that the game defined by (2) has a unique symmetric Nash equilibrium:

x * := -λ x 2 (γ 11 + (n -1) γ 12 ) .

We assume that the denominator is non-zero so that the symmetric Nash equilibrium is well-defined. By applying a parallel transformation y i := x i -x * , (2) becomes:

u i = λ t x + x t Γx = λ t y y + y t Γy + c
where Y -i = j =i y j , y = (y i Y -i ) t , λ y = (0 λ Y ) t , and λ Y and c are independent of y.

As the terms λ t y y = λ Y Y -i and c have no strategic consequence on player i's behavior (i.e. the best-reply function of player i is unaffected), we can assume λ Y = 0 and c = 0 without loss of generality. Therefore, in the following, we focus our attention on the games with the payoff function:

u i = y t Γy, (3)
with γ 11 < 0 and γ 11 + (n -1) γ 12 = 0 (as in [START_REF] Angeletos | Efficient use of information and social value of information[END_REF]). Notice that there is a unique symmetric Nash equilibrium y * i = 0 for all i. The first-order condition of player i is:

∂u i ∂y i = 2γ 11 y i + 2γ 12 Y -i .
When player i holds a stochastic belief over the strategies of the other players, the aggregate of the other players' strategies is a random variable Ỹ-i . Since the first-order condition is linear in Y -i in quadratic games, the best reply against a mixed-strategy profile coincides with the best reply against the aggregate strategy which takes deterministically the expected value of the random variable:

BR i Ỹ-i = - γ 12 γ 11 E Ỹ-i . (4)
In order to describe asymptotic properties, we consider a sequence of linear quadratic games in which the number of players increases. More precisely, let

G (n) = n, R, (u n i ) n i=1
be a normal-form game with n players where the set of pure strategies is fixed as the set of real numbers R,33 and u n i is the payoff function of player i which satisfies (3). We analyze asymptotic properties of the strategies under the sequence of games {G (n)} ∞ n=2 . Remember that the three models under our scrutiny here, L, CH and ICH, differ only in the assumption imposed on players' beliefs on the types of the other players. For each model, the strategy in each level is defined in the same way as in Definition 1. The only difference is that the frequency g k (h), assigned in the belief of a level-k player to the event in which each of the other players should be level-h, is specified by the equation (ICH) in Section 2 in the ICH model, but it is replaced by (L) (resp. (CH)) in the L (resp. CH) model. It is worth emphasizing that our results in this section do not hinge on the Poisson assumption concerning the underlying distribution f k . We consider a sequence of levelsymmetric strategies σ = (σ k ) k≥0 where for each k ≥ 1, σ k maximizes the expected payoff under the belief g k (h) (Definition 1).

For each game G (n) and in each of the three models, the level-0 strategy σ 0 is exogenously given, allowing the possibility of a mixed strategy. In order to make the comparison explicit across the models for k ≥ 1, we add a superscript which represents the model, such as σ L k , σ CH k , and σ ICH k . Note that, by (4), σ M k (n) are all pure strategies for k ≥ 1 for each model M ∈ {L, CH, ICH}.

We assume that the following limit exists, allowing infinity:

A := lim n→∞ γ 12 γ 11 n ∈ R ≥0 ∪ {∞} .
Remember that -γ 12 γ 11 is the slope of the best-reply function (4). Since Ỹ-i is the sum of the strategies of the other players, A is the limit of the slope of player i's best-reply function, as a function of the average of the other players.

First, we consider the case A = ∞. In such games, we say that the sequence of the games is asymptotically expanding, denoting the property that the sensitivity of one's strategy to the average strategy of the other players increases without a bound. We show that the strategies diverge from the Nash equilibrium in the L and the CH models, while it is bounded in the ICH model.

Theorem 1 Consider a sequence of games {G (n)} ∞ n=2 in which the payoff functions satisfy (3) for each n. Consider any σ 0 and let µ :

= E [σ 0 ]. Suppose A = ∞. For any µ = 0, lim n→∞ σ L k (n) = ∞ and lim n→∞ σ CH k (n) = ∞, while lim n→∞ σ ICH 1 (n) < ∞ and lim n→∞ σ ICH k (n) = 0 for all k ≥ 2.
Among the examples described above, the sequence {G (n)} ∞ n=2 is asymptotically expanding (A = ∞) in the simple quadratic game (Example 1) and in the linear Cournot competition (Example 2). A common feature of these games is that the aggregate of all players' strategies enters into each player's payoff in a way that the aggregate term does not dissipate for large n. When A = ∞, we show that the behaviors in the ICH model show a stark contrast with those in the L or in the CH model. The presence of the inclusiveness condition thus leads to an intrinsic difference in the prediction. Moreover, we show that the ICH strategy converges to the Nash equilibrium for any level k ≥ 2.

We interpret these properties of the ICH model as accurately capturing the phenomenon which is frequently observed in the behavioral data. In addition to a naïve strategy (level-0), we often observe sophisticated behaviors, with possibly heterogeneous degrees of sophistication. What is implied by Theorem 1 for asymptotically expanding games is that there are fundamentally three degrees of strategic sophistication: naïve (level-0), partially sophisticated (level-1), and highly sophisticated (level-2 or more). Since the strategies of level-2 or higher all converge to the Nash equilibrium, behaviors in this class of games fall into one of the following three classes asymptotically: (i) naïve strategy which does not maximize the expected payoff, (ii) level-1 strategy which maximizes the payoff but under an inconsistent belief, and (iii) fully sophisticated strategy, which maximizes the payoff under the consistent belief. Now, consider a sequence of games which satisfies the same conditions assumed in Theorem 1, except for that in A.

Theorem 2 Suppose A < ∞. For any µ, σ L k (n) , σ CH k (n) and σ ICH k (n) are all bounded as n → ∞, for all k ≥ 1.
In the standard Keynesian beauty contest games (Example 3), we have:

A = lim n→∞ 1 -p n p n n -1 -p n 2 = p < ∞.
In the games with A < ∞, the slope of the best-reply function is bounded as n goes to infinity. Hence, even in a game with a large number of players, the optimal strategy of a player does not diverge. In the beauty contest games, we see that the aggregate term is relevant in each player's payoff to the degree of the average of all players.

Most of the remarkable results in the literature of strategic thinking have considered the games with A < ∞, such as the beauty contest games. Other examples include finite games, such as market entry games, coordination games, or centipede games, to name a few. Even for the finite games, as long as the game is dominance solvable, we can consider that the game falls into the class of A < ∞, in the sense that the infinite iteration of applying the best-reply function leads to a convergence to the Nash equilibrium.

It is rather straightforward to see why dominance-solvable games have been the most frequently chosen object of research in the literature. When the iterative application of the best-reply function leads to the unique Nash equilibrium, it corresponds to the high-level strategies converging to the Nash equilibrium. Such a property is often considered to fit well to the idea that the deviation from the Nash equilibrium caused by the limited cognitive ability of the low-level players dissipates as the cognitive limit goes to infinity.

Implied by Theorem 2 is that the inclusiveness condition has little relevance asymptotically in the games with A < ∞. It does not follow, however, that the inclusiveness condition makes no difference in the prediction power of the models. 34Table 3 shows the comparison of the L, CH, and ICH models fitting to the data in beauty contest games. The data sets are exactly the ones studied in [START_REF] Bosch-Domènech | One, Two, (Three), Infinity, ...: Newspaper and Lab Beauty-Contest Experiments[END_REF]. 35 Following [START_REF] Camerer | A cognitive hierarchy model of games[END_REF], the Poisson coefficient τ is estimated by a method of moment, matching the mean. By comparing the second moment, we observe that the ICH model fits slightly better to the data than L and CH.

Conclusion

We have introduced a cognitive hierarchy model which allows inclusiveness in the belief over the cognitive levels of other players. We find that asymptotic properties of the group decisionmaking, especially asymptotic efficiency, exhibit a stark contrast depending on whether the inclusiveness condition is admitted in the model or not. Results from our laboratory experiment provide evidence for (i) systematic deviations from Nash equilibrium behavior, and (ii) a better fit to the data under the model with inclusiveness.

Our theoretical analysis implies that the asymptotic property of the slope of the bestreply function is the key ingredient to determining whether the asymptotic properties differ between the models with and without inclusiveness. Even with an increasing sensitivity of the best-reply function to the beliefs, inclusiveness prevents divergence of the strategies from fully rational behavior. Since the same property is shared by the best-reply function of the collective decision-making studied in this paper, the presence of the inclusiveness prevents strategies from diverging away from the symmetric Nash equilibrium, and hence provides asymptotic efficiency as the group size increases.

Most of the games studied with cognitive hierarchy models share the property that the best-reply function is asymptotically non-expansive (i.e. A < ∞). In such games, our analysis implies that the presence of inclusiveness has little relevance, at least asymptotically. Therefore, inclusiveness does not imperil the insights obtained in the existing cognitive hierarchy models, even though its explanatory power may vary as a function of model settings and parameters. A major example is the classical Keynesian beauty contest game. The data provides us with evidence that the model with inclusiveness fits the data toward the same direction as those without inclusiveness. Similar insights are inherited in the games with best-reply functions that are 'contractive' in a broader sense, such as dominance-solvable games, coordination games, and market entry games, among others. We hence do not insist on an intrinsic improvement of the predictive power of the cognitive hierarchy model with the presence of the inclusiveness condition in this class of games. The main message of this paper is that there are games in the other class in which the presence of inclusiveness matters. We think there are interesting games in this class, e.g. Cournot competitions, that are worth pursuing in further analysis.

Our interests go beyond the analytical results obtained in this paper. A crucial difference induced by the presence of inclusiveness is the existence of 'sophisticated' players. A highest-level player in the ICH model best replies holding the correct belief concerning the distribution of the levels of other players. This is not the case in the cognitive hierarchy models without inclusiveness. Players are supposed to be Savage rational, but full consistency of their beliefs is not postulated even for the highest level. In that sense, simply the existence of fully-sophisticated players may suffice to convey our message. However, our model consists of players who are naïve (level-0), best-replying but with inconsistent beliefs (level-1), and sophisticated (level-2). [START_REF] Rubinstein | A typology of players: Between instinctive and contemplative[END_REF] proposes a typology of players with two types, e.g. "instinctive" and "contemplative". The three types emerging from our approach based on cognitive hierarchy theory can be seen as a richer typology in the same vein that allows for an intermediary type. The beauty of the cognitive hierarchy models lies, we believe, in the heterogeneous degrees of belief inconsistency that can be explicitly accommodated. We would like to further understand the role of heterogeneous degrees of inconsistent beliefs under the existence of fully-sophisticated players. We leave this for future research.

For the sake of induction, assume lim n→∞ σ ICH h (n) =:

s h < ∞ for 1 ≤ h ≤ k -1. Then, for k ≥ 2, lim n→∞ σ ICH k (n) ≤ A k-1 h=0 g ICH k (h) s h 1 -Ag ICH k (k) < ∞.

Figure 1 :

 1 Figure 1: Responses in the post-experimental questionnaire.

  Level-1 and level-2 strategies. Note that CH1 = L1.

Figure 2 :

 2 Figure 2: A numerical example.

Figure 3 :

 3 Figure 3: Histogram of the estimated cutoff strategies, 140 observations.

Table 1 :

 1 Comparison of level-0 specifications. Maximum log-likelihood under the ICH model with n = 5.

		9 2.5	4.45	4.28	-190.7
	σ 0 = 5	10 1.6	4.17	4.23	-222.9
	σ 0 = 10	10 1.6	3.84	4.21	-225.6
	σ 0 =unif[0, 10] 4.3 2.3	4.00	4.20	-213.3

Table 2 :

 2 Table2summarizes the comparison of the models by maximum likelihood estimation. Comparison of the models by maximum log-likelihood. Pooled across sessions, N = 140. σ 0 = 0. Grid search for both the Poisson coefficient τ and the error coefficient ρ, from 0.1 to 10 with increment 0.1.

			τ *	ρ *	σ 1	σ 2	LL
		L	0.1 0.4 10	0	-415.3
	n = 5	CH 8.0 0.6 10 1.76 -366.4
		ICH 3.9 2.5 4.45 4.28 -190.7
		L	0.1 0.4 10	0	-416.3
	n = 9	CH 10.0 0.7 10 2.52 -334.3
		ICH 5.0 2.3 4.91 4.67 -210.6
		L	0.1 0.4 10	0	-414.8
	n = 19 CH 9.1 0.7 10 2.40 -338.0
		ICH 6.7 2.4 5.11 4.84 -210.7

Table 3 :

 3 MoM estimation of beauty contest games. The first moment (mean) is matched.

			Data	L			CH		ICH
	Name	N	mean s.d.	τ	s.d.	τ	s.d.	τ	s.d.
	Lab	86	35.1 19.6 1.06 12.4 1.31 9.6 0.90 12.9
	Classroom	138	26.8 17.7 1.90 12.7 2.53 8.6 1.51 14.1
	Take-home	119	25.2 17.0 2.11 12.5 2.85 8.3 1.65 14.2
	Theorists	54	17.8 24.3 3.46 10.4 5.15 6.0 2.42 14.3
	Conference	92	16.8 20.1 3.72 9.9 5.64 5.6 2.54 14.3
	E-mail	150	22.2 20.7 2.56 11.9 3.58 7.5 1.93 14.3
	Newspapers 7893 23.1 20.2 2.41 12.1 3.34 7.7 1.85 14.3

[START_REF] Crawford | Structural models of nonequilibrium strategic thinking: Theory, evidence, and applications[END_REF] provide a fine review of these models and applications.

Self-awareness is probably not the best term here, since what is meant is the capacity for projecting the self to others, which has little to do with the capacity for introspection often meant by self-awareness in, for example, psychology.

In their experiment, 50 to 80% of the participants behaved non-strategically when voting was simultaneous, and they find that non-optimal behavior is typically due to difficulty in extracting information from hypothetical events.

 4 As[START_REF] Camerer | Behavioral game theory: Experiments in strategic interaction[END_REF] stresses (Chapter 7), the effect of group size on behavior in strategic interactions is a persistent phenomenon, especially towards coordination.

[START_REF] Camerer | Overconfidence and Excess Entry: An Experimental Approach[END_REF] report on experimental evidence for overconfident behavior in the case of the market entry game. When ability is a payoff-relevant variable in a strategic interaction, evidence shows that players tend to be overconfident. See[START_REF] Benoît | Apparent overconfidence[END_REF]. On the other hand,[START_REF] Azmat | Can't hurt you? A field experiment on relative performance feedback in higher education[END_REF] find an underestimation of students' grades in the absence of feedback.

We assume f i > 0 for all i ≤ K. For the truncated distribution to be well-defined, it is sufficient to assume f 0 > 0, but we restrict ourselves to the cases where all levels are present with a positive probability.

The assumption of symmetric prior is without loss of generality, since we allow the payoffs of the two types of right decisions to be heterogeneous. Although the preferences are often represented by a loss function for wrong decisions in the standard CJT models, we equivalently use a gain function for right decisions in accordance with our experiment, which awards positive points to right decisions, rather than subtracting points for wrong decisions.

Even though a number of CJT models assume a binary signal space, we believe that this is not the right assumption for information aggregation problems. Even under the binary state space, there are uncountably many ways to update the prior belief, and thus the set of possible beliefs spans a continuous space. Assuming the signal space to be binary implies that there are only two ways of Bayesian update, which is far from innocuous.

Both the error term and the Poisson coefficient that are estimated for the experimental data in Section 4 are calibrated here as the average of the estimated values.

Both the z-Tree program[START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF]) and the website for participant registration were developed by Sri Srikandan, to whom we are very much grateful.

The online appendix can be found at http://sites.google.com/site/ozkesali.

In the phase with n = 9, two groups of 9 randomly-chosen members were formed at each period. Having 20 participants in total in each session, 2 randomly-chosen participants were 'on hold' during the period. The same method is applied in the phase with n = 19. A group of 19 was formed and thus one randomly-chosen participant was on hold.

We also conducted pilot sessions with the rewards of 800 ECUs and 300 ECUs. As there was less of a marked contrast in the observed deviations from Nash behavior, we decided to run the rest of the sessions with the rewards 900 ECUs and 200 ECUs.

Participants were told both verbally and through info sheets that in the case where the lottery picked a period for remuneration in which a participant had been on hold, the payoff in that phase for this participant was set at 500 ECUs, which is about the average of the winning points.

Behaviors close to the informative strategy are indeed observed in our experiments with unbiased payoffs. See histogram in Figure 3(a).

In the post-experiment questionnaire, a few participants expressed reasonings which seemed to have no clear connection with any Bayesian update, such as "I chose yellow when I saw three or more yellow cards aligned in a row, since I thought it was a strong sign that the box is yellow." We assume that such deviations from rationality are accounted for by the error term.

Remember that K is the highest level under consideration. Our estimation sets K = 2 due to a heavy computational burden for K ≥ 3 and large n.

σ k could be a mixed strategy, as is often assumed for k = 0.

Similarly, we observed that τ * hits the boundary value at σ 0 = 5 and σ 0 = 10 for n = 9 and n = 19, as well as under the L and the CH models, which implies that the prediction power of the models is quite limited under the assumption of σ 0 = 5 or σ 0 = 10.

Note that the best reply is always well-defined, since the probability of being pivotal is always non-zero, given the probabilistic voting action of each player specified by the logistic error function (1).

As a robustness check, the session-wise estimation shows that the difference of σ 2 is not statistically significant between n = 5 and n = 9. We observe that an increase from n = 9 to n = 19 is significant at the p = 0.10 level by the t-test, although such an increase is not observed in the histogram of all estimated strategies in Figure3.

This observation is robust with the session-wise estimations. The differences are statistically significant at the p = 0.01 level under the Wilcoxon test. The t-statistics for ICH1 are 13.9 and 6.94, and for ICH2 are 9.65 and 9.74, from n = 5 to n = 9 and from n = 9 to n = 19, respectively, implying that all differences are statistically significant at the p = 0.01 level.

Obtained insights could be extended to a game with more general payoff functions, to the extent that the second-degree Taylor expansion of the payoff function with respect to the aggregate strategy provides an approximation.

The assumption of the one-dimensional, unbounded strategy space allows us to obtain clear insights on the convergence and/or divergence of the strategies. In the games with a compact, one-dimensional strategy space, these insights could be inherited with some adjustments, e.g. divergence corresponds to a bang-bang corner solution.

Papers in the literature provide comparative studies over different models, e.g.[START_REF] Breitmoser | Strategic reasoning in p-beauty contests[END_REF].

We express our deepest gratitude to Rosemarie Nagel for providing us with the data.

In the proof, we write the (possibly mixed) strategy of a level-0 player as σ 0 = µ, identifying it with its expected value, since expectation is the only relevant term which determines the best reply in the linear quadratic games.

Remember that g k is the truncated distribution induced by f , the underlying distribution over levels defined in Definition 1.

/Labex Ecodec/ANR-11-LABX-0047 and IDEX ANR

A Mathematical Appendix

A.1 Proof of Theorem 1

Proof : Let α n := -γ 12 γ 11 (n -1) .

By (4), α n is the slope of the best-reply function with respect to the average of the other players' strategies. Using α n , we can explicitly write the level-k strategy under each of the three models, L, CH, and ICH. Note that these models differ only in the belief held by each player, specified in equations (L), (CH) and (ICH) in Section 2. 36 By definition, lim n→∞ |α n | = A.

In the L model, the strategy of the level-(k + 1) player is defined as the best reply to the level-k player. By ( 4) and (L),

In the CH model, by ( 4) and (CH),

For the sake of induction, assume that σ CH h (n) is a polynomial of degree h with respect to α n for h ≤ k -1. Then, by (5), σ CH k (n) is a polynomial of degree k with respect to α n . Therefore, we have:

where ϕ k (•) is a polynomial of degree k. Therefore, for any µ = 0 and any k ≥ 1, we have

In the ICH model, by ( 4) and (ICH),

.

.

As lim n→∞ |α n | = ∞, we have:

The first two terms in the bracket cancel out, since

A.2 Proof of Theorem 2

Proof : Suppose A < ∞. Then by ( 6), for k ≥ 1,

.

Especially, for k = 1,

< ∞.