Discontinuous Payoff Functions Research Articles

Asymptotic Normality for EMS Option Price Estimator with Continuous or Discontinuous Payoff Functions

Empirical martingale simulation (EMS) was proposed by Duan and Simonato (Duan, J.-C., J.-G. Simonato. 1998. Empirical martingale simulation for asset prices. Management Sci. 44(9) 1218–1233) as an adjustment to the standard Monte Carlo simulation to reduce simulation errors. The EMS price estimator of derivative contracts was shown to be asymptotically normally distributed in Duan et al. (Duan, J.-C., G. Gauthier, J.-G. Simonato. 2001. Asymptotic distribution of the EMS option price estimator. Management Sci. 47(8) 1122–1132) when the payoffs are piecewise linear and continuous. In this paper, we extend the asymptotic normality result to more general continuous payoffs, and for discontinuous payoffs we make a conjecture.

Existence of pure Nash equilibria in discontinuous and non quasiconcave games

In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny’s theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence (I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper).

Asymptotic Properties of Monte Carlo Estimators of Derivatives

We study the convergence of Monte Carlo estimators of derivatives when the transition density of the underlying state variables is unknown. Three types of estimators are compared. These are respectively based on Malliavin derivatives, on the covariation with the driving Wiener process, and on finite difference approximations of the derivative. We analyze two different estimators based on Malliavin derivatives. The first one, the Malliavin path estimator, extends the path derivative estimator of Broadie and Glasserman (1996) to general diffusion models. The second, the Malliavin weight estimator, proposed by Fournié et al. (1999), is based on an integration by parts argument and generalizes the likelihood ratio derivative estimator. It is shown that for discontinuous payoff functions, only the estimators based on Malliavin derivatives attain the optimal convergence rate for Monte Carlo schemes. Estimators based on the covariation or on finite difference approximations are found to converge at slower rates. Their asymptotic distributions are shown to depend on additional second-order biases even for smooth payoff functions.

A Note on the Existence of Nash Equilibrium in Games with Discontinuous Payoffs

This paper generalises the approach taken by Dasgupta & Maskin (1986) and Simon (1989) and provides necessary and sufficient conditions for the existence of pure and mixed strategy Nash equilibrium in games with continuous strategy spaces and discontinuous payoff functions. The conditions can be applied widely, and examples for existence of pure strategy and monotonic equilibria in First-Price auctions are provided. The conditions are also appropriate for ensuring that computer generated equilibrium solutions can be extended to continuous strategy spaces.

Fixed set theory for closed correspondences with applications to self-similarity and games

This paper sketches a highly elementary order-theoretic approach which allows one to derive general existence theorems concerning the largest and minimal fixed and almost-fixed sets of closed (and condensing) families of commuting set-valued self-maps defined on a compact topological (complete and bounded metric, resp.) space. Certain converses of these results are also established, thereby providing new characterizations of compact (complete, resp.) spaces. We also study the asymptotic behavior of fixed sets sequences associated with a uniformly convergent sequence of closed correspondences, and give two applications. The first application provides two theorems on the existence of a compact self-similar set with respect to infinite families of continuous functions. The second application uses the present fixed set theory to derive a result on the existence of rationalizable outcomes in normal-form games with possibly discontinuous payoff functions.

Auction Games and Pricing Process

It's considered a multi-item first price closed auction with asymmetric bidders and one seller under competition. This base auction can be modeled as a non-zero sum game with discontinuous payoff functions. It's shown that in this game exist unique Nash equilibrium in mixed strategies when antimonopoly and competition conditions are accomplished. Then it's investigated also a multistage repeated auction with described base auction as a single step. It's derived sufficient conditions when a cooperative cartel solution is time consistent. To construct the imputation in models of profit allocation and to compute transaction (cost allocation as a dual problem) is used the solution of the base auction game.

Existence of equilibrium in abstract economies with discontinuous payoffs and non-compact choice spaces

This paper proves the equilibrium existence for abstract economies with non-compact infinite- dimensional strategy spaces, infinitely many agents, and discontinuous payoff (utility) functions by using the quasi-variational inequality approach. The motivations come from economic applications showing that payoff functions are discontinuous in many cases and the set of feasible allocations generally is not compact in a given topology of the commodity space, a typical situation in infinite dimensional vector space. It will be noted that our results also extend a foundational quasi-variational inequality by relaxing the compactness and concavity conditions. Thus many existence theorems in the quasi-variational inequalities literature can also be generalized by our results.

The Existence of Equilibrium in Discontinuous Economic Games, II: Applications

In part I of this study we established two existence theorems for mixed-strategy equilibrium in games with discontinuous payoff functions (see Theorems 4(4*) and 5(5*) in Dasgupta and Maskin (1986); hereafter D-M (1986)). In this second part we use the second of these two theorems to prove the existence of mixed-strategy equilibrium in some wellknown discontinuous economic games that fail to have pure-strategy equilibria. The models we analyse are the Bertrand-Edgeworth example of price setting duopolists with capacity constraints (see Section 2.2); price competition among firms producing differentiated products (Hotelling (1929) and d'Aspremont, Gabszewicz and Thisse (1979); see Section 2.3); spatial competition (Eaton and Lipsey (1975) and Shaked (1975); see Section 3); and models of market dependent information (Rothschild and Stiglitz (1976) and Wilson (1977); Section 4). The paper concludes (Section 5) with a classification of these models.

The sequence method for finding solutions to infinite games: A first demonstrating example

A noncooperative infinite game can be approached by a sequence of discrete games. For each game in the sequence, a Nash solution can be found as well as their limit. This idea and procedure was used before as a theoretical device to prove existence of solutions to games with continuous payoffs and recently even for a class of games with discontinuous ones (Dasgupta and Maskin, 1981). No one, however, used the method for the actual solution of a game. Here, an example demonstrates the method's usefulness in finding a solution to a two-person game on the unit square with discontinuous payoff functions.