Continuous Payoff Research Articles

Agency and investment with triggered time-inconsistent preferences

Individual’s time preferences may be influenced by a variety of factors such as economic conditions. In this paper, we consider the time inconsistent preferences triggered by the switches in firm’s profitability states and apply them to the model of dynamic agency and to the q theory of investment. And we investigate effects of triggered time-inconsistent preferences on the optimal contract between investors and manager, Tobin’s q and firm investment. We find that time-inconsistent preferences increase the value loss of investors in both states. Owning to the manager’s time-inconsistent preferences, the contract should make more adjustments for the manager’s continuation payoff conditional on a jump from one state to the alternative state. Meanwhile, the firm prefers paying the manager with cash in advance due to the triggered time-inconsistent preferences. It also shows that the manager’s time inconsistency induces underinvestment and reduces the sensitivity of investment, firm’s average q and marginal q in both states.

Continuous Positional Payoffs

What payoffs are positionally determined for deterministic two-player antagonistic games on finite directed graphs? In this paper we study this question for payoffs that are continuous. The main reason why continuous positionally determined payoffs are interesting is that they include the multi-discounted payoffs. We show that for continuous payoffs, positional determinacy is equivalent to a simple property called prefix-monotonicity. We provide three proofs of it, using three major techniques of establishing positional determinacy -- inductive technique, fixed point technique and strategy improvement technique. A combination of these approaches provides us with better understanding of the structure of continuous positionally determined payoffs as well as with some algorithmic results.

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.

A partial identification framework for dynamic games

Empirical work on strategic interactions is often subject to the critique that equilibrium selection assumptions drive the results. We develop a framework for partially identifying parameters of dynamic games without equilibrium selection assumptions. Our framework relies on incentive compatibility constraints that incorporate game theoretical results on equilibrium payoff sets to bound the unknown continuation payoffs. We apply this framework to identify cost parameters in three dynamic games where collusion is a potential outcome. The identified set demonstrates the ease of sustaining collusion with patient firms, in low demand and when monitoring is perfect, and can also be used to detect collusion.

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

Fictitious Play in Zero-Sum Stochastic Games

We present fictitious play dynamics for stochastic games and analyze its convergence properties in zero-sum stochastic games. Our dynamics involves players forming beliefs on opponent strategy and their own continuation payoff (Q-function), and playing a greedy best response using estimated continuation payoffs. Players update their beliefs from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update of the beliefs on strategies. We show both in the model-based and model-free cases (without knowledge of player payoff functions and state transition probabilities), the beliefs on strategies converge to a stationary mixed Nash equilibrium of the zero-sum stochastic game.

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.

Self-accessibility and repeated games with asymmetric discounting

We propose a simple geometric construct called ‘self-accessibility’ to study discounted infinitely repeated games with perfect monitoring. Self-accessibility delivers payoffs by computing action paths that keep all continuation payoff vectors close to the target payoff. It unifies the analysis of games with symmetric and asymmetric discounting, dispenses with public randomization, and generates pure on-path strategies. We first use it to simplify Fudenberg and Maskin (1986, 1991) and generalize their results to asymmetric discounting. Next, we use self-accessibility to find all payoff profiles that are realizable via some path at some (possibly asymmetric) discount vector. Finally, we offer an easily-verifiable sufficient and almost necessary condition for a payoff profile to arise in a subgame-perfect equilibrium, and use self-accessibility to explicitly construct the corresponding equilibrium strategies. We show that achieving cooperation via intertemporal trade among unequally patient players may require very different arrangements from those proposed earlier by Lehrer and Pauzner (1999) for two players using public randomization.

Record-Keeping and Cooperation in Large Societies

Abstract We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”

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.

On games without approximate equilibria

This note shows that the work by Simon and Tomkowicz (Israel J Math 227(1):215–231, 2018) answers another outstanding open question in game theory in addition to the non-existence of approximate Harsányi equilibrium in Bayesian games: it shows that strategic form games with bounded and separately continuous payoffs need not possess approximate equilibria.

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.

On the statistical differences between binary forecasts and real-world payoffs

We map the difference between (univariate) binary predictions, bets and “beliefs” (expressed as a specific “event” will happen/will not happen) and real-world continuous payoffs (numerical benefits/harm from an event) and show the effect of their conflation and mischaracterization in the decision-science literature. We also examine the differences under thin and fat tails. The effects: [A] Spuriousness of many psychological results, particularly those documenting that humans overestimate tail probabilities. We quantify such conflations. [B] Being a “good forecaster” in binary space doesn’t lead to having a good actual performance, and vice versa, especially under nonlinearities. A binary forecasting record is likely to be a reverse indicator under some classes of distributions or deeper uncertainty. [C] Machine Learning: Some nonlinear payoff functions, while not lending themselves to verbalistic expressions, are well captured by ML or expressed in option contracts. Fattailedness: The difference is exacerbated in the power law classes of probability distributions.

A control problem related to the parabolic dominative [formula omitted]-Laplace equation

We show that value functions of a certain time-dependent control problem in Ω×(0,T), with a continuous payoff F on the parabolic boundary, converge uniformly to the viscosity solution of the parabolic dominative p-Laplace equation 2(n+p)ut=Δu+(p−2)λn(D2u), with the boundary data F. Here 2<p<∞, and λn(D2u) is the largest eigenvalue of the Hessian D2u.

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.

Repeated implementation: A practical characterization

We characterize the social choice functions that are repeatedly implementable. The necessary and sufficient condition is formulated in terms of the equilibrium payoff set of an associated repeated game. It follows that the implementability of a function can be tested numerically by approximating the equilibrium payoff set. Additionally, with the help of our characterization, we demonstrate that an efficient function is implementable if and only if it satisfies a weaker version of Maskin monotonicity. As an application, we prove that utilitarian social choice functions are implementable by showing that continuation payoff promises effectively play the role of side-payments, which are needed for implementation in static setups.

Stochastic games with hidden states

This paper studies infinite‐horizon stochastic games in which players observe actions and noisy public information about a hidden state each period. We find a general condition under which the feasible and individually rational payoff set is invariant to the initial prior about the state when players are patient. This result ensures that players can punish or reward their opponents via continuation payoffs in a flexible way. Then we prove the folk theorem, assuming that public randomization is available. The proof is constructive and uses the idea of random blocks to design an effective punishment mechanism.

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.

Imperfect public monitoring with a fear of signal distortion

This paper proposes a model of signal distortion in a two-player game with imperfect public monitoring. We construct a tractable theoretical framework where each player has the opportunity to distort the true public signal and each player is uncertain about the distortion technologies available to the other player. We show that when players evaluate strategies according to their worst-case guarantees—i.e., are ambiguity averse over certain distributions in the environment—perceived continuation payoffs endogenously lie on a positively sloped line. We then provide examples showing that, counterintuitively, identifying deviators can be harmful in enforcing a strategy profile; moreover, we illustrate how the presence of such signal distortion can sustain cooperation when it is impossible in standard settings. We show that the main result and examples are robust to a number of natural modifications to our setting. Finally, we extend our model to a repeated game where our concept is a natural generalization of strongly symmetric equilibria. In this setting, we prove an anti-folk theorem, showing that payoffs under our equilibrium concept are under general conditions bounded away from efficiency.