Using Game Theory To Encourage Cooperation in Levee System Planning

Delta Island Flooding

A fragile levee system threatens to flood Delta islands. Credit: USACE, Sacramento District

By Rui Hui, Jay Lund and Kaveh Madani

Levees protect land from floods, but not perfectly. Different levees on a river often are controlled by different agencies or groups. A landowner on one riverbank sees the levee system differently from a landowner on the opposite bank or downstream. Each landowner, or elected levee board, is likely to make levee planning decisions considering only local benefits and costs, and not those to other levee districts. Collaborating with the other levee owners might provide system-wide benefits, perhaps sometimes sacrificing a little local benefit for larger gains elsewhere.

Our study, recently published in Water Resources Research, applies game theory to this levee system problem with risk-based analysis of individual and collaborative levee decisions. As expected, it finds that collaborative planning can be economically best for the overall system. But, collaboration among many levee authorities often is impractical without more centralized authority or appropriate compensation to some local areas. Rational and self-interested landowners on opposite riversides can be myopic and independently optimize their own levees, resulting in a less efficient system (a “tragedy of commons”). Game theory can provide solutions to encourage cooperation.

Flood protection with levees

Levees can increase channel capacity to protect adjacent areas from floods, but they can fail by overtopping and other causes (Figure 1). Flood risk to economic activity is the likelihood of flooding times the magnitude of losses, and usually is measured by economic impacts. Levees decrease, but cannot eliminate the likelihood of flooding and flood risk.

Alternative levee plans distribute flood risks differently. Flooding in a leveed river system depends largely on levee heights. A symmetric levee system has identical levees (and failure probabilities) on opposite riversides (Figure 2(a)), while an asymmetric levee system has a lower levee more likely to fail (Levee 1 in Figure 2(b), Levee 2 in Figure 2(c)). Total flood risk could be lowered by transferring risk from a high-cost urban side to a lower-valued rural side with a higher urban levee, which can increase rural costs. So better system-wide solutions are not necessarily acceptable for all stakeholders, and compensation might be required. Various transaction costs and legal and political barriers often prevent such compensation.


Figure 2. Profile view of (a) symmetric levees with the same height, (b) asymmetric levees with lower Levee1, and (c) asymmetric levees with lower Levee2

Historic non-cooperation in levee systems

In flood-prone river basins, individual landholders sometimes lack incentives to cooperate in levee planning with other landholders upstream, downstream, or across the river. Historically, non-cooperation often causes damaging outcomes and conflicts.

In California, levees have been used since the mid-1800s. From 1867 to 1880, levee districts along the Sacramento River raced one another to build higher levees on each riverside. Flood-prone landholders raised their levees to force floodwater onto their neighbors. The resulting escalation of levees in the Sacramento Valley became ineffective and economically inefficient, and ultimately led to violence as it became less expensive to demolish the opposing levee than to strengthen one’s own (Kelley 1989).

Similar levee battles have occurred elsewhere. During a major flood, flood risk transfer through lower levees also can occur by breaching the levee on the lower-valued side or raising the levee on the opposite riverside. Such a “levee battle” happened in the Mississippi floodplain during the post-Civil War boom near New Orleans. Due to many breaks in adjoining areas (Plaquemines Parish), rumors gradually arose that levees were purposefully weakened to save more valuable city property on the opposite river bank. A worse unexpected situation appeared after the 1849 flood on the Mississippi River that broke the levee at River Ridge, where uptown residents thought of strengthening the levee on their side, but those living on the opposite side threatened to prevent this by armed force.

Game theory analysis

Game theory, which examines how independent and self-interested individuals interact, can help in analyzing each landholder’s levee strategy. Non-cooperative game theory helps examine short-sighted decision-making and potential ways to guarantee a better system-wide solution. Each landholder would decide its own levee height, without considering the system-wide economic costs and impacts on others. Payoffs for game outcomes are the total average annual cost for each landholder, including average annual damage and annualized construction cost. These payoffs of individual decisions drive decisions and individual strategy.

An example illustrates game theory application to levee system planning for various institutional conditions. Both riversides are assumed rural areas for the symmetric river channel system, while Riverside 1 is rural and Riverside 2 is urban for the asymmetric system.

Table 1 summarizes different institutional and flood damage cases and results.


The system-wide least-cost plan has the minimum total of levee construction cost and overall average flood damage. However, without interference (e.g. authority or compensation) to support collaboration, economically inefficient plans are likely (a “tragedy of commons”). A rational landholder may have no equilibrium best strategy, get stuck in cycling best response decisions (Figure 3) or accept inferior converged stable heights. A farsighted landholder who foresees inferior end results may take the “strategic loss” to reduce conflict and attain better long-term benefits. A system-wide decision-maker/regulator that understands the effects of short-sightedness can design mechanisms to create an overlap between stability and optimality.


Figure 3. Best response (BR) curves of levee heights and changing trends for each landowner in response to the other’s previous height decision, in the multiple-shot leveed river system planning where decisions are reversible

A cooperative game can determine how to allocate the benefit from cooperation, where appropriate compensation or authority can incentivize or force collaboration among players. Institutions and compensation schemes should give all parties the greatest support for joining a grand coalition supporting the best overall solution.

By examining game outcomes representing different institutional arrangements, decision-makers, funders, and regulators can better assess how to lead and organize river system plans for better and more stable overall outcomes, which usually require cooperation.

Dr. Rui Hui is an analyst in the Program Management Office in California ISO and recently completed her dissertation on levee system risk analysis and game theory.  Jay Lund is a Professor of Civil and Environmental Engineering at the University of California – Davis.  Kaveh Madani is a Senior Lecturer in Environmental Policy at Imperial College, London.

Further reading

Hui, R., J. R. Lund, and K. Madani (2015), Game theory and risk-based leveed river system planning with noncooperation, Water Resour. Res., 51, doi:10.1002/2015WR017707.

Barry, J. M. (1997). Rising tide: The great Mississippi flood of 1927 and how it changed America. Simon and Schuster. TOUCHSTONE, Rockefeller Center, 1230 Avenue of the Americas, New York, NY 10020.

Croghan, L. (2013). Economic Model for Optimal Flood Risk Transfer. Master’s Thesis. Civil and Environmental Engineering Department, UC Davis.

Gordon, H. S. (1954). The Economic Theory of a Common-Property Resource: The Fishery. Journal of Political Economy, 62, 124-142

Hanak, E., Lund, J., Dinar, A., Gray, B., Howitt, R., Mount, J., Moyle, P., & Thompson, B. (2011). Managing California’s water: From conflict to reconciliation. Public Policy Instit. of CA.

Kelley, R. (1989). Battling the Inland Sea: Floods, Public Policy and the Sacramento River. Univ. of Cal. Press: Berkeley, CA.

Madani, K. (2010). Game theory and water resources. Journal of Hydrology, 381(3), 225-238.

Madani, K., & Hipel, K. W. (2012). Non-cooperative stability definitions for strategic analysis of generic water resources conflicts. Water Resources Management, 25(8), 1949-1977.

Pinter, N., Huthoff, F., Dierauer, J., Remo, J. W., & Damptz, A. (2016). Modeling residual flood risk behind levees, Upper Mississippi River, USA. Environmental Science & Policy, 58, 131-140.

Read, L., Madani, K., & Inanloo, B. (2014). Optimality versus stability in water resource allocation. Journal of environmental management, 133, 343-354.

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2 Responses to Using Game Theory To Encourage Cooperation in Levee System Planning

  1. Mike Hamblett says:

    In Britain all levee management is by one authority and heights each side are normally equal. New thinking considers the deliberate flooding of agricultural land to relieve the towns, bearing in mind that crop production in the floodplain is innappropriate. Especially now that more land is used to produce feed for animals than directly for human consumption. Finally thought is being given to the affect of massive deforestation in the catchment on ability to hold back water as in a natural system; instead of trying to raise levee further, investigate land-use changes as an increased of floodlevels. Some land must be permanently sacrificed and returned to its natural function as flood plain, as in the delta photo above.

  2. Martijn van Staveren says:

    Interesting analysis using game theory. In the Netherlands, levee heights on both sides of rivers are usually equal. Debates about risks and risk transfer usually concerns upstream areas in relation to downstream (and vice versa). In several cases, levees are lowered for river widening purposes, allowing for flood diversion/retention in selected locations. The effect of such projects is discussed only in relation to up/downstream effects such as lowered peak water levels, not particularly about what happens on the immediate ‘other side’ of the river.

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