L Bovo
email: l.bovo@ucl.ac.uk

L D C Jaubert

C W Holdsworth

S T Bramwell

P C W Holdsworth

Crystal Shape-Dependent Magnetic Susceptibility and Curie Law Crossover in the Spin Ices Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7

Spin ice materials [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF][START_REF] Bramwell | Frustration in Ising-Type Spin Models on the Pyrochlore Lattice[END_REF][START_REF] Ramirez | Zero-Point Entropy in Spin Ice[END_REF][START_REF] Bramwell | Spin Ice State in Frustrated Magnetic Pyrochlore Materials[END_REF] are generally discussed in terms of two approximate models: the 'near neighbour' model introduced in Ref. [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF][START_REF] Bramwell | Frustration in Ising-Type Spin Models on the Pyrochlore Lattice[END_REF] and the 'dipolar model' introduced in Ref. [START_REF] Siddharthan | Ising Pyrochlore Magnets: Low-Temperature Properties, "Ice Rules," and Beyond[END_REF][START_REF] Hertog | Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets[END_REF] and later refined in Ref. [START_REF] Yavorskii | Dy 2 Ti 2 O 7 Spin Ice: A Test Case for Emergent Clusters in a Frustrated Magnet[END_REF]. The main difference between the two models is treatment of the magnetic dipole-dipole interaction that is truncated to near neighbour in the former model but retained in its full (classical) form in the latter [START_REF] Bramwell | Spin Ice State in Frustrated Magnetic Pyrochlore Materials[END_REF]. The essential physics of spin ice is contained already in the near neighbour model, with the notable exception of magnetic Coulomb interaction [START_REF] Castelnovo | Magnetic Monopoles in Spin Ice[END_REF] between emergent 'monopole' defects [START_REF] Castelnovo | Magnetic Monopoles in Spin Ice[END_REF][START_REF] Ryzhkin | Magnetic Relaxation in Rare-Earth Pyrochlores[END_REF][START_REF] Jaubert | Signature of Magnetic Monopole and Dirac String Dynamics in Spin Ice[END_REF]. However it should be noted that quantum effects are neglected in both approaches and these may yet prove to be important at low temperature (T < 1 K) [START_REF] Shannon | Quantum Ice: A Quantum Monte Carlo Study[END_REF][START_REF] Benton | Seeing the Light : Experimental Signatures of Emergent Electromagnetism in a Quantum Spin Ice[END_REF].

The experimental magnetic susceptibility of spin ice materials is a property of considerable interest [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF][START_REF] Bramwell | Bulk Magnetization of the Heavy Rare Earth Titanate Pyrochlores -a Series of Model Frustrated Magnets[END_REF][START_REF] Snyder | Low-Temperature Spin Freezing in the Dy 2 Ti 2 O 7 Spin Ice[END_REF][START_REF] Petrenko | Titanium Pyrochlore Magnets: How Much Can be Learned From Magnetization Measurements?[END_REF][START_REF] Matsuhira | Spin Dynamics at Very Low Temperature in Spin Ice Dy 2 Ti 2 O 7[END_REF][START_REF] Quilliam | Dynamics of the Magnetic Susceptibility Deep in the Coulomb Phase of the Dipolar Spin Ice Material Ho 2 Ti 2 O 7[END_REF]. It was first reported in Ref. [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF][START_REF] Bramwell | Bulk Magnetization of the Heavy Rare Earth Titanate Pyrochlores -a Series of Model Frustrated Magnets[END_REF] and analysed in terms of a Curie-Weiss law. Later theoretical developments [START_REF] Ryzhkin | Magnetic Relaxation in Rare-Earth Pyrochlores[END_REF][START_REF] Isakov | Magnetization Curve of Spin Ice in a [111] Magnetic Field[END_REF][START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF] showed that the near neighbour model has a 'Curie Law crossover' in the isothermal susceptibility:

χ T = C(T ) T . (1)
Here the effective Curie 'constant' C crosses over from the paramagnetic value C = C (≈ 4 K for Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 ) as T → ∞, to exactly twice that number C = 2C in the low temperature limit. The detailed theoretical study of Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF] calculated the crossover analytically for a Husimi tree approximation to near neighbour spin ice. It showed that this crossover is an extraordinarily slow function of increasing temperature that is not complete until temperature of order 100 × the characteristic energy scale J of the interactions in the near neighbour model. This surprising result may be ascribed to the long range propagation of local constraints on spin configurations, leading to effectively amplified interactions that are captured in the infinite range summation of the Husumi tree approximation. The predicted slow Curie Law crossover was found to agree very well with the experimental susceptibility of Ho 2 Ti 2 O 7 measured at temperatures T > 10 K by bulk magnetometry and by neutron scattering at both at the Brillouin zone centre 002 and the Brillouin zone boundary 001 [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF]. At lower temperatures the susceptibility measured by magnetometry still agreed with theory, as did that measured at the zone boundary, while that measured by neutron scattering at the zone centre showed systematic deviations. The present study was aimed in part at investigating this puzzling disagreement between bulk susceptibilities estimated by two different experimental methods. The resolution of this disagreement, presented here, will facilitate the task of understanding the difference between experiment and theory, possible reasons for which are discussed in Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF].

The basic principle of magnetic thermodynamics is well known, but worth repeating here. We assume a sample that may be described in the cubic crystal system, as appropriate to spin ice. In this case the susceptibility tensor χ αβ is diagonal with equal components (its representation quadric is a sphere). Hence, the isothermal susceptibility tensor may be replaced by a single scalar quantity, χ T . If a magnetic field is applied to a sample of general shape by a solenoid then the incremental work done to change the magnetization of the sample is:

dW = µ 0 [H 0 (r) + H 1 (r)] • dM(r)dr, (2)
where r is a position within the sample, H 0 is the applied (solenoidal) field and H 1 is the (irrotational) field arising from magnetic moment density within the sample. Homogeneity and parallel alignment of the magnetization and field are guaranteed only for ellipsoidal samples, in which case the above equation integrates to:

dW = µ 0 V H int dM, (3)
where H int = H 0 -DM is the internal H-field, D is the demagnetizing factor, and M and H are the magnitudes of the vector quantities. For a spherical sample D = 1/3. Assuming a linear medium in sufficiently small field the isothermal susceptibility χ T may be defined as :

χ T = M H int . (4)
The earliest experimental measurements of the bulk susceptibility of single crystals spin ice [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF][START_REF] Bramwell | Bulk Magnetization of the Heavy Rare Earth Titanate Pyrochlores -a Series of Model Frustrated Magnets[END_REF] took careful account of the very strong demagnetizing fields [START_REF] Bramwell | Spin ice Book Editor(s): Diep, HT Source: Frustrated Spin Systems[END_REF]. However the crystals used were not only flux grown and certainly more defective than more recently produced image furnace grown crystals, but also of an irregular block-like shape, rather than the ideal ellipsoid. These factors may have introduced systematic errors into the experimental estimation of the isothermal susceptibility of pristine spin ice.

The work described in this paper used spherical crystals made by commercial handcutting of larger blocks which were provided by D. Prabhakaran [START_REF] Prabhakaran | Crystal Growth of Spin-Ice Pyrochlores by the Floating-Zone Method[END_REF]. Fig. 1 shows photographs of several such crystals, including at least one that is perfectly spherical to human perception and others less so. We measured the susceptibility of two nominally spherical crystals of each of Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 in order to estimate the effect of departures from an ideal spherical shape. Magnetic susceptibility was measured using a Quantum Design SQUID magnetometer. Crystals were positioned in a cylindrical plastic tube to ensure a uniform magnetic environment. Three measurements were made on each sample: low field susceptibility (at µ 0 H 0 = 0.005, 0.01 and 0.02 T) and fieldcooled (FC) versus zero-field-cooled (ZFC) susceptibility. Also magnetic field sweeps at fixed temperature were performed in order to evaluate the susceptibility accurately and confirm the linear approximation. The FC versus ZFC susceptibility measurements involved cooling the sample to 1.8 K in zero field, applying the weak magnetic field, measuring the susceptibility whilst warming up to 300 K, cooling to 1.8 K again and finally re-measuring the susceptibility while warming. Before switching the magnetic field off, field scans with small steps were performed in order to estimate the absolute susceptibilities. Furthermore, data taken at different fields were compared, such that the high temperature data (above 150 K) should correspond for each crystal. As previously reported [START_REF] Bramwell | Bulk Magnetization of the Heavy Rare Earth Titanate Pyrochlores -a Series of Model Frustrated Magnets[END_REF], no difference between field-cooled and zero-field cooled magnetization was observed down to 1.8 K for both materials.

The measured susceptibility at high temperature was found to evidence the expected temperature-independent term χ ti that accounts for combined effects of diamagnetism and Van-Vleck susceptibility. The measured susceptibility displayed in this paper represents the Langevin contribution estimated by correcting the measured data for this term. The correction factors used were χ ti (Dy

2 Ti 2 O 7 ) = 1.5 × 10 -3 , χ ti (Ho 2 Ti 2 O 7 ) = 1.2 × 10 -3 .
Fig. 2 shows our main result, where the experimentally determined susceptibilities in the form χ T T /C are compared with theory for coupling parameters appropriate to Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 . Our theory is based on the Husimi tree method [START_REF] Husimi | Note on Mayer's theory of cluster integrals[END_REF] where calculations are done on a lattice made of corner-sharing tetrahedra, similar to a pyrochlore lattice in absence of any closed loops of spins beyond a tetrahedron [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF]. It is reminiscent of a Bethe lattice where each vertex would be a tetrahedron. The strength of this mean field method is to allow exact recursive results while correctly incorporating the correlations within each frustrated unit brick. It has been successfully compared to Monte Carlo simulations on the true 3-dimensional pyrochlore lattice [START_REF] Jaubert | Three-Dimensional Kasteleyn Transition : Spin Ice in a [100] Field[END_REF][START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF]. From linear expansion in a field, one finds

χ T = 2C T • 1 + e 2J eff /T 2 + e 2J eff /T + e -6J eff /T , (5)
where J eff is the effective nearest neighbour coupling constant defined in [START_REF] Hertog | Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets[END_REF] and

C = N µ 0 (g J Jµ B ) 2 3k ( 6
)
Here µ 0 , µ B and k are, respectively, the vacuum permittivity, Bohr magneton and Boltzmann constant; N is the number of magnetic ions (R = Dy, Ho) per unit volume, J is the the quantum number for total electron angular momentum of R 3+ and g J is the Landé g-factor.

A note about Eqn. 6 is in order. The Langevin susceptibility for a single rare earth ion in a state defined by quantum number J may be written:

χ single T = N µ 0 g 2 J µ 2 B kT m J m 2 J p(m J ) (7)
where m J is the magnetic quantum number and p is the Boltzmann probability. In writing the Curie constant in the form of Eqn. 6 it is assumed that the single ion crystal field ground state is a magnetic doublet composed entirely of the free ion states m J = ±J and that this doublet is well separated in energy from other states. In addition, an extra factor of 3 is necessary in the denominator of Eqn. 6 to account for summing the single ion susceptibility tensor over four sets of ions in the pyrochlore structure of spin ice, each defined by a local trigonal quantisation axis belong to the cubic 111 set. Another way to think about this factor of 3 is to consider that the single ion susceptibility has a single term in its susceptibility tensor while the cubic crystal has three equal diagonal terms. The factor 3 then guarantees that the trace of the tensor is invariant under rotations. Eqn. 6 is reminiscent of the Curie constant of a free ion, C = N µ 0 g 2 J J(J + 1)µ 2 B /3k, on account of the restoration of symmetry as one passes from trigonal point symmetry to cubic space symmetry. The assumption that the ground state doublet is a pure m J = ±J state is an approximation [START_REF] Rosenkranz | Crystal-Field Interaction in the Pyrochlore Magnet Ho 2 Ti 2 O 7[END_REF][START_REF] Cao | Ising Versus XY Anisotropy in Frustrated R 2 Ti 2 O 7 Compounds as "Seen" by Polarized Neutrons[END_REF][START_REF] Malkin | Static Magnetic Susceptibility, Crystal Field and Exchange Interactions in Rare Earth Titanate Pyrochlores[END_REF], as symmetry requires admixtures of other m J states, but it is believed to be an accurate one [START_REF] Rosenkranz | Crystal-Field Interaction in the Pyrochlore Magnet Ho 2 Ti 2 O 7[END_REF] and this is born out by our results. Note however, that while Fig. 2 has been obtained without any fitting parameter, in the comparison of Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF] the magnetic moment of the rare-earth ion had to be adjusted by 4% for comparison with bulk measurements. Referring to Fig. 2, we may draw the following firm conclusions. First, different crystals of the same compound give essentially the same susceptibility χ T (T ), showing that small departures from a perfect spherical shape are unimportant. Second, the experimental estimates of χ T for both spin ice compounds agree very closely with theory at T > 10 K, but show systematic deviations at lower temperatures. Thus the Curie Law crossover reaches a maximum of C ≈ 1.4C, significantly below the theoretical limit of C = 2C. Third, the differences in the susceptibility between the two spin ice compounds accurately mirror the differences in their basic energy scales. In the standard dipolar spin ice model, the parameter J eff , which comprises dipole and exchange contributions, has been evaluated as 1.8 K in Ho 2 Ti 2 O 7 [START_REF] Bramwell | Spin Correlations in Ho 2 Ti 2 O 7 : a Dipolar Spin Ice System[END_REF] and 1.1 K in Dy 2 Ti 2 O 7 [START_REF] Hertog | Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets[END_REF]. Ho 3+ and Dy 3+ having nearly the same magnetic moment and lattice constant, their dipolar interaction is fixed. Their nearest neighbour exchange coupling can vary though; if antiferromagnetic, it is responsible for making the creation of topological defects or "monopoles" less costly in energy. For the dumbbell model, in which the point dipoles of the dipole model are extended into needles which touch at the centres of the tetrehedral cells [START_REF] Castelnovo | Magnetic Monopoles in Spin Ice[END_REF], this translates into different values of the monopole chemical potential µ. Creating a single monopole in the absence of intermonopole interactions costs -µ ≈ 4.35, 5.7 K in Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 respectively, although the numerical result for the dipolar spin ice model is slightly higher that this (4.6 K for Dy 2 Ti 2 O 7 ) [START_REF] Jaubert | Signature of Magnetic Monopole and Dirac String Dynamics in Spin Ice[END_REF]. The parameters J eff and -µ in the dumbbell model are related by [START_REF] Castelnovo | Magnetic Monopoles in Spin Ice[END_REF]:

-µ = 2J eff + √ 2C 4 √ 6 -3 /6π. (8)
It is therefore clear from our results that susceptibility measurements on single crystal samples can be used as a general method of establishing either J eff or µ in spin ice materials.

The result, Fig. 2., for Ho 2 Ti 2 O 7 differs from that reported in Ref. [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF] and exhibited in Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF] in that it deviates from theory at temperatures below ∼ 10 K. It behaves similarly to the neutron scattering susceptibility at zero wavevector (see Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF]). Given plausible experimental uncertainties in the latter estimation we may conclude that there is is no longer a significant discrepancy between the zero wavevector neutron scattering susceptibility and the bulk susceptibility. However, this opens up a new puzzle: why did the originally measured bulk susceptibility agree so closely with theory down to a low temperature?

To investigate this question we applied the measurement technique described above to four different cuboid crystals of Dy 2 Ti 2 O 7 applying the standard demagnetizing correction for rectangular parallelepipeds [START_REF] Sato | Simple and Approximate Expressions of Demagnetizing Factors of Uniformly Magnetised Rectangular Rod and Cylinder[END_REF]. In particular, we chose a cubic [111] (nominally D z = 1/3.0), two squeezed [111] and [100] cuboidal shape (nominally D z = 1/1.80 and D z = 1/2.0, respectively) and an elongated [111] one (nominally D z = 1/5.04) where [hkl] indicates the orientation of the crystal with respect of the applied field -representatives sketches of the crystals are shown in Fig. 3 panels a and b. The measured susceptibility in the form χT /C is shown in Fig. 3, where a significant discrepancy with the result for the spheres is noted. The cuboid results are closer to the theory, but can also be projected onto the spherical result with effective demagnetizing factors of D z = 1/3.20(1), D z = 1/2.00(1), D z = 1/2.17 [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF] and D z = 1/6.10(1), respectively, or about 8 -12 % higher than the nominal ones. In view of this, it seems likely that the non-ellipsoidal shape of the sample used in Ref. [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF] contributed to the agreement with theory reported in Ref. [START_REF] Jaubert | Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice[END_REF].

The fact that the experimental data for the cuboidal crystals differs from that of the spherical crystals and also falls on the theory raises a number of issues. The demagnetizing correction used here is the 'magnetometric' demagnetizing factor, which is spatially averaged over the whole cuboidal sample. If the sample fields were homogenous (as they are in a sphere), then the magnetometric demagnetizing factor would give the correct magnetostatic energy [START_REF] Aharoni | Introduction to the Theory of Ferromagnetism[END_REF] and presumably a consistent equation of state and susceptibility. The observed insufficiency of the magnetometric demagnetizing factor implies that either the sample fields are not homogenous in the temperature range discussed, or that Dy 2 Ti 2 O 7 exhibits shape dependent physics in this range, or a combination of these factors. In Table 1 we compare the magnetometric demagnetizing factor with the 'ballistic' (or 'fluxmetric') demagnetizing factor, which is found by averaging over a median plane in the sample [START_REF] Joseph | Demagnetizing factors in nonellipsoidal samples: a review[END_REF][START_REF] Chen | Demagnetizing factors for rectangular prisms[END_REF], and with the 'experimental' value needed to project the cuboidal data on to the spherical result, as described above. It is seen that the experimental value falls systematically in between the two theoretical estimates, suggesting that the method of averaging does play a role. It appears that all cuboidal samples studied here require a definite demagnetizing factor, which is neither magnetometric nor ballistic, suggesting a new systematic result for cuboids. It may seem surprising that a well-studied problem such as the demagnetizing factor could yield such a new result, but we note that spin ice offers a particularly tough test of the theory of demagnetizing factor as it combines extremely high magnetization density with negligible magnetic stiffness. On the other hand the fact that use of the magnetometric demagnetizing factor projects the susceptibility onto our theory is intriguing. One possible explanation is that the true susceptibility differs from the Husimi susceptibility because of local dipolar terms that be represented, in a mean field sense, as a correction to the demagnetising factor. However, we should also note that the effect of boundaries on the very low temperature susceptibility of spin ice has recently been discussed [START_REF] Revell | Evidence of Impurity and Boundary Effects on Magnetic Monopole Dynamics in Spin Ice[END_REF]. In the light of these observations it is not possible to decide at present whether the systematics described above apply to all magnets, or just to a restricted class that includes Dy 2 Ti 2 O 7 , or whether the projection of the cuboidal susceptibility onto theory contains any definite physics. It represents an interesting future challenge to decide between these possibilities.

In conclusion, we have established the true bulk magnetic susceptibility of spin ice down to 1.8 K. There is a significant deviation from the Curie Law crossover expected on the basis of the Husimi tree approximation. In the future it will be useful to extend these results to yet lower temperatures and to compare them with more detailed microscopic models of spin ice that incorporate long range dipolar interactions [START_REF] Hertog | Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets[END_REF][START_REF] Yavorskii | Dy 2 Ti 2 O 7 Spin Ice: A Test Case for Emergent Clusters in a Frustrated Magnet[END_REF] and quantum fluctuations [START_REF] Shannon | Quantum Ice: A Quantum Monte Carlo Study[END_REF][START_REF] Benton | Seeing the Light : Experimental Signatures of Emergent Electromagnetism in a Quantum Spin Ice[END_REF]. More generally we have shown that the use of spherical samples in magnetometry experiments on spin ice is generally highly desirable, to minimise systematic errors. This conclusion may well extend to several other rare earth magnets with the pyrochlore structure. 1. Numerical values of demagnetizing factors. q denotes the aspect ratio of the sample length in the field direction to the orthogonal length (z/x). 'Magnetometric' and 'ballistic' estimates of D z are taken from [START_REF] Joseph | Demagnetizing factors in nonellipsoidal samples: a review[END_REF][START_REF] Chen | Demagnetizing factors for rectangular prisms[END_REF]; the experimental D z correspond to the corrected value used to project the experimental data onto the spherical result.

Fig. 1 .

 1 Fig. 1. Spin ice spherical crystals. From right to left Dy 2 Ti 2 O 7 sample A and sample B; Ho 2 Ti 2 O 7 sample C and sample D. Each sphere has a diameter ranging between 3 to 4 mm.

Fig. 3 .

 3 Fig. 3. Shape-dependent susceptibility in Dy 2 Ti 2 O 7 . Solid line (purple): calculated χT /C (see [19] and text for details). Bulk magnetometry data of a) cubic[111] (black spots) and squeezed[111] (grey spots) crystals; b) squeezed[100] (black triangles) and elongated[111] (grey triangles). Demagnetizing correction has been performed for cuboids according to Ref. [28] (full symbols). Relatively small correction of about 10% (see text for details) can project the data (open symbols) onto the spherical result (pink crosses, corresponding to sample B).

  Sampleq = z/x D z ballistic D z magnetometric D z experimental

	cubic[111]	1	0.2587	0.3333	0.312(1)
	squeezed[100]	0.5	0.406	0.5050	0.461(2)
	squeezed[111]	0.4	0.5073	0.5482	0.500(3)
	elongated[111]	2	0.1109	0.1983	0.163(1)
		2.0
		1.8
	χT/C	1.4 1.6
		1.2
		1.0
		10 -1		10 0	10 1	10 2	10 3
					T (K)

Fig. 2. Temperature dependence of the dimensionless quantity χT /C calculated usung the expected J eff (see [19] and text for details) for Dy 2 Ti 2 O 7 (purple line) and Ho 2 Ti 2 O 7 (green line). The points represent bulk magnetometry data (see text for details) on spherical-shape single crystals of Dy 2 Ti 2 O 7 -sample A (magenta) and sample B (pink) -and of Ho 2 Ti 2 O 7 -sample C (blue) and sample D (light blue).

Table

Acknowledgments

It is a pleasure to thank D. Prabhakaran and A. Boothroyd for supplying the crystals for this study and for related collaborations. This research was supported by EPSRC Grant EP/I034599/1. PCWH thanks the Institut Universitaire de France for financial support.


Our understanding of organizations has greatly progressed v ith the realization that one of their essential internal mechanisms is constituted by voluntary informal transactions betveen members. This important truth vould not, perhaps, be considered as completely nev in other disciplines (see Padioleau 119861), but, in economics, it does strike by its novelty and, in fact, it is far from being fu lly assimilated by the profession. In my case at least, it vas the book of Breton and Wintrobe [1982] that proved illuminating. Voluntary exchanges of informal services constitute precisely the central part of their analysis.

From the centre, ve can move upvards or dovnvards. Moving upvards, ve encounter the problem of explaining on vhat basis the development of exchanges of informal services can be founded vhen third-party enforcing agencies, and markets, do not exist. Going dovnvards, ve find the predictions that can be derived from analysing the bureaucracy as a locus of exchanges betveen individuals. We cannot expect the vtole argumentation, from the foundations part dovn to the predictions, to be completely binding since auxiliary assumptions have to be injected at various stages. But ve can require the story to be consistent. The story proposed by Breton and Wintrobe is very consistent. On the upper level, the strong assumption is made that informal transactions rely exclusively on trust -vieved as an endogenous variable-and take place v ith in trust-based netvorks that are substitutes for markets. On the bottom level, the strong prediction is derived that vertical trust (trust betveen a superior and his subordinates) is efficiency-enhancing from the point of v ie v of the superior, vh lle horizontal trust (trust betveen subordinates) Is inefficient1

In this paper, both contentions are qualified. But the ambition of the paper is certainly not to change substantially Breton and Wintrobe's story. We can be sure that many of the different pieces that they have forged and assembled in their book v i l l be replaced or modified in the course of time. This is the fate of all interesting theories. But only time-consuming and collective efforts can achieve this. The only ambition of this paper is a small beginning in that direction.

Section II of the paper compares some of the mechanisms, others than trust, vhich have been proposed to explain self-enforcing exchanges. Then, Breton 8nd Wint robe's modelling of trust and trust accumulation is looked at c ritic a lly in Section III. Section IV presents an alternative mechanism, inspired by the Popperian philosophy of the grovth of knovledge. Up to that point, we are concerned only v ith the foundations part of the story. However, in Section Y, partly as an application of ideas presented in the previous sections and partly on the basis of some other ideas introduced at that stage, I analyze a particular kind of network, the 'transbureau" network, based on horizontal trust. Since transbureau networks are compatible with vertical trust, and moreover are efficient whenever there is enough of this vertical trust, an important prediction derived by Breton and Wi ntrobe is qualified.

II. Alternatives to trust as substitutes for third-partu enforcement.

Explanations of the observation that individuals do enter into agreements that are not enforced by a third-party agency can be divided into five categories. The explanations can be founded on the expectation of repeat business, on strategic equilibrium, on reputation, on trust, and on enlarged rationality. I shall not discuss the somewhat too numerous possibilities opened by the enlargement of rationality (see for instance Akerlof 11984]), and shall turn to trust only in the next section. None of the three remaining types of explanations is completely satisfactory in itself.

A. The perspective of repeat business.

Expectations of benefits from repeat business are an important element in most, if not a ll, attempts to account for self-enforcing agreements but they cannot constitute the sole basis of a consistent story. This point is particularly clear i f we look at an attempt such as [START_REF] Telser | A Theory of Self-Enforcing Agreements[END_REF] 's to rely exclusively on repeat business. Telser's analysis focusses on a two-persons relationship and we shall interpret his world as the one of Robinson Crusoe and Friday, and the future benefits he refers to 8S the traditional gains from trade. In a nutshell, his argument is the following: "A party to a self-enforcing agreement calculates whether his gain from violating the agreement is greater or less than the loss from future net benefits that he would incur as a result of the detection of his violation and the consequent termination of the agreement by the other party" (p. 28). Clearly, a crucial assumption here is that it w ill be in the other party's interest to terminate the agreement i f it has been violated by the firs t party. This assumption is contradictory v ith the other assumptions made in Telser's paper, in the world of this author, bygones are bygones, goodwill is irrelevant, "rational behavior by the parties to an agreement requires that the probability of continuing their relation does not depend on their past experience with each other" (p. 36). But i f this is the situation, a violation cannot affect the calculations that are made one second after it has occurred. At this point of time, it is in the other party's interest to accept any new promise made by the violator that he w ill respect his obligations in the future (that is, to accept it exactly to the same degree as he accepted the same promise made before the violation).

Thus, under Telser's assumptions, what is compared to the short-run gains of violation is definitely not the consequence of the termination of the agreement. Alternatively, if , for semantic reasons, one wishes to define a self-enforcing agreement as one that is terminated as the consequence of a violation (as done, for instance, by Yarbrough and Yarbrough 11986]), what is not compared with the benefit of the violation is the total interruption of an interaction with the other party. What it is exactly that is compared cannot be said within Telser's framework.

Introducing the new considerations that are needed leads us to the theories of optimal strategies, to the theory of reputation or to the theory of trust.

B. Strategic interaction.

The foregoing objection to Telser's reasoning is clearly related to the problem of non-credible threats, which is well known from the literature on strategic interaction 2. There, the basic assumption is that the real objects of choice are strategies, i* . rules or plans that indicate what are the policies to to be followed at any time in function of what the others are expected to do or have done. Within a strategy, and contrarily to Telser's main assumption, past experience does have an influence on actions at any point of time. The history of the interaction counts. Whether a strategy produces self-enforcing cooperation ( if that is the objective), depends very much on the strategies v ith vhich it interacts. When these are not knovn, the simple strategy of reciprocity, or "tit for tat" (start by cooperating and then do whatever the other did on the previous move), does very v e il on average for that purpose (see Axelrod [ 1984]).

Reciprocity, or other strategies, can certainly e lic it self-enforcing cooperation in some circumstances. But, in the context of interpersonal interaction vhich is the one ve are concerned v ith in the case of bureaucratic netvorks, information plays a more subtle role than the one it is given by most i f not all models of strategic interaction. Many models are concerned v ith firm s, vhose objectives can be assumed to be something like long-term profits or the value of their equity, or, more generally, v ith situations in vhich pay-offs can be specified. When ve interact v ith a person (even for given, limited, purposes), to know the pay-offs he draws from this relationship, or hov he values them, ve vould have to knov his u tility function, or some aspects of it. At the same time, ve are w illing to learn in the course of our relationship about that person.

In particular, ve are ready to take notice (immediately) of a ll sorts of nev information (not only the one provided by the history of the strategic interaction) and of nev circumstances. Thus, our strategy, inasmuch as it can be said that ve have one, should be tentative and change in the course of time 3. The same applies to the person v ith whom ve interact. Strategic interaction models, by segmenting the relationship (see belov), can deal only v ith one aspect of this complexity.

Strategic interaction is hardly avoidable in contexts in vhich relatively isolated tvo-persons relationships are implied but, like repeat business, can only be an element in the story that ve seek.

C. Reputation.

There are several vays to introduce reputation. One can speak of reputation even in a tvo persons framevork by segmenting the relationship: the reputation established in early periods influences the strategy adopted by the other party in later periods. For instance, concern of one actor v ith Ms or her reputation can play a role in such long-term relationships (considered in isolation) as the one between a wife and her husband, the one between The United States and the Soviet Union or the one between a bureaucrat and his or her secretary. Treated in this way, reputation is only a particular aspect (when we allow strategies to change in the course of time as a result of the history of the interaction) of the strategic interaction already considered.

More frequently, reputation is introduced in models in which more than two decision-makers is assumed. Among these models, however, a distinction must be made between those that invoke reputation (or 'brand name") but do not give it a really decisive role and those that do. Perhaps surprisingly, the models of [START_REF] Shapiro | Premiums for High Quality Products as Returns to Regulations[END_REF] and, less clearly, of Klein and Leffler 119811 (see also [START_REF] Klein | Contract Costs and Administered Prices: An Economic Theory of Rigid Wages[END_REF]) belong to the firs t group. Let us, for example, look more closely at the structure of Shapiro's model. X is a firm which sells a high-quality product and a, b , ..., n are its customers. They know that the product is high-quality because X has built up a non-salvageable asset, "reputation", on which it earns a quasi-rent. If the quality of the product is reduced, a ll the consumers w ill know this after a lag and they w ill not pay the quasi-rent any more. By violating its im p licit commitment to sell a high-quality good, X would make an immediate gain but would dissipate ("m ilk") its "reputation". Since they understand this, consumers are confident that the product they buy is high-quality.

In this model, the fact that there are many consumers is completely irrelevant. We can replace them by one consumer, say A. Thus, the main relationship is a two-persons one between X and A. Hence, it resembles the relationship in Telser's model. However, in Shapiro's model, the immediate termination of the relationship after a violation of the im p licit commitment is well founded while in Telser's model we saw that it is not. The difference results from two assumptions made by Shapiro. The firs t one is that firm X is in competition with other firm s selling the high-quality product (hence, Shapiro's world cannot be reduced to an isolated two-persons relationship, but not for the reason which is generally thought of). As a consequence of this assumption, once the cost of the violation has been assumed, there is no opportunity cost for consumer A involved in turning to firm s Y or Z. The gains from trade, or consumer surplus, that consumer A makes from the possibility of buying the high-quality good is not destroyed by the termination of the relationship. InTelser'sworld (i.e. in a world limited to Robinson and Friday), it is. In other words, in Telser's model, both parties earn a quasi-rent from the relationship, while, in Shapiro's model, only the would-be violator does.

Let us assume that no other firm can sell the high-quality good. Then terminating the relationship with X implies for A the loss of his consumer surplus. But can he avoid this? In Shapiro's model, the answer is no, as a consequence of the second assumption we referred to: a "reputation" is definitively destroyed by a violation 4 . If we assume instead that the promise by X to sell a high-quality product, provided a quasi-rent can be earned ( a price premium can be charged), has some non-zero value for the consumer, then the termination of the relationship as a consequence of the violation is not warranted any more. We are not exactly back to the problem encountered when analyzing Telser's model, because we do not have any more to assume that past violations have no effects on present decisions, but we are s till faced with the need to make more explicit what it is that deters violations of agreements.

Most sociologists or anthropologists who refer to reputation as a powerful mechanism for the enforcement of agreements or, more generally, of rules of behaviour, would not recognize what they have in mind in the preceding discussion. They would not accept the assertion made above, in the context of Shapiro's model, that the fact that there is more than one consumer is unimportant.

The reason for this is that the violations of agreements they are concerned with are violations of a two-parties agreement by one party engaged (or potentially engaged) into other two-parties agreements. For instance, X is engaged (or potentially engaged) into agreements with a, b, ...,j,..., n and violates only his agreement with j. In this case, again, i f X has no competitor and "reputation" is not destroyed completely by one violation, there is no compelling reason to think that j w ill terminate his relationship with X. But, although not directly affected by the violation, the relationship of X with a,b,...,n might be so indirectly, as a result of the reassessment by these individuals of the re lia b ility of the promises that X made or could make to them. The total cost to X of these reassessments (we shall see more precisely later in what it consists) might be relatively high, in particular i f information circulates rapidly, while the benefit from the violation of one single agreement would normally be relatively small. Thus, reputation, in this sense, provides a powerful incentive against violating a single agreement (an agreement with one person), especially in societies where information can be assumed to circulate at little cost.

With the possible exception of what was said at the beginning on segmented strategies, the whole discussion of reputation up to now has been concerned with the reputation of the would-be violator. But we have seen that, in many contexts, an important difficulty for understanding self-enforcing agreements lies in the fact that the "victims" draw quasi-rents from their relationships with violators and consequently cannot be expected to terminate these relationship after violations agreements have occurred. When this applies, the threats they can formulate to deter violations are not credible. The reputation of the victim offers a way to make threats of termination credible. If j , the potential victim of the violation of an agreement with X, is also engaged into agreements with other parties, then having the reputation of terminating at any cost whatever agreement has been violated acts as a deterrent against cheating by these others parties (as a deterrent against cheating by X as w ell).

If we consider reputation in its various aspects, it is tempting to conclude from the preceding analysis that it is potentially a very powerful mechanism in favour of the respect of agreements that are not enforceable by third-party agencies. If we want to compare, however, the explanatory power of reputation with the explanatory power of trust such a conclusion would be unwarranted for a very simple reason: trust is already im plitely included in the mechanisms through which reputation exerts it influence. In other words, what we have discussed is not in fact pure reputation as an alternative to trust: no theory of self-enforcing agreements founded exclusively on reputation (without the intervention of trust) is viable. But the reverse proposition is not true: reputation plays no role in many situations where trust is necessary.

III. Trust.

The compelling reference here is to the analysis of Breton and Wintrobe (hereafter BW) (see their [19821, chapter 4, and [1986-a] for a slightly modified formulation). But I shall present things somewhat differently.

Largely inspired by BW's definition, I shall define trust as the probability assigned by one person to the respect of an un-enforceable (often im p licit) promise made to him by another person or, by extension, to the respect of a set of (sim ilar) promises made to him or that might be made to him in the future by that person5 .

Armed v ith th is nev instrument, we can return to the isolated bilateral relationship ve considered in the context of our discussion of Telser's and Shapiro's models. The effect of the violation of a promise made by X to Y is simply the lowering of the probability Y assigns to promises of the same kind made to him by X, i.e. a decrease in the degree to which Y trusts X for this kind of promises. In its turn, this w ill entail a decrease in yYx , the certainty equivalent of the present value of the services Y expects to receive from X. Normally, this entails that less w ill be offered in exchange for these services (but see below). Thus x Yy diminishes also. If we disregard the possibility of in d ivisib ilitie s, termination w ill occur at the initiative of Y only i f the degree of trust of Y in X has been brought to zero. Thus, the cost to X involved in reneging on the promise he made to Y is a decrease in the present value of the net benefit he expects to draw from his relationship with Y. This cost is compared with the benefit of the violation.

In a world in which X has perfect competitors, we saw that it is not the consumer surplus but the quasi-rent drawn by the consumers from their relationship with X that counts. This quasi-rent is equal to zero, already when X is respecting his promise. When it reneges on it, its certainty equivalent becomes negative as a result of a decrease in trust, however small that is, and the consumers terminate their relationship with X. In a more realistic setting, for instance with costs of transaction, the quasi-rent drawn by the consumers from their relationship with X is positive as long os X respects its promise. When he reneges on it , the certainty equivalent of this quasi-rent decreases. It may or it may not become negative and thus termination may or may not follow.

Finally, in the case of reputation in the traditional sense, the knowledge by a number of individuals interacting v ith X that he has reneged on the promise he made to one of them decreases the probability they assign to the respect of the promises he made, or could make in the future, to them. Thus, the value they give to these promises is reduced, they offer less in exchange for them.

The cost to X of reneging on his promise to one individual is a reduction in the gains he dravs from his relationship v ith all the individuals informed of his violation (including the direct victim of the violation). Thus, it is clear in this case that reputation vorks through trust, vh ile , in a bilateral relationship, ve can do vithout reputation and reason directly in terms of trust.

The foregoing analysis of trust seems straightforvard enough, in fact, it raises tvo main problems. The firs t problem results from the fact that the level of trust ve have in X's promise is positively related to x Yy . We have argued that, after a violation of his promise, XVY decreases.

However, increasing it might be a vay for the victim to make new violations less profitable (in Breton and Wintrobe's terminology, the less ve trust somebody, the more ve have to invest in trust). x Yy is a function both of trust (of Y in X) and of an autonomous decision by Y as to the quasi-rents (price premium) given to X. In its turn, trust (of Y in X) is a function ofxYY. Thus a violation by X automatically reduces the trust Y has in X, but this can be restored by a higher price paid to X. But there is certainly a problem in having X revarded for reneging on his promise.

I may be vrong but I think that this problem is somevhat masked but exists all the same in Breton and Wintrobe's modelling of trust (see the chapter 4 of their book). They say that both X and Y invest in the asset (Y trusts X) X invests by renouncing the short-run gains of cheating and Y invests by (in our terminology) giving a quasi-rent to X (in their example, a reduction in interest rate). The problem is that a violation by X is a partial dissipation of the asset, but this asset can be restored by an increase in the qu8si-rent. Observation confirms that people often do react in this strange vay to violations of commitments by the other party. Far from retaliating, they offer more for keeping promises in the future, in an 18th century novel that has remained famous in France, Manon Lescaut. poor Chevalier des Grieux behaves exactly in this vay v ith the very attractive but unreliable heroin.

A way to interpret such situations is to assume that X has some monopoly power while Y not only has none but h8s a demand that is inelastic (in the sense in which a Marshall offer-demand curve is inelastic) in the relevant range. Then, X unilaterally increases the price of his services (defined by y Yx /xYy ) by violating promises and the victim responds by supplying more services. When the price reaches a level at which Y's demand becomes elastic, Y responds to further increases in price by reducing the quantity of services he is w illing to give. But the optimal price for X w ill be reached only whenX's marginal rate of substitution between yYx and

x Yy is equal to the slope of Y*s demand curve. Of course, i f both parties are perfectly informed and are fu lly rational, this equilibrium should be reached directly.

Why would X but not Y enjoy some monopoly power? In fact, in the context of buraucracies, it can be argued that all vertical ties are potentially characterized by monopolies in the absence of horizontal trust and cooperation. To make the story as convincing as possible, without cost in plausibility, let us assume firs t that all bureaucrats are paid efficiency wages (see for instance Yellen [198 4]), so that they have a strong incentive not to quit bureaucracy altogether, second that the subordinates of X cannot quit the particular bureau whose chief is X without his permission, and third that they find exchanging informal services with X important, for instance because he has the power to promote some of them to interesting positions or to assign each of them to very pleasant or unpleasant tasks. In such -not untypical-situations, horizontal ties between subordinates would be very valuable for them but w ill be very difficult to establish. It is clear then that X w ill tend to act as a monopolist and that it might not be unusual at all to observe the response of a subordinate (say j) to the violation of a promise that X made to him as consisting in increased quasi-rents promised to X ( j works hardin view of a promotion promised to him by X, he is not promoted and works even harder).

However, it is also often the case that both X and Y have some power. In this case, Y would retaliate i f X behaved in the way depicted 8bove. This might lead to a "tariff war" and the disparition of exchange. Think of the daily relationship between a bureaucrat and Ms secretary when both are tied together for some gears as a result of organizational rules plus union supervision (again, not an untypical situation at all in the public bureaucracies which I know). If there is some asymmetry, it is usually in favour of the secretary. But, generally speaking, the context of this relationship is very much a Robinson and Friday one. Whether substantial cooperation w ill obtain is 8 question that brings us back to a large extent to the analysis of strategic interaction (a reciprocity rule might be effective in this case, but, thinking that they know better, both actors w ill probably start with more complicated strategies).

The second and even more important problem with trust (8nd hence with reputation), however, lies in understanding the way it is built up or dissipated. In BW [ 19841, the stylized story is the following. A lends 1 $ to B to with a small reduction in interest rate (compared to the one he would ask for loans in the same risk category); B repays the loan; as noted before, both have thus invested in the asset "A trusts B". Then, the operation is repeated with a supplementary reduction in interest rate, and so on. Each time the operation is repeated, the asset "A trusts B" is increased, although with diminishing returns. "After a certain point, they can trade with each other and both can be better off than in the absence of the asset "A trusts B" (p. 68)6.

There are several problems with this analysis. The firs t, less serious one, concerns the separation of the exchanges that are necessary for the building-up of trust and the "real" exchanges that are permitted by trust once it is built-up. This separation facilitates exposition but investment in trust should possibly be modelled more like learning by doing. The most serious problem is that, as depicted by BW, the accumulation of trust is blatantly inductive and thus subject to David Hume's famous characterization of induction 8S irrational. At one time, the dominant philosophy (logical empiricism, in particular as expressed by Rudolf Carnap) tried to save induction by the use of probability theory, an inductive proposition being "confirmed" (the probability of its truth being increased) by each positive instance and "disconfirmed" by each negative instance. This attempt is widely considered as having aborted 8nd current philosophy is admittedly somewhat in disarray. However, a significant number of philosophers have adopted the Popperian view that we can do without induction, and in fact never use induction but instead conjectures and refutations. Even if one does not fu lly adopt the Popperian v ie v, it seems that one needs really a lot of confidence in induction to accept BW's story as a rational foundation for the accumulation of trust.

Since the existence of the phenomenon of accumulation of trust seems hardly a matter of discussion, I shall attempt in the next section to tell a story vhich could account for it. But, and this is the third problem in BW'a account of building up trust, the tvo authors are not clear on vhy ve need to bother v ith the accumulation of trust. They argue forcefully that ve need to incorporate trust in our analyses of bureaucracy as an endogenous variable. Aral it is clear that most i f not all the interesting results produced by their theory (see in particular their [ 1986-aJ and [ 1986-b], as v e il as the work made v ith Gianluigi Galeotti) require an endogenous concept of trust. But this could mean that ve need only to spell out the conditions for trust to be "high" or "lov", or "higher" or "lover" (for the "organization of trust", as suggested by [START_REF] Coleman | Introducing Social Structure in Economic Analysis[END_REF]).

Zero trust in all circumstances clearly leads to s illy behaviour, even v ith perfect strangers. But ve must knov vhat are the settings vhich favour a high level of trust (e.g. the prospect of repeated and profitable exchanges) or a lo v level (e.g. very high incentives to compete). Trust could be endogenous only in this sense. Although they deny it (in particular by assuming an in itia l level of tru st), BW sometimes give the impression that they vant to ju stify the existence of trust in a vorld of narrow egoists ( "takers"). inasmuch as this is the case, tvo criticism s can be made.

One is that an explanation of the existence of trust should probably be mainly biological. The other is that, inasmuch as trust is a conjecture ( a theory), it is in need of no justification, as argued convincingly in my opinion by Popperians (see for instance Andersson 11984]).

Section iV. Interpersonal conjectures and the strengthening of ties I tell here a story vhich accounts for the variation of trust although 1t is founded on the assumption that bureaucrats (as everybody else) use the hypothetico-deductive method to increase their knovledge (for a sim ilar attempt in another context, see [START_REF] Boland | The Foundations of Economic Method[END_REF]Boland [ ][ 1986]]).

Basically, this story treats trust as one product among many of a general mechanism, in his interpersonal relations, vhat a person needs is to predict hov the other person(s) w ill react to specified circumstances (including his ovn actions in these circumstances). The predictions are about trust i f the person is interested in the probability of breaches of unenforceable promises; but hi3 main concern m8y be different and, for example, call for predictions on reactions to threat, to coercion or to persuasion. In this section, I address tvo questions. Hov does trust vary as a consequence of a process of conjectures and refutations? Hov, in the context of bureaucracies, can one interpret the accumulation of trust? As ve shall see, the tvo questions, although related, are not identical.

A. The variation of trust as a consequence of conjectures and refutations.

Bureaucrat A contemplates exchanging informal services v ith bureaucrat B. A forms in his mind ("hov" does not matter) a conjecture or theory. I rule out pure curiosity and assume that the only thing that A is interested in is the prospect of exchanging informal services v ith B. Hence, the content of the conjecture is directly related to the nature of the exchanges that A vants to make v ith B Its ultimate objective is to deliver predictions about reactions of B that are directly relevant for the realization of these exchanges. Thus the conjecture can be called instrumental, from vhich does not follow, as it has been argued by authors such as Boland [1979], that its truth is irrelevant [see [START_REF] Salmon | Du pluralisme méthodologique en économie[END_REF] and Mingat, [START_REF] Ningat | Méthodologie économique[END_REF], chapter 6).

Certainty about the truth of the conjecture is out of reach, but conjectures are necessarily falsified (by the modus tollens mechanism) i f one of their implications is asserted to be refuted (the assertion is in its turn fa illib le in this uncertain and complicated vorld). If some implications are tested and come out "not refuted", the conjecture is corroborated. Through a process of conjectures, refutations, revised conjectures and so on, better (truer) conjectures are obtained. In a context of action (as opposed to a context of pure knovledge), it is advisable to act on the basis of the best corroborated conjectures available (some philosophers argue that there is a "v h iff of inductivism' in this proposition, but see [START_REF] Watkins | Science and Scepticism[END_REF], chapter 9). Consequently, although the predictions that interest him are those that have a bearing on his expected exchanges v ith B (and this largely determines vhat should be in the conjecture), A is also interested in predictions that have no 3uch bearing but makes it possible to test the conjecture. In other words, an instrumental conjecture should be neither too wide (this would be a waste of time for A), nor too narrow ( it could not easily be tested, or only by testing the predictions on which A bases his decisions and that he certainly would not like to be refuted).

What kinds of (conjectural) assumptions should we find in A's conjecture? Without trying to be exhaustive or rigorous, let us single out three particularly relevant categories of assumptions (there are also assumptions on human behaviour in general, etc.). First, there are assumptions about the "personality" of B (ejL his objectives, his degree of risk-aversion, his morality, his time preference). Then, there are assumptions about what we can call the "non-personality resources" of B (e j. his position in the bureaucracy, the networks he is in , his enemies, the time he disposes of, the technical knowledge he has acquired). Finally, there are what may be called AB intercognitive assumptions: answers to questions such as "how does B see A?" or even such 83 "what does B think about how does A sees B?". To deal with the problem of endogenity involved in such questions (A can influence the way B sees him and the way B thinks that A sees him ), I assume that in "how B sees A" (ss well 8s in the more complicated case), there are exogenous components (A can do nothing about it) and endogenous ones. I include only the first in the conjecture.

We can call "principal conjecture* the conjecture discussed so far. To it must be added auxiliary conjectures or assumptions. In particular, there 8re conjectures on the st8te of the world (about, for instance, the services that can be exchanged within and outside the particular bureau, the regulations, sanctions, level of monitoring prevailing in this bureau). Together, the principal and the auxiliary conjectures yield conditional predictions about B's behaviour These conditional predictions have the form: "If circumstances are c , , c^,..., cn, then the probabilities that B w ill do actions a1, ^ are x, Jx2*-/Xm". They can be derived exclusively for the purpose of testing the principal conjecture or they can be directly the ones that are needed for A's action; in the second case, ve have already mentioned that no refutation is seeked (although after the action has taken place they w ill serve as tests). Since discussing the difficult methodological problem of testing (in particular vhen the implications to be tested are expressed in terms of probabilities) vould bring us too much afar, let us consider only the problem raised by the second category. The simplest case is vhen A contemplates making a single offer to B, v ith vhom he has never had much relation before and has no intention to establish a more enduring relationship.

Then, this offer can be simply included in the circumstances, say as c^ But before making this offer, A can consider alternatives, say one alternative Cj. In that case, the prediction including Cj and the one including Cj are compared, and A selects the course of action vhich seen» best.

If the relationship between A and B is to be more enduring, things become more complicated. The effects on A s and B's mutual perceptions of the history of the interaction must nov be included in the conjectures made by each. But since these effects can be, to some extent, anticipated, they must be alloved somehov to influence the selection of the firs t moves. We shall not attempt to go into the details of the complicated dynamics that follovs but v ill stress a fev points F irst, each actor can try to mislead the other (influence the conjectures made by the other), by vord and deed. This complicates matters but does not make less relevant the usefulness for each to get to the truth about the other. Word and deed of B are for A observations that his conjectures should account for; hence, they serve as minimum tests for A's conjectures (conjectures must at least "3ave the phenomena", although they should do more).

Second, in a sim ilar vay, hov B behaves v ith other individuals is an important source of observations that the conjectures made by A on B should account for. If B behaves in a vay that is "surprising", this constitutes a "refutation" of these conjectures, in the sense that they should be modified. In its turn, this affects the predictions on vhich A founds his behaviour tovards B. Thus the reputation of B is i mportant to A in the sense that it helps A discover the "true" personality of B (a point stressed by [START_REF] Colson | Tradition and Contract: The Problem of Order[END_REF]).

Third, the foregoing analysis can explain vhy trust changes in the course of the relationship between the two individuals. But trust can change in both directions, it can be reduced as well as increased. Violations of agreements reduce trust i f they are morefrequent or serious than predicted 8nd increase trust in the opposite case. The analysis can explain why trust of A in B accumulates only i f we assume that A underestimates sytematically B's trustfulness (although it can be argued that, in a competitive environment, individuals who are relatively un-trustworthy tend to be eliminated from exchange relationships and that, consequently, the long-lasting relationships that we observe are on average established between individuals who are relatively trustworthy). In other words, what the preceding analysis can explain is that, in the course of time, the two individuals have theories on each other that are better corroborated, but what it cannot explain is why they attribute higher probabilities to the fulfilment of their mutual promises.

In particular, the mere repetition of sim ilar exchanges in a stable environment, while it does produce better corroborated conjectures (when implications have to be tested statistically), constitutes nevertheless a rather weak instrument for that purpose, superseded by the experience of more varied exchanges when it can be obtained. It is when we have observed an individual in different circumstances and moreover have exchanged with him varied services, that we feel we "know" him relatively well.

B. The accumulation of trust.

To explain the building-up of trust, we must complete the foregoing analysis by taking into account the quasi-rents that are created by the bilateral relationship in the course of time.

Quasi-rents are mainly created because the cognitive process described above is both time-consuming and productive. Being an investment in specific knowledge, it builds up an asset which is clealy non-salvageable: if A terminates his relationship with B and turns to C, whom he does not know, his knowledge of B w ill be lost and new knowledge w ill have to be acquired.

Why is knowledge valuable? Firstly, better knowledge of B enables A to discover valuable services that B can render trim easily, as v e il as services which he can render B also at relatively little cost although they are particularly attractive and useful to B. Thus, thanks to improved knowledge of B, the productivity of mutual exchanges is increased. B's knowledge of A increases it further. Trust is not directly involved in this firs t reason for the usefulness of knowledge.

But, secondly, better knowledge of B gives A the possibility to make better and more detailed conditional predictions about B's reactions, and thus to desaoreoate t rust. Thus, after a while, A w ill know what are the services which should not be asked to B, even if the latter promises to render them (for instance no one who knows me well would ask me to post 8 relatively urgent letter, although I would always promise to do so on the firs t occasion). B w ill also learn in the course of time what are the kinds of violations of commitments to which A is particularly sensitive and thus w ill abstain from making such promises when he is not sure to respect them.

Finally, A w ill be able to predict what are the changes in the external circumstances which may induce B to renege on his promises and w ill have the possibility to decide on this basis to seek only conditional promises. In sum, violations w ill become more predictable and thus w ill tend to be avoided. As a consequence, trust w ill be increased in the sense that promises for which B is not to be trusted w ill tend to be neither seeked nor seriously made.

The increase in the course of time of quasi-rents (or the growth of the non-salvageable asset of which they are the counterpart) entails an increase for both A and B in the cost of the termination of their relationship. It also increases trust for the reason just explained. Thus, the cognitive mechanism exposed above seems more able th8n the non-explicitely cognitive mechanism proposed by Breton and Wintrobe (especially when presented in the form of their 1$ loans example) to account for the accumulation of trust. However, it must be noted that increased quasi-rents, especially when symmetrical, decrease the credibility of the threat of a termination of the relationship. By itself, this encourages violations. With the increase in expected quasi -rents, the relationship between A and B resembles more and more the "Robinson and Friday"

relationship whose problems we have already analyzed. To some extent, our cognitive mechanism suggests a way in which violations may be more easily deterred, when termination is not credible.

Increased knowledge of B allows A to make threats that are more exactly proportionate to the possible violations, for instance in the form of a refusal to enter into exchanges whose value to B he can predict. Of course, this argument has its lim its. Counter-threats are also improved by B's knowledge of A and both exploitation and escalation remain possible (no analysis that excludes them should be accepted since they are so frequently observed).

The foregoing analysis has led to some changes in the foundations of exchanges of informal services in bureaucracies. In particular, reputation has been related to trust and recognized as potential important. Trust, and its accumulation, have been given a different explanation from the one proposed by Breton and Wintrobe. What are the consequences of these changes on the derivation of the main empirical propositions that confer so much relevance to the analysis of these authors?

To answer this question, we would have to turn our attention to the varied nature of exchanges that take place in bureaucracies. Within the lim its of this paper, it is impossible to do so seriously.

However, in the next section, I shall consider, in the light of the preceding analysis, the particular case of "transbureau networks' .

Y. The ambivalence of transbureau horizontal ties.

I call transburau ties and networks those that are established between some subordinates in one bureau and some subordinates in other bureaus, or between the head of one bureau and some of the subordinates in other bureaus that are situated on the same hierarchical level. A characteristic of these ties or networks is that, despite the fact that they are clearly horizontal, the individuals who are involved in them cannot normally be in strong competition or riv a lry with one another. In this sense, the establishment of transbureau ties is not subject to the main obstacle encountered by the establishment of other horizontal ties.

Breton and Wintrobe s most powerful and striking proposition is that horizontal trust is inefficient from the point of view of the superior. Does this apply to transbureau ties? In this section, I address fir s t this question. Then, I spell out the characteristics of an important class of exchanges v ith in bureaucracy and suggest that they are particularly lik e ly to take place vith in transbureau networks. Finally, I illustrate the importance of transbureau links by a 3hort analysis of the transbureau networks that are found in the French higher bureaucracy.

A. The efficiencu of horizontal trust as a function of vertical trust.

A problem is Breton and Wintrobe's thesis is that, from the point of view of a superior, there are clearly some gains as well as some costs involved in the cooperation of his subordinates.

Think of the relationship between a sales department and a production department within a firm . If there is only riv a lry and no cooperation between these two departments, the sales department makes promises to customers (e.g. on the date of delivery, or on tailored-made characteristics of the good) which can easily be sabotaged by the production department (see Padioleau [1986],

pp.162-163, and the literature he cites). If there is high horizontal trust -and thus many exchanges of informal services-between these two departments or their chiefs, services to consumers can be improved. On the other hand, competition between the two departments or their chiefs (for resources and for the promotion of the chiefs) is a means for the superior to stimulate the efforts of both and to avoid a collusion between the two chiefs which could be directed againt him. in Breton and Wintrobe's theory, horizontal trust (trust between subordinates) and vertical trust (trust between the superior and each of his subordinates) are essentially antagonic, since the fir s t im plies the absence of competition that is required by the second . Although horizontal trust is necessarily ambivalent, Breton and Wintrobe show convincingly that it is on the whole inefficient, mainly because of its incompatibility with competition.

Let us assume that A and B work in two bureaus that are in a sense complementary (e.g. an accounting and a production department). The chiefs of these two departments are G and H. Their common superior is K There is nobody above K, which means that what is efficient from his point of view is efficient from the point of view of the organization as a whole. A and B establish a network, founded on high mutual trust, and exchange informal services. This is not directly antagonic with competition (as it would be in the case of a network between G and H) since competition cannot exist anyway between A and B. The policies of G and H can be of two ki nds. Some policies are competitive or conflicting policies, in the sense that they are instruments designed by the one against the other. Other policies are cooperative; they yield mutual gains to G and H. In their turn, the services that are exchanged between A and B can be divided into four categories. In From the point of view of K, there are four cases.

( i) The level of vertical trust is high all the way down, that is between K and G or H, between G and A and between H and B. Cooperation between A and B is efficiency-enhancing.

(ii)The level of vertical trust is high at the top (between K and G or H) but low at the bottom transbureau ties Figure 1.

consequence of the analysis that has been developed in the previous section on the accumulation of trust. If A knows B very well, he can recognize, when it occurs, the occasion of doing a lot of good or a lot of bad to B at little cost for himself. Typically, A takes the initiative; he gives without being asked. Let us make the extreme assumption that, within a given network, no services whose costs are high for those who render them are ever exchanged. This would not imply that no high-valued services are exchanged, i f there are occasions. But, then, they would have the characteristics spelled out above. It is true of course that favours are asked. If B also knows A very w ell, this knowledge w ill help him in shaping his demand. But the cost to A w ill s t ill, typically, be higher than when A takes the initiative; it might easily be too high. In other words, rendering services that are asked is usually more costly than rendering services that are not asked, although their value for the person to whom they are rendered is not necessarily higher.

Secondly, the cost involved in rendering services is often very low in bureaucracies because the bul k of it is externalized and because decisions are often collective. Let us consider the case of promotions. Promotions are not generally decided by one single superior who knows very well the persons concerned. They are made by committees or are made by decision-makers higher in the hierarchy than the immediate superior. They often involve moving from one bureau to another (the ambassador in Montevideo is promoted as firs t counsellor in The Hague). Now, let us assume that the question of a job to be filled is raised within a committee or in the context of a conversation in the office of a high-level bureaucrat. Then, for A, one of the persons who are present, just mentioning the name of B, usually entails little costs indeed (but see below), although it may change the whole carrier of 6. If there is a cost involved in the appointment of B rather than C, the cost is born by other potential candidates and by the bureaucracy as a whole ( if 6 proves in retrospect less efficient in his new job than C would presumably have been). The cost of expressing a doubt on B's capacity ( it can be done without a single word) is usually not (much) more costly for A and may have the same drastic effect on B. Another example is information on contemplated policies (leaks). It has more or less the same characteristics. In both cases, of course, the initiative can be B's. But it is often the case that B does not even know that a policy is in preparation or that the appointment to a job that suits him w ill appear as an item on the agenda of a meeting in which A w ill be present. In addition, as we saw, it often happens that A w ill do B a favour only i f he thinks that its cost is low: whether this is the case, he is the only or the main one to know.

It is clear that the foregoing analysis is not very plausible when B is the subordinate of A.

Then, the main private cost to A of rendering a service to B is the opportunity cost of not rendering that service to C or D, on the same level as B. This opportunity cost can be high. Thus, I think that the peculiar characteristics of the services considered in this paragraph are mainly (but not exclusively) to be observed in the case of transbureau exchanges (in particular those that correspond to category ( ii) in the preceding paragraph).

Even within transbureau networks, many exchanges do not have the characteristics presented above. But services which do have these characteristics generally remain in the background of the stronger relationships, as a possibility of which the two parties in the most costly agreement are well aware. This a powerful additional force against the termination of cooperative relationships. One never knows, in the hard world of bureaucracy, whether the other party might not be sometimes in the position, at practically no cost to himself, to do one a lot of good or a lot of bad. Thus, although the services whose characteristics we have emphasized tend to favour the establishment of weak (and numerous) ties (on weak ties, see Granovetter [ 1974?] and Boorman [ 1975]), they also affect the shaping of strong, more exclusive, ties.

C. The example of transbureau networks in the French higher administration.

In France, the recruitment of high-level bureaucrats is largely based on a severe Other features of GCs must be mentioned. Many members leave the public bureaucracy after some years for jobs in politics and in public or private firm s. Hovever, they remain members of the GCs (in practice if not officially). When they are in the public bureaucracy, they usually do not stay for extended periods of time in the same bureau and not even in the same m inistry (although a GC has a natural habitat or stronghold in one m inistry, to vhich members can alvays return i f they v ish ). Mobility is very high. Higher level positions in the public bureaucracy are occupied partly by members of GCs and partly by otter bureaucrats (nothing is required for becoming a director). Many qualifications and additional features should be introduced for the sake of accuracy and of completeness, but this v i l l do for our purpose.

GCs are typical transbureau netvorks, very much alike the dispersed clans that are analyzed by some anthropologists (see, for instance, vhat [START_REF] Colson | Tradition and Contract: The Problem of Order[END_REF] says on cross-cutting ties). The obligations of a member are the folloving. F irstly , he must, vhenever he can, help other members find a revarding job. In practice, this means that veil-placed members must help less veil-placed members to find jobs(not necessarily as subordinates). Secondly, each member must render small services to other members if they do not cost much (e.g. answering calls or transmitting information). Thirdly, the GC itse lf must be defended i f it is menaced (by increased numbers or by restriction of its prerogatives). Finally, each member must concur in sanctions against "traitors". It must be noted that reputation, as in the small clans studied by the anthropologists, is a very important mechanism here.

From the standpoint of its members, GCs are very efficient networks. They have the optimal size (and are kept so). If they were sm aller, it could not be assumed that there is always a sufficient number of members who are well-placed and consequently the security the GCs offer their members would be diminished. But they are small enough for members to know each other fa irly well; rendering services to others is not too fastidious for those who are well-placed; competition for rewarding jobs takes place essentially with non-members since, in comparison to the size of each GC and even to the agregated size of the five GCs, there are plenty rewarding jobs available in the bureaucracy as a whole, in politics and in firm s. The efficiency of the GCs would be reduced i f they attempted to defend particular policies or i f their political loyalties were not divided. They would not be tolerated by politicians or by the rest of the bureaucracy. But, actually, political loyalties are sp lit among the main political parties and freedom of thinking on matters of substance is allowed ( if members of a GC often tend to think along the same lines on many matters, it is because they usually have the same background and because their training gives them a bias towards conformism, but dissenters are well tolerated). The fact that all the main political loyalties can be found in a GC means that there w ill be well-placed members whatever the majority in power. Thus, when the government changes, some well-placed members lose their position but other members get to the top (not necessarily in the same jobs).

Are the GC networks efficient fom the point of view of superiors? Since these GCs have been there for many centuries, during which France has gone through numerous revolutions and political regimes, it is probable that superiors find advantages in their existence. Actually, all depends on the level of vertical trust that is established between superiors and those among their subordinates who are members of GCs. An obstacle to vertical trust lies in the great independence that is provided by membership in a GC. Members are like "barons" in Tullock [19651*3 account of bureaucracy. They do not risk much, whatever happens. However, normally, a superior can easily get rid of a member of a GC. The GC does not fight to maintain its members in their present jobs. The superior can even hope to replace the member of a GC by another member of the same GC.

In any case, the superior Ins a strong incentive to have members of GCs among his subordinates and to establish vertical trust with them. If he succeeds, he benefits from direct links with almost all the important bureaus and public or private firm s, as v e il as v ith some politicians. In practice, all ministers try to have members of GCs among their close collaborators, snd if possible one of each GC (only the most important ministers can get that). Thus, except for the short periods of time that fo llo v the coming into pover of completely fresh and innocent politicians (and, v ith them, of many equally innocent directors), vertical trust is typically high between superiors and members of GCs, and consequently GCs are efficient.

VI. Conclusion.

Three main points have been made in this paper. To start v ith the less important, it has been argued that reputation can be a poverful mechanism for the support of self-enforcing transactions v ith in bureaucracies, but that this effectiveness does not diminish the role played by trust since reputation implies trust. Then, a mechanism for the building-up of trust or the strengthening of ties inspired by the Popperian theory of knovledge has been proposed as an alternative to the model present«! in Chapter 4 of Breton and Wintrobe [1982]. Finally, efficiency-enhancing horizontal trust has been shovn to be possible in the case of transbureau ties, and most lik e ly to prevail if vertical trust is strong. As revealed by the title of the paper, it is not argued that these various contributions are perfectly interrelated, but some links exist.

Notes definitive termination of the relationship is not realistic since the firm can re-establish its reputation in the course of time, or do it indirectly by transforming itse lf into another firm .

5 An inconvenient of this definition is that it implies that ve cannot decide to trust.

6 We can note in passing that the assertion formulated by BW that nothing is implied by this mechanism as to the ssset "B trusts A" is not completely convincing : the sacrifice (investment) made by B for the sake of building up the asset "A trusts B" is giving up cheating; it is made on the basis of the prospect of the quasi-rents that he v i l l earn in the future i f he remains trustful, hence on the basis of an im plicit promise by A to make more business with him. Consequently B must trust A from the moment vhen he gives up cheating on the firs t loan and there is not much reason to understand vhy his trust in A does not grov, with a lag of one period, parallelly with the trust of A in him.

  selection at the graduate level. Bright young graduates are admitted as c iv il servants to a number of very exclusive Grandes Ecoles. They spend there a couple of years and are admitted as members to one of the Grands Corns (hereafter GCs).The most prestigious among these are the Ingénieurs des Ponts et Chaussées, the Ingénieurs des Mines (misleading names in both cases), the Inspecteurs des Finances, the members of the Cour des Comptes and the members of the Conseil d'Etat. In the fir s t tvo cases, undergraduate studies take place at (the Ecole) Polutechnlaue. itse lf highly selective, and those vho come out of Polutechnlque among the firs t generally choose to go to the tvo "applied'' schools. In the case of the last three GCs, graduate studies take place at the Ecole Nationale d'Admi ni st ration -o r ENA-(itse lf highly selective), and those vho come out among the firs t usually choose one of the three GCs. Thus, the firs t tvo GCs are in reality nested in the much larger Polytechniciens network, vhile the three others are nested in the much larger ENA netvork. Hovever numbers count. The total annual recruitment in the five GCs is probably only a tenth of the annual flovs coming out of Polytechnique and ENA. I assume in vhst follows that, typically, a GC has a fev hundred members.

  category ( i) are services that divert resources of the organization in favour of A and B. I assume that category ( i) is unimportant since A and Bare in two different bureaus. In category ( ii) are services giving an advantage to A or B over their colleagues in the same bureau, say A' and B'. I assume that these services are important (see paragraph B below) but that G and H (and K) do not care. In category ( iii) are services (for instance leaks) that undermine the conflicting policies of G and H. In category (iv ) are services that enhance the cooperative policies of G and H.Let us look at the reactions of G and H to the cooperation between A and B. If G and H cooperate rather than compete, I assume that they have only cooperative policies. Cooperation between A and B is efficient from their point of view ( it saves time). If G and H compete, they have conflicting policies. But they also have cooperative policies. To see this, one must remember that competition between G and H is associated with high vertical trust between each of them and K.

	Hence, G and H are assumed to serve ICs interest. But K's interest is that a certain amount of
	cooperation exists between G and H despite their riva lry . Thus, cooperation between A and B w ill
	be ambivalent from the point of view of G and H. Conflicting policies might be undermined but
	cooperative policies w ill be made more efficient. Now, i f the level of vertical trust between G and A
	and/or between H and B is high, G and/or H w ill assume that A and/or B w ill know what should be
	exchanged between themselves and what should not. Consequently, vertical trust at this level w ill
	mean that transbureau cooperation is encouraged by G and/or H. If the level of vertical trust at
	this level is low, the opposite is implied.

On this, in addition toBreton and Wintrobe [1982], see[START_REF] Breton | Is Proportional Representation Alvays the Best Electoral Rule ?[END_REF],Breton and Wintrobe [ 1986-a] and [ 1986-b], and[START_REF] Galeotti | An Economic Theory of Political Parties[END_REF].

See, for instance,[START_REF] Friedman | OliqoooluTheoru[END_REF], pp. 124-135 and 2 2 2 -2 2 6 , for a simple explanation of the nature of the problem of non-credible threats in the context of oligopoly theory and of its relationship v ith the notion of perfect equilibrium introduced by Selten in 1975. 5 In the field of macroeconomic policy, although recent experience constitutes an incentive to think in terms of policy rules rather than policy decisions, a crucial objection Is precisely that ve cannot rule out the influence of nev knovledge or nev circumstances.

 (between 6 and A and between H and 6). Cooperation between A and B is also efficiency-enhancing.

It is inefficient from the point of view of G and H because it undermines the conflicting policies they design against each other more than it favours the cooperative policies that serve K's interest, but the value for K of the conflicting policies is zero or negative (they are either indifferent or represent the cost that he has to pay for the benefit he draws from the competition between G and H), while the value of the cooperative policies is positive.

( iii) The level of vertical trust is low at the top (there is collusion between G and H) but high at the bottom. Cooperation between A and B is inefficient. It reinforces collusion between G and H.

(iv)The level of vertical trust is low all the way down. Cooperation between A and B is also inefficient. It cannot undermine conflicting policies between G and H since there are none.

Inasmuch as it serves their cooperating policies, it reinforces collusion.

In sum, i f there is a high level of vertical trust between the superior and the bureau chiefs, transbureau networks are efficient, from the point of view of the superior. They are not i f the level of vertical trust between the superior and the bureau chiefs is low. If we assume that the level of vertical trust at the top and the strength of transbureau ties are continuous variables, the foregoing propositions may suggest the shape of the Isoquants (defined as identical levels of efficiency) represented on Figure 1.

[Insert Figure 1 about here] B. Characteristics of an important class of services exchanged within transbureau networks.

I

suggest that many informal services within bureaucracies (not only within transbureau networks) have the following characteristics:

( i) The service cannot be predicted (neither its content nor the date of its occurrence, i f ever).

( ii) It costs very little to the one rendering it.

( iii) lt is of high value to the one receiving it.

(iv)The same characteristics apply to disservices (negative services).

Why such apparently strange characteristics? F irstly , to a large extent, they are a


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