Continuous Payoff Functions Research Articles

European and Asian Greeks for Exponential Lévy Processes

In this paper we give easy-to-implement closed-form expressions for European and Asian Greeks for general L2-payoff functions and underlying assets in an exponential Lévy process model with nonvanishing Brownian motion part. The results are based on Hilbert space valued Malliavin Calculus and extend previous results from the literature. Numerical experiments suggest, that in the case of a continuous payoff function, a combination of Malliavin Monte Carlo Greeks and the finite difference method has a better convergence behavior, whereas in the case of discontinuous payoff functions, the Malliavin Monte Carlo method clearly is the superior method compared to the finite difference approach, for first- and second order Greeks. Reduction arguments from the literature based on measure change imply that the expressions for the Greeks in this paper also hold true for generalized Asian options in particular for fixed and floating strike Asian options.

Games over Probability Distributions Revisited: New Equilibrium Models and Refinements

This article is an overview of recent progress on a theory of games, whose payoffs are probability distributions rather than real numbers, and which have their equilibria defined and computed over a (suitably restricted yet dense) set of distributions. While the classical method of defining game models with real-valued utility functions has proven strikingly successful in many domains, some use cases from the security area revealed shortcomings of the classical real-valued game models. These issues motivated the use of probability distributions as a more complex object to express revenues. The resulting class of games displays a variety of phenomena not encountered in classical games, such as games that have continuous payoff functions but still no equilibrium, or games that are zero-sum but for which fictitious play does not converge. We discuss suitable restrictions of how such games should be defined to allow the definition of equilibria, and show the notion of a lexicographic Nash equilibrium, as a proposed solution concept in this generalized class of games.

Symmetric games with only asymmetric equilibria: examples with continuous payoff functions

Symmetric games with only asymmetric equilibria: examples with continuous payoff functions

Non-emptiness of the fuzzy core in a finite production economy with infinite-dimensional commodity space

In this paper, we prove the non-emptiness of the fuzzy core in a finite coalition production economy under some standard assumptions which generalizes the results by Boehm [Rev Econ Stud (1974) 41: 429-436] and Predtetchinski [Economic Theory (2005) 26: 717-724], where we allow the utility functions to be possibly discontinuous. We also derive the non-emptiness of α-cores for production economies with possibly discontinuous payoff functions which implies the well-known α-core existence theorem by Scarf [Journal of Economic Theory (1971) 3: 169-181], and derive the non-emptiness of fuzzy α-cores for production economies with continuous payoff functions.

Where to stand when playing darts?

This paper analyzes the question of where one should stand when playing darts. If one stands at distance d > 0 and aims at alpha is an element of R-n, then the dart (modelled by a random vector X in Rn) hits a random point given by alpha + dX. Next, given a payoff function f, one considers sup(alpha) Ef (alpha + dX) and asks if this is decreasing in d; i.e., whether it is better to stand closer rather than farther from the target. Perhaps surprisingly, this is not always the case and understanding when this does or does not occur is the purpose of this paper. We show that if X has a so-called selfdecomposable distribution, then it is always better to stand closer for any payoff function. This class includes all stable distributions as well as many more. On the other hand, if the payoff function is cos(x), then it is always better to stand closer if and only if the characteristic function vertical bar phi(X) (t)vertical bar is decreasing on [0, infinity). We will then show that if there are at least two point masses, then it is not always better to stand closer using cos(x). If there is a single point mass, one can find a different payoff function to obtain this phenomenon. Another large class of darts X for which there are bounded continuous payoff functions for which it is not always better to stand closer are distributions with compact support. This will be obtained by using the fact that the Fourier transform of such distributions has a zero in the complex plane. This argument will work whenever there is a complex zero of the Fourier transform. Finally, we analyze if the property of it being better to stand closer is closed under convolution and/or limits.

Lipschitz Continuity and Approximate Equilibria

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in L_1 and L_2^2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

A Semi-Potential for Finite and Infinite Games in Extensive Form

We consider a dynamic approach to games in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite sequential games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies. For infinite games in extensive form we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are Delta ^0_2-sets.

A New Approach to Noncooperative Games Under Uncertainty

The novelty of the approach presented in this paper is that each player of a conflict seeks not only to increase his payoff, but also to reduce his risk, taking into account a possible realization of any uncertainty from a given admissible set. A new concept, the so-called strongly guaranteed Nash equilibrium in payoffs and risks, is introduced and its existence in mixed strategies is proved under standard constraints for noncooperative games, i.e., compact sets of players’ strategies and continuous payoff functions.

A Semi-Potential for Finite and Infinite Sequential Games (Extended Abstract)

We consider a dynamical approach to sequential games. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies. For infinite sequential games we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are Delta^0_2 sets.

Pareto-optimal Nash equilibrium: Sufficient conditions and existence in mixed strategies

This paper considers the Nash equilibrium strategy profiles that are Pareto optimal with respect to the rest Nash equilibrium strategy profiles. The sufficient conditions for the existence of such pure strategy profiles are established. These conditions employ the Germeier convolutions of the payoff functions. For the non-cooperative games with compact strategy sets and continuous payoff functions, the existence of the Pareto optimal Nash equilibria in mixed strategies is proved.

On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities

The nonemptiness of the α-core of games with continuous payoff functions was proved by Scarf (1971) for nontransferable utilities and by Zhao (1999) for transferable utilities. In this paper we present generalizations of their results to games with possibly discontinuous payoff functions. Our handling of discontinuity is based on Reny's (1999) better-reply-security concept. We present examples to show that our generalizations are nonvacuous.

A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value

We give an example of a zero-sum stochastic game with four states, compact action sets for each player, and continuous payoff and transition functions, such that the discounted value does not converge as the discount factor tends to 0, and the value of the n-stage game does not converge as n goes to infinity.

Static Replication of Forward-Start Claims and Realized Variance Swaps

The goal of this work is to examine the static replication of path-dependent derivatives such as realized variance swaps, using more standard products such as forward-start binary (i.e. digital) double calls and puts. We first examine, following Carr and Madan (2002), the static replication of path-independent claims with continuous and discontinuous payoff functions. Subsequently, the static replication of forward-start claims with payoffs given by a bivariate function of finite variation is examined. We postulate that certain forward-start binary (or barrier) options are traded. The work concludes by an application of our general results to the static hedging of a realized variance swap with forward-start binary (or barrier) options.

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.

Brown-Von Neumann-Nash Dynamics: The Continuous Strategy Case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown--von Neumann--Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS *) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS * is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS * by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS *. We show by an example that IESDS * may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS * preserves the set of Nash equilibria.

Iterated Strict Dominance in General Games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS by means of a stability criterion, and offer a sufficient and necessary epistemic condition for IESDS. We show by an example that IESDS may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS preserves the set of Nash equilibria.

Essential stability in games with infinitely many pure strategies

We study the notion of essential stability of equilibria in games with compact metric spaces of pure strategies and continuous payoff functions. We first prove that the games whose equilibria are all essential form a residual set in the sense of Baire Category. Then we prove that each of the games has one minimal essential set and one essential component of the set of equilibria. Finally, we investigate relations between essential stability and strategic stability (Al-Najjar 1995), and show that every essential equilibrium is perfect and that every minimal essential set contains a stable set. We deduce that each of the games has such an essential component that contains a minimal essential set containing a stable set.

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium. The joint continuity of the expected payoff functions is not obvious at all. A general proof was presented by Glycopantis and Muir [D. Glycopantis, A. Muir, 2000. Continuity of the payoff functions, Economic Theory 16, 239–244]. An alternative proof was obtained by Zarichnyi [M. Zarichnyi, 2004. Continuity of the payoff function revisited, Economics Bulletin 3, 1–4]. A proof can also be obtained by using Billingsley [P. Billingsley, 1968. Convergence of Probability Measures, Wiley, New York, Theorem 3.2, p. 21] or by invoking a result of Balder [E.J. Balder, 1988. Generalized equilibrium results for games with incomplete information, Mathematics of Operations Research 13, 265–276, Theorem 3.1, p. 273]. In this note an alternative proof of the joint continuity of the expected payoff functions is given that is based on measure theoretic techniques.

Representing von Neumann–Morgenstern Games in the Situation Calculus

Sequential von Neumann–Morgernstern (VM) games are a very general formalism for representing multi-agent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent game-theoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of planning and multi-agent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions.