Emergence of firms in (d+1) -dimensional work space
G. Weisbuch, D. Stauffer, D. Mangalagiu, R. Ben-Av, S. Solomon
aa r X i v : . [ q -f i n . GN ] J a n Emergence of firms in ( d + 1) -dimensional work space G Weisbuch , D Stauffer , D Mangalagiu , R Ben-Av , S SolomonRacah Institute of Physics, Hebrew University, IL-91904 Jerusalem, Israel Visiting from Laboratoire de Physique Statistique, Ecole Normale Sup´erieure, F-75231 Paris, France [email protected] Visiting from Institute for Theoretical Physics,Cologne University, D-50923 K¨oln, Euroland Management and Strategy Department, Reims Management School,59, rue Pierre Taittinger, F-51061 Reims Cedex, France Department of Software Engineering, Jerusalem College of Engineering(JCE), Israel
Standard micro-economics concentrate on the description of markets but isseldom interested in production. Several economists discussed the conceptof a firm, as opposed to an open labour market where entrepreneurs wouldrecrute workers on the occasion of each business opportunity. Coase [1] isone of them, who explains the existence of firms as institution because theyreduce the transaction costs with respect to an open labour market.Whatever the rationale proposed by economists to account for the ex-istence of firms, their perspective is based on efficiency and cost analysis.Little attention is paid to the dynamics of emergence and evolution of firms.The aim of the present manuscript is to check the global dynamical prop-erties of a very simple model based on bounded rationality and reinforcementlearning. Workers and managers are localised on a lattice and they choosecollaborators on the basis of the success of previous work relations. Thechoice algorithm is largely inspired from the observation and modeling oflong term customer/sellers relationships observed on perishable goods mar-kets discussed in Weisbuch etal[2] and Nadal etal[3].The model presented here is in no way an alternative to Coase. Wedescribe the build-up of long term relationships which do reduce transaction Laboratoire associ´e au CNRS (UMR 8550), `a l’ENS et aux Univ. Paris 6 et Paris 7
Let us imagine a production network of workers: we use the simplest structureof a lattice: at each node is localised a ”worker” with a given production ca-pacity of 1. Business opportunities of size Q randomly strike ”entrepreneur”sites at the surface of the lattice.The work load received by the entrepreneur is too large to be carried outby her: she then then distributes it randomly to her neighbours upstream;let us say that these neighbours are her nearest neighbours upstream. Wepostulate two mechanisms here: a probabilistic choice process according topreferences to different neighbours, and the upgrading of preferences by as afunction of previous gains. The probability of choosing neighbour j is givenby the logit function: p j = exp( βJ j ) / nb X k =1 exp( βJ k ) (1)where the sum extends to all neighbours of the node. The preferences J j are updated at each time step according to: J j ( t ) = (1 − γ ) J j ( t −
1) + q j ( t ) (2)where q j ( t ) is the work load attributed to node j .One time step corresponds to the distribution of work loads across theset of collaborators of the entrepreneur who received the work load.A series of work loads strike the entrepreneur at successive time steps. Wewant to characterise the asymptotic structure generated by a large numberof work loads presented in succession to the entrepreneur. Workers are placed on a ( d + 1)-dimensional hyper-cubic lattice of height L z . Each hyper-plane line = 1 , , . . . , L z is a lattice of linear dimension L with L d sites and periodic (helical) boundary conditions. A workload Q is2istributed from the top level (hyper-plane) line = 1 upstream, in steps fromlevel line to level line + 1, until Q different workers (sites) i each have a localworkload q i = 1. All local workloads q i are integers 1 , , . . . , Q .One iteration corresponds to the downward distribution of one workloadand proceeds as follows: Initially all sites have workload zero. A new work-load arrives at the central site of the top level. Thereafter, each site i onlevel line having a local workload q i > q i − n nb = 2 d + 1 nearest and next-nearest neighbours j on the lower level line + 1, in unit packets q i → q i −
1. For this purpose it selects, again andagain, randomly one such neighbour j and transfers to it with probability p ,given by equation (1) one unit of workload, increasing by one unit the pref-erence J ij storing the history of work relations. After site i has distributedits workload in this way to the lower level of hierarchy and has only a re-maining unit workload, the algorithm moves to the next site having a localworkload bigger than unity. The whole iteration stops when the lowest level line = L z is reached or when no site has a local workload above unity. Thenall workloads q i are set back to zero, all stored preferences J are diminishedby a factor 1 − γ , and a new iteration starts, influenced by the past historystored in the preferences J ij . Fig.1 shows that for the chosen parameters some stationary equilibrium isobtained between the increase of sum of all J ij , called the flow, due to newwork, and the decrease of the flow due to the forgetting parameter. The depthof the load pattern in the lattice also increases and finally reach saturationas the flow. These dynamics are similar in lower (1+1) and higher (1+4)dimensions. According to β , γ and load values, after many iterations two dynamicalregimes are observed: a quasi-deterministic regime such that only one linkout of 2 d + 1 is systematically chosen resulting in a ”snake” portion of thework pattern, and a random regime where all 3 links are used, resulting in a3blob” portion of the work pattern. The interface between the two regimescorresponds to β ( q ( z ) − /γ = Constant (3)where q ( z ) − q ( z ) at depth z . Because the work load to be distributed, q ( z ) − z , there is a given depth where the interface between the deterministicregime and the random regime is located.Figure 2 displays workloads obtained in a (1+1)-dimensional lattice. Onthis figure the snakes extends from the initial load of 20 to the load of 7followed by a small blob of height 3. Parameters for this simulation were γ = 0 . , β = 0 . z such that: β ∗ ( q ( z ) − γ = 2 d + 1 (4)For larger values of q ( z ), all the work load is transfered to a single neigh-bour with a preference coefficient of ( q ( z ) − /γ , and all other coefficientsare 0. For lower values of q ( z ), all preference coefficients are small, withpossible fluctuations around the interface. These predictions are verified infigure 3 computed for simulations in 1+1, 1+2 and 1+3 dimensions.Figure 4 is a more systematic test in 1+1 dimension of this dynamics. Wehere plotted in the mean square distance of the positions of the workers ineach hyper-plane = line from the position of the highest worker concentration(more precisely: from the center of mass of their distribution). In the tailthis squared width is exactly zero (left part), while in the blob (right partfor each curve) it has a peak. The peak position shifts from small depths(close top plane, no snake) to large depths (close to bottom plane at 60, longsnake), when beta increases. Similar plots of the snake lengths were obtainedfor d = 2 and 3 (not shown); also one test for d = 4 , L = 29 displays aninterface between zero and positive width.We plot in Fig.5 the average (over ten samples of 10,000 iterations each)of the position of the lowest hyper-plane touched by the work distributionprocess as a function of β . Although the statistics obtained are a clearindication that the depth of the system follows the same trend from 1+1 to1+3 dimensions they are not directly interpreted. The measured depth is4n fact the sum of the length of the snake part plus the height of the blobpart. Both parts vary with β . The snake length is obtained from equation(3) since q ( z ) is simply Q − z . But the blob height depends upon the chargeat the interface and we don’t have any simple analytic expression for it.Figure 6 shows the total number N t of sites which were involved in atleast one of the 10,000 iterations, i.e. the total work force with long-timeemployment, and also at the fluctuations N f in the work force from oneiteration to the next. (Thus N f is the number of sites which are used atiteration t and not at time t −
1, or the other way around.) We see that forlarge β the fluctuations are diminished (the same snake tail again and againpasses on the work), but this decrease is accompanied by an increase in N t ,an effect which helps the labour market but not the company.In the above version, also the managers who distributed work to theirlower neighbours took over one work unit each for themselves. If instead theygive on all the workload given to them (provided it was at least two units),then each iteration requires not only Q people as above, but Q workers plusa fluctuating number of managers. Moreover, if the snake hits the bottomline at depth L z = Q , part of the work is never finished. This is hardly anefficient way to run a business, but Figure 7 and 8 show a much sharpertransition, from a localised cluster at low β to delocalized snake tails at high β . The simple reinforcement learning presented here does end-up in a metastablepath in the worker space represented here by the snake + blob picture, whichwe interpret as a firm. On the other hand we would rather imagine firms ashierarchical structures such as trees [4, 5] . Because of the blob-snake sharptransition as a function of z , we never observe a well balanced tree with aselection at each node of several preferred collaborators, but rather either anearly complete preference for one neighbour or roughly equal preference forall. In conclusion, the present model explains the metastability of employmentrelations in the firm, but something has to be added to it to explain the moreefficient workload repartition observed in real firms.The present manuscript was written during GW and DS stay at thephysics department of the Hebrew University in Jerusalem which we thank5or its hospitality. It was supported by GIACS, a Coordination Action of theEuropean communities. References [1] R. Coase, Economica, 4, 386-3405, (1937).[2] G. Weisbuch, A. Kirman, and D. Herreiner, The Economic Journal 110,411 (2000).[3] J.-P. Nadal, G. Weisbuch, O. Chenevez, and A. Kirman. In: JacquesLesourne and Andr´e Orlean, editors, Advances in Self-Organization andEvolutionary Economics, pages 149–159. Economica, (1998).[4] G. Toulouse and J. Bok, Revue fran¸caise de Sociologie, 19:3, 391 (1978).[5] D. Braha, Phys. Rev. E 69, 016113 (2004)D. Helbing, H. Ammoser and C. Kuhnert, Physica A 363, 149 (2006)D. Stauffer and P.M.C. de Oliveira, Int. J. Mod. Phys. C 17, 1367 (2006)6
10 100 1000 1 10 100 1000 10000 f l o w , dep t h iterationsFlow (up), depth (down), d=1, L=29 10 100 1000 1 10 100 1000 10000 f l o w , dep t h iterationsFlow (up), depth (down), d=4, L=29 Figure 1: Top: Equilibration of the sum of all preference coefficients (topcurve), and depth at β = 0 . , γ = 0 . , , Q = 60, averagedover 10,000 iterations. Bottom: Same parameters except for d = 4.7 Figure 2: One instance of the work load repartition from an initial load of20 at the top site until the lower line. The load is initially distributed with astrong preference for one neighbour out of three and is then more uniformlyfrom load 7. β = 0 . , γ = 0 . , Q = 20.8 p r e f e r en c e J depthSnake tails for beta = gamma = 0.03, Q=20 0 20 40 60 80 100 120 140 160 180 200 14 15 16 17 18 p r e f e r en c e J depthExpanded plot of blob region, all data Figure 3: Evolution in 1+ d dimensions of the preference coefficient with dis-tance from the ”surface”. For smaller depths, in the tail region, the preferencecoefficients are either strong (( q ( z ) − /γ ) and independent of the dimension d , or zero. A transition is observed around resp. charges of 5, 4 and 3 ratherthan for 2 d +1 = 7, 5 and 3 resp. as predicted by the mean field theory (equa-tion (3). Top part: Overall picture emphasising the snake. Bottom part: En-largement of blob region. Q = 20 , β = 0 . , γ = 0 . , d = 1(+) , x ) , ∗ ).9 s qua r e s i g m a positionbeta=.01, .03, .05, .1, .2, .4 left to right Figure 4: Variation with vertical position of the square width of the workingregion within a horizontal hyper-plane (= line), averaged over the last 5,000of 10,000 iterations. γ = 0 . , d = 1 , Q = 60 Similar results were obtainedfor d = 2 and 3. D ep t h betaTen 60*30^d samples d=1(top), 2(middle), 3(bottom) Figure 5: Variation with β of the vertical position of the lowest working planefor several dimensions (1+1,1+2,1+3). Enlargement to L = 300 or L z = 600gives no significant changes. 10 f l u c t ua t i on , t o t a l betaTotal(up), fluctuation(down); d=1,2,3 bottom to top Figure 6: Variation with β of the total number of different people who workedduring at least one of the 10,000 iteration (three top curves), and of thefluctuations in the work force (three bottom curves); L = 30 , L z = 60, onesample. Date are averages over the last 1000 iterations.11