Coalition-proof Nash Equilibrium Research Articles

N-Defendant Litigation and Settlement: The Impact of Joint and Several Liability

Abstract This article generalizes the analysis of settlement under joint and several liability from lawsuits involving one plaintiff and two defendants to those involving n ≥ 2 defendants. We demonstrate that, depending upon the correlation of outcomes among the defendants, but regardless of the distribution of liability shares, joint and several liability may encourage plaintiffs to select some defendants for litigation while settling with those remaining. In particular, under joint and several liability, when each defendant is grouped with others sharing correlated outcomes, and the plaintiff’s probability of success against any defendant is sufficiently high, we show that, in all Nash equilibrium, the plaintiff litigates against the defendant from each group with the smallest share and settles with all other defendants. There are a continuum of equilibria but they are all payoff-equivalent up to permutation. Plaintiff’s payoff is identical in all equilibria; only the payoffs of the members with the smallest share within a group are permuted. We also show that, for sufficiently high plaintiff’s probability of success, this result holds even when the defendants are allowed to form coalitions, and derive the set of offers in the behaviorally unique coalition proof Nash equilibrium; as before, these offers induce litigation against the defendant with the smallest share in each group and settlement with all other defendants.

Coalition-proof stable networks

We propose the notion of coalition-proof stability for predicting the networks that could emerge when group deviations are allowed. A network is coalition-proof stable if there exists no coalition which has a credible group deviation. A coalition is said to have a credible group deviation if there is a profitable group deviation to some network and there is no subcoalition of the deviating players which has a subsequent credible group deviation. Coalition-proof stability is a coarsening of strong stability. We emphasize the importance of coalition-proof stability by considering four models where a strongly stable network fails to exist while a coalition-proof stable network does exist. We provide an easy to verify condition for the existence of a coalition-proof stable network while a strongly stable network may not exist. There is no relationship between the set of coalition-proof stable networks and the set of networks induced by a coalition-proof Nash equilibrium of Myerson’s linking game.

The interactions of R&D investments and horizontal mergers

In a homogenous good industry in which firms choose their cost-reducing R&D investments and make merger proposals prior to competing à la Cournot, we identify conditions under which there are coalition-proof Nash equilibria involving horizontal mergers as well as non-integration. Mergers arise whenever the R&D technology is sufficiently effective. Moreover, if firms’ R&D investments are substitutes (complements) and the bargaining power is unevenly (evenly) distributed among merger participants, a merger is the unique coalition-proof Nash equilibrium. Antitrust policy guidelines are simple when the welfare standard is the consumer surplus and R&D investments are substitutes, but are more complex when the standard is total surplus and/or R&D investments are complements.

Organizational refinements of Nash equilibrium

Strong Nash equilibrium (see Aumann, 1959) and coalition-proof Nash equilibrium (see Bernheim et al., 1987) rely on the idea that players are allowed to form coalitions and make joint deviations. Both of these notions consider cases in which any coalition can be formed. Accordingly, there may arise “conflicts of interest” that prevent a player from choosing an action that simultaneously meets the requirements of two coalitions to which he or she belongs. Here, we address this observation by studying an organizational framework such that the coalitional structure is (i) motivated by real-life examples where players cannot form some coalitions and (ii) formulated in such a way that no conflicts of interest remain. We define an organization as an ordered collection of partitions of the player set such that any partition is coarser than the partitions that precede it. For any given organization, we introduce the notion of organizational Nash equilibrium. We analyze the existence of equilibrium in a subclass of games with strategic complementarities and illustrate how the proposed notion refines the set of Nash equilibria in some examples of normal form games.

Implications of multilateral tariff bindings on the extent of preferential trade agreement formation

Using an endogenous preferential trade agreement (PTA) formation model, we examine the effects of continual reduction in tariff bindings on the extent of PTA formation and its role in attaining global free trade. When countries are completely symmetric, no country has an incentive to free ride from global free trade while exclusion incentives arise when tariff bindings are sufficiently tight. Due to the relatively flexible nature of the free trade area (FTA) formation, such exclusion incentives go unexercised and free trade always obtains as the coalition-proof Nash equilibrium (CPNE) of the FTA game. However, since customs union (CU) expansion leads directly to global free trade and member countries are able to veto CU expansion, they exercise the exclusion incentive and thus free trade fails to be a CPNE. We then consider a scenario where countries are asymmetric with respect to their comparative advantage. The country with a weaker comparative advantage has an incentive to free ride on trade liberalization of the other two countries and lower tariff bindings discipline this incentive via constraining the ability to set optimal tariffs. As a result, continual reduction in tariff bindings facilitates FTA formation in attaining global free trade.

Coalition-Proof Stable Networks

We propose the notion of coalition-proof stability for predicting the networks that could emerge when group deviations are allowed. A network is coalition-proof stable if there exists no coalition which has a credible group deviation. A coalition is said to have a credible group deviation if there is a profitable group deviation to some network and there is no subcoalition of the deviating players which has a subsequent credible group deviation. Coalition-proof stability is a coarsening of strong stability. There is no relationship between the set of coalition-proof stable networks and the set of networks induced by a coalition-proof Nash equilibrium of Myersonis linking game. Contrary to coalition-proof stability, coalition-proof Nash equilibria of Myersonis linking game tend to support unreasonable networks.

Undominated coalition-proof Nash equilibria in quasi-supermodular games with monotonic externalities

I introduce an undominated coalition-proof Nash equilibrium that is unique and is able to select undominated Nash equilibria in a quasi-supermodular game with a strict single-crossing property and monotonic externalities.

A note on budget constraints and outside options in common agency

Bernheim and Whinston (Q J Econ 101:1–31, 1986) show that, in a common agency problem without budget constraints, the set of Nash equilibria with truthful strategies (TNE), the set of coalition-proof Nash equilibria (CPNE), and the principal-optimal core of the underlying coalitional game are non-empty and all equivalent in payoff space. We show that, with budget constraints, none of Bernheim and Whinston’s (Q J Econ 101:1–31, 1986) results hold: (i) a CPNE may not exist, (ii) a TNE may not exist even when a CPNE exists, (iii) a TNE may not be a CPNE, and (iv) both TNE and CPNE payoffs are core allocations but are not necessarily principal-optimal. However, when principals have outside options but no budget constraints, (i), and (iii) continue to hold but not for (ii) and (iv). In particular, a TNE always exists but the core may be empty.

R&D Investments Fostering Horizontal Mergers

We study a homogenous good triopoly in which firms first choose their cost-reducing R&D investments and consider alternative merger proposals, and then compete a la Cournot in the ensuing industry. We identify conditions under which both horizontal mergers and non-integration are sustained by Coalition-Proof Nash equilibria (CPNE). These conditions involve the effectiveness of the R&D technology, as well as the distribution of the bargaining power between the acquirer and the acquiree, which determine the allocation of the incremental profits generated by the merger. We show that whether firms follow duplicative or complementary research paths, sustaining a merger generally requires a sufficiently effective R&D technology that creates endogenous cost asymmetries and renders the merger profitable, and a moderate distribution of bargaining power that allows to spread the benefits of the merger. We examine the welfare effects of mergers and obtain clear policy guidelines.

Common Agency with Budget Constraints

Bernheim and Whinston (1986) show that, in a common agency problem without budget constraints, the set of Nash equilibria with truthful strategies (TNE), the set of coalition-proof Nash equilibria (CPNE), and the principal-optimal core of the underlying coalitional game exist and are all equivalent in payoff space. We show that, with budget constraints, none of Bernheim and Whinston's (1986) results hold: (i) a CPNE may not exist, (ii) a TNE may not exist even when a CPNE exists, (iii) a TNE may not be a CPNE, and (iv) both TNE and CPNE payoffs are core allocations but are not necessarily principal-optimal.

External trade diversion, exclusion incentives and the nature of preferential trade agreements

In a game of endogenous trade agreements between three countries, we show that while the pursuit of customs unions (CUs) prevents global free trade from emerging as a coalition-proof Nash equilibrium, the pursuit of free trade agreements (FTAs) does not. This result reflects the relatively flexible nature of FTAs: whereas each FTA member can independently undertake further trade liberalization with respect to the non-member, CU members must do so as a group due to their common external tariff. By diverting members' exports away from the non-member, both types of trade agreements induce the non-member to voluntarily lower its import tariffs.

Overlapping International Environmental Agreements

The perfectly coalition-proof Nash equilibrium (PCPN) concept is extended to allow for the emergence of overlapping coalitions in equilibrium. We study the efficiency and stability properties of environmental agreements to control emissions of correlated continental and global pollutants. We show that set of PCPN equilibria includes perfectly strong Nash (PSN) equilibria if the national damage from continental pollution is sufficiently large relative to the national damage from global pollution. We also show that: (i) continental agreements may be perfectly coalition-proof under much less restrictive circumstances; (ii) perfect Nash equilibria for fully overlapped agreements may be superior to the Grand Coalition's optimal allocation in the presence of coalitional operation costs; and (iii) PSN equilibria may still emerge in a larger world if coalition formation yields R&D spillover benefits.

Perfectly Coalition-Proof Overlapping Agreements: With an Appliation to International Environmental Agreements

The Perfectly Coalition-Proof Nash equilibrium (PCPN) concept is extended to allow for the emergence of overlapping coalitions in equilibrium. We apply the extended concept to study the efficiency and stability properties of environmental agreements to control emissions of correlated continental and global pollutants. We show that set of PCPN equilibria includes Perfectly Strong Nash (PSN) equilibria if the national damage from continental pollution is sufficiently large relative to the national damage from global pollution. We also show that: (i) continental agreements may be perfectly coalition-proof under much less restrictive circumstances; and (ii) perfect Nash equilibria for fully overlapped agreements may be superior to the Grand Coalition’s optimal allocation in the presence of coalitional operation costs.

Coalition-Proof Core in Adverse Selection Economies

We reexamine the core in the adverse selection insurance economy first studied by Rothschild and Stiglitz (1976). Defining blocking in a way that takes private information into account, the core is sometimes empty. We define the coalition-proof core as the set of allocations which are blocked only by allocations which are themselves blocked by coalition-proof allocations. This definition is closely related to Coalition Proof Nash Equilibrium, introduced by Bernheim, Peleg and Whinston (1987). We prove that the coalition-proof core consists of the Miyazaki allocation-- the Pareto-optimal allocation most preferred by low-risk agents. The coalition-proof core is always nonempty in this economy and has attractive continuity properties. Cross- subsidization can occur in the coalition-proof core.

A 'Coalition Proof' Equilibrium for a Private Information Credit Economy

This paper examines an economy in which agents with private information about their own productive capabilities seek to raise capital to fund their investment projects. We employ an equilibrium concept which is closely related to Coalition Proof Nash Equilibrium. In equilibrium, all agents who succeed in raising capital (entrepreneurs) are pooled; they all receive the same contract or consumption schedule. Entrepreneurs, however, are separated from those who fail to raise capital. This separation results in productive efficiency for the economy. If the economy has no viable alternative investment opportunity (other than agents' projects) then equilibrium allocations can be supported by a (non-intermediated) securities market. If there is a viable alternative, the equilibrium allocations can only be supported through the formation of a form of financial intermediary coalition.

The Stable Equilibrium of Preferential Trade Agreements under Technology Asymmetry

The Stable Equilibrium of Preferential Trade Agreements under Technology Asymmetry-Preferential Trade Agreement, Technology Asymmetry, Tariff Complementary, Cournot Oligopoly, Coalition-Proof Nash Equilibrium

Coalition-proofness under weak and strong Pareto dominance

This article examines the structure of pure strategy coalition-proof Nash equilibria under weak and strong Pareto dominance in games with strategic complementarities and isotone externalities. The analysis is particularly focused on the following issues: (i) the inclusion relation between the set of coalition-proof Nash equilibria under weak Pareto dominance and the set of coalition-proof Nash equilibria under strong Pareto dominance; (ii) the conditions for the coincidence of the strong (resp. weak) Pareto dominance refinement and the coalition-proofness refinement under strong (resp. weak) Pareto dominance; (iii) the conditions for the uniqueness of a coalition-proof Nash equilibrium under both notions of Pareto dominance. Dual results are stated for games with antitone externalities.

Weakest-link contests with group-specific public good prizes

We examine the equilibrium effort levels of individual players and groups in contests in which n group compete to win a group-specific public good prize, individual players choose their effort levels simultaneously and independently, and each group's probability of winning the prize follows a weakest-link rule or weakest-link contest success function. In our basic model, we show that the lowest-valuation players in each group play decisive roles in determining the Nash equilibria of the game. There are multiple pure-strategy Nash equilibria in the game but there is a unique coalition-proof Nash equilibrium at which neither any player nor any group does not have an incentive to coordinate and deviate from the equilibrium. No free riding problem exists in equilibrium. As an example of our basic model, we consider a simple contest where two groups with two players compete against, and find that the high-valuation players in each group have incentives to subsidize the low-valuation players in their group. Finally, we examine the equilibrium subsidy rates of the groups in a contest where first the high-valuation players in each group decide how much to subsidize low-valuation players in their group and then the individual players in the contest choose their effort levels simultaneously and independently.

Mixed Duopoly with Price Competition

This paper examines coalition-proof Nash equilibria (CPNE) of a mixed duopoly with price competition where the public firm meets all the demand coming to it. If the private firm is free to supply less than demand, then the unique CPNE involves the competitive price. If however the private firm also has to supply all its demand, then the set of CPNE prices turns out to be an interval, with prices ranging from the socially optimal one, to the price under complete privatization.

Contributing or free-riding? Voluntary participation in a public good economy

We consider a (pure) public goods provision problem with voluntary participation in a quasi-linear economy. We propose a new hybrid solution concept, the free-riding-proof core (FRP-Core), which endogenously determines a contribution group, public good provision level, and its cost-sharing. The FRP-Core is always nonempty in public good economies but does not usually achieve global eciency. The FRP-Core has support from both cooperative and noncooperative games. In particular, it is equivalent to the set of perfectly coalition-proof Nash equilibrium (Bernheim, Peleg, and Whinston, 1987) of a dynamic game with players’ participation decisions followed by a common agency game of public goods provision. We illustrate various properties of the FRPCore with an example. We also show that the equilibrium level of public good shrinks to zero as the economy is replicated.