A note on sequential auctions with multi-unit demand

Within the independent private value paradigm, this note first analyzes two-round sequential first-price auctions with multi-unit demand. We show that the expected price in the first round is strictly lower than that in the second round due to the “extraction effect”. We then compare the revenues for the sequential auctions and the simultaneous auctions. We show that the discriminatory auction, the Vickrey auction, and the sequential second-price auctions generate the same revenue for the seller, followed in order by the sequential first-price auctions, and by the uniform-price auction.     

Ranking asymmetric auctions

We compare the expected revenue in first- and second-price auctions with asymmetric bidders. We consider “close to uniform” distributions with identical supports and show that in the case of identical supports the expected revenue in second-price auctions may exceed that in first-price auctions. We also show that asymmetry over lower valuations has a stronger negative impact on the expected revenue in first-price auctions than in second-price auctions. However, asymmetry over high valuations always increases the revenue in first-price auctions.     

All-pay auctions with risk-averse players

We study independent private-value all-pay auctions with risk-averse players. We show that: (1) Players with low values bid lower and players with high values bid higher than they would bid in the risk neutral case. (2) Players with low values bid lower and players with high values bid higher than they would bid in a first-price auction. (3) Players’ expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-averse players and the seller’s expected revenue. In particular, we show that in all-pay auctions the seller’s expected payoff in the risk-averse case may be either higher or lower than in the risk neutral case.     

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.     

On Sears’s general transformation formula for basic hypergeometric series

We present a new proof of a general transformation formula for basic hypergeometric series that was discovered by D.B. Sears (Proc. Lond. Math. Soc. 53(2): 181–191, 1951) and discuss some special cases. Next we apply Sears’s general transformation to give an extension of a result of Andrews et al. (Duke Math. J. 108: 395–419, 2001).     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

How to bid in unified second-price auctions when requests are duplicated

In display advertising auctions, a unique display opportunity may trigger many bid requests being sent to the same buyer. Bid request duplication is an issue: programmatic bidding agents might bid against themselves. In a simplified setting of unified second-price auctions, the optimal solution for the bidder is to randomize the bid, which is quite unusual. Our results motivate the recent switch to a unified first-price auction by showing that a unified second-price auction could have been detrimental to all participants.     

Charity auctions for the happy few

Recent literature has shown that all-pay auctions raise more money for charity than either winner-pay auctions or lotteries. We demonstrate that first-price and second-price winner-pay auctions have a better revenue performance than first-price and second-price all-pay auctions when bidders are sufficiently asymmetric. Lotteries can also provide higher revenue than all-pay auctions. To prove this, we consider a framework with complete information. Complete information is helpful and may reflect events that occur, for instance, in a local service club (such as a voluntary organization) or at a show-business dinner.     

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

One-dimensional chemotaxis kinetic model

In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar–Dunbar–Alt system (Othmer in J Math Biol 26(3):263–298, 1988). This version was exhibited in Calvez in Amer Math Soc, pp 45–62, 2007 for the macroscopic well-known Keller–Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009, Chalub, Math Models Methods Appl Sci, 16(7 suppl):1173–1197, 2006, Chalub, Monatsh Math, 142(1–2):123–141, 2004, Chalub, Port Math (NS), 63(2):227–250, 2006. The model we study here behaves in a similar way to the original model in two dimensions with the spherical symmetry assumption on the initial data which is described in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009. We prove the existence and uniqueness of solutions for this model, as well as a convergence result for a family of numerical schemes. The advantage of this model is that numerical simulations can be easily done especially to track the blow-up phenomenon.     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287–349, [2005]] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), [2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, [1985]] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, [1987]] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, [1965], [1974]] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245–246, [1976]] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, [1976]]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951–976, [1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.      

A Generalized Comparison Theorem for BSDEs and Its?Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).     

Components of the Fundamental Category II

In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom. Différentielle Catég., 44: 281–316, 2003; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003), we show in this paper a similar theorem for component categories (conjectured in Fajstrup et al. (APCS, 12(1): 81–108, 2004). This proves useful for inductively computing the component category in some circumstances, for instance, in the case of simple PV mutual exclusion models (Goubault and Haucourt, A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517, Springer, Berlin Heidelberg New York, 2005), corresponding to partially ordered subspaces of R ⁿ minus isothetic hyperrectangles. In this last case again, we conjecture (and give some hints) that component categories enjoy some nice adjunction relations directly with the fundamental category.      

The buy price in auctions with discrete type distributions

This paper considers second-price, sealed-bid auctions with a buy price where bidders’ types are discretely distributed. We characterize all equilibria in which bidders whose types are less than the buy price bid their own valuations. Budish and Takeyama (2001) analyze the two-bidder, two-type framework. They show that if bidders are risk-averse, then the seller can obtain a higher expected revenue from the auction with a certain buy price than from the auction without a buy price. We extend their revenue improvement result to the n-bidder, two-type framework. In case of three or more types, however, bidders’ risk aversion is not a sufficient condition for a revenue improvement. We point out that even if bidders are risk-averse, the seller cannot always obtain a higher expected revenue from the auctions with a buy price.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3:162–170, 1980) of adaptive type based on the Newton–Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17:207–217, 1991; ACM Trans. Math. Softw. 17:218–232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner’s ones (Computation of numerical quadrature formulas. In: Rice, J.R. (ed.) Mathematical Software, 229–259. Academic, Orlando, FL, 1971).      

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).