Corruption versus efficiency in water allocation under uncertainty: is there a trade-off?

ABSTRACT In the absence of a cooperative solution to the problem of rights over shared water, water allocation through third-party intervention is most commonly used. This paper considers water allocation within a federal set-up with the requisite legal institutions to enforce third-party adjudication and tries to capture the politically charged motivations that often guide such allocations. It compares two mechanisms generally used by central planners to allocate water between up- and downstream regions, namely fixed and proportional allocation rules. By considering a corrupt central planner, this paper models the underlying political manoeuvring that drives the assignment of water rights. It is found that the politically pliable central planner’s choice of allocation rule depends on the expected state of nature. Interestingly, the corrupt central planner’s equilibrium choice of allocation rule turns out to be efficient, unless the problem of severe water scarcity is expected to occur.


INTRODUCTION
Shared water, almost everywhere, remains a source of conflict, which becomes notoriously intractable under arid conditions.It is well argued that under conditions of pure conflict, a negotiated solution is not possible as the initial allocation of rights itself is at stake (Richards & Singh, 2002).In the absence of negotiation between disputing parties, third-party intervention seems to be the most immediate way out, particularly in the case of within-country water disputes. 1Allocation of water by a central authority remains a frequently used means of assigning disputed water, especially in developing countries that face difficulties in allowing the free market to take over.This is because high transaction costs and the lack of contract enforcement obstruct the bargaining process over water (Richards & Singh, 2001).Further, for water to be efficiently allocated by the free market, there has to be a system of pure private property rights, which is largely absent from the character of any of the water doctrines followed around the world, namely, riparian rights, public allocation and prior allocation (Sampath, 1992).In the absence of any possibility of a negotiated solution or the institutional framework necessary for water markets, an enforceable public allocation system appears to be the most plausible way of water sharing under pure conflict within a country.
In this paper, we analyse the issue of water sharing between an upstream region and its downstream counterpart in a federal arrangement considering that dispute resolution by a central planner is binding upon the parties involved. 2The legal provisions in various federations show that the necessary institutional framework exists to render binding any centralized third-party intervention in case of interstate conflicts.For example, there are three ways of resolving interstate conflicts in the United States: by congressional Act, by the formation of an interstate compact approved by Congress or by an 'equitable apportionment' by the Supreme Court (Bennett et al., 2000).The constitution of India, via entry 56 in the Union List and article 262, enables the federal government to legislate and intervene effectively in the case of interstate disputes and gives it primacy over the Supreme Court, even though water issues fall within the jurisdiction of the states as in the United States (Richards & Singh, 2002).The Interstate River Water Disputes Act of 1956 in India and its subsequent amendments provide for the establishment of tribunals by the central government in the event of negotiations failing.Such tribunals are often considered to be an extension of the federal government and are not beyond influence, political or otherwise, and corruption (Katz & Moore, 2011;Richards & Singh, 2002).
It is well argued that political favouritism may be shown more directly through the biased allocation of government resources and favourable regulatory laxity in regions represented by politicians from the ruling party (Asher & Novosad, 2017).Beg (2019) documents that in the arid Indus basin, a region's alignment with the political party in power leads to favourable access to water during scarcity and pre-empting of flood risk during excess.The issue of corruption driving inefficiency in water allocation has been treated by an evolving strand of literature that largely tracks empirical evidence.The number of case studies on the distortionary effects of corruption also abounds.For example, Esteban et al. (2019) argue that in the Jucar River basin in Spain, water users of both up-and downstream regions spend considerable resources to organize and lobby for greater water allocation to their respective regions.In Jordan, patronage in the form of cheap water to certain shadow state agents yields political support to the Hashemite rulers (Hussein, 2018).A similar case may be made in respect of areas in Northern India where large farmers enjoying immense political heft have for long been able to retain access to cheap or free electricity and, thereby, unbridled control over groundwater, in the face of blatant over-exploitation of water.In a decentralized public irrigation system in Southern Punjab of Pakistan, it is found that corruption in water allocation involve not only the economically and politically powerful farmers but also sharecroppers and other lower social segments of rural society.Since this is not confidential information, corruption here happens to be a well-established working rule (Rinaudo, 2002).In the same Indus basin, lobbying in a canal system by farmers is found to induce both inequality and inefficiency (Jacoby & Mansuri, 2020).Corruption in water also threatens political stability and regional security in developing as well as industrialized countries (Transparency International, 2008).Wenzel (2021) argues that this risk of corruption in allocation of funds is heightened in the wake of drought or severe scarcity of water.However, to the best of our knowledge, this issue of influence of corruption has not received much attention in the existing theoretical literature on water disputes.
This paper considers a situation in which water flow in a river is uncertain and two regions, an upstream region and a downstream region, of a country have disputes concerning water rights.The conflict between the two regions is defined over the entire range of water flow, so that flooding is as problematic as scarcity.The demand for water could be driven by a host of exogenous factors such as historical use or complementary investment in infrastructure.
To keep the analysis focused, it is assumed that ancillary investments have already been undertaken, determining the contours of the conflict ex-ante.The dispute over water sharing between the two regions is resolved through intervention by a third party (henceforth, the central planner or planner).
The central planner is considered to be corruptible and both up-and downstream regions try to influence the planner's choice of water allocation through contributions.The contributions can be thought of as encompassing a gamut of political influence-building tactics (monetary or otherwise) undertaken by the disputing regions. 3The central planner favours one region over the other if the former contributes more than the latter; otherwise, the central planner's water allocation decision is free from any bias.Each region is interested in maximizing its own benefit from water usage net of its cost of contribution, while the central planner's objective is to maximize total contribution.It is assumed that the preferences of each of the three players, that is, of the central planner and the two disputing regions, and the implication of contributions by region(s) are exogenously determined and are common knowledge.
This paper considers a three-stage sequential move game.In the first stage of the game, the central planner chooses one of the two most frequently used water allocation rules -fixed allocation or proportional allocation.Next, in the second stage, the upstream region and the downstream region simultaneously and independently decide their respective contribution levels.Finally, in the third stage, the central planner decides the fixed amount or the proportion of total water flow to be allocated to the downstream region depending on the allocation rule decided in the first stage, before uncertainty regarding total water flow is resolved.Considering the sub-game perfect Nash equilibrium (SPNE) water allocation of this game to be legally binding and enforceable, this paper demonstrates the following.
It shows that the proportional allocation rule leads to higher total benefits from water usage by the two regions and, thus, is more efficient than the fixed allocation rule, under uncertainty.A benevolent central planer always chooses the efficient allocation rule.Interestingly, corruption does not necessarily lead to inefficiency.The corrupt central planner chooses the more (less) efficient allocation rule, that is, proportional (fixed) allocation rule, if at the average level of water flow, the problem of severe water scarcity does not (does) occur.This is because each region contributes more in the equilibrium under the proportional allocation rule compared with that under the fixed allocation rule, if the expected flow of water is sufficiently large; otherwise, the opposite occurs.It implies that whether efficiency will be compromised in the case of corruption or not depends on the state of nature.However, note that the corrupt central planner never aims to achieve efficiency: it occurs in some cases as mere coincidence.
This paper also carries out a set of robustness checks of the main result by considering a number of different scenarios.First, it analyses the implications of alternative fixed allocation rules that implicitly make one of the two regions face the entire uncertainty in water flow.Second, it examines the implications of sequential choices of contributions by regions (leader-follower).Third, it extends the analysis by allowing for more than one downstream region.Fourth, the issue of credibility of the central planner's announced rule of allocation and its implications are discussed.Finally, it examines possible implications of timings of moves by reversing the ordering of the first two stages of the original game.Interestingly, the main result of this paper goes through in each of these alternative scenarios.Further, it demonstrates that, despite being identical in terms of benefit and cost functions, conflicting regions contribute different amounts and the social planner allocates water in a biased manner in the equilibrium, provided that either the two conflicting regions choose their contributions sequentially or there is more than one downstream region.
We note here that this paper is closely related to Bennett et al. (2000).Considering fixed and proportional water allocation rules, they compare the efficiency of interstate water compacts based on two alternative allocation rules and evaluate their sensitivities to mean water flow and variance.While they also allow for the central planner to be biased towards one of the two regions, in their model the extent of bias is exogenously determined.On the contrary, in this paper, possible bias of the central planner is endogenously determined.Further, unlike Bennett et al., this paper assesses implications of alternative allocation rules on corruption and the effect of corruption on equilibrium allocation.
This paper presents a simple model of behaviour often found in the actions of agents bound up in a tangle of political necessities and reciprocities.The planner here is as much a rational agent as the rest of the players, and so it is futile to expect him not to take care of his own interests.This sort of scenario fits quite well into the landscape of federal politics where the central government invariably seeks out support from lower levels of government and, in return, bestows upon them favourable verdicts.The eventual nature of alignments or coalitions is determined through nothing else but the calculus of relative benefits drawn from various partners.
There is a growing literature on distributive politics, which seeks to parse out politically charged motivations that drive public good provisioning.Golden and Min (2013) find that distributive politics can be studied in terms of effects on core or swing voters, political favouritism to population subgroups or in terms of political business cycles that map the timing of fiscal allocations with electoral cycles.Dixit and Londregan (1996) find that parties tend to direct partisan allocations towards swing voters than core political constituencies of rival parties who need even larger transfers to switch loyalties.Arulampalam et al. (2009), in a model of an opportunistic central government that transfers resources to states, use data from India to show that an aligned and swing state is likely to get 16% more grants than a state which is unaligned and non-swing.Livert and Gainza (2018) show that in Chile municipalities with mayors belonging to the party ruling in the national government receive more funds than other municipalities.In Brazil, Chile and Colombia, transfer distributions are found to be mediated by electoral concerns (Gainza & Livert, 2021).Cadot et al. (2006) observe that infrastructure investment in the transportation sector in French regions between 1985 and 1992 was driven by lobbying and electoral incentives.Writing on political budget cycles, Shi and Svensson (2006) observe that public expenditure before elections is beneficial to the incumbent when voters fail to distinguish between election-driven fiscal policy manipulations and incumbent competence.Rodríguez-Pose et al. (2016) find that during the period 1975-2009, Greek governing parties tended to reward constituencies that returned them to office.In India, state governments run by parties that represent lower castes tend to devote larger fractions of government spending towards the benefit of these castes (Teitelbaum & Thachil, 2010).This paper complements this stream of the empirical literature by offering a theoretical analysis of pursuance of self-interest by central planner, unproductive resource utilization by regions for the purpose of influence building, and allocative efficiency in allocation of watera critical good for human welfare (Hansen, 2022).
The paper also provides a template for studying the choice between competing rules of water allocation, which can be extended beyond the two considered here, in light of institutional path dependence and other historical influences that shape regional preferences for water allocation rules (Punjabi & Johnson, 2019).This assumes significance given the rising spectre of regional conflicts over water between cities and rural areas (Meinzen-Dick & Pradhan, 2005;Saleth & Dinar, 2004) and the pre-eminence of cities that leads to peri-urban blindness of policymakers and loss of rural access to water in terms of quantity and quality (Narain et al., 2013).
The rest of the paper is organized as follows: section 2 presents the model, characterizes equilibrium outcomes under alternative rules of water allocation, and analyses the corrupt planner's optimal choice of the rule.Analysis of the efficient allocation rule and its comparison with the corrupt planner's optimal choice is presented in section 3. Section 4 discusses further issues.Section 5 concludes.Proofs are presented in Appendix A in the supplemental data online.

THE SET-UP
There are two regions, upstream (U ) and downstream (D), with conflicting interests over a shared river.These two regions and the river are located within the boundary of a single country, which has a federal set-up.We mention here that regions in this model could be interpreted as firms or any productive entities with claims to a shared river.Water flow, W ( ≥ 0), in the river is random and is assumed to have the following distribution: where W h and W l denote, respectively, high and low water flow: W h .W l ≥ 0. Water flow W is fully allocated between the two regions.It is assumed that there is no free disposal of water, that is, we do not ignore water involved due to flooding in case of excess water flow.Region i's benefit from water usage is given by B i (W i ), where W i (≥ 0) denotes the amount  Johnson et al. (1981) argue that for efficient allocation of water, property rights must be defined in terms of consumptive use and not diversion.Following this argument, we assume that the entire water allocation is used for consumptive purposes.Now, if W l , W * i , W h , i = U , D, the probability of the event of water scarcity and the probability of the event of flood are both strictly positive for each region in case that region receives the entire water flow (W ).Further note that if W , W * U + W * D , there is the problem of scarcity.Alternatively, if total water flow W . W * U + W * D , there is the problem of flooding.
We assume (a , which implies that each situationwater scarcity and floodingoccurs with positive probability, regardless of whether water is shared between the two regions or only one region receives the entire flow.In either of the two situations, scarcity or flooding, there are disputes between the two regions over water sharing, which they fail to resolve by themselves through cooperation.The central planner, who is a third party, intervenes in the process of dispute settlement and resolves the dispute once and for all.Allocation of water prescribed by the central planner is considered to be legally binding and enforceable. In order to resolve the dispute, the central planner first chooses the allocation rule, which can be either a fixed allocation rule or a proportional allocation rule.In case of fixed allocation rule, the central planner chooses a fixed amount of water W 0 [ (0, W h ) such that the downstream region will receive (a) W 0 amount of water, if W 0 , W , or (b) the entire water W , if W 0 ≥ W , and the upstream region will receive the remaining water, if any.On the other hand, in case of proportional allocation rule, the central planner decides the proportion b [ (0, 1) of total water flow W to be allocated to the downstream region, which implies that the upstream region will receive (1 − b)W amount of water.The central planner chooses W 0 [ (0, W h ) or b [ (0, 1), depending on the predefined allocation rule, so that the expected value of a weighted sum of regions' benefits from water usage is maximum, which is as follows: (2) where l(≥ 0) is the weight assigned by the central planner to the downstream region's benefit à la Bennett et al. (2000).The weight parameter l measures the bias of the central planner.The central planner is said to be biased towards the downstream (upstream) region, if l . 1 (l , 1).However, unlike Bennett et al., C U , the central planner will be biased towards the downstream region (l .1); while the opposite occurs, if C D , C U .It is assumed that the central player's bias is given by l = l(C D , C U ), where and Z are assumed to be common knowledge. 4 Each region incurs cost to make contribution.Let region i's cost to contribute C i be given by cC 2 i 2 , where c .0 is the cost parameter; i = U , D.
There are three stages of the game as follows: . Stage 1: The central planner chooses the water allocation rule, fixed allocation versus proportional allocation, which maximizes the sum of the contributions from the two regions O = C D + C U . .Stage 2: The two regions decide their respective levels of contribution they make to the planner, simultaneously and independently. .Stage 3: The central planner decides the fixed amount (W 0 ) or the proportion (b), depending on the allocation rule decided in the first stage, of water to be allocated to the downstream region, such that the expected weighted benefit E[Z] is maximized.
We solve this game by backward induction by considering for simplicity, where a ( .0) and b ( .0) are benefit parameters; i = U , D. We assume that: and W . a b , where W = E(W ), which ensures the existence of unique interior equilibrium and stability of the equilibrium in each of the two cases, fixed and proportional, considered in this paper.Note that in stage 3 the central planner's objective is to maximize the expected weighted benefits, where weight l depends on the contributions of the two regions.l increases in the downstream region's contribution and decreases in the upstream region's contribution.That is, the corrupt central planner allocates water based on a biased objective function.
We first solve stage 3 and then stage 2 by considering fixed allocation rule and proportional allocation rule separately.

Fixed allocation rule
First consider that fixed allocation rule has been chosen by the central planner in the first stage of the game.In this case, the central planner decides the fixed amount of water W 0 [ (0, W h ) in stage 3, which implies that 108 Rupayan Pal et al.

REGIONAL STUDIES
water flows in the downstream region and in the upstream region are, respectively, as follows: Therefore, expected weighted benefits from water usage by the two regions under fixed allocation rule is as follows, where subscript F indicates fixed allocation rule.
Solving the problem of the central planner in stage 3, max , we obtain the following.
Lemma 2.1: Suppose that the fixed allocation rule is in force.
Then, for any given bias of the central planner l (≥ 0), the central planner's optimal choice of the fixed amount of water is given by: and the corresponding equilibrium gross benefits from water usage of the downstream region and the upstream region are, respectively, as follows: where Proof: See Appendix A in the supplemental data online.
It is easy to check that: Therefore, the fixed allocation to the downstream region is . So, if the planner's bias towards the downstream region is more, he allocates more water to the downstream region in case water scarcity is expected W , 2a b , while he allocates less water to the downstream region in case flood is expected to occur W . 2a b .Further, from (5b) it follows that each region gets more water when the expected water flow is higher, but the positive effect of expected flow on downstream (upstream) region's share decreases (increases) with the increase in central planner's bias towards the downstream region.It implies that the central planner serves the interest of the downstream region (by tempering the award in a situation of overabundance), at the expense of the upstream region, when he is biased towards the downstream region.Under fixed allocation rule, efficient fixed allocation (W E 0 ) maximizes the total benefit from water usage by the two regions and, thus, W 0 = W E 0 equates downstream region's expected marginal benefits from water usage to that of upstream region: In contrast, in the present scenario the central planner's optimal choice of W 0 is given by: that is, at the central planner's optimal choice, the upstream region's expected marginal benefit is less (greater) than the downstream region's marginal benefit when the central planner is biased towards (against) the upstream region, that is, when l , 1 (l .1).It follows that if the central planner is biased towards (against) the upstream region, its optimal choice of the fixed amount of water for the downstream region is less (more) than the efficient level.
Let E[G i, F ] denote the expected net benefit from water usage of region i(= U , D) under fixed allocation where

is easy to
Corruption versus efficiency in water allocation under uncertainty: is there a trade-off?109 REGIONAL STUDIES observe the following.
. 0 (6a) . 0 (6b) From ( 6a) and (6b), it follows that it is optimal for each of the two regions to make positive contributions to the central planner.The reason is as follows: the central planner's bias towards the downstream region is increasing (decreasing) in downstream (upstream) region's contribution: mean of total water flow is less than a critical level, (a) the central planner allocates higher (lower) amount of water to the downstream (upstream) region in case he is more biased towards the downstream region; and (b) a region's expected gross benefit from water usage is increasing in expected water flow in that region; otherwise, opposites are true. 5 Further, it is always optimal for the downstream region to increase its contribution in response to an increase in the upstream region's contribution, that is, the downstream region's contribution-reaction function is always upward sloping in the C D C U plane, that is, the downstream region considers its own contribution and the upstream region's contribution to be strategic complements.On the other hand, corresponding to an increase in the downstream region's contribution, the upstream region's best response is to reduce its contribution unless the downstream region's contribution is sufficiently low, as depicted in Figure 1.That is, the upstream region considers that the two regions' contributions are strategic complements (sub- A in the supplemental data online for the proof).
In stage 2, problem of region i is max Solving these two problems simultaneously, we obtain the following.
Lemma 2.2: In the case of fixed allocation rule under uncertainty, the downstream region and the upstream region contribute equally in order to align with the central planner in the equilibrium and the equilibrium contribution of each region is equal to: . 0.
Proof: See Appendix A in the supplemental data online.
First, note that each region contributes a positive amount to influence the central planner's allocation decision in the equilibrium, although it incurs a sufficiently high cost in doing so.Second, given the mean share of water ( W ) and the rate of increase in marginal cost of contribution (c), the equilibrium contribution of a region is increasing in the probability of actual water flow to be greater than the fixed amount of water allocated to the downstream region (n).Implying that water flow variability affects the equilibrium contribution levels.Third, it is interesting to observe that, though the upstream region alone bears the entire risk due to uncertainty in water flow, it makes the same level of contribution as that of the downstream region in the equilibrium.The reason is as follows.
Regions are identical in terms of their cost and benefit functions.Further, the marginal effect of an increase in a region's contributions on its expected amount of water is the same for each of the two regions (conditions 5a), ceteris paribus, since ∂ l ∂C U = − ∂l ∂C D .Therefore, starting from scenario with C D = C U = 0, (a) each region's marginal net benefit of its own contribution is the same when the other region does not contribute; and (b) each region's marginal net loss due to other region's contribution is the same, when it does not contribute.As a result, each region ends up contributing the same amount in the equilibrium.Now since each region contributes the same amount, the central planner remains unbiased in the equilibrium: It implies that each region is worse off in the equilibrium when the central planner is corrupt compared with the scenario in which the central planner is not corruptible.That is, from the regions' perspective, this is a prisoner's dilemma case.Further, note that the sum of the contributions made by two regions to a corrupt social planner is his bribe income, and being corrupt, the social planner is likely to use this amount for his private benefit.It implies that social welfare (i.e., total welfare minus bribe amount) is less under corruption than that under no corruption.From Lemmas 2.1 and 2.2, the SPNE fixed amount chosen by the central planner (W * 0 ), expected water flow E[W i, F ] * to region i(= D, U ) and the corrupt central planner's payoff O * F are, respectively:

Proportional allocation rule
We now turn to analyse the equilibrium under proportional allocation rule.When proportional allocation rule is in force, in stage 3 of the game, the central planner decides the proportion, b [ (0, 1), of water flow to be allocated to the downstream region.It implies that the downstream region will receive W D = bW amount of water, while the upstream region will receive W U = (1 − b) W amount of water, where W denotes the total flow of water in the river.Thus, expected benefits from water usage of the downstream region and the upstream region are, respectively, as follows: Therefore, expected weighted benefit from water usage by the two regions under proportional allocation rule is given by: where bW )] are given by (7b) and (7a), respectively, and subscript P denotes proportional allocation rule.Now, solving the central planner's problem in stage 3, max b[(0, 1) E[Z] P , we obtain the equilibrium proportions of total water flow for the two regions, given their contributions, as follows.
Lemma 2.3: Suppose that proportional allocation rule is in place for water allocation.Then, given the central planner's bias (l), in the equilibrium the proportion of water accruing to the downstream region is given by: and that to the upstream region by: , where e is a small positive number.
Proof: See Appendix A in the supplemental data online.
It is easy to check the following: where Assuming that b(l) [ (0, 1), from (9a) we can state the following.An increase in the central planner's bias towards the downstream region, that is, an increase in l, never leads to greater proportion of water accruing to the down- , that is, if there is water abundance at the mean flow or if variation in water flow is sufficiently large.In other words, greater bias of the central planner towards the downstream region serves the downstream region's interests to a greater extent at the expense of the upstream region.Further, condition (9b) implies that the proportion of water allocated to the downstream region rises, while that to the upstream region falls, with increase in mean water flow when the coefficient of variation is > 1 and the planner cares less for the downstream region.
When the mean water flow falls, ceteris paribus, the proportion allocated to the downstream region falls and that to the upstream rises.This essentially means that the planner, when he cares less about the downstream region, makes it bear a greater amount of risk in the face of highly variable water flow by reducing the downstream proportion with fall in mean flow.
Note that the central planner optimal choice of b is given by: whereas efficient level of b (which maximizes joint benefit of water usage by the two regions) satisfies the condition of marginal benefit equalization across regions: Clearly, the corrupt central planner's optimal choice b * is not efficient.If the central planner is biased towards the downstream region (l .1), the downstream (upstream) regions marginal benefit from water usage is over (under) emphasized by the central planner.
Let E[G i, P ] denote the expected net benefit from water usage of region i(= U , D) under proportional allo- where E(B i ) P = E[B i (W ) |b = b(l)] (assuming interior solution).Then, we have the following: 6 . 0 (10a) Clearly from (10a) and (10b) it follows that (i) a region can induce the corrupt central planner to be biased in its favour by making contributions; and (ii) each region would make positive contribution in the equilibrium under proportional allocation rule, as observed in the case of fixed allocation rule.Moreover, the downstream region considers that the contributions of the two regions are strategic complements.On the other hand, the upstream region considers that the contributions of the two regions are strategic complements (substitutes) if: (see Appendix A in the supplemental data online for the proof).That is, the strategic nature of regions' contributions does not depend on the rule of allocation, fixed or proportional.Thus, the implications of strategic nature of contributions on equilibrium outcomes in the case of proportional allocation rule will be the same as that in the case of fixed allocation rule.Now, solving the two regions' problems in stage 2, max taneously, we derive the equilibrium contribution of each region under proportional allocation rule.
Lemma 2.4: Suppose that the corrupt central planner allocates water between the two regions according to the proportional allocation rule.Then it is optimal for each region to contribute the amount C P to influence the central planner's choice of proportions, where: ) .
Proof: See Appendix A in the supplemental data online.
Lemma 2.4 states that each region contributes equally to the central planner in the equilibrium under proportional allocation rule, as under fixed allocation rule (Lemma 2.2).It follows that in the equilibrium, the central planner gives equal weight to each region's expected benefit from water usage, regardless of the rule of allocation.From Lemmas 2.3 and 2.4, under the proportional allocation rule, the SPNE proportion of total water flow allocated to the downstream region (b * ), expected water flow E[W i, P ] * to region i(= D, U ) and the corrupt central planner's payoff O * P are, respectively: , and ) .

Corrupt planner's choice of allocation rule: fixed versus proportional
Now the question is: What is the corrupt central planner's optimal choice of water allocation rule -fixed or proportional, in stage 1 of the game?Comparing the central planner's equilibrium payoff, which is given by total contribution received by the central planer from the two regions in the equilibrium, under the fixed allocation rule with that under the proportional allocation rule, we obtain the following.
Proposition 2.1: In the SPNE, the corrupt central planner's choice of water allocation rule under uncertainty is as follows: • It is optimal for the corrupt central planner to choose the fixed allocation rule, if severe water scarcity is expected, that is, if W [ (W l , W ), where: • The corrupt social planner prefers the proportional allocation rule over the fixed allocation rule, if expected water flow in the river is either excessive W . Proof: See Appendix A in the supplemental data online.
Note that in the absence of uncertainty in water flow, the proportional allocation rule is synonymous with the fixed allocation rule and, thus, regions contribute the same amount under alternative allocation rules.To illustrate it further, note that, whenever s = 0, we must have W l = W h = W and: Therefore, the corrupt social planner is indifferent between the two allocation rules in case water flow is certain, regardless of whether there is water scarcity or water abundance.This together with Proposition 2.1 implies that existence of uncertainty in water flow has a significant bearing on the corrupt social planer's equilibrium choice of the allocation rule.

EFFICIENT ALLOCATION RULE: HONEST SOCIAL PLANNER
An allocation rule is said to be efficient, if that allocation rule results in maximum total expected benefit from water usage by the two regions.If the social planner is honest (i.e., not corruptible), he always strictly prefers the more efficient allocation rule.That is, the honest social planner will prefer the fixed allocation rule over the proportional allocation rule, if the sum of expected benefits from water usage, , is greater under fixed allocation rule compared with that under proportional allocation rule.In such a scenario, the possibility influencing the social planner's decisions through contributions does not exist, and thus l = 1 holds true always.Now, under the fixed allocation rule, the honest social planner chooses the fixed amount W 0 by solving the following problem: Solving the above problem, we get On the other hand, under proportional allocation rule, the honest social planner chooses the proportion b by solving the following problem: Solving the above problem, we get b = 1 2 = b * * .Thus: Note that: It follows from the above discussion that the honest social planner always chooses the proportional allocation rule to allocate water under uncertainty, that is, the proportional allocation rule is more efficient than the fixed allocation rule.The intuition is as follows: note that: and thus: It implies that in the absence of uncertainty, the fixed allocation rule and the proportional allocation rule are equally efficient.Under uncertainty, the two disputing regions equally share the risk involved in the case of proportional allocation rule, whereas one of the two regions (the upstream region in the present analysis) bears disproportionately greater share of the risk in the case of fixed allocation rule.Since regions' benefit functions are considered to be the same, risk sharing is more efficient in the case of proportional allocation rule compared with that in the case of fixed allocation rule.Thus, when there is uncertainty in water flow, the proportional allocation rule turns out to be more efficient than the fixed allocation rule.From Proposition 2.1 and the above discussion, it is evident that when there is uncertainty in water flow, the corrupt social planner chooses the inefficient rule of water allocation only if severe water scarcity is expected to occur (W l , W , W ); otherwise, the SPNE under corruption is efficient.
Since (a) uncertainty in water flow is a prevalent phenomenon in the real world and (b) the proportional allocation rule is more efficient than the fixed allocation rule under uncertainty in water flow, a benevolent social planner should choose the proportional allocation rule over the fixed allocation rule.This observation together with Proposition 2.1 seems to imply the following.Instances of fixed allocation rules in river compacts, as observed in the case of the Colorado River Compact in 1922 (Bennett et al., 2000) and in most allocation regimes in Organisation for Economic Co-operation and Development (OECD) countries (OECD, 2015), suggest possibilities of corruption/lobbying in the process of water dispute resolutions.In other words, it is more likely that corruption/lobbying has taken place in case conflicts over water have been settled through a fixed allocation rulebased river compact compared with a proportional allocation rule-based river compact.The higher likelihood of corruption in the case of fixed allocation rule-based river compacts might also be a reason for higher intensity of non-compliance in case of fixed allocation rule compared with that in case of proportional allocation rule, a phenomenon well documented by Bennett and Howe (1998).We note here that a detailed empirical analysis is necessary to test this hypothesis, which is beyond the scope of the paper, and we leave it for future research.
It is interesting to observe that while both water scarcity and excessive water flow are undesirable to each of the two regions, corruption leads to inefficiency in allocation only if the problem of severe water scarcity occurs at the average water flow.

FURTHER ISSUES
In this section we extend the analysis by considering alternative scenarios.In particular, unlike as in section 2, we have considered: (a) an alternative fixed allocation rule which assigns a fixed amount of water to the upstream region; (b) sequential decision making by the regions wherein the downstream region decides its level of contribution before the upstream region; and (c) water allocation among three regionsone upstream and two downstream regions, separately.We also discuss the issue of credibility of the central planner's commitment regarding the water allocation rule.Further, we discuss how reversing stages 1 and 2 of the main analysis impacts the results.

Fixed water allocation to upstream regions
Consider that, under fixed allocation rule, the central planner in stage 3 decides the fixed amount W 0 [ (0, W h ) of water to be allocated to the upstream region, instead of the downstream region, and the remaining water goes to the downstream region.In that case, the water flows in the two regions are as follows: Analogous to the case of fixed allocation rule in sub-section 2.1, solving the central planner's maximization problem in stage 3, we get the central planner's optimal choice to be allocated to the upstream region, for any given bias, as follows: It can easily be observed that if the central planner's bias l towards the downstream region is more, he allocates more water to the upstream region in case flood is expected to occur: W . 2a b and he allocates less water to the upstream region in case water scarcity is expected W , 2a b . 7This is consistent with the case when the fixed amount of water is allocated to the downstream region.
Further, solving the regions' maximization problems in stage 2, we get the same results as Lemma 2.2 (in sub-section 2.2).The central planner remains unbiased in the equilibrium, and regions' contributions are given by: Therefore, the optimal water allocation to the regions and the central planner's payoff are same, whether the central planner allocates the fixed amount of water to the downstream region or the upstream region.It follows that our results are not sensitive to which region, up-or downstream, gets the fixed amount of water under fixed allocation rule.

Sequential contributions by two regions
In section 2, we analysed the situation when the upstream region and the downstream region choose their respective contributions in stage 2, simultaneously and independently.In this sub-section, we discuss the implications of sequential contributions by the two regions.
We consider that the downstream region chooses its contribution level first, and then observing its contribution level, the upstream region chooses its contribution level.Therefore, the stages of the game are as follows: . Stage 1: The central planner chooses the water allocation rule, fixed allocation versus proportional allocation, which maximizes the sum of the contributions from the two regions O = C D + C U . .Stage 2: The downstream region chooses its contribution level C D , to maximize its expected benefit. .Stage 3: Observing the contribution level of the downstream region C D , the upstream region chooses its contribution level C U , to maximize its expected benefit. .Stage 4: The central planner decides the fixed amount (W 0 ) or the proportion (b), depending on the allocation rule decided in the first stage, of water to be allocated to the downstream region, such that the expected weighted benefit E[Z] is maximized.
We solve the game by backward induction, by considering fixed allocation rule and proportional allocation rule separately, and obtain the following result.The downstream region contributes more than the upstream region in the equilibrium, regardless of the water allocation rule -fixed or proportional.The reason behind this asymmetric contribution in the equilibrium is as follows: It is optimal for the upstream region (the follower) to contribute a lower amount in response to an increase in contribution by the downstream region (the leader).This is because when the downstream region increases its contribution, the upstream region's benefit from an increase in its own contribution is less than the increase in cost of contribution.Thus, being the leader, the downstream region can benefit more by setting a higher level of contribution compared with the upstream region.It implies that the central planner will be biased in the equilibrium towards the downstream region, unlike as in the case of simultaneous choice.Nonetheless, the relative comparison of the total contribution (i.e., bribe income of the central planner) under alternative water allocation rules remains the same as in the case of simultaneous choices by the regions (i.e., as in Proposition 2.1) (see Appendix A in the supplemental data online for details).

Water allocation among three regions
Consider the following set-up.There are three regions, two downstream (D 1 and D 2 ) and one upstream (U ), with conflicting interests over a shared river.The distribution of water flow and the regions' benefit functions from the water usage are same as in the model with two regions.
Similar to the case with two regions, the central planner, in order to resolve the dispute, chooses either a fixed allocation rule or a proportional allocation rule.Under fixed allocation rule, the central planner chooses fixed amounts of water W D 1 and W D 2 which will go to the downstream regions D 1 and D 2 , respectively, and the upstream region will receive the remaining water.In order to keep the analysis tractable without compromising on the qualitative results, we assume W D 1 + W D 2 , W , where W is the total water flow, so that the upstream region will receive (W − W D 1 − W D 2 ) amount of water.In case of proportional allocation rule, the central planner decides the proportions b 1 and b 2 (b 1 , b 2 [ (0, 1)) of total water flow W to be allocated to the downstream regions D 1 and D 2 , respectively, and the upstream region will receive (1 − b 1 − b 2 )W amount of water.The central planner chooses W D 1 and W D 2 or b 1 and b 2 , depending on the chosen allocation rule, in order to maximize the expected value of a weighted sum of regions' benefits from water usage, which is as follows: where W D i = W D i (b i W ) under fixed allocation rule (proportional allocation rule).l 1 and l 2 (≥ 0) are the weights assigned by the central planner to the benefits of downstream regions D 1 and D 2 , respectively, and are defined as follows, given that contribution level of region i is The parameter l i (i = 1, 2) measures the central planner's relative bias towards the downstream region D i .l i is determined endogenously, similar to the analysis with two regions.
Each region i's cost to contribute is given by c C 2 i 2 , and benefit by There are three stages of the game, which are given as follows: . We solve the game by backward induction, by considering fixed allocation rule and proportional allocation rule separately, and obtain the following result.Regardless of the allocation rule, each downstream region makes more contribution than the upstream region in the equilibrium.The intuition is as follows: The weight functions l 1 and l 2 are such that corresponding to one unit increase in contribution by the downstream region i(= 1, 2), each of the other two regions needs to contribute by one unit in order to keep l i unchanged.That is, one unit of contribution by a downstream, region needs to be matched by two units of contributions from other two regions in order to make the central planner to treat that downstream region and the upstream region equally.As a result, each downstream region has a greater incentive to contribute compared with that of the upstream region.It follows that in the equilibrium the corrupt central planner is biased towards the downstream regions, while the extent of the bias is same for the two downstream regions.However, the relative comparison of the total contribution of the regions (i.e., the corrupt central planner's payoff), under fixed allocation rule and proportional allocation rule, remains qualitatively the same as in the case of two regions with simultaneous contributions (i.e., Proposition 2.1 holds true qualitatively even in the case with three regions) (see Appendix A in the supplemental data online for the details).

Credibility of commitment
Given the stages of the game in section 2, a natural question arises: will the central planner stick to his announced allocation rule in stage 1?We argue, in this sub-section, that there does not arise the issue of credibility of the central planner's commitment in the present context.
To illustrate this, let us consider the case where the central planner announces fixed allocation rule in stage 1.Suppose that the regions believe that the central planner in stage 3 will implement the announced allocation rule, with probability p [ [0, 1].
Solving stage 3 and, subsequently, stage 2 of the game, we get the following equilibrium contribution level of the two regions (details are provided in Appendix A in the supplemental data online): Differentiating the equilibrium contribution level with respect to p, we obtain: It is easy to verify that: where O * F and O * P are the central planner's equilibrium payoffs under fixed allocation rule and proportional allocation rule, respectively.
Therefore, using Proposition 2.1, we can say that (i) whenever it is optimal for the central planner to choose the fixed allocation rule, regions' contribution levels increase with the extent of credibility of the central planner's announcement of the fixed allocation rule in stage 1; and (ii) by a similar argument, whenever the central planner's payoff is higher under proportional allocation rule, regions' contribution levels increase with the extent of credibility of announcement of proportional allocation rule in stage 1.It implies that the central planner has no incentive to deviate from his announced policy ex-post.In other words, in the present context, there does not exist any credibility issue regarding whether the central planner will actually stick to his announced allocation rule or not.Therefore, results of this paper hold true in a dynamic game of repeated interactions as well.
Further, suppose that in the event of severe scarcity, the government directly intervenes and induces the central planner to choose proportional rule with probability t(0 , t , 1), that is, the planner cannot stick to his announced rule ex-post with probability t, in case of severe scarcity.Then, anticipating this possibility of deviation by the central planner, regions will decide their contributions accordingly.It is easy to observe that this case is similar to the case where regions believe that the planner's announced rule is credible with probability t 0 (say).Then, if W [ (W l , W ), and central planner chooses the fixed allocation rule, then the central planner's payoff can be written as (1 − t 0 )O * F + t 0 O * P , which is greater than O * P for all t 0 [ [0, 1].Thus, results of this analysis hold true even in case the credibility of commitment issue arises due to possible interventions by the higher up authority or any other exogenous factor.

Reversing stages 1 and 2 of the main analysis
Consider that the regions make their respective contributions first, then the central planner announces the allocation rule -fixed or proportional.That is, the stages of the game are as follows: . Stage 1: The two regions decide their respective levels of contribution they make to the planner, simultaneously and independently. .In stage 3, the central planner maximizes the expected weighted benefits of the regions, given the contributions of the regions (i.e., for any given bias).Now, suppose that the regions believe that fixed allocation rule will be announced, and, subsequently, will be chosen, with a probability q [ [0, 1], and proportional allocation rule with a probability (1 − q).Then, the expected benefit of region where E[G i,F ] and E[F i,P ] are region i's expected net benefit under fixed allocation rule and proportional allocation rule, respectively.
The regions' maximization problems are similar to that in the case of credibility of commitment, discussed in subsection 4.4.Therefore, we can say that whenever severe water scarcity is expected, it is optimal for the regions to contribute considering that the fixed allocation rule will be chosen, and in case excessive or moderately scarce water flow is expected, the regions will contribute considering that the proportional allocation rule will be chosen.That is, Proposition 2.1 holds true in this case as well.

CONCLUSIONS
In this paper, we have developed a simple model to understand the implications of corruption on efficiency in water allocation between two conflicting regions by a third party (central planner) in a federal arrangement.Considering that the conflicting regions are symmetric, we have shown that, given the rule of water allocation -fixed or proportional, even under uncertainty, the corrupt planner opts for efficient allocation of water between conflicting regions in the equilibrium.However, corruption has significant distortionary effects on the planner's choice of the rule of allocation when water flow is uncertain, unlike as in the case of certain water flow.In the equilibrium, the corrupt planner opts for inefficient rule of allocation in a scenario in which severe water scarcity occurs on average.In other words, a corrupt planner compromises with 116 Rupayan Pal et al.

REGIONAL STUDIES
efficiency for his own private benefit when severe water scarcity is expected to occur.This is a robust result.Empirically, our findings suggest that a policy choice of an inefficient rule of allocation could possibly point towards an underlying corrupt mechanism.In broader terms, this result seems to suggest that while corruption may not portend inefficiency in resource-rich nations, prevalence of corruption does hurt efficiency of allocation in resource-poor nations.Note that the corrupt social planner chooses the efficient rule of allocation only when it is aligned with his own private gain.In other words, deadweight loss due to corruption occurs even when the chosen rule of allocation is efficient.Thus, while in some cases allocative efficiency may not suffer due to corruption, socially wasteful utilization of resources by regions due to corruption continues to hinder their development and growth prospects.Empirical evidence of such negative effects of corruption abounds.For example, Lewis (2017) documents that in Indonesia outcomes of local government spending on health, education and infrastructure services are poorer in more corrupt regions; analysing a rich panel data of Italian regions, Lisciandra and Millemaci (2017) find that there is a significant negative effect of corruption on long-term economic growth of each Italian region; Del Monte et al. ( 2022) complements the findings of Lisciandra and Millemaci (2017) by demonstrating that poor quality of institutions results in delay in project completion (see also Gillanders, 2014;Charron et al., 2014).The present analysis attempts to account for corruption induced losses of regions by considering 'costs of contribution', which enters additively in regions' net benefit functions.While such consideration simplifies the analysis to a large extent, it ignores the possible negative effect of corruption at the central level on quality of local governance and, hence, on utilization of water at the local level.
In this analysis we have assumed that conflicting regions are symmetric, which helps to clearly identify the effects of uncertainty in water flow and corruption.Intuitively we can say that the distortionary effect of corruption will be more pronounced in case conflicting regions are asymmetric in terms of their benefits from water usages and/or costs of engaging in corrupt activities.This is because, in case of asymmetric regions, for any given rule of allocation, equilibrium contributions made by the two conflicting regions are likely to be different from each other, which will induce the corrupt planner to deviate from efficient allocation of water.Thus, in the case of asymmetric regions, the corrupt planner's choice of both (a) the rule of allocation and (b) given the rule of allocation, the levels of water allocated to different regions are likely to be distorted, even in the case of two conflicting regions and simultaneous contribution choices by them.Nonetheless, it seems to be interesting to characterize the equilibrium in the case of asymmetric regions.We also note here that this paper sidesteps the issue of endowment heterogeneity and associated constraints of regions to contribute for lobbying.However, in reality, regions differ in terms of their abilities to influence the central authority, due to regional differences in terms of revenue potential, population share (and, hence, vote shares), ethnolinguistic fragmentation, etc.
As discussed above, several empirical studies have attributed such endowment heterogeneity across regions as an important source of bias of central authorities, which further accentuates socio-economic inequality across regions.Therefore, it seems to be important to examine, both theoretically and empirically, implications of endowment heterogeneities in the context of the present analysis.We leave these for future research.Clearly, u U (C U , C D ) and u D (C U , C D ) can be interpreted as contest success functions of up-and downstream regions, respectively (Skaperdas, 1992(Skaperdas, , 1996;;Baik, 1998;Ansink & Weikard, 2009).Now, for any given C U and C D : and, thus, maximization of E( Z) with respect to W 0 (orb) is equivalent to the maximization of E(Z) with respect to W 0 (orb) where: cannot be of the same sign.
Stage 1: The central planner chooses the water allocation rule, fixed allocation versus proportional allocation, which maximizes the sum of the contributions from the three regionsO = C D 1 + C D 2 + C U .. Stage 2: The three regions decide their respective levels of contribution they make to the central planner, simultaneously and independently. .Stage 3: The central planner decides the fixed amounts (W D 1 and W D 2 ) or the proportions b 1 and b 2 , depending on the allocation rule decided in the first stage, of water to be allocated to the downstream regions, such that the expected weighted benefit E[Z] is maximized.
Stage 2: The central planner chooses the water allocation rule, fixed allocation or proportional allocation, which maximizes the sum of the contributions from the two regions O = C D + C U . .Stage 3: The central planner decides the fixed amount (W 0 ) or the proportion (b), depending on the allocation rule decided in the second stage, of water to be allocated to the downstream region, such that the expected weighted benefit E[Z] is maximized.
suppose that:Z = u U (C U , C D )B U (W U ) + u D (C U , C D )B D (W D ),where u U (C U , C D ) and u D (C U , C D ) are the weights give to benefits of up-and downstream regions, respectively.Further,u U (C U , C D ) + u D (C U , C D ) = 1, u i (C i , C j ) ∀i, j [ {U , D}and i = j.

Z(
= E(B U (W U ))+ u D (C U , C D ) u U (C U , C D ) E(B D (W D )), u D (C U , C D ) u U (C U , C D ) = l.Note that l = 1+C D −C U (the functional form considered in this analysis) is supported by contest success functions: It is easy to check that:u D (C, C) = u U (C, u D (C U , C D ) + u U (C U , C D ) = 1.) 5.If W , 2a b , we have: ∂W 0 ∂l .0, ∂( W − W 0 ) ∂l ,can be checked that either both: ∂E(B D ) P ∂b and ∂b(l) ∂l are positive, or both are negative.On the other hand, ∂E(B U ) P ∂(1 − b) and ∂(1 − b) ∂l .