{"title": "Causal Strategic Inference in Networked Microfinance Economies", "book": "Advances in Neural Information Processing Systems", "page_first": 1161, "page_last": 1169, "abstract": "Performing interventions is a major challenge in economic policy-making. We propose \\emph{causal strategic inference} as a framework for conducting interventions and apply it to large, networked microfinance economies. The basic solution platform consists of modeling a microfinance market as a networked economy, learning the parameters of the model from the real-world microfinance data, and designing algorithms for various computational problems in question. We adopt Nash equilibrium as the solution concept for our model. For a special case of our model, we show that an equilibrium point always exists and that the equilibrium interest rates are unique. For the general case, we give a constructive proof of the existence of an equilibrium point. Our empirical study is based on the microfinance data from Bangladesh and Bolivia, which we use to first learn our models. We show that causal strategic inference can assist policy-makers by evaluating the outcomes of various types of interventions, such as removing a loss-making bank from the market, imposing an interest rate cap, and subsidizing banks.", "full_text": "Causal Strategic Inference in Networked\n\nMicro\ufb01nance Economies\n\nMohammad T. Irfan\n\nDepartment of Computer Science\n\nBowdoin College\n\nBrunswick, ME 04011\n\nmirfan@bowdoin.edu\n\nLuis E. Ortiz\n\nleortiz@cs.stonybrook.edu\n\nDepartment of Computer Science\n\nStony Brook University\nStony Brook, NY 11794\n\nAbstract\n\nPerforming interventions is a major challenge in economic policy-making. We\npropose causal strategic inference as a framework for conducting interventions\nand apply it to large, networked micro\ufb01nance economies. The basic solution\nplatform consists of modeling a micro\ufb01nance market as a networked economy,\nlearning the parameters of the model from the real-world micro\ufb01nance data, and\ndesigning algorithms for various causal questions. For a special case of our model,\nwe show that an equilibrium point always exists and that the equilibrium interest\nrates are unique. For the general case, we give a constructive proof of the exis-\ntence of an equilibrium point. Our empirical study is based on the micro\ufb01nance\ndata from Bangladesh and Bolivia, which we use to \ufb01rst learn our models. We\nshow that causal strategic inference can assist policy-makers by evaluating the\noutcomes of various types of interventions, such as removing a loss-making bank\nfrom the market, imposing an interest rate cap, and subsidizing banks.\n\n1\n\nIntroduction\n\nAlthough the history of micro\ufb01nance systems takes us back to as early as the 18th century, the\nfoundation of the modern micro\ufb01nance movement was laid in the 1970s by Muhammad Yunus,\na then-young Economics professor in Bangladesh.\nIt was a time when the newborn nation was\nstruggling to recover from a devastating war and an ensuing famine. A blessing in disguise may it be\ncalled, it led Yunus to design a small-scale experimentation on micro-lending as a tool for poverty\nalleviation. The feedback from that experimentation gave Yunus and his students the insight that\nmicro-lending mechanism, with its social and humanitarian goals, could successfully intervene in\nthe informal credit market that was predominated by opportunistic moneylenders. Although far from\nexperiencing a smooth ride, the micro\ufb01nance movement has nevertheless been a great success story\never since, especially considering the fact that it began with just a small, out-of-pocket investment\non 42 clients and boasts a staggering 100 million poor clients worldwide at present [27]. Yunus and\nhis organization Grameen Bank have recently been honored with the Nobel peace prize \u201cfor their\nefforts to create economic and social development from below.\u201d\nA puzzling element in the success of micro\ufb01nance programs is that while commercial banks dealing\nwith well-off customers struggle to recover loans, micro\ufb01nance institutions (MFI) operate without\ntaking any collateral and yet experience very low default rates! The central mechanism that MFIs use\nto mitigate risks is known as the group lending with joint-liability contract. Roughly speaking, loans\nare given to groups of clients, and if a person fails to repay her loan, then either her partners repay\nit on her behalf or the whole group gets excluded from the program. Besides risk-mitigation, this\nmechanism also helps lower MFI\u2019s cost of monitoring clients\u2019 projects. Group lending with a joint-\nliability contract also improves repayment rates and mitigates moral hazard [13]. Group lending\nand many other interesting aspects of micro\ufb01nance systems, such as ef\ufb01ciency and distribution of\n\n1\n\n\fintervening informal credit markets, failure of pro-poor commercial banks, gender issues, subsidies,\netc., have been beautifully delineated by de Aghion and Morduch in their book [9].\nHere, we assume that assortative matching and joint-liability contracts would mitigate the risks of\nadverse selection [13] and moral hazard. We further assume that due to these mechanisms, there\nwould be no default on loans. This assumption of complete repayment of loans may seem to be\nvery much idealistic. However, practical evidence suggests very high repayment rates. For example,\nGrameen Bank\u2019s loan recovery rate is 99.46% [21].\nNext, we present causal strategic inference, followed by our model of micro\ufb01nance markets and our\nalgorithms for computing equilibria and learning model parameters. We present an empirical study\nat the end. We leave much of the details to the Appendix, located in the supplementary material.\n\n2 Causality in Strategic Settings\n\nGoing back two decades, one of the most celebrated success stories in the study of causality, which\nstudies cause and effect questions using mathematical models of real-world phenomena, was the\ndevelopment of causal probabilistic inference. It was led by Judea Pearl, who was later awarded the\nACM Turing prize in 2011 for his seminal contribution. In his highly acclaimed book on causal-\nity, Pearl organizes causal queries in probabilistic settings in three different levels of dif\ufb01culty\u2014\nprediction, interventions, and counterfactuals (in the order of increasing dif\ufb01culty) [22, p. 38]. For\nexample, an intervention query is about the effects of changing an existing system by what Judea\nPearl calls \u201csurgery.\u201d We focus on this type of query here.\nCausal Strategic Inference. We study causal inferences in game-theoretic settings for intervention-\ntype queries. Since game theory reliably encodes strategic interactions among a set of players,\nwe call this type of inference causal strategic inference. Note that interventions in game-theoretic\nsettings are not new (see Appendix B for a survey). Therefore, we use causal strategic inference\nsimply as a convenient name here. Our main contribution is a framework for performing causal\nstrategic inference in networked micro\ufb01nance economy.\nAs mentioned above, interventions are carried out by surgeries. So, what could be a surgery in a\ngame-theoretic setting? Analogous to the probabilistic settings [22, p. 23], the types of surgeries we\nconsider here change the \u201cstructure\u201d of the game. This can potentially mean changing the payoff\nfunction of a player, removing a player from the game, adding a new player to the game, changing\nthe set of actions of a player, as well as any combination of these. We discuss other possibilities in\nAppendix A. See also [14].\nThe proposed framework of causal strategic inference is composed of the following components:\nmathematically modeling a complex system, learning the parameters of the model from real-world\ndata, and designing algorithms to predict the effects of interventions.\nReview of Literature. There is a growing literature in econometrics on modeling strategic scenarios\nand estimating the parameters of the model. Examples are Bjorn and Vuong\u2019s model of labor force\nparticipation [5], Bresnahan and Reiss\u2019 entry models [6, 7], Berry\u2019s model of airline markets [4],\nSeim\u2019s model of product differentiation [24], Augereau et al.\u2019s model of technology adoption [2].\nA survey of the recent results is given by Bajari et al. [3]. All of the above models are based on\nMcFadden\u2019s random utility model [18], which often leads to an analytical solution. In contrast, our\nmodel is based on classical models of two-sided economies, for which there is no known analytical\nsolution. Therefore, our solution approach is algorithmic, not analytic.\nMore importantly, although all of the above studies model a strategic scenario and estimate the\nparameters of the respective model, none of them perform any intervention, which is one of our main\ngoals. We present more details on each of these as well as several additional studies in Appendix B.\nOur model is closely related to the classical Fisher model [12]. An important distinction between\nour model and Fisher\u2019s, including its graphical extension [16], is that our model allows buyers (i.e.,\nvillages) to invest the goods (i.e., loans) in productive projects, thereby generating revenue that can\nbe used to pay for the goods (i.e., repay the loans). In other words, the crucial modeling parameter of\n\u201cendowment\u201d is no longer a constant in our case. For the same reason, the classical Arrow-Debreu\nmodel [1] or the recently developed graphical extension to the Arrow-Debreu model [15], does not\ncapture our setting. Moreover, in our model, the buyers have a very different objective function.\n\n2\n\n\f3 Our Model of Micro\ufb01nance Markets\n\nWe model a micro\ufb01nance market as a two-sided market consisting of MFIs and villages. Each MFI\nhas branches in a subset of the villages, and each branch of an MFI deals with the borrowers in that\nvillage only. Similarly, each village can only interact with the MFIs present there.\nWe use the following notation. There are n MFIs and m villages. Vi is the set of villages where MFI\ni operates and Bj is the set of MFIs that operate in village j. Ti is the \ufb01nite total amount of loan\navailable to MFI i to be disbursed. gj(l) := dj + ejl is the revenue generation function of village\nj (parameterized by the loan amount l), where the initial endowment dj > 0 (i.e., each village has\nother sources of income [9, Ch. 1.3]) and the rate of revenue generation ej \u2265 1 are constants. ri\nis the \ufb02at interest rate of MFI i and xj,i is the amount of loan borrowed by village j from MFI i.\nFinally, the villages have a diversi\ufb01cation parameter \u03bb \u2265 0 that quanti\ufb01es how much they want their\nloan portfolios to be diversi\ufb01ed. 1 The problem statement is given below.\nFollowing are the inputs to the problem. First, for each MFI i, 1 \u2264 i \u2264 n, we are given the total\namount of money Ti that the MFI has and the set Vi of villages that the MFI has branches. Second,\nfor each village j, 1 \u2264 j \u2264 m, we are given the parameters dj > 0 and ej > 1 of the village\u2019s\nrevenue generation function 2 and the set Bj of the MFIs that operate in that village.\nMFI-side optimization problem. Each MFI i wants to set its interest rate ri such that all of its loan\nis disbursed. This is known as market-clearance in economics. Here, the objective function is a\nconstant due to the MFIs\u2019 goal of market-clearance.\n\nmax\n\nri\n\nsubject to ri\n\n1\n\n\uf8eb\uf8edTi \u2212(cid:88)\n(cid:88)\n\nj\u2208Vi\nxj,i \u2264 Ti\n\n\uf8f6\uf8f8 = 0\n\nxj,i\n\n(PM )\n\nj\u2208Vi\nri \u2265 0\n\nVillage-side optimization problem. Each village j wants to maximize its diversi\ufb01ed loan portfolio,\nsubject to its repaying it. We call the second term of the objective function of (PV ) the diversi\ufb01cation\nterm, where \u03bb is chosen using the data. 3 We call the \ufb01rst constraint of (PV ) the budget constraint.\n\n(cid:88)\nsubject to (cid:88)\n\nxj =(xj,i)i\u2208Bj\n\nmax\n\ni\u2208Bj\n\ni\u2208Bj\nxj \u2265 0\n\n(cid:88)\n\ni\u2208Bj\n\nxj,i + \u03bb\n\nxj,i log\n\n1\nxj,i\n\nxj,i(1 + ri \u2212 ej) \u2264 dj\n\n(PV )\n\ni for each MFI i and a vector x\u2217\n\nFor this two-sided market, we use an equilibrium point as the solution concept. It is de\ufb01ned by an\ninterest rate r\u2217\nj,i)i\u2208Bj of loan allocations for each village j such\nthat the following two conditions hold. First, given the allocations x\u2217, each MFI i is optimizing the\nprogram (PM ). Second, given the interest rates r\u2217, each village j is optimizing the program (PV ).\nJusti\ufb01cation of Modeling Aspects. Our model is inspired by the book of de Aghion and Mor-\nduch [9] and several other studies [20, 26, 23]. We list some of our modeling aspects below.\n\nj = (x\u2217\n\n1For simplicity, we assume that all the villages have the same diversi\ufb01cation parameters.\n2When we apply our model to real-world settings, we will see that in contrast to the other inputs, dj and ej\nare not explicitly mentioned in the data and therefore, need to be learned from the data. The machine learning\nscheme for that will be presented in Section 4.2.\n\n3Note that although this term bears a similarity with the well-known entropic term, they are different, be-\n\ncause xj,i\u2019s here can be larger than 1.\n\n3\n\n\fObjective of MFIs. It may seem unusual that although MFIs are banks, we do not model them as\npro\ufb01t-maximizing agents. The perception that MFIs make pro\ufb01ts while serving the poor has been\ndescribed as a \u201cmyth\u201d [9, Ch. 1]. In fact, the book devotes a whole chapter to bust this myth [9, Ch.\n9]. Therefore, empirical evidence supports modeling MFIs as not-for-pro\ufb01t organizations.\nObjective of Villages. Typical customers of MFIs are low-income people engaged in small projects\nand most of them are women working at home (e.g., Grameen Bank has a 95% female customer\nbase) [9]. Clearly, there is a distinction between customers borrowing from an MFI and those bor-\nrowing from commercial banks. Therefore, we model the village side as non-corporate agents.\nDiversi\ufb01cation of Loan Portfolios. Empirical studies suggest that the village side does not maximize\nits loan by borrowing only from the lowest interest rate MFI [26, 23]. There are other factors, such\nas large loan sizes, shorter waiting periods, and \ufb02exible repayment schemes [26]. We added the\ndiversi\ufb01cation term in the village objective function to re\ufb02ect this. Furthermore, this formulation is\nin line with the quantal response approach [19] and human subjects are known to respond to it[17].\nComplete repayment of loans. A hallmark of micro\ufb01nance systems worldwide is very high repay-\nment rates. For example, the loan recovery rate of Grameen Bank is 99.46% and PKSF 99.51% [21].\nDue to such empirical evidence, we assume that the village-side completely repays its loan.\n\n3.1 Special Case: No Diversi\ufb01cation of Loan Portfolios\n\nits loan, i.e.,(cid:80)\nthat offer the lowest interest rate. That is, (cid:80)\n\nIt will be useful to \ufb01rst study the case of non-diversi\ufb01ed loan portfolios, i.e., \u03bb = 0. In this case, the\nvillages simply wish to maximize the amount of loan that they can borrow. Several properties of an\nequilibrium point can be derived for this special case. We give the complete proofs in Appendix C.\nProperty 3.1. At any equilibrium point (x\u2217, r\u2217), every MFI i\u2019s supply must match the demand for\nj,i = Ti. Furthermore, every village j borrows only from those MFIs i \u2208 Bj\nx\u2217\ni \u2212 ej) = dj for any MFI\nx\u2217\nj,i(1 + r\u2217\nk > r\u2217\nj,k = 0 for any MFI k such that r\u2217\n.\n\nj\u2208Vi\nmj \u2208 argmini\u2208Bj r\u2217\nProof Sketch. Show by contradiction that at an equilibrium point, the constraints of the village-side\nor the MFI-side optimization are violated otherwise.\nWe next present a lower bound on interest rates at an equilibrium point.\nProperty 3.2. At any equilibrium point (x\u2217, r\u2217), for every MFI i, r\u2217\n\ni > maxj\u2208Vi ej \u2212 1.\n\ni\u2208Bj ,r\u2217\n\ni =r\u2217\nmj\n\nProof Sketch. Otherwise, the village-side demand would be unbounded, which would violate the\n\ni , and x\u2217\n\nmj\n\nMFI-side constraint(cid:80)\n\nj,i \u2264 Ti.\nx\u2217\n\nj\u2208Vi\n\nFollowing are two related results that preclude certain trivial allocations such as all the allocations\nbeing zero at an equilibrium point.\nProperty 3.3. At any equilibrium point (x\u2217, r\u2217), for any village j, there exists an MFI i \u2208 Bj such\nthat x\u2217\n\nj,i > 0.\n\nProof Sketch. In this case, j satis\ufb01es its constraints but does not maximize its objective function.\nProperty 3.4. At any equilibrium point (x\u2217, r\u2217), for any MFI i, there exists a village j \u2208 Vi such\nthat x\u2217\n\nj,i > 0.\n\nProof Sketch. The \ufb01rst constraint of (PM ) for MFI i is violated.\n\n3.2 Eisenberg-Gale Formulation\n\nWe now present an Eisenberg-Gale convex program formulation of a restricted case of our model\nwhere the diversi\ufb01cation parameter \u03bb = 0 and all the villages j, 1 \u2264 j \u2264 m, have the same revenue\ngeneration function gj(l) := d + el, where d > 0 and e \u2265 1 are constants. We \ufb01rst prove that\nthis case is equivalent to the following Eisenberg-Gale convex program [11, 25], which gives us the\nexistence of an equilibrium point and the uniqueness of the equilibrium interest rates as a corollary.\nBelow is the Eisenberg-Gale convex program [11, p. 166].\n\n4\n\n\fm(cid:88)\nsubject to (cid:88)\n\nmin\n\nj=1\n\nz\n\n(cid:88)\n\ni\u2208Bj\n\nzj,i\n\n\u2212 log\n\nzj,i \u2212 Ti \u2264 0, 1 \u2264 i \u2264 n\n\n(PE)\n\nj\u2208Vi\nzj,i \u2265 0,\nWe have the following theorem and corollary.\nTheorem 3.5. The special case of micro\ufb01nance markets with identical villages and no loan portfolio\ndiversi\ufb01cation, has an equivalent Eisenberg-Gale formulation.\n\n1 \u2264 i \u2264 n, j \u2208 Vi\n\nProof Sketch. The complete proof is very long and given in Appendix C. We \ufb01rst make a connection\nbetween an equilibrium point (x\u2217, r\u2217) of a micro\ufb01nance market and the variables of program (PE).\nIn particular, we de\ufb01ne x\u2217\ni in terms of certain dual variables of (PE). Using\nthe properties given in Section 3.1, we show that the equilibrium conditions of (PM ) and (PV ) in\nthis special case are equivalent to the Karush-Kuhn-Tucker (KKT) conditions of (PE).\nCorollary 3.6. For the above special case, there exists an equilibrium point with unique interest\nrates [11] and a combinatorial polynomial-time algorithm to compute it [25].\n\nj,i and express r\u2217\n\nj,i \u2261 z\u2217\n\nAn implication of Theorem 3.5 is that in a more restricted case of our model (with the additional\nconstraint of Ti being same for all MFI i), our model is indeed a graphical linear Fisher model where\nall the \u201cutility coef\ufb01cients\u201d are set to 1 (see the convex program 5.1 [25] to verify this).\n\n(PV ) can be written as (cid:80)\n\n3.3 Equilibrium Properties of General Case\n\n\u03bb(cid:80)\n\ni\u2208Bj\n\nIn the general case,\n\nthe objective function of\n\nxj,i \u2212\nxj,i log xj,i. While the \ufb01rst term wants to maximize the total amount of loan, the sec-\nond (diversi\ufb01cation term) wants, in colloquial terms, \u201cnot to put all the eggs in one basket.\u201d If \u03bb is\nsuf\ufb01ciently small, then the \ufb01rst term dominates the second, which is a desirable assumption.\nAssumption 3.1.\n\ni\u2208Bj\n\n0 \u2264 \u03bb \u2264\n\n1\n\n2 + log Tmax\n\nwhere Tmax \u2261 maxi Ti and w.l.o.g., Ti > 1 for all i.\nThe following equilibrium properties will be used in the next section.\nProperty 3.7. The \ufb01rst constraint of (PV ) must be tight at any equilibrium point.\nProof Sketch. Otherwise, the village can increase its objective function slightly.\nWe de\ufb01ne ei\nProperty 3.8. At any equilibrium point, for each MFI i, ei\n\nmax \u2261 maxj\u2208Vi dj and obtain the following bounds.\n+ ei\n\nmax \u2261 maxj\u2208Vi ej and di\n\nmax\n\nProof Sketch. The proof of ei\nis derived from the maximum loan a village j can seek from the MFI i at an equilibrium point.\n\nmax \u2212 1 < r\u2217\n\nmax \u2212 1 < r\u2217\n\nmax \u2212 1.\ni is similar to the proof of Property 3.2. The upper bound\n\ni \u2264 |Vi|di\n\nTi\n\n4 Computational Scheme\n\nFor the clarity of presentation we \ufb01rst design an algorithm for equilibrium computation and then\ntalk about learning the parameters of our model.\n\n4.1 Computing an Equilibrium Point\n\nWe give a constructive proof of the existence of an equilibrium point in the micro\ufb01nance market\nde\ufb01ned by (PM ) and (PV ). The inputs are \u03bb > 0, ej and dj for each village j, and Ti for each MFI\ni. We \ufb01rst give a brief outline of our scheme in Algorithm 1.\n\n5\n\n\fmax \u2212 1.\n\nfor all MFI i do\n\nwhile Ti (cid:54)=(cid:80)\n\nAlgorithm 1 Outline of Equilibrium Computation\n1: For each MFI i, initialize ri to ei\n2: For each village j, compute its best response xj.\n3: repeat\n4:\n5:\n6:\n7:\n8:\n9:\n10: until no change to ri occurs for any i\n\nj\u2208Vi\n\nxj,i do\n\nend while\n\nend for\n\nChange ri as described after Lemma 4.3.\nFor each village j \u2208 Vi, update its best response xj re\ufb02ecting the change in ri.\n\nBefore going on to the details of how to change ri in Line 6 of Algorithm 1, we characterize the best\nresponse of the villages used in Line 7.\nLemma 4.1. (Village\u2019s Best Response) Given the interest rates of all the MFIs, the following is the\nunique best response of any village j to any MFI i \u2208 Bj:\n\n(cid:18) 1 \u2212 \u03bb \u2212 \u03b1\u2217\n\n(1)\n\n(2)\n\nwhere \u03b1\u2217\n\nx\u2217\nj,i = exp\nj \u2265 0 is the unique solution to\n\n(cid:18) 1 \u2212 \u03bb \u2212 \u03b1\u2217\n\n(cid:88)\n\ni\u2208Bj\n\nexp\n\n(cid:19)\n\nj (1 + ri \u2212 ej)\n\u03bb\n(cid:19)\n\nj (1 + ri \u2212 ej)\n\u03bb\n\n(1 + ri \u2212 ej) = dj.\n\nProof Sketch. Derive the Lagrangian of (PV ) and argue about optimality.\nTherefore, as soon as ri of some MFI i changes in Line 6 of Algorithm 1, both x\u2217\nj,i and the Lagrange\nj change in Line 7, for any village j \u2208 Vi. Next, we show the direction of these changes.\nmultiplier \u03b1\u2217\nLemma 4.2. Whenever ri increases (decreases) in Line 6, xj,i must decrease (increase) for every\nvillage j \u2208 Vi in Line 7 of Algorithm 1.\nProof Sketch. Rewrite the expression of x\u2217\nj,k for some k \u2208 Bj. Use the two expressions for \u03b1\u2217\nx\u2217\nThe next lemma is a cornerstone of our theoretical results. Here, we use the term turn of an MFI to\nrefer to the iterative execution of Line 6, wherein an MFI sets its interest rate to clear its market.\nLemma 4.3. (Strategic Complementarity) Suppose that an MFI i has increased its interest rate at\nthe end of its turn. Thereafter, it cannot be the best response of any other MFI k to lower its interest\nrate when its turn comes in the algorithm.\n\nj,i given in Lemma 4.1 in terms of \u03b1\u2217\n\nj to argue about the increase of ri.\n\nj . Do the same for\n\nj for j \u2208 Vi cannot increase.\n\nProof Sketch. The proof follows from Lemma 4.2 and Assumption 3.1. The main task is to show\nthat when ri increases \u03b1\u2217\nIn essence, Lemma 4.2 is a result of strategic substitutability [10] between the MFI and the village\nsides, while Lemma 4.3 is a result of strategic complementarity [8] among the MFIs. Our algorithm\nexploits these two properties as we \ufb01ll in the details of Lines 6 and 7 next.\nLine 6: MFI\u2019s Best Response. By Lemma 4.2, the total demand for MFI i\u2019s loan monotonically\ndecreases with the increase of ri. We use a binary search between the upper and the lower bounds\nof ri given in Property 3.8 to \ufb01nd the \u201cright\u201d value of ri. More details are given in Appendix D.\nLine 7: Village\u2019s Best Response. We use Lemma 4.1 to compute each village j\u2019s best response x\u2217\nto MFIs i \u2208 Bj. However, Equation (1) requires computation of \u03b1\u2217\nj,i\nj , the solution to Equation (2).\nWe exploit the convexity of Equation (2) to design a simple search algorithm to \ufb01nd \u03b1\u2217\nj .\nTheorem 4.4. There always exists an equilibrium point in a micro\ufb01nance market speci\ufb01ed by pro-\ngrams (PM ) and (PV ).\n\nProof Sketch. Use Lemmas 4.3 and 4.1 and the well-known monotone convergence theorem.\n\n6\n\n\f4.2 Learning the Parameters of the Model\n\nThe inputs are the spatial structure of the market, the observed loan allocations \u02dcxj,i for all village j\nand all MFI i \u2208 Bj, the observed interest rates \u02dcri and total supply Ti for all MFI i. The objective of\nthe learning scheme is to instantiate parameters ej and dj for all j. We learn these parameters using\nthe program below so that an equilibrium point closely approximates the observed data.\n\n(cid:88)\n\ni\n\ni \u2212 \u02dcri)2\n(r\u2217\n(cid:88)\ni \u2212 ej) \u2264 dj\n\ni\u2208Bj\n\nxj,i + \u03bb\n\nxj,i log\n\n1\nxj,i\n\n(3)\n\n(cid:88)\n\n(cid:88)\n\nj\u2208Vi\n\ni\n\nfor all j,\n\nmin\ne,d,r\n\nsuch that\n\nj,i \u2212 \u02dcxj,i)2 + C\n(x\u2217\n(cid:88)\n\ni\u2208Bj\n\nxj,i(1 + r\u2217\n\nj \u2208 arg maxxj\nx\u2217\n\ns. t. (cid:88)\n\ni\u2208Bj\nxj \u2265 0\nej \u2265 1, dj \u2265 0\n\n(cid:88)\n\nx\u2217\nj,i = Ti, for all i\n\nj\u2208Vi\nri \u2265 ej \u2212 1, for all i and all j \u2208 Vi\n\nThe above is a nested (bi-level) optimization program. The term C is a constant. In the interior\noptimization program, x\u2217 are best responses of the villages, w.r.t. the parameters and the interest\nrates r\u2217. In practice, we exploit Lemma 4.1 to compute x\u2217 more ef\ufb01ciently, since it suf\ufb01ces to search\nfor Lagrange multipliers \u03b1j in a much smaller search space and then apply Equation (1). We use the\ninterior-point algorithm of Matlab\u2019s large-scale optimization package to solve the above program.\nIn the next section, we show that the above learning procedure does not over\ufb01t the real-world data.\nWe also highlight the issue of equilibrium selection for parameter estimation.\n\n5 Empirical Study\n\nWe now present our empirical study based on the micro\ufb01nance data from Bolivia and Bangladesh.\nThe details of this study can be found in Appendix E (included in the supplementary material).\nCase Study: Bolivia\nData. We obtained micro\ufb01nance data of Bolivia from several sources, such as ASOFIN, the apex\nbody of MFIs in Bolivia, and the Central Bank of Bolivia. 4 We were only able to collect somewhat\ncoarse, region-level data (June 2011). It consists of eight MFIs operating in 10 regions.\nComputational Results. We \ufb01rst choose a value of \u03bb such that the objective function value of the\nlearning optimization is low as well as \u201cstable\u201d and the interest rates are also relatively dissimilar.\nUsing this value of \u03bb, the learned ej\u2019s and dj\u2019s capture the variation among the villages w.r.t. the\nrevenue generation function. The learned loan allocations closely approximate the observed alloca-\ntions. The learned model matches each MFI\u2019s total loan allocations due to the learning scheme.\n\nIssues of Bias and Variance. Our dataset consists of a single sample. As a result, the traditional\napproach of performing cross validation using hold-out sets or plotting learning curves by varying\nthe number of samples do not work in our setting. Instead, we systematically introduce noise to\nthe observed data sample. In the case of over\ufb01tting, increasing the level of noise would lead the\nequilibrium outcome to be signi\ufb01cantly different from the observed data. To that end, we used two\nnoise models\u2013Gaussian and Dirichlet. In both cases, the training and test errors are very low and the\nlearning curves do not suggest over\ufb01tting.\n\n4http://www.aso\ufb01nbolivia.com; http://www.bcb.gob.bo/\n\n7\n\n\fEquilibrium Selection.\nIn the case of multiple equilibria, our learning scheme biases its search\nfor an equilibrium point that most closely explains the data. However, does the equilibrium point\nchange drastically when noise is added to data? For this, we extended the above procedure using\na bootstrapping scheme to measure the distance between different equilibrium points when noise is\nadded. For both Gaussian and Dirichlet noise models, we found that the equilibrium point does not\nchange much even with a high degree of noise. Details, including plots, are given in Appendix E.\nCase Study: Bangladesh\nBased on the micro\ufb01nance data (consisting of seven MFIs and 464 villages/regions), dated Decem-\nber 2005, from Palli Karma Sahayak Foundation (PKSF), which is the apex body of NGO MFIs in\nBangladesh, we have obtained very similar results to the Bolivia case (see Appendix E).\n\n6 Policy Experiments\n\nFor a speci\ufb01c intervention policy, e.g., removal of government-owned MFIs, we \ufb01rst learn the pa-\nrameters of the model and then compute an equilibrium point, both in the original setting (before\nremoval of any MFI). Using the parameters learned, we compute a new equilibrium point after the\nremoval of the government-owned MFIs. Finally, we study changes in these two equilibria (before\nand after removal) in order to predict the effect of such an intervention.\nRole of subsidies. MFIs are very much dependent on subsidies [9, 20]. We ask a related question:\nhow does giving subsidies to an MFI affect the market? For instance, one of the Bolivian MFIs\nnamed Eco Futuro exhibits very high interest rates both in observed data and at an equilibrium point.\nEco Futuro is connected to all the villages, but has very little total loan to be disbursed compared to\nthe leading MFI Bancosol. Using our model, if we inject further subsidies into Eco Futuro to make\nits total loan amount equal to Bancosol\u2019s, not only do these two MFIs have the same (but lower than\nbefore) equilibrium interest rates, it also drives down the interest rates of the other MFIs.\nChanges in interest rates. Our model computes lower equilibrium interest rate (around 12%) for\nASA than its observed interest rate (15%). It is interesting to note that in late 2005, ASA lowered its\ninterest rate from 15% to 12.5%, which is close to what our model predicts at an equilibrium point. 5\nInterest rates ceiling. PKSF recently capped the interest rates of its partner organization to 12.5%\n[23], and more recently, the country\u2019s Micro\ufb01nance Regulatory Authority has also imposed a ceiling\non interest rate at around 13.5% \ufb02at. 6 Such evidence on interest rate ceiling is consistent with the\noutcome of our model, since in our model, 13.4975% is the highest equilibrium interest rate.\nGovernment-owned MFIs. Many of the government-owned MFIs are loss-making [26]. Our model\nshows that removing government-owned MFIs from the market would result in an increase of equi-\nlibrium interest rates by approximately 0.5% for every other MFI. It suggests that less competition\nleads to higher interest rates, which is consistent with empirical \ufb01ndings [23].\nAdding new branches. Suppose that MFI Fassil in Bolivia expands its business to all villages. It\nmay at \ufb01rst seem that due to the increase in competition, equilibrium interest rates would go down.\nHowever, since Fassil\u2019s total amount of loan does not change, the new connections and the ensuing\nincrease in demand actually increase equilibrium interest rates of all MFIs.\nOther types of intervention. Through our model, we can ask more interesting questions such as\nwould an interest rate ceiling be still respected after the removal of certain MFIs from the market?\nSurprisingly, according to our discussion above, the answer is yes if we were to remove government-\nowned MFIs. Similarly, we can ask what would happen if a major MFI gets entirely shut down? We\ncan also evaluate effects of subsidies from the donor\u2019s perspective (e.g., which MFIs should a donor\nselect and how should the donor distribute its grants among these MFIs in order to achieve some\ngoal). 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