Axiomatizing core extensions

We give an axiomatization of the aspiration core on the domain of all TU-games using a relaxed feasibility condition, non-emptiness, individual rationality, and generalized versions of the reduced game property (consistency) and superadditivity. Our axioms also characterize the C-core (Guesnerie and Oddou, Econ Lett 3(4):301?C306, 1979; Sun et?al. J Math Econ 44(7?C8):853?C860, 2008) and the core on appropriate subdomains. The main result of the paper generalizes Peleg??s (J Math Econ 14(2):203?C214, 1985) core axiomatization to the entire family of TU-games.     

The covering values for acyclic digraph games

We introduce a novel covering method to compute values for acyclic digraph games, and we call the values obtained by this method the covering values. These values may be considered as natural extensions of the component efficient solutions for line-graph games studied by van den Brink et?al. (Econ Theory 33:349?C364, 2007), and the tree values studied by Khmelnitskaya (Theory Decis 69(4):657?C669, 2010a). With the new method, we reinterpret the tree values proposed by Khmelnitskaya (2010a). Besides, we propose the covering values in the digraph game with general acyclic digraph structures presenting flow situations when some links may merge while others split into several separate ones. We give axiomatizations of these values, and interpret these values in terms of dividend distributions.     

A cognitive hierarchy model of behavior in the action commitment game

We apply the cognitive hierarchy model of Camerer et al. (Q J Econ 119(3):861–898, 2004)—where players have different levels of reasoning—to Huck et al. (Games Econ Behav 38:240–264, 2002) discrete version of Hamilton and Slutsky (Games Econ Behav 2:29–46, 1990) action commitment game—a duopoly with endogenous timing of entry. We show that, for an empirically reasonable average number of thinking steps, the model rules out Stackelberg equilibria, generates Cournot outcomes including delay, and outcomes where the first mover commits to a quantity higher than Cournot but lower than Stackelberg leader. We show that a cognitive hierarchy model with quantal responses can explain the most important features of the experimental data on the action commitment game in (2002). In order to gauge the success of the model in fitting the data, we compare it to a noisy Nash model. We find that the cognitive hierarchy model with quantal responses fits the data better than the noisy Nash model.     

Modularity and monotonicity of games

The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) and provide a condition under which for a game \(v\) , its Möbius inverse is equal to zero within the framework of the \(k\) -modularity of \(v\) for \(k \ge 2\) . This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, 2007). Second, we provide a condition under which for a game \(v\) , its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of \(k\) -monotone games. Furthermore, this paper shows that the modularity of a game is related to \(k\) -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443–467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, 2011).     

Adjustment Coefficient for Risk Processes in Some Dependent Contexts

Following Müller and Pflug (Insur Math Econ 28:381?C392, 2001) and Nyrhinen (Adv Appl Probab 30:1008?C1026, 1998; J Appl Probab 36:733?C746, 1999), we study the adjustment coefficient of ruin theory in a context of temporal dependency. We provide a consistent estimator for this coefficient, and perform some simulations.     

Imitation, local interaction, and coordination

This paper analyzes players’ long-run behavior in evolutionary coordination games with imitation and one-dimensional local interaction. Players are assumed to interact with their two neighbors and to imitate actions with the highest average payoffs. We find that the payoff-dominant equilibrium survives in the long run with positive probability. The results derive the conditions under which both risk-dominant-strategy and payoff-dominant-strategy takers co-exist in the long run. The risk-dominant equilibrium is the unique long-run equilibrium for the remaining cases. This study extends and complements the analyses of Eshel et al. (Am Econ Rev 88:157–179, 1998) and Vega-Redondo (Evolution, games, and economic behaviour, 1996). Combining Alós-Ferrer and Weidenholzer’s (Econ Lett 93:163–168, 2006; J Econ Theory 14:251–274, 2008) and our results, we conclude that players’ long-run behavior varies with imitation rules and information collecting modes. Finally, we show the convergence rate to all the long-run equilibria.     

On cooperative games with large monotonic core

Cooperative games with large core were introduced by Sharkey (Int. J. Game Theory 11:175–182, 1982), and the concept of Population Monotonic Allocation Scheme was defined by Sprumont (Games Econ. Behav. 2:378–394, 1990). Inspired by these two concepts, Moulin (Int. J. Game Theory 19:219–232, 1990) introduced the notion of large monotonic core giving a characterization for three-player games. In this paper we prove that all games with large monotonic core are convex. We give an effective criterion to determine whether a game has a large monotonic core and, as a consequence, we obtain a characterization for the four-player case.     

The General Solution of Abel-Type Functional Equations

The general measurable solution of (A) was found by Stamate [8]. Aczél [3] and Lajkô [6] proved that the general solution of (A) for unknown functions ψ, g, h: ? → ? are (1), (2) and (3), respectively. Filipescu [5] found the general measurable solution of (B). We establish an elementary prof for the general solution of equation (A) (Theorem 1.). Our method is suitable for finding the general solution of (B) (Theorem 2.).     

Monotonic stable solutions for minimum coloring games

For the class of minimum coloring games (introduced by Deng et al. Math Oper Res, 24:751–766, 1999) we investigate the existence of population monotonic allocation schemes (introduced by Sprumont Games Econ Behav 2:378–394, 1990). We show that a minimum coloring game on a graph $G$ has a population monotonic allocation scheme if and only if $G$ is $(P_4,2K_2)$ -free (or, equivalently, if its complement graph $\bar{G}$ is quasi-threshold). Moreover, we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme.     

On the Dalgaard-Strulik Model with Logistic Population Growth Rate and Delayed-Carrying Capacity

Recently Dalgaard and Strulik have proposed (in Resour. Energy Econ. 33:782–797, 2011) an energy model of capital accumulation based on the mathematical framework developed by Solow-Swan and coupled with Cobb-Douglas production function (Solow in Q. J. Economics 70:65–94, 1956; Swan in Econ. Rec. 32(63):334–361, 1956). The model is based on a constant rate of population growth assumption. The present paper, according to the analysis performed by Yukalov et al. (Physica D 238:1752–1767, 2009), improves the Dalgaard-Strulik model by introducing a logistic-type equation with delayed carrying capacity which alters the asymptotic stability of the relative steady state. Specifically, by choosing the time delay as a bifurcation parameter, it turns out that the steady state loses stability and a Hopf bifurcation occurs when time delay passes through critical values. The results are of great interest in the applied and theoretical economics.     

Adjustable and fixed interest rates mortgage markets modelling

Due to recent developments in credit markets, the interdependencies among fixed rate mortgages (FRMs) and adjustable rate mortgages (ARMs) markets are analysed in the study of the interactions within credit markets since it appears very suitable and interesting. A meaningful and complete database of information on Financial Institutions in Italy (1997:q1–2011:q4) hold by the Banca d’Italia shows that the relative importance of these markets recently displayed significant fluctuations since in 2005 fixed interest rate mortgage loans were about 10 % while in 2009 they raised up to 70 %. In the context of the FRMs and ARMs characteristics in Italian markets as well as the available database, among the proposed models for the study of interconnected markets (see Brock and Hommes in Econometrica 65(5):1059–1095, 1997, J Econ Dyn Control 22:1235–1274, 1998; Dieci and Westerhoff in Appl Math Comput 215:2011–2023, 2009; J Econ Behav Organ 75(3):461–481, 2010 and related literature cited therein), the models recently developed by Casellina et al. (Comput Econ 38:221–239, 2011) is applied to test its capacity to capture the dynamics of the observed data. It is worth stressing that the involved real data (volume of contracts and average interest rate in the FRMs and ARMs markets) are not sample information but they are evaluated on the entire population. The obtained findings point out the good level to fit the interest rates dynamics. Moreover, the model captures the switching mechanism and it catches the structural breaks when they occurs.     

The reverse self-dual serial cost-sharing rule

In this study we define a cost-sharing rule for cost-sharing problems. This rule is related to the serial cost-sharing rule defined by Moulin and Shenker (Econometrica 60:1009–1037, 1992). We give some formulas and axiomatic characterizations for the new rule. The axiomatic characterizations are related to some previous ones provided by Moulin and Shenker (J. Econ. Theory 64:178–201, 1994) and Albizuri (Theory Decis. 69:555–567, 2010).     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

The Shapley value for shortest path games: a non-graph-based approach

The shortest path games are considered in this paper. The transportation of a good in a network has costs and benefits. The problem is to divide the profit of the transportation among the players. Fragnelli et al. (Math Methods Oper Res 52: 251–264, 2000) introduce the class of shortest path games and show it coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further five characterizations of the Shapley value (Hart and Mas-Colell’s in Econometrica 57:589–614, 1989; Shapley’s in Contributions to the theory of games II, annals of mathematics studies, vol 28. Princeton University Press, Princeton, pp 307–317, 1953; Young’s in Int J Game Theory 14:65–72, 1985, Chun’s in Games Econ Behav 45:119–130, 1989; van den Brink’s in Int J Game Theory 30:309–319, 2001 axiomatizations), and conclude that all the mentioned axiomatizations are valid for the shortest path games. Fragnelli et al. (Math Methods Oper Res 52:251–264, 2000)’s axioms are based on the graph behind the problem, in this paper we do not consider graph specific axioms, we take $TU$ axioms only, that is we consider all shortest path problems and we take the viewpoint of an abstract decision maker who focuses rather on the abstract problem than on the concrete situations.     

An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203–214, 1995; d’Aspremont et?al. in J Mach Learn Res 9:1269–1294, 2008; d’Aspremont et?al. SIAM Rev 49:434–448, 2007; Jolliffe in J Appl Stat 22:29–35, 1995; Journée et?al. in J Mach Learn Res 11:517–553, 2010; Jolliffe et?al. in J Comput Graph Stat 12:531–547, 2003; Moghaddam et?al. in Advances in neural information processing systems 18:915–922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015–1034, 2008; Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006), Pitprops (Jeffers in Appl Stat 16:225–236, 1967), and gene expression data (Chin et?al in Cancer Cell 10:529C–541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.     

Awareness-dependent subjective expected utility

We develop awareness-dependent subjective expected utility by taking unawareness structures introduced in Heifetz et al. (J Econ Theory 130:78–94, 2006; Games Econ Behav 62:304–324, 2008; Unawareness, beliefs, and speculative trade. University of California, Davis, 2011a) as primitives in the Anscombe–Aumann approach to subjective expected utility. We observe that a decision maker is unaware of an event if and only if her choices reveal that the event is “null” and the negation of the event is “null”. Moreover, we characterize “impersonal” expected utility that is behaviorally indistinguishable from awareness-dependent subject expected utility and assigns probability zero to some subsets of states that are not necessarily events. We discuss in what sense probability zero can model unawareness.     

Burgers type equations, Gelfand’s problem and Schumpeterian dynamics

Burgers?? equations have been introduced to study different models of fluids (Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill andWhitham, 1955, etc.). The difference-differential analogues of these equations have been proposed for Schumpeterian models of economic development (Iwai, 1984, Polterovich and Henkin, 1988, Belenky, 1990, Henkin and Polterovich, 1999, Tashlitskaya and Shananin, 2000, etc.). This paper gives a short survey of the results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Proofs of some new results are given. This paper is an extension and an improvement of (Henkin, 2007, 2011).     

A probabilistic position value

In this article, we generalize the position value, defined by Meessen (Master??s thesis, 1988) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations (Gómez et al. in European Journal of Operational Research 190:539?C556, 2008). We provide two characterizations of this new allocation rule. Following in Slikker??s (International Journal of Game Theory 33:505?C514, 2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (International Game Theory Review 7:473?C489, 2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule.