Consistent cost sharing

A new concept of consistency for cost sharing solutions is discussed, analyzed, and related to the homonymous and natural property within the rationing context. Main result is that the isomorphism in Moulin and Shenker (J Econ Theory 64:178–201, 1994) pairs each additive and consistent single-valued mechanism with a corresponding monotonic and consistent rationing method. Then this answers the open question in Moulin (Econometrica 68:643–684, 2000; Handbook of social choice and welfare. Handbooks in economics, pp 289–357, 2002) whether such notion for cost sharing exists. The conclusion is that renown solutions like the average and serial cost sharing mechanisms are consistent, whereas the Shapley–Shubik mechanism is not. Average cost sharing is the only strongly consistent element in this class. The two subclasses of incremental and parametric cost sharing mechanisms are further analyzed as refinement of the main result.     

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, 2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, 1975). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, 2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, 1975) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria.     

Decreasing serial cost sharing: an axiomatic characterization

The increasing serial cost sharing rule of Moulin and Shenker (Econometrica 60:1009–1037, 1992) and the decreasing serial rule of de Frutos (J Econ Theory 79:245–275, 1998) are known by their intuitive appeal and striking incentive properties. An axiomatic characterization of the increasing serial rule was provided by Moulin and Shenker (J Econ Theory 64:178–201, 1994). This paper gives an axiomatic characterization of the decreasing serial rule.     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

A simple model of corporate international investment under incomplete information and taxes

This paper extends the theory of corporate international investment in Choi (J. Int. Bus. Stud. 20: 145–155, 1989) in an environment where the segmentation of international capital markets for investors or the presence of agency costs provide some independence to corporate decisions. The model shows that the real exchange risk, the competition between firms in different markets and diversification gains affect corporate international investment. By accounting for the role of information as defined in the models of Merton (J. Finance 42: 483–510, 1987), Bellalah (Int. J. Finance Econ. 6: 59–67, 2001a) and Bellalah and Wu (Int. J. Theor. Appl. Finance 5(5): 479–495, 2002), the model embodies different existing explanations based on economic and behavioral variables. We show in a “two-country” firm model that real exchange risk, diversification motives and information costs are important elements in the determination of corporate international investment decisions. The dynamic portfolio model reflects the main results in several theories of foreign direct investment. Our model accounts for the role of information in explaining foreign investments. It provides simple explanations which are useful in explaining the home bias puzzle in international finance. Using the dynamical programming principle method, we provide the general solution for the proportion of firm’s total capital budget. We also use a new method to get explicit solutions in some special cases. This new method can be applied to solve other financial control problems. The simulating results are given to show our conclusion and the influence of some parameters to the optimal solution. The economic results can be seen as a generalization of the model in Solnik (J. Econ. Theory 8: 500–524, 1974).     

Efficiency and equilibria in games of optimal derivative design

In this paper, optimal derivative design when multiple firms compete for heterogenous customers is studied. Ties in the agents’ best responses generate discontinuous payoffs. Efficient tie-breaking rules are considered: In a first step, the model presented by Carlier et al. (Math Financ Econ 1:57–80, 2007) is extended, and results of Page and Monteiro (J Math Econ 39:63–109, 2003, J Econ Theory 134:566–575, 2007, Econ Theory 34:503–524, 2008) are used to prove the existence of (mixed-strategies) Nash equilibria. In a second step, the case of risk minimizing firms is studied. Socially efficient allocations are introduced, and their existence is proved. In particular, the entropic risk measure is considered.     

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.     

Local and parallel algorithms for fourth order problems discretized by the Morley–Wang–Xu element method

This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley–Wang–Xu (MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881–909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas in (Xu and Zhou (Math Comp 69:881–909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al. (Appl Numer Math 37:519–533, 2001); Huang et al. (Sci China Ser A 49:109–120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes of methods are of the sizes O(h + H ²) and O(h + H ²(H/h)^(d−1)/2), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.     

Bidding and sequential coalition formation with externalities

We study coalition formation games with externalities where each agent’s payoff depends on the entire partition. As in Bloch (Games Econ Behav 14:90–123, 1996) and Ray and Vohra (Games Econ Behav 26:268–336, 1999), we assume that coalitions form sequentially and agreements are irreversible. Instead of a fixed order protocol, we use a “bidding mechanism” (see Pérez-Castrillo and Wettstein in Am Econ Rev 92(5):1577–1587, 2002) to determine proposals and transfers among the agents. We show that such a mechanism facilitates the attainment of efficiency; in particular, our game admits a Markov perfect equilibrium with the property of full dynamic efficiency. In addition, the aggregate equilibrium payoffs of our game exhibit monotonicity in time. Nevertheless, inefficient equilibria can also emerge.     

Multicriteria games and potentials

In this note we study how far the theory of strategic games with potentials, as reported by Monderer and Shapley (Games Econ Behav 14:124–143, 1996), can be extended to strategic games with vector payoffs, as reported by Shapley (Nav Res Logist Q 6:57–61, 1959). The problem of the existence of pure approximate Pareto equilibria for multicriteria potential games is also studied.      

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

The evolution of fairness under an assortative matching rule in the ultimatum game

This paper studies how a matching rule affects the evolution of fairness in the ultimatum game. Gale et al. (Games Econ Behav 8: 56–90, 1995) show that under the random matching rule, a partially fair imperfect Nash equilibrium in which all proposers are fair but some responders are selfish is asymptotically stable in the limit as noise in learning vanishes if responders are noisier than proposers. This paper shows that, under an assortative matching rule, a mutually fair imperfect Nash equilibrium in which all proposers are fair and all responders are reciprocal is limit asymptotically stable as noise due to committed agents vanishes.     

Conditional minimum volume ellipsoid with application to multiclass discrimination

In this paper, we present a new formulation for constructing an n-dimensional ellipsoid by generalizing the computation of the minimum volume covering ellipsoid. The proposed ellipsoid construction is associated with a user-defined parameter β∈[0,1), and formulated as a convex optimization based on the CVaR minimization technique proposed by Rockafellar and Uryasev (J. Bank. Finance 26: 1443–1471, 2002). An interior point algorithm for the solution is developed by modifying the DRN algorithm of Sun and Freund (Oper. Res. 52(5):690–706, 2004) for the minimum volume covering ellipsoid. By exploiting the solution structure, the associated parametric computation can be performed in an efficient manner. Also, the maximization of the normal likelihood function can be characterized in the context of the proposed ellipsoid construction, and the likelihood maximization can be generalized with parameter β. Motivated by this fact, the new ellipsoid construction is examined through a multiclass discrimination problem. Numerical results are given, showing the nice computational efficiency of the interior point algorithm and the capability of the proposed generalization.     

An exact algorithm for the 0–1 linear knapsack problem with a single continuous variable

The 0–1 linear knapsack problem with a single continuous variable (KPC) is an extension of the binary knapsack problem (KP). It is an NP-hard problem. In this paper, we show that KPC can be reduced to KP and a pseudo-knapsack problem (PKP), which is similar to the traditional knapsack problem except that the profits of items may be non-positive, and the weight sum has two sided limits on capacity. We use the Dynamic Programming algorithm COMBO (Martello et al., Manag Sci 45(3):414–424, 1999) to solve KP, and modify the branch and bound method EXPKNAP (Pisinger, Eur J Oper Res 87:175–187, 1995) for KP to solve the PKP. Numerical experiments show the efficiency of our method.     

A new smoothing Broyden-like method for solving nonlinear complementarity problem with a <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-function

In this paper, we propose a new smoothing Broyden-like method for solving nonlinear complementarity problem with P ₀ function. The presented algorithm is based on the smoothing symmetrically perturbed minimum function φ(a, b) = min{a, b} and makes use of the derivative-free line search rule of Li et al. (J Optim Theory Appl 109(1):123–167, 2001). Without requiring any strict complementarity assumption at the P ₀-NCP solution, we show that the iteration sequence generated by the suggested algorithm converges globally and superlinearly under suitable conditions. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical results are also reported in this paper.     

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Arnold, Falk, and Winther recently showed (Bull. Am. Math. Soc. 47:281–354, 2010) that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article (arXiv:), we extended the Arnold–Falk–Winther framework by analyzing variational crimes (à la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace–Beltrami operator on 2- and 3-surfaces, due to Dziuk (Lecture Notes in Math., vol. 1357:142–155, 1988) and later Demlow (SIAM J. Numer. Anal. 47:805–827, 2009), as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold–Falk–Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.     

On the Union of Cylinders in Three Dimensions

We show that the combinatorial complexity of the union of n infinite cylinders in ℝ³, having arbitrary radii, is O(n ^2+ε ), for any ε>0; the bound is almost tight in the worst case, thus settling a conjecture of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000), who established a nearly-quadratic bound for the restricted case of nearly congruent cylinders. Our result extends, in a significant way, the result of Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000), in particular, a simple specialization of our analysis to the case of nearly congruent cylinders yields a nearly-quadratic bound on the complexity of the union in that case, thus significantly simplifying the analysis in Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000). Finally, we extend our technique to the case of “cigars” of arbitrary radii (that is, Minkowski sums of line-segments and balls) and show that the combinatorial complexity of the union in this case is nearly-quadratic as well. This problem has been studied in Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000) for the restricted case where all cigars have (nearly) equal radii. Based on our new approach, the proof follows almost verbatim from the analysis for infinite cylinders and is significantly simpler than the proof presented in Agarwal and Sharir (Discrete Comput. Geom. 24:645–685, 2000).     

A consistent talmudic rule for division problems with?multiple references

We consider an extension of the classic division problem with claims, division problems with multiple references. We show that the theory of cooperative games is able to provide a single-valued allocation rule for this class of problems. Moreover, this rule can be related with the Talmud rule for the classic division problem, as in Aumann and Maschler (J. Econ. Theory 36, 195?C213, 1985). Finally, we establish the consistency and other basic properties of the rule.     

Variational problem in the non-negative orthant of ℝ<Superscript>3</Superscript>: reflective faces and boundary influence cones

In this paper we consider the variational problem in the non-negative orthant of ℝ³. The solution of this problem gives the large deviation rate function for the stationary distribution of an SRBM (Semimartingal Reflecting Brownian Motion). Avram, Dai and Hasenbein (Queueing Syst. 37, 259–289, 2001) provided an explicit solution of this problem in the non-negative quadrant. Building on this work, we characterize reflective faces of the non-negative orthant of ℝ ^d , we construct boundary influence cones and we provide an explicit solution of several constrained variational problems in ℝ³. Moreover, we give conditions under which certain spiraling paths to a point on an axis have a cost which is strictly less than the cost of every direct path and path with two pieces.