Payoff Structures Research Articles

Circulant games

We study a class of two-player normal-form games with cyclical payoff structures. A game is called circulant if both players’ payoff matrices fulfill a rotational symmetry condition. The class of circulant games contains well-known examples such as Matching Pennies, Rock-Paper-Scissors, as well as subclasses of coordination and common interest games. The best response correspondences in circulant games induce a partition on each player’s set of pure strategies into equivalence classes. In any Nash Equilibrium, all strategies within one class are either played with strictly positive or with zero probability. We further show that, strikingly, a single parameter fully determines the exact number and the structure of all Nash equilibria (pure and mixed) in these games. The parameter itself only depends on the position of the largest payoff in the first row of one of the player’s payoff matrix.

Labor Unemployment Risk and CEO Incentive Compensation

In this paper, we investigate the impact of workers’ unemployment risk, a salient attribute of labor market frictions, on the design of CEO incentive compensation. Exploiting state-level changes in unemployment insurance benefits as a source of variation in the unemployment costs faced by workers, we find that, after unemployment insurance benefits increase, boards adjust the compensation structure and provide managers with more stock option grants that result in more convexity payoffs. Such a change is consistent with the view that CEO’s risk-taking incentives are amplified to take advantage of lower unemployment costs. The increase in convexity payoff structures is more pronounced in labor-intensive industries and industries that have higher layoff propensity, but is significantly attenuated when strong labor unions are present. The results are also stronger when CEO wealth is more closely tied to firm performance. Finally, we show that the increase in convexity payoff from option-based compensation induces firms to undertake more risky investments that deliver improved operating performance after the increase in unemployment insurance benefits. These results suggest that firms respond to unemployment risk by changing their risk taking behavior, and one channel through which they do so is executive compensation.

Nash equilibrium and evolutionary dynamics in semifinalists' dilemma.

We consider a tournament among four equally strong semifinalists. The players have to decide how much stamina to use in the semifinals, provided that the rest is available in the final and the third-place playoff. We investigate optimal strategies for allocating stamina to the successive matches when players' prizes (payoffs) are given according to the tournament results. From the basic assumption that the probability to win a match follows a nondecreasing function of stamina difference, we present symmetric Nash equilibria for general payoff structures. We find three different phases of the Nash equilibria in the payoff space. First, when the champion wins a much bigger payoff than the others, any pure strategy can constitute a Nash equilibrium as long as all four players adopt it in common. Second, when the first two places are much more valuable than the other two, the only Nash equilibrium is such that everyone uses a pure strategy investing all stamina in the semifinal. Third, when the payoff for last place is much smaller than the others, a Nash equilibrium is formed when every player adopts a mixed strategy of using all or none of its stamina in the semifinals. In a limiting case that only last place pays the penalty, this mixed-strategy profile can be proved to be a unique symmetric Nash equilibrium, at least when the winning probability follows a Heaviside step function. Moreover, by using this Heaviside step function, we study the tournament by using evolutionary replicator dynamics to obtain analytic solutions, which reproduces the corresponding Nash equilibria on the population level and gives information on dynamic aspects.

Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives

Developing a liquid longevity market requires reliable and well-designed financial instruments. An index-based longevity swap and a cap are analyzed in this paper under a tractable stochastic mortality model. The model is calibrated using Australian mortality data and analytical formulas for prices of longevity derivatives are provided. Hedge effectiveness is examined under a hypothetical life annuity portfolio subject to longevity risk. The paper presents various hedging features exhibited by a longevity swap and a cap based on different assumptions underlying the market price of longevity risk, the term to maturity of hedging instruments, as well as the size of the underlying annuity portfolio. The results are demonstrated to have important implications for the optimal use of longevity hedging instruments with linear and nonlinear payoff structures.

Contagion and uninvadability in local interaction games: The bilingual game and general supermodular games

In a setting where an infinite population of players interact locally and repeatedly, we study the impacts of payoff structures and network structures on contagion of a convention beyond 2×2 coordination games. First, we consider the “bilingual game”, where each player chooses one of two conventions or adopts both (i.e., chooses the “bilingual option”) at an additional cost. For this game, we completely characterize when a convention spreads contagiously from a finite subset of players to the entire population in some network, and conversely, when a convention is never invaded by the other convention in any network. We show that the Pareto-dominant (risk-dominant, resp.) convention is contagious if the cost of bilingual option is low (high, resp.). Furthermore, if the cost is in a medium range, both conventions are each contagious in respective networks, and in particular, the Pareto-dominant convention is contagious only in some non-linear networks. Second, we consider general supermodular games, and compare networks in terms of their power of inducing contagion. We show that if there is a weight-preserving node identification from one network to another, then the latter is more contagion-inducing than the former in all supermodular games.

Choice deferral can arise from absolute evaluations or relative comparisons.

When choosing among several options, people may defer choice for either of 2 reasons: because none of the options is good enough or because there is uncertainty regarding which is the best. These reasons form the basis of the 2-stage, 2-threshold (2S2T) framework, which posits that a different kind of processing corresponds to these 2 reasons for choice deferral: absolute evaluations and relative comparisons, respectively. Three experiments are reported in which each type of processing was triggered in different conditions either via different payoff structures or different degrees of attribute knowledge. The effects of the 3 main independent variables (the size of the choice set, the utility of the best option, and the number of competitive options) differed depending on the payoff structure or attribute knowledge conditions in ways predicted by the 2S2T framework. Implications for consumer decision making, marketing, and eyewitness identification are discussed.

Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes

We develop efficient fast Fourier transform algorithms for pricing and hedging discretely sampled variance products and volatility derivatives under additive processes (time-inhomogeneous Levy processes). Our numerical algorithms are non-trivial versions of the Fourier space time stepping method to nonlinear path dependent payoff structures, like those in variance products and volatility derivatives. The exotic path dependency associated with the discretely sampled realized variance is captured in the numerical procedure by updating two path dependent state variables across monitoring dates. The time stepping procedure between successive monitoring dates can be performed using fast Fourier transform calculations without the usual tedious time stepping calculations in typical finite difference algorithms. We also derive effective numerical procedures that compute the hedge parameters of variance products and volatility derivatives. Numerical tests on pricing various variance products and volatility derivatives were performed that illustrate efficiency, accuracy, reliability and robustness of the proposed Fourier transform algorithms.

Robust Option Pricing with Characteristic Functions and the B-Spline Order of Density Projection

This paper extends and refines the method of option pricing by frame projection of risk-neutral densities to incorporate general B-splines, including the cubic basis, and general payoff structures. We introduce a coefficient stabilization method which substantially improves convergence for higher order splines. This stabilization has subsequently provided robust implementations for exotic option extensions of the method. After demonstrating the greater robustness of a fixed-width truncation rule over generally accepted cumulant based approaches, we devise a novel Hilbert transform based approach for the selection of truncated density supports, applicable to any density approximation method, which facilitates greater control over realized pricing errors. Robustness of B-spline frame projection is demonstrated by an extensive set of experiments which guide the selection of splines of various orders and subsequent parameter decisions. Finally, we provide formulas for digital and forward starting options.

Dynamics of spatial traveler's dilemma games

By examining payoff structures, we elucidate the traveler's dilemma (TD) game. It can be said that the TD game as the 2-player & multi-strategy game is a variant of the donor & recipient game, which itself is a subclass of the 2-player & 2-strategy (2 × 2) prisoner's dilemma, in the sense that the Nash equilibrium and fair Pareto optimum appear on mutual defection and mutual cooperation, respectively. Exploring spatial TD games, we observed that the network reciprocity of TD games can be universally scaled with a single dilemma parameter as can be performed in usual 2 × 2 games. Furthermore, we observed a local peak event reflecting a resurging cooperation fraction with increasing dilemma strength after degradation to a low level. We successfully demonstrate that this is caused by the specific dynamics of TD games created by their specific payoff structure.

Robustness of public equilibria in repeated games with private monitoring

A repeated game with private monitoring is “close” to a repeated game with public monitoring (or perfect monitoring) when (i) the expected payoff structures are close and (ii) the informational structures are close in the sense that private signals in the private monitoring game can be aggregated by some public coordination device to generate a public signal whose distribution is close to the distribution of the public signal in the public monitoring game. We provide a sufficient condition for the set of uniformly strict perfect public equilibria for a public monitoring game to be robust in nearby private monitoring games in the sense that they remain equilibria with respect to the public signal that is generated by such public coordination devices with truthful reporting. Our sufficient condition requires that every player is informationally small in a well-defined sense.

Book-Tax Conformity and Capital Structure

We examine the effect of increased book-tax conformity on corporate capital structure. Prior studies document a decrease in the informativeness of accounting earnings for equity markets resulting from higher book-tax conformity. We argue that the decrease in earnings informativeness impacts equity holders more than debt holders because of the differences in payoff structures between debt and equity investments such that increases in book-tax conformity lead to increases in firms’ reliance on debt capital. We exploit a natural experiment in the U.S. and find that firms facing an increase in required book-tax conformity increase leverage relative to other firms. We also provide evidence of an increase in the cost of equity (but not of debt) capital for firms facing an increase in required book-tax conformity relative to control firms and that these increases in cost of equity capital are positively associated with an increase in leverage. Our findings are consistent with firms substituting away from equity and towards more debt in the presence of higher book-tax conformity.

Optimal Partial Proxy Method for Computing Gammas of Financial Products with Discontinuous and Angular Payoffs

We extend the limit optimal partial proxy method to compute second order sensitivities of financial products with discontinuous or angular payoffs by Monte Carlo simulation. The methodology is optimal in terms of minimizing the variance of likelihood ratios terms. Applications are presented for both equity options and interest rate products with discontinuous payoff structures. The first order optimal partial proxy method is also implemented to calculate the Hessians of insurance products with angular payoffs. Numerical results are presented which demonstrate the speed and efficacy of the method.

SUBGAME CONSISTENT COOPERATIVE SOLUTIONS FOR RANDOMLY FURCATING STOCHASTIC DYNAMIC GAMES WITH UNCERTAIN HORIZON

In cooperative stochastic dynamic games a stringent condition — subgame consistency — is required for a dynamically stable solution. A cooperative solution is subgame consistent if an extension of the solution policy to a situation with a later starting time and any feasible state brought about by prior optimal behavior would remain optimal. This paper considers subgame consistent cooperative solutions in randomly furcating stochastic discrete-time dynamic games with uncertain horizon. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. Novel forms of Bellman equations and Hamilton–Jacobi–Bellman equations for solving intertemporal problems with randomly furcating payoffs and random horizon are developed. This is the first time that subgame consistent solution for games with stochastic dynamics, uncertain future payoff structures and random horizon is obtained.

Gambling primates: reactions to a modified Iowa Gambling Task in humans, chimpanzees and capuchin monkeys.

Humans will, at times, act against their own economic self-interest, for example, in gambling situations. To explore the evolutionary roots of this behavior, we modified a traditional human gambling task, the Iowa Gambling Task (IGT), for use with chimpanzees, capuchin monkeys and humans. We expanded the traditional task to include two additional payoff structures to fully elucidate the ways in which these primate species respond to differing reward distributions versus overall quantities of rewards, a component often missing in the existing literature. We found that while all three species respond as typical humans do in the standard IGT payoff structure, species and individual differences emerge in our new payoff structures. Specifically, when variance avoidance and reward maximization conflicted, roughly equivalent numbers of apes maximized their rewards and avoided variance, indicating that the traditional payoff structure of the IGT is insufficient to disentangle these competing strategies. Capuchin monkeys showed little consistency in their choices. To determine whether this was a true species difference or an effect of task presentation, we replicated the experiment but increased the intertrial interval. In this case, several capuchin monkeys followed a reward maximization strategy, while chimpanzees retained the same strategy they had used previously. This suggests that individual differences in strategies for interacting with variance and reward maximization are present in apes, but not in capuchin monkeys. The primate gambling task presented here is a useful methodology for disentangling strategies of variance avoidance and reward maximization.

Cultural evolution in more than two dimensions: Distinguishing social learning biases and identifying payoff structures

Bentley et al.'s two-dimensional conceptual map is complementary to cultural evolution research that has sought to explain population-level cultural dynamics in terms of individual-level behavioral processes. Here, I qualify their scheme by arguing that different social learning biases should be treated distinctly, and that the transparency of decisions is sometimes conflated with the actual underlying payoff structure of those decisions.

Dynamic market for lemons with endogenous quality choice by the seller

We analyze a dynamic market for lemons in which the quality of the good is endogenously determined by the seller. Potential buyers sequentially submit offers to one seller. The seller can make an investment that determines the quality of the item at the beginning of the game, which is unobservable to buyers. At the interim stage of the game, the information and payoff structures are the same as in the market for lemons. Our main result is that the possibility of trade does not create any efficiency gain if (i) the common discounting is low, and (ii) the static incentive constraints preclude the mutually agreeable ex-ante contract under which the trade happens with probability one. Our result does not depend on whether the offers by buyers are private or public.

What does “clean” really mean? The implicit framing of decontextualized experiments

It is standard in experimental economics to use decontextualized designs where payoff structures are presented using neutral language. Here we show that cooperation in such a neutrally framed Prisoner’s Dilemma is equivalent to a PD framed as contributing to a cooperative endeavour. Conversely, there is substantially less cooperation in a PD framed as a competition. We conclude that in a decontextualized context, our participants by default project a cooperative frame onto the payoff structure.

Differential Effects of Billing Styles on Workers’ Overbilling

In four studies, we examine whether billing style (itemized vs. bulk bill) can influence instances of overbilling. Study 1, a field experiment, supported the hypothesis that an itemized billing style results in less unethical behavior. Specifically, we found that asking auto mechanics to itemize labor and parts to perform repairs rather than asking them for a lump sum cost figure led to estimates that were lower and closer to the industry standard recommended by AutoMD. Study 2, a laboratory study, demonstrated that asking participants to itemize their bills based on the number of units of work completed rather than the total amount of money earned decreased overbilling when the billing structure was a function of units completed. Studies 3 and 4, showed that asking participants to itemize their bills based on the number of minutes spent on the task rather than the bulk money earned decreased overbilling when the billing structure was by the minute. Moreover, felt accountability mediated this effect. Together, the results of these studies using different methods and participant populations demonstrate that billing style, in identical work and payoff structures, can influence ethical behavior. We discuss the implications of our findings for theory and practice.

Agent or Borrower? An Incentive of Moral Hazard with China Commercial Guarantee Company

This paper studies the behavior of commercial guarantee company in the game theory framework. Different payoff structures with guarantee company behavior for banks, borrowers, and the guarantors are formed in the presence of credit rationing. We derived and explained the surviving rule of different types of guarantee companies, and we further conclude the condition in which guarantee companies are likely to be involved in a moral hazard problem. We conclude that, in the presence of regulatory arbitrage opportunity, the guarantee company has incentive to switch its role from bank agent to the borrower, being a second-hand lender. When the macroeconomic risk is high, the switching of this role will transfer the risk back to the banks.

Explicit approximations of multi-asset option prices including Greeks

In this article, we give a lower bound approximation to a stochastic programme that is typical for many complex financial derivatives, e.g., Bermudan options, Rainbow options and swing contracts. The approximation is based on an algorithm that does not require any simulation but uses a well-known spread option pricing formula. Since the method explicitly computes lower bounds for the claim’s value, sensitivities such as deltas and gammas are available as well as derivatives with respect to correlations and other inputs. Our resulting procedure is a powerful and practical tool for the pricing and risk analysis of exotic payoff structures. As such, it is useful in the day-to-day environment of financial institutions.