Implications of “victim pays” infeasibilities for interconnected games with an illustration for aquifer sharing under unequal access costs



[1] This paper considers application of interconnected game theory to modeling of bilateral agreements for sharing common pool resources under conditions of unequal access. Linking negotiations to issues with reciprocal benefits through interconnected game theory has been proposed in other settings to achieve international cooperation because it can avoid outcomes that are politically unacceptable due to the “victim pays” principle. Previous studies have not considered adequately the critical nature of this political infeasibility, if it exists, in determining advantages of interconnection. This paper investigates how game structure and benefits suggested by interconnected game theory are altered when victim pays strategies are removed from the feasibility set. Linking games is shown to have greater advantages than when the structural implications of eliminating victim pays strategies are not considered. Conversely, a class of cases exists where the full cooperation benefits of interconnection are attainable without linking through isolated component games when victim pays outcomes are feasible.

1. Introduction

[2] Many water resource problems involve unidirectional externalities. For example, downstream users along a river may be at the mercy of upstream users with respect to both water quality and quantity. While binding agreements or regulations can be enforced by governments within national jurisdictions to address these externalities, neither regulation nor enforcement of agreements is typically possible at the international level. Rather, international agreements must be voluntarily sustainable. Participation in voluntary agreements related to unidirectional externalities is hard to achieve because at least one party has an incentive to defect. Even when cooperation on a one-sided issue is possible, a “victim pays” outcome is often required.

[3] Many regard victim pays outcomes as politically unacceptable or undesirable and some have investigated the associated implications [Maler, 1989, 1990; Markusen, 1975; Folmer and van Mouche, 2000], although a thorough underpinning and formal empirical evidence of this principle has not been offered. Alternatively, situations are occasionally observed where a country enters an agreement where cooperation appears to reduce its welfare without compensation. For example, the United States is in the upstream position with respect to salinity control on the Colorado River, and could ignore its “polluter pays” responsibility toward the downstream country, Mexico. Nevertheless, the United States committed by way of a 1972 agreement with federal funding through 2015 to keep the salinity level at the Mexican border below a certain level, apparently reducing U.S. welfare without compensation (see Bennett [2000] for further discussion). As another example, in a 1994 agreement Israel gave Jordan an additional 50 million cubic meters per year of drinkable water from the Yarmouk and Jordan Rivers even though Israel faces severe water scarcity and produces some high cost water through desalination (see the Israel-Jordan Peace Treaty, Annex II, Water Related Matters). If not viewed in conjunction with other agreements, this action appears to reduce Israel's welfare without compensation. Such agreements cannot be understood in the context of a single issue. Rather, countries are often willing to lose on one agreement in return for a gain on another.

[4] Game theory has become a common tool for understanding international agreements. In problems of unidirectional externalities, the characteristics of a prisoner's dilemma are often present whereby the most attractive actions for one county are detrimental to a neighboring country. The emphasis of game theory on individual rationality and strategic behavior has underscored the potential nonsustainability of cooperative agreements when players have significant differences in preferences and bargaining positions [Thomson and Lensberg, 1989; Osborne and Rubinstein, 1994]. However, Folmer et al. [1993] introduced interconnected game theory to capture the complexity of negotiations involving two or more issues when players have possibly offsetting preferences and positions. Interconnected game theory provides a way to expand the set of strategies available to players by connecting the independent issues. For example, a downstream country in a weak bargaining position related to water pollution or withdrawal may, by linking the water issue to another issue in which the downstream country has an offsetting advantage, be able to gain sufficient leverage to achieve cooperation without a monetary victim pays bribe. For example, while the U.S.-Mexico 1944 International Boundary Water Treaty appeared to leave the U.S. disadvantaged by giving up Colorado River water, Ragland [1995] used interconnected game theory to show that the U.S. was willing to give up water from the Colorado River in exchange for water from the Lower Rio Grande River for which the headwaters are controlled by Mexico.

[5] To date, interconnected games have been applied to understanding international water agreements primarily for the case of rivers with upstream-downstream issues [Dinar and Wolf, 1994; Ragland, 1995; Bennett et al., 1998]. This paper presents an example of how interconnected games offer similar possibilities for analyzing agreements under asymmetric access to common pool water resources. Countries may have unequal access costs for an international aquifer due to the geology, hydrology, and geography of the aquifer. With a common pool resource, each player may generate externalities that affect the other. However, the relative impact may be much greater in one direction than the other under unequal access, so that asymmetries similar to upstream-downstream problems are generated. As a result, the negotiating positions may be highly asymmetric and cooperation on aquifer sharing alone may be possible only through victim pays strategies. Also, cooperation may reduce the total amount of water pumped, for example, when salinization is caused by overpumping. If victim pays infeasibilities prevent cooperation, then the outcome is a typical prisoner's dilemma where the Nash (dominant) equilibrium is inefficient. Alternatively, joint cooperation on multiple issues with reciprocal benefits may achieve sustainable efficiency through interconnected game theory.

[6] While several studies of interconnected game theory have highlighted outcomes that do not require side payments [Dinar and Wolf, 1994; Ragland, 1995; Bennett et al., 1998], the implications of systematically removing strategies that involve side payments from the feasible set has not been examined. The purpose of this paper is to explore the structural implications of eliminating victim pays strategies in interconnected game theory. We show that removing these strategies can have substantive effects on the benefits attributable to interconnection. Isolated games may capture most or all of the benefits that are available when victim pays outcomes are politically feasible whereas the feasible set in the interconnected game may be considerably reduced when all strategies involving side payments are eliminated. These results are demonstrated in a stylized application of interconnected game theory to the Israeli-Palestinian sharing of the Mountain Aquifer in Israel. This example is presented with hypothetical data because, while agreement on numerical data is difficult in the highly contested negotiations between Israel and the Palestinian Authority, qualitative benefits are relatively clear conceptually and can be sufficient to determine the structure of the dominant interconnected agreement.

[7] Section 2 briefly reviews the apparent political infeasibility or, at least, political undesirability of victim pays agreements that has emerged in the literature. Section 3 casts the Israeli-Palestinian problem of sharing its Mountain Aquifer as a prisoner's dilemma when considered in isolation. Section 4 outlines how the benefits of interconnected games depend on the structure of the isolated games. Section 5 applies the conclusions of section 4 to identify an alternative isolated issue that has reciprocal benefits and structure suitable for linking to the Mountain Aquifer sharing problem. Section 6 establishes the benefits of the aggregated isolated games, which are critical for determining the magnitude of additional benefits achieved by interconnected game theory. Section 7 demonstrates that interconnected games do not always generate additional benefits over playing isolated games because, under a broad range of circumstances, both games have the same full cooperation point even though the set of feasible outcomes is expanded. On the other hand, section 8, by focusing on the implications of removing victim pays strategies, shows that the primary advantage of interconnection may be the achievement of full cooperation without using victim pays strategies, as opposed to achieving full cooperation alone. Section 9 presents conclusions.

2. Potential Political Infeasibility of Victim Pays Agreements

[8] Both monetary and in-kind side-payments have been suggested as mechanisms to compensate countries that become worse off from an international agreement [e.g., Maler, 1989, 1990; Markusen, 1975]. However, side payments are rare in practice. Maler [1990] argues that victim pays agreements are politically unacceptable for two reasons. First, the victim pays principle is associated with the notion of a weak negotiator and can negatively affect the victimized player in future negotiations. Second, countries are engaged in other unrelated conflicts and negotiations so that “in-kind side-payments” can be a superior alternative to monetary side-payments. Instead of side-payments, countries that share common conflicts of contrasting advantages can reciprocate concessions in order to reach agreement. Thus a loss associated with an agreement on one issue can be offset by a gain from an agreement on another issue. Thus, instead of treating conflicts in isolation and resolving them by monetary side-payments, conflicts can be interconnected and side-payments can be avoided by reciprocation.

[9] Folmer et al. [1993] were first to demonstrate the advantages of modeling international cooperation by means of interconnected games. Initially, they considered a game where an international environmental problem is resolved by side-payments. Then, noting that victim pays side payments are often viewed as undesirable compared to the internationally more acceptable polluter pays principle, they suggest the alternative potential of interconnected games where the environmental issue is linked to a trade issue. Their results demonstrate that the threat of a country to defect from a second agreement if the other country defects from the first can be sufficient to avoid defection from either, so that indefinite cooperation can be achieved in repeated play. While Folmer et al. [1993] showed how countries can add new incentives and sustain cooperation through conditioning actions in one arena on outcomes in another (interconnection), Folmer and van Mouche [2000] have demonstrated additional advantages related to Pareto optimality, social welfare, and encouraging cooperation. However, these studies of interconnected games do not examine the implications of eliminating strategies that involve victim pays side payments. Rather, cooperation is achieved as a Nash equilibrium of the combined structure where, for example, in the work by Folmer et al. [1993], payoffs are carefully chosen so that an outcome without side payments is the equilibrium strategy of the interconnected game. With different payoffs, this may not be the case.

[10] Many studies have identified sustainable international gains through application of interconnected game theory following Folmer et al. [1993]. Carraro and Siniscalco [1994] link international agreements concerning abatement and R&D spillover. Barrett [1994] shows that linking environmental policy to international trade in the context of the Montreal Protocol attains sustainability based on the threat of imposing trade restrictions. Hauer and Runge [1999] examine application of interconnected games to international environmental linkages suggested by (1) Cataldo's [1992] example of U.S.-Canadian joint cooperation on SO2 emissions and Persian Gulf war policy, (2) Runge et al.'s [1994] example of U.S.-Mexican joint cooperation on environmental policy and the North American Free Trade Agreement (NAFTA), and (3) Pearce et al.'s [1995] example of the potential for joint cooperation on rain forest preservation and debt forgiveness. Kroeze-Gil and Folmer [1998] further use interconnected games in a cooperative setting to show that repeated games linking environmental problems to nonenvironmental problems can not only lead to international cooperation but assure sustainable cooperation. These studies highlight how interconnection can improve cooperation and sustainability by expanding the feasible set.

[11] To date, only a few studies have applied interconnected games to examine international cooperation on water resources. Ragland [1995] applies interconnected game theory to the U.S.-Mexico 1944 International Boundary Water Treaty. Bennett et al. [1998] apply it to unidirectional externalities along international rivers in central Asia and the Middle East. Dinar and Wolf [1994] link upstream-downstream water issues through trading water for water-saving technology. These studies show that international agreements on upstream-downstream water conflicts can be negotiated and understood through interconnected game theory when the negotiated outcome does not appear rational based on the water issue alone. However, because of the stark unidirectional flow of externalities in upstream-downstream conflicts, these applications underscore the motivation of using interconnected game theory to permit cooperation without using victim pays strategies.

[12] To our knowledge, only the research on which we are reporting explicitly applies interconnected games to common pool water resource problems, such as international aquifers. The application demonstrates that unequal access costs can cause the problem to approximate the unidirectional externalities of the upstream-downstream case, and thus similarly underscores the potential infeasibility of victim pays strategies.

3. Israeli-Palestinian Sharing of the Mountain Aquifer: A Prisoner's Dilemma

[13] The common pool Mountain Aquifer (MA) of Israel serves as a useful application to demonstrate that benefits of interconnection can depend primarily on the political infeasibility of victim pays strategies. The MA is a broad aquifer that covers much of Northern Israel and the West Bank. From a water divide that roughly bisects the West Bank in a north-south direction, water flows primarily west toward the Mediterranean Sea or east to the Jordan River, although a triangular region in the north flows northward to the Sea of Galilee. Access is unequal because the West Bank is a high-elevation area with a deep water table and high pumping costs, while much of the water percolates rapidly downhill to two major springs in Israel toward the Mediterranean Sea, making Israel's extraction cost negligible. Having no other water sources, the Palestinian Authority (PA) depends on Israel's water sharing both for water availability and to control pumping cost. Israeli water sharing takes the form of simply extracting less water because the PA can pump from parts of the MA in the West Bank, but only at a very high cost that is increasing in the amount of Israeli water extraction, which lowers the water table in the mountains.

[14] This water-sharing problem is illustrated in the context of game theory in Figure 1. Each cell contains two values. The left-hand value represents Israel's payoff and the right-hand value represents the PA's payoff. The strategies that yield each cell's payoff are indicated on the left side for Israel and across the top for the PA. To simplify the discussion, we consider only one alternative of Israeli water sharing for which only one victim pays bribe by the PA is potentially suitable to both players even though many quantitative sharing and side-payment arrangements are possible. The payoffs are hypothetical although numerical values are used for conceptual clarity. Use of hypothetical or conjectural payoffs is common in this literature. For example, Bennett et al. [1998] use this approach merely to suggest an order of preference rather than to represent the actual value of outcomes that arise under the various strategies. For interconnected games where good data and careful empirical work can avoid conflicts in quantitative assessment of payoffs, hypothetical analysis is not necessary and greater clarity can be provided. However, Botteon and Carraro [1998] appear to stand alone as a case where such careful empirical assessment has been feasible.

Figure 1.

The water sharing game.

[15] In another study [Just and Netanyahu, 2000], by representing payoffs algebraically, we demonstrate that the structure of solutions to games of the form considered here is invariant for wide differences in quantitative payoffs as long as qualitative relationships are preserved. Because of the highly politicized nature of this particular aquifer sharing problem, the data for precise estimation of payoffs is not only lacking, but available estimates are subject to considerable controversy, especially with respect to the “what if” payoffs. By considering conjectural payoffs for which only qualitative relationships are critical, agreement on the structure of a solution is more likely, and the debate can focus on broader issues than specific estimation techniques or practices. We note, however, that the invariance property that permits characterization of outcomes based on qualitative relationships of payoffs is typically less applicable for problems involving more than two players or more than two conflicts.

[16] In this game, Israel's strategies are water sharing or no water sharing, and the PA's strategies are to make the payment or make no payment. The outcome of no water sharing and no payment represents the status quo, and thus generates a zero payoff for both parties. If Israel provides water sharing but the PA makes no payment, then the PA is better off than the status quo because of increased water availability and/or reduced pumping cost, but Israel is worse off because of reduced water use. If the PA makes the payment but Israel provides no water sharing, then the PA is worse off and Israel is better off than the status quo by the amount of the payment (assumed equal to 4 in Figure 1). Finally, if Israel provides water sharing and the PA makes the payment, then Israel is not as well off as if it provides no water sharing when the PA makes a payment nor is the PA as well off as if it makes no payment when Israel provides water sharing. However, both are better off than in the status quo. For Israel, the cooperative strategy of water sharing reduces its welfare relative to the noncooperative no water sharing strategy given either PA strategy. For the PA, choosing the cooperative strategy of making the payment reduces its welfare relative to the noncooperative strategy of making no payment given either Israeli strategy.

[17] A game where both players are better off taking the noncooperative action given either action that can be chosen by the other player, but yet both are better off with full cooperation than in the status quo, is a standard prisoner's dilemma (PD) game. When a PD game is played as a one-shot game, it results in the less desired outcome of universal noncooperation, i.e., no water sharing by Israel and no payment by the PA. When a PD game is played with infinite repetition, full cooperation becomes an equilibrium possibility from the standpoint of game theory because both players can come to a mutual understanding of the gains that are possible (although infinite repetition is not necessary for full cooperation as demonstrated by Folmer and van Mouche [1994]). However, full cooperation in the game in Figure 1 may be political unacceptable because it involves direct victim pays side payments for water sharing. Thus cooperation on Israeli-Palestinian water sharing of the MA may be unlikely in isolation even if both players can envision the potential of mutual gains.

4. Dependence of Interconnection Benefits on Structure

[18] Before choosing an issue to connect to the water-sharing issue, the properties that make interconnection advantageous must be considered. While a number of studies have identified cases where interconnection appears to be advantageous, analytical understanding of the properties that make interconnection advantageous has developed in piecemeal fashion. The early work of Chayes and Chayes [1991] and Folmer et al. [1993] provides insights. Chayes and Chayes [1991] argue that by creating a linkage between unrelated issues with reciprocity (where each country has an advantage over the other on at least one issue), countries not only avoid monetary side-payments but also form a credible threat that serves as an effective enforcement mechanism.

[19] Folmer et al. [1993] argue that when player A cooperates on an issue in which player B defects and when player A defects on another issue in which player B cooperates, then mixed interconnected strategies (i.e., mixtures of the pure strategies in the interconnected game) result in the largest total gain and can therefore potentially be sustained. In the case of unidirectional externalities, Bennett et al. [1998] suggest that interconnected game theory is advantageous when players' asymmetric strength is of comparable magnitude. For example, if player A has a definite advantage over player B in one game, then that game is best interconnected to another game with reciprocal externalities of comparable magnitude where player B has the advantage. Unless the order of magnitude of the externalities is similar, the advantage of one player on one issue cannot be roughly compensated by the advantage of the other player on the interconnected issue, and thus the most preferred outcome may still be politically infeasible because of the victim pays principle. Kroeze-Gil [2003] has recently introduced the concept of modified minor games, which highlights this problem.

[20] Bennett et al. [1998] and Hauer and Runge [1999] illustrate a further potential motivation for interconnection. They demonstrate that not only can interconnection provide a counter-balanced enforcement mechanism, but the feasible set is expanded relative to the aggregate payoffs of the component isolated games. The feasible set is expanded because outcomes that are infeasible in isolated games due to individual rationality constraints (no player will rationally agree to an outcome worse than the status quo) become feasible when compensated by offsetting gains related to an interconnected issue. Thus expanded incentives for cooperation can be realized.

[21] In summary, interconnected games are attractive when (1) each player has an advantage over the other player in at least one issue, (2) the asymmetry of advantages is sufficiently comparable in magnitude, and (3) interconnection expands the set of feasible strategies. In practical terms, interconnection is attractive when mechanisms of formal enforcement are not available, victim pays side payments are not attractive or feasible, and mutually beneficial agreements are sustainable only by identifying issues with symmetric and reciprocal benefits. With this background, we now turn to identifying an issue that is suitable for linking to the Israeli-Palestinian water-sharing problem with its unequal access costs.

5. Interconnected Issue

[22] In the context of the Israeli-Palestinian example, nonwater issues such as agricultural trade, economic development, tourism, land, and refugee control, to name a few, have the potential to be linked to an agreement on management and sharing of aquifer water. Following Folmer et al. [1993, p. 316], however, an issue suitable for consideration with interconnected game theory must be chosen so that the “isolated games are strategy and payoff independent.” In other words, the strategy choice of a player and resulting payoffs in one isolated game must not restrict the strategy choice or affect the resulting payoffs in the other isolated game.

[23] For example, suppose control of Palestinian sewage contamination of the aquifer were considered as the interconnected game. Palestinian contamination tends to affect Israeli users and not vice versa because of the direction of flow in the aquifer. However, the choice of strategies and resulting payoffs in Israeli water sharing and Palestinian sewage control would not be independent of one another because the transmission of sewage contamination takes place through the flow of water to Israeli springs. Thus the benefits of choosing a water sharing strategy in one game would depend on water quality and therefore on whether a strategy of sewage control is chosen in the other game. Likewise, the benefits of choosing a sewage control strategy in the other game would depend on the quantity of water sharing chosen in the first game.

[24] Interconnected game theory loses tractability when issues are not independent. Note, however, that water sharing and sewage control could be combined into a single game and considered for linking to another issue unrelated to either. We consider this prospect in another study. As might be expected, the dependence of quality on quantity complicates the problem. Also, PA control of contamination gives the PA a credible threat in the joint quantity-quality game. In reality, however, the PA sewage problem is dominated by the water quantity problem so that the quantity problem alone captures the major issues. The environmental impact of sewage on the PA is being addressed alternatively through international aid.

[25] One issue which complements the MA water-sharing issue well for purposes of linking negotiations is control of illegal trade (smuggling) of PA agricultural products into Israel. While the PA is interested in Israeli cooperation concerning water sharing, Israel is interested in PA cooperation concerning illegal agricultural trade. The PA has a clear advantage relative to Israel in enforcing laws against illegal agricultural trade because Israel would have to build a complex system for control and inspection whereas the PA can more efficiently control potential sources of smuggling. Smuggling is a major issue in trying to keep the Israeli-PA border open because of cheap, low-quality agricultural goods that are both produced in and imported into the PA from neighboring Arab countries. Because the advantages and externalities in controlling smuggling tend to counter balance those of the water-sharing issue, they offer an attractive opportunity for interconnection. Israel is the benefactor of smuggling control at the PA's expense, so it would likely have to offer a side-payment to entice Palestinian cooperation if smuggling control were considered in isolation.

[26] A game theoretic representation of PA control of smuggling of agricultural produce is shown in Figure 2. Israel's strategies are to provide compensation or provide no compensation. With compensation, the PA is made better off by the same cost Israel incurs. The Palestinian strategies are to provide control of smuggling or provide no control. If the PA chooses to control, Israel is better off by the same amount under either of its potential choices (i.e., the same side payment equal to 3 in Figure 2 applies in either case). Again, hypothetical payoffs are used for illustration even though any set of payoffs with the same qualitative relationships yields similar results. This permits considerable flexibility in interpreting the interconnected issue. It could represent any alternative conflict for which Israel gains at the expense of the PA independent of water sharing but may be secured only by a side payment, e.g., control of terrorist activities.

Figure 2.

The control-of-smuggling game.

[27] According to the payoffs in Figure 2, Israel's dominant strategy is the noncooperative approach of providing no compensation because the alternative cooperative strategy of compensation is costlier and reduces its welfare given either Palestinian action. For the PA, the noncooperative strategy of no control dominates the costly cooperative strategy of control given either Israeli action. Combining the Israeli and PA dominant strategies yields the Nash (dominant) equilibrium of no control and no compensation, which provides both entities with zero payoffs (the status quo). This outcome is clearly inferior to the outcome of full cooperation whereby the PA provides control and Israel provides compensation even though this outcome does not provide as great a payoff for either player as its noncooperative strategy given that the other player chooses the cooperative strategy. Thus this control-of-smuggling game is also a PD game. While a full-cooperation outcome can be achieved in theory if the game is infinitely repeated, the resulting necessity of victim pays side payments is likely to make cooperation politically infeasible.

6. Aggregated Isolated Games

[28] This section considers the aggregated possibilities of the isolated component games for the purpose of providing a standard of comparison for the interconnected game in the following section. Interconnection is advantageous only if it attains outcomes preferred by each player to the sum of payoffs that would be attained in the component isolated games. The critical implication of victim pays infeasibilities becomes clear through this comparison in section 8.

[29] If the water-sharing and control-of-smuggling games are played as one-shot games, both result in the less desired outcome of no cooperation even though the cooperative outcomes of water sharing, payment, compensation, and control are clearly superior to the outcomes with no cooperation. In both games, both players prefer cooperative behavior of the other player over noncooperative alternatives. That is, Israel gains from PA control whether or not it provides compensation or water sharing to the PA, and the PA gains from water sharing whether or not the PA makes a payment or provides smuggling control. However, a noncooperative strategy by one player triggers a noncooperative strategy by the other player in all cases. With one-shot play, players unable to secure a binding agreement because of absence of third-party enforcement thus choose a noncooperative strategy, which yields the PD outcome of mutual noncooperation.

[30] With repeated play, the chances of agreeing on cooperation are better in theory but mixed strategies also become possible. A mixed strategy merely amounts to attaining, in effect, a convex combination of the pure strategies (defined above) by playing each with probabilities that sum to one. For example, a player may cooperate part of the time to induce the other player to cooperate but then take advantage by defecting part of the time. Figures 3 and 4 present the feasible and equilibrium payoffs available to players in each game when mixed strategies are possible. The points in Figures 3 and 4 correspond to the outcomes in Figures 1 and 2. The line segments between pairs of points in Figures 3 and 4 represent the average payoffs that are possible by choosing the outcome represented at either end of the line segment part of the time and the outcome at the other end of the line segment otherwise.

Figure 3.

Potential payoffs from the water sharing game.

Figure 4.

Potential payoffs from isolated games.

[31] Figure 1 shows all four quadrants applicable to the water sharing game. The four points correspond to the four outcomes in Table 1. However, the feasible payoffs under individual rationality are restricted to the positive quadrant where each player receives a positive payoff. The outcomes in other quadrants are represented by dashed lines because they represent infeasible strategies (strategies to which rational players would not agree). Rational infeasibilities in the isolated games are critical to achieving gains by linking. As shown by Ragland [1995], interconnected games are effective when the isolated (independent) games have at least one payoff for at least one player outside the set that is feasible in the interconnected game.

Table 1. Aggregation of the Isolated Games
Payoffs in Water Sharing GamePayoffs in Control of Smuggling GameAggregate PayoffsOther Payoffs Dominating This Payoff
(0, 4.33)(0, 1.8)(0, 6.13) 
(0, 4.33)(2, 1)(2, 5.33) 
(0, 4.33)(3.5, 0)(3.5, 4.33) 
(1, 3)(0, 1.8)(1, 4.8)(2, 5.33)
(1, 3)(2, 1)(3, 4)(3.5, 4.33)
(1, 3)(3.5, 0)(4.5, 3) 
(2.28, 0)(0, 1.8)(2.28, 1.8)(3.5, 4.33)
(2.28, 0)(2, 1)(4.28, 1)(4.5, 3)
(2.28, 0)(3.5, 0)(5.78, 0) 

[32] The feasible payoffs under individual rationality in the case of the water-sharing game consist of (1, 3), where both parties receive a positive payoff; all convex combinations of (1, 3) and (0, 4.33), which represents all convex combinations of (1, 3) and (−2, 7) that have positive payoffs for both parties; and all convex combinations of (1, 3) and (2.28, 0), which represents all convex combinations of (1, 3) and (4, −4) that have positive payoffs for both parties. Accordingly, all similar subsequent graphs show only the positive quadrant. Figure 4 depicts the frontier from the positive quadrant of Figure 3 for the water-sharing game and also shows the corresponding positive quadrant of the frontier of the control-of-smuggling game.

[33] The first two columns of Table 1 list the payoff points that define the feasible sets (in the positive quadrants) of the two respective isolated games shown in Figure 4. That is, the points where one coordinate is zero are points along the axes, as shown in Figure 3, and thus correspond to mixed strategies of playing the cooperative strategy only part of the time in repeated play. These possibilities are described by the three feasible points, (1, 3), (0, 4.33), and (2.28, 0), that define the feasible set of the water-sharing game in Figure 3, and the corresponding three points that define the feasible set in the control-of-smuggling game, (2, 1), (0, 1.8), and (3.5, 0). The first two columns of Table 1 give all nine possibilities of combining these two sets of three points.

[34] The third column of Table 1 lists the aggregate payoffs of the two games. For example, aggregating the full-cooperation outcomes from both games, (1, 3) and (2, 1), gives an aggregate payoff of (3, 4). Alternatively, aggregating the partial cooperation outcome (0, 4.33) from the water-sharing game and the partial cooperation point (3.5, 0) from the control-of-smuggling game gives a higher aggregate payoff of (3.5, 4.33). The forth column of Table 1 lists the points that are dominated by other points in the feasible set (both players are better off with another outcome). For example, the full cooperation point is shown to be dominated by the combination of partial cooperation strategies that yields an aggregate payoff of (3.5, 4.33). The nondominated points are then used to draw the convex hull of the aggregated isolated games as shown by the dashed line frontier in Figure 4.

[35] The three frontier points in the interior of the positive quadrant points in Figure 4 correspond to the following outcomes: (0, 4.33) in the water-sharing game with (2, 1) in the control-of-smuggling game; (0, 4.33) in the water-sharing game with (3.5, 0) in the control-of-smuggling game; and (1, 3) in the water-sharing game with (3.5, 0) in the control-of-smuggling game. All strategies on the aggregated isolated frontier make both players better off than no cooperation (represented by zero payoffs at the origin). The full cooperation point (3, 4), which corresponds to full cooperation in both games, is inside the frontier of the aggregated isolated games and is thus dominated by mixed strategies where combinations of strategies corresponding to adjacent frontier points are each played part of the time in repeated play.

[36] An important consideration, not exploited in previous studies involving victim pays strategies, is that every point on the frontier of the aggregated isolated games thus involves side payments that are potentially politically infeasible if (1) one of the strategies in each isolated game involves side payments and (2) choice of the noncooperative strategy causes a negative payoff for the cooperative strategy to the other player in each case in each isolated game. Previous studies have either structured strategies in the isolated games so they do not involve victim pays side payments or have ignored the potential political infeasibility of victim pays side payments in the resulting interconnected game [e.g., Hauer and Runge, 1999]. In this case, all mixed strategies in both isolated games involve victim pays side payments at least part of the time so that no cooperation is possible when such payments are politically infeasible. More specifically, the only strategies other than the status quo that are politically feasible in either isolated game in this case are mixtures of the status quo with a rationally infeasible outcome, all of which are rationally infeasible. If so, then the possibility of interconnection becomes critical in moving beyond the status quo of universal noncooperation on either issue. While, upon a little reflection, these points are abundantly clear, they have important implications as developed in section 8 below.

7. Interconnected Game

[37] Now consider the opportunities available by linking issues. The interconnected game offers four possible strategies for each of the players. Players may choose to cooperate on both, to cooperate on the first (second) and not on the second (first), or not to cooperate on either. The payoffs from all strategies in the two isolated games are added together and presented in Figure 5. That is, Figure 5 corresponds to consideration of all 16 strategies represented by combining the 4 possible outcomes of each of the two games in Figures 1 and 2. In each case, the entry in Figure 5 corresponds to the aggregated payoff. For example, if the water-sharing game has an outcome with water-sharing and a payment, which generates a payoff of (1, 3), and the control-of-smuggling game has an outcome with compensation and control, which has a payoff of (2, 1), then the aggregate payoff is (3, 4), which is the upper left hand entry in Figure 5. Alternatively, if the water-sharing outcome is water sharing and no payment, which generates a payoff of (−2, 7), and the control-of-smuggling game has an outcome of no compensation and control, which generates payoff (5, −1), then the aggregate payoff is (3, 6), which is the outcome in the second row and third column of Figure 5.

Figure 5.

The interconnected game.

[38] Depending on the characteristics of the two component isolated games, the dominant equilibrium of the interconnected game with one-shot play can be the noncooperative outcome. For example, Cesar and de Zeeuw [1994] have argued that a cooperative outcome of an asymmetric PD one-shot game cannot be achieved unless the game is interconnected to another PD game where the two games form a symmetric PD game. Barrett [1995] suggests that the PD problem is solved by this result and Van Damme's [1989] result, which shows that the full cooperation outcome of an infinitely repeated symmetric PD game is a subgame perfect equilibrium (see Friedman [1990] for further discussion). Hauer and Runge [1999], however, demonstrate that Cesar and de Zeeuw's [1994] scope is inappropriately narrow. They consider linking trade and environmental issues in three cases: (1) the trade game is an assurance problem and the environmental game is a PD problem, (2) both games are assurance games, and (3) the trade game has a dominant-strategy open-trade equilibrium and the environmental game is either a PD or an assurance problem. Their results demonstrate that interconnecting games with offsetting gains and concessions can be advantageous with games involving other than PD structures.

[39] Barrett [1995] emphasizes the crucial nature of repetitive interaction as a means of attaining sustainability in interconnected games [see also Folmer et al., 1993; Folmer and van Mouche, 1994]. Even when an effective balance of punishments and rewards can be achieved in the interconnected game, the Nash equilibrium in the one-shot game can still be universal noncooperation. With repetition, a country is less likely to gain by defecting on one issue as soon as it captures its benefits in another. That is, when each country is provided a sufficient gain by one of the games to overcome its loss in the other, then defection from a losing game can cause future defection by the other country from its winning game, resulting in greater overall welfare loss. Thus repetition can be a critical aspect of achieving cooperation in interconnected games just as in the isolated games. This requirement of repetition seems to apply to most problems of international resource management because of the critical need to facilitate a feasible self-enforcement mechanism.

[40] Repetition also motivates possibilities of playing mixed strategies. In contrast, for example, Hauer and Runge [1999] consider only the cases of pure interconnected strategies, i.e., full cooperation and universal noncooperation, ignoring the possibilities of playing mixed interconnected strategies in the interconnected game. This is a strong assumption, which should be investigated by game theoretic analysis rather than imposed by assumption. With repetition, mixed interconnected strategies are important outcomes of the interconnected game that facilitate additional opportunities in negotiations. In fact, Just and Netanyahu [2000] show for a wide variety of game structures that the feasible payoff set of the interconnected game weakly dominates the aggregated isolated games only because of mixed strategies involving partial cooperation.

[41] In the Israeli-Palestinian game of this paper, the interconnected game may be regarded as a repeated game because the same water and smuggling issues are recurring. When the game is infinitely repeated, a subgame perfect equilibrium occurs where Israel and the PA have positive average payoffs. However, in the interconnected game, not all cases of mutual positive payoffs involve potentially politically infeasible victim pays side-payments. Any outcome that involves the compensation or payment strategies would involve these side payments. However, at least one outcome, the sharing-no-compensation-no-payment-control outcome, offers positive payoffs to both players (3 to Israel and 6 to the PA) without side payments (without compensation and payment strategies). Thus the interconnected game provides possibilities for cooperation without victim pays outcomes whereas the aggregated isolated games do not.

8. Critical Nature of Victim Pays Infeasibilities

[42] Previous studies that apply interconnected game theory have not considered how excluding strategies that involve side payments changes game structure and the consequent gains from interconnection. For the case where side payments are not considered infeasible, the outcomes that are not dominated by others (i.e., no other outcome offers as great or greater payoffs for both parties) among those in Figure 5 are identified by underlining. Figure 6 represents the feasible strategies that are possible by mixing these outcomes with a solid line frontier. The payoff outcome (−5, 10) is not shown in Figure 3 because it is outside the feasible set satisfying individual rationality and the next point on the frontier with which it could be mixed at (0, 8) is on the boundary of feasibility (of individual rationality). The other payoff outcome that is outside the feasible set, (9, −5), however, plays a role in defining the feasible frontier because some mixtures of it with the next point on the frontier at (6, 2) are inside the feasible set.

Figure 6.

Feasible and equilibrium payoffs for the interconnected and aggregated games.

[43] For purposes of comparison, the frontier of the aggregated isolated games is also shown in Figure 6 by the dashed line frontier, which is identical to Figure 4. Figure 6 thus illustrates the typical result claimed by interconnected game theory that strategies with greater payoffs for both parties are attained by the interconnected game compared to the aggregated possibilities of playing the two component games in isolation. The aggregated payoff that corresponds to full cooperation in both isolated games is strictly dominated in the interconnected game, as it is in the aggregated isolated games. In this case, the payoff frontier of the interconnected game strictly dominates the payoff frontier of the aggregated isolated games, i.e., they have no common points. Both players can be made better off compared to every individual possibility on the frontier of the aggregated isolated games.

[44] These results are typically used to argue that interconnecting two games can expand the available equilibrium outcomes relative to the case where the games are played independently. The additional benefits attained by the domination of interconnected outcomes over aggregated isolated outcomes thus add to the self-enforcing incentives to sustain any equilibrium agreement provided by reciprocity [Folmer et al., 1993]. That is, not only does each party receive benefits on one issue by making concessions on another, but the overall gains are greater than can be attained by negotiating the issues in isolation. Thus, for example, PA defection in the control-of-smuggling game would cost the PA gains from water allocation that are possible in the water-sharing game. As a result, Israel could be assured in such an agreement that the PA would not have incentives to defect if it agrees to share water. Given a similar consideration in reverse for the other issue, the sovereign countries can self-enforce and thus sustain an agreement in absence of a third-party enforcer and achieve the mutual benefits of a win-win outcome.

[45] Folmer et al. [1993] and Cesar and de Zeeuw [1994] have asserted that interconnected asymmetric PD games strictly dominate the aggregated isolated games. However, Just and Netanyahu [2000] show that this is neither a necessary nor a sufficient condition. Considering cases of interconnection where isolated games involve any combination of PD, assurance, iterated dominance, or chicken games, they show that interconnection is advantageous when it is necessary to achieve cooperation, but that interconnection does not enhance players' welfare over cooperation in the aggregated isolated games when full cooperation is the preferred outcome under interconnection. Alternatively, interconnection dominates the aggregated isolated games when the preferred outcome under interconnection involves mixed strategies composed of partial cooperation.

[46] A significant remaining issue, however, is that many of the frontier strategies of the interconnected game may involve side payments that may be politically infeasible. In fact, the only point on the interconnected frontier in Figure 6 that does not involve some mix of such side payments is at the point (3, 6). This is evident from examining Figure 5 where the only strategies that do not involve side payments are those in both the no payment columns and no compensation rows. In the case where victim pays outcomes are not politically feasible, the frontier of the politically and rationally feasible set must be considered as a convex hull of only the corresponding four cells of Figure 5. The politically feasible set is thus described by the points {(3, 6), (−2, 7), (5, −1), (0, 0)}. The politically and rationally feasible set is found by excluding points from the negative quadrants of this set. The frontier of politically and rationally feasible outcomes is thus represented in Figure 6 by the dotted lines.

[47] Considering that none of the points on the frontier of the aggregated isolated games are feasible if side payments are politically infeasible (they all represent mixtures of side payments), several observations follow. First, some points in the interconnected game that are politically and rationally feasible dominate only parts of the frontier of the aggregated isolated game where side payments are assumed feasible, because the dotted frontier is partially outside of the frontier of the aggregated isolated games. Second, the gains from interconnection are much greater in the case where side payments are politically infeasible. That is, the domination of the origin by the dotted line frontier of Figure 6 is much greater than the domination of the dashed line frontier by the solid line frontier. Third, when side payments are politically infeasible, the interconnected game will always strictly dominate the aggregated isolated games if (1) one of the strategies in each isolated game involves side payments, (2) choice of the noncooperative strategy causes a negative payoff for the cooperative strategy to the other player in each case in each isolated game, and (3) at least one outcome produces positive benefits for both players without side payments.

[48] Because the numeric values in Figures 1 and 2 are hypothetical and the advantages of interconnection are dependent on structure, a brief sensitivity analysis is of value. Note in Figure 1 that given a choice of water sharing, Israel does not benefit from the PA payment by as much as the payment itself (compare entries in the first row of Figure 1). This is perhaps a likely outcome when water sharing involves typical political and economic inefficiencies. Inefficiencies such as studied in welfare economics are common when distortionary policies are implemented for political purposes of benefiting target groups in society or when constraints due to sovereignty, security, religious, and other social concerns restrict economic efficiency. Also, given a PA choice to control smuggling, the PA does not benefit by as much as the compensation paid by Israel (compare entries in the first column of Figure 2), which may represent similar political and economic inefficiencies. To consider another possibility, suppose these differences are the same as the amount of the payment. For example, if the Israeli payoff in Figure 1 with water sharing and a PA payment is equal to 2 instead of 1, and the PA payoff in Figure 2 with control and compensation is 2 instead of 1, then the results in Figure 6 are replaced by those in Figure 7.

Figure 7.

Feasible and equilibrium payoffs for an alternative case.

[49] Figure 7 demonstrates that while the interconnected game expands the feasible set of outcomes relative to the aggregated isolated games when either isolated game involves negative and thus irrational outcomes for either player, the interconnected feasible set does not always strictly dominate the aggregated isolated games. Figure 7 represents one case found to be typical by Just and Netanyahu [2000] where the frontiers of the two cases coincide at the point of full cooperation (in both isolated games) and along frontier segments on either side of the full cooperation point. In these cases, the feasible set of the interconnected game weakly dominates the aggregated isolated games only because of the shoulders added to the frontier close to either axis. For this reason, if full cooperation is on the frontier of the interconnected game, then the interconnected frontier does not dominate the frontier of the aggregated isolated games near the full cooperation point. In this case, the frontier of the interconnected game is preferred only when the structure of payoffs of the interconnected game is inequitable in favor of one player so that a sustainable agreement is likely to be reached on one of the shoulders.

[50] Considering potential political infeasibility of side payments, in this case, the story is somewhat different. The point of full cooperation in the isolated games is politically infeasible because full cooperation in either isolated game involves politically infeasible side payments. With interconnection, however, the same payoff can be secured for both players in Figure 7 without side payments. That is, the point (3, 6) is made possible by combining the water-sharing-no-payment outcome in Figure 1 with the no-compensation-control outcome of Figure 2 (note that these two sets of payoffs are not altered for the case of Figure 7), which attains the (3, 6) payoff without side payments (with no compensation and no payment). The irrationality of this outcome in the case of isolated play is overcome by interconnection because, in combination, both players can be made better off than the status quo without side payments. Again, no point in Figure 7 is both politically and rationally feasible with isolated play other than the origin (the frontier of the aggregated isolated games is politically infeasible in its entirety), whereas the dotted-line frontier, which is both politically and rationally feasible with interconnection, offers much of the benefits without side payments. This is particularly true when the corresponding point on the frontier is equitable, i.e., would be the chosen equilibrium if side payments were feasible.

[51] For the case where side payments are politically infeasible, the results here underscore the requirements for interconnection suggested by Ragland [1995]. She shows that interconnecting two isolated PD games can result in either dominance or partial dominance but that strict dominance is achieved when, in each of the isolated asymmetric PD games, there is at least one pair of payoffs associated with a set of mixed strategies in which the gain for one player is greater (in an absolute sense) than the loss of the other player. Upon reflection, one can see that this is exactly the requirement of the two strategies from Figures 1 and 2 that leads to attaining a desirable point in Figure 7 that is both politically and rationally feasible. However, our results show that strict dominance is always achieved when (1) one of the strategies in each isolated game involves side payments, and (2) choice of the noncooperative strategy causes a negative payoff for the cooperative strategy to the other player in each case of each isolated game.

9. Conclusions

[52] This paper examined a conceptual framework for exploring the potential of linking issues for negotiation in cases of international common pool water resource with unequal access costs. The problem has been demonstrated using the conflict regarding Israeli-Palestinian water sharing of the Mountain Aquifer in Israel. A stylized example has been developed that shows how an agreement can be reached only by linking to another issue with reciprocal advantages when the victim pays principle is unacceptable. As in previous literature on interconnected game theory, the approach is shown to hold potential for achieving sustainability of the agreement by enabling a self-enforcement mechanism associated with the characteristics of the issue chosen for the interconnected agreement. However, the strength of this self-enforcement mechanism is amplified if victim pays side payments are considered politically infeasible.

[53] In contrast, previous literature on interconnected game theory has not considered how elimination of side payments alters the structure of the games. If side payments are considered feasible, some studies have falsely claimed strict dominance of interconnected games over isolated games in cases of certain structures whereas other studies have shown that only weak dominance holds generally. In fact, if side payments are feasible and full cooperation is the ideal outcome, then interconnection does not offer improved possibilities for many cases even though the feasible set is expanded. The results here for the case where side payments are infeasible demonstrate in contrast that no cooperation may be feasible in the isolated games. If so, strict dominance of interconnection is achieved because the isolated issues offer mutual gains only through infeasible side payments. Thus, when victim pays strategies are politically infeasible, interconnection makes a much greater difference.

[54] Although advances in the Israeli-Palestinian peace process have broken down over the past few years, prior to that time occasional announcements were made by both sides that progress on one issue could be achieved only if progress were made on another issue. The allocation of water between Israel and the PA has been one of the major issues of negotiation, as has been smuggling and border control. The words “linkage” and “reciprocity” have arisen frequently in these negotiations, implicitly suggesting that the parties understood that gains may be possible from linking issues. In this setting, an effort to identify opportunities for application of interconnected game theory and to separate the fruitful opportunities from the unfruitful ones is useful. Also, because strict dominance of interconnected games is not necessary to make interconnection attractive, the focus should be more on how the set of equilibrium strategies is expanded by interconnection (beyond the case of isolated negotiations) rather then whether it is expanded. For example, opportunities are more likely enhanced when equity is a major issue. A general implication of this paper is that other issues of political infeasibility should also be considered substantively in conceptual understanding negotiation possibilities. In particular, when inequities are strong and require politically infeasible side payments for resolution, then a study of how games are affected by such political infeasibilities may be critical to finding the best issues for linkage.