Actual Payoff Research Articles

Two-stage strategy update rule based on learning cost in weak prisoner's dilemma.

When players are dissatisfied with their actual payoffs, they will change the actuality by learning strategy of neighbors. The more effort players put in, the more likely they are to succeed in learning. Inspired by this, this paper proposes a two-stage strategy update rule based on learning cost. The players first decide whether to learn strategy according to the updating willingness. If the players imitate the strategy of neighbors, they need to pay the learning cost. Results show that for the well-mixed population, if the updating willingness is homogeneous and remains unchanged, reducing the updating willingness or increasing the learning cost can extend the life cycle of cooperators. If the updating willingness is heterogeneous and dynamically adjusted based on the difference between the actual payoff and the expected payoff, increasing aspiration value and learning cost promotes cooperation. For the structured population, if the updating willingness is homogeneous and remains unchanged, the moderate learning cost is beneficial for cooperators to resist the temptation of defection, and reducing updating willingness makes the system maintain cooperation within a larger parameter range. If the updating willingness is heterogeneous and dynamically adjusted, the larger learning cost and the appropriate aspiration value promote cooperation. This study highlights the complex dynamics of cooperation in paid strategy learning, contributing to the theory of cooperation in the evolutionary game.

A Rubinstein bargaining experiment in continuous time

I conduct a bargaining experiment with discounting in continuous time. At any point in time, each of the two players can make a proposal. However, after making an offer, a player must wait for a specific duration of “waiting time” before she can make a new offer or accept the other's offer. In addition, the player's share is discounted by her discount rate for every unit of time. In the equilibrium, the agreement is reached immediately, and a player's share increases with her waiting time and decreases with her discount rate. Results show that a longer waiting time or a lower discount rate alone increases a player's proportion of the actual payoff, but the combination of the two factors has a weaker effect.

Price discrimination with robust beliefs

This paper considers the problem of second-degree price discrimination when the type distribution is unknown or imperfectly specified by means of an ambiguity set. As robustness measure we use a performance index, equivalent to relative regret, which quantifies the worst-case attainment ratio between actual payoff and ex-post optimal payoff. We provide a simple representation of this performance index, as the lower envelope of two boundary performance ratios, relative to beliefs that lie at the boundary of the ambiguity set. A characterization of the solution to the underlying robust identification problem is given, which leads to a robust product portfolio, for which we also determine the worst-case performance over all possible consumer types. For a standard linear-quadratic specification of the robust screening model, a worst-case performance index of 75% guarantees that the robust product portfolio exhibits a profitability that lies within a 25%-band of an ex-post optimal product portfolio, over all possible model parameters and beliefs. Finally, a numerical comparison benchmarks the robust solution against a number of alternative belief heuristics.

Self-interest is intuitive during opportunity (in)equity: Evidence from multivariate pattern analysis of electroencephalography data

Fairness is a remarkable preference for human society, involving both outcome and opportunity equity. Most previous studies have explored whether fairness itself or self-interest is intuitive during outcome (in)equity. However, intuition during outcome (in)equity can be affected by both fairness level and actual payoff. Since opportunity (in)equity is only affected by the fairness level, we explored only intuition during fairness by measuring event-related potential responses to opportunity (in)equity. Participants played a social non-competitive two-person choice game with advantage opportunity inequity (AI), opportunity equity (OE), and disadvantage opportunity inequity (DI). The behavioral results suggested an opportunity inequity bias, with greater feelings of fairness and pleasantness during OE than during AI and DI. However, multivariate pattern analysis of the event-related potential (ERP) data suggested that AI, OE, and DI can be significantly distinguished from each other in relatively early windows overlapping with early positive negativity (EPN), and AI and DI can be significantly further distinguished during a relatively late window overlapping with late positive potential (LPP). Moreover, the conventional ERP analysis found that EPN amplitudes were more negative for AI than for OE and DI, as well as for OE than for DI, suggesting a pleasure bias for increased self-interest. LPP amplitudes were greater for DI than for AI and OE, suggesting enhanced sensitivity to DI. These results suggest that self-interest is intuitive during opportunity (in)equity.

A Path to Data-Driven Health Care Enforcement

The Department of Justice (“DOJ”) has a long-stated goal of encouraging companies to engage in what the author refers to as “compliant behaviors”—maintenance of an effective pre-existing compliance program, post-enforcement adoption of an effective compliance program, cooperation with a government investigation, and self-disclosure of misconduct. Substantial DOJ guidance over the past two decades, along with the concrete incentive structure of the United States Sentencing Guidelines, have increasingly made clear to organizations when and how such behaviors will be rewarded in criminal matters. Recently, DOJ has made transparency and clarity regarding the benefit of compliant behaviors a priority in calculating and announcing criminal resolutions. With respect to civil False Claims Act (“FCA”) settlements, however, meaningful formal DOJ guidance on the effect of compliant behavior only arrived in 2019. More significantly, DOJ’s treatment of compliant behavior in civil cases, in contrast to that in the case of criminal resolutions, has appeared inconsistent and certainly opaque. This lack of clear implementation philosophy is particularly problematic in the area of health care, an industry for which the FCA is the primary tool for government action in response to misconduct. While much has been written on the systemic efficiencies—or lack thereof—that DOJ’s enforcement practices bring to the activities of regulated industries, this is the first article to use data to ask whether DOJ has a governing practice concerning civil settlements, or whether instead, its settlement practices fail to match its stated principles. For decades, even as resources devoted to health care compliance by market participants have skyrocketed and DOJ has pressed corporate entities to engage in compliant behaviors, the health care industry and the defense bar have expressed skepticism regarding the actual payoff they might realize by engaging in those behaviors. DOJ’s response has been a series of public statements amounting to, “trust us, they matter.” Until now, the health care industry has been without any mechanism to test those assurances. In response to changes in the tax code, however, DOJ made adjustments in 2018 to its practice of disclosing information regarding FCA settlements—changes that have shaken loose data that provides an opportunity to test DOJ’s claims of rewarding compliant behaviors in civil cases. The author is the first and thus far the only person to have identified and analyzed this newly available data. Examination of this data demonstrates that wide-ranging, structural changes are necessary. The data raises substantial questions about: the quantum of credit given for cooperation; conduct DOJ values in resolving FCA cases; and the degree of consistency in cases settled by U.S. Attorney’s Offices across the country. For example, analysis of the data reveals inconsistent benefits for cooperation. Cases where defendants self-disclosed misconduct or cooperated were often not treated more leniently than cases where defendants did not self-disclose or cooperate, and a review of more than 100 settlements did not find a single instance in which DOJ purported to give an entity a reduction based on its pre-existing compliance program. At the same time, DOJ appears to be greatly rewarding defendants for agreeing to settle—highlighting concerns both with regard to whether DOJ is achieving adequate deterrence and with regard to whether the FCA’s potential penalties are coercing settlements. And the data appears to show significant variation in settlement positions depending on the identity of enforcing DOJ component, with cases handled by DOJ’s Civil Division in Washington, D.C. treated more leniently than cases handled solely by U.S. Attorney’s Offices across the country. With the data now public—and at a moment in time when DOJ is emphasizing transparency and clarity in rewarding compliant behaviors in the resolution of criminal cases—DOJ’s corporate health care enforcement regime is at a crossroads. Without change in this area, DOJ risks undercutting its efforts at encouraging compliant behaviors in one of DOJ’s primary enforcement areas. No longer left in the dark about the impact of compliant behaviors in calculating FCA resolutions, the health care industry may be less likely to continue investing in compliance programs at the same rate, to cooperate with government investigations, and especially to self-disclose misconduct to the government. The analysis reveals that a detailed structure of DOJ’s calculations in FCA settlements, with calculations transparent in each FCA resolution, is needed to accomplish DOJ’s goal of encouraging cooperation and investment in compliance programs, as well as to provide an assurance that like cases are treated alike.

Evolution of cooperation through aspiration-based adjustment of interaction range in spatial prisoner’s dilemma game

Neighborhood size is an important factor in the spatial prisoner’s dilemma game. Traditional static interaction range does not adapt to the dynamic social environment. Therefore, it is necessary to study the adaptive adjustment mechanism of the interaction range. In order to enhance the adaptability of individuals to the dynamic social environment, in this paper we propose an interaction range adjustment mechanism based on the aspiration of payoff. Specifically, the interaction range tends to expand according to a certain probability when the current actual payoff reaches or exceeds the aspired payoff. Otherwise, the interaction range will shrink with a certain probability. Simulation results show that such a mechanism of interaction range adjustment can remarkably promote cooperation. The probability of adjusting the interaction range has a great impact on cooperation evolution. Cooperation can be effectively established in a population in which individuals are with higher shrinkage and lower expansion probabilities. Moreover, the aspiration level is an important factor to affect the effectiveness of the interaction range adjustment. Larger aspiration level is more advantageous to promoting cooperation when the aspiration level is less than 1, while cooperators cannot survive when the aspiration level is larger than 1. The underlying mechanism is furthermore explored, revealing that the ‘isolation zones’ composed of individuals with zero interaction radius play an important role in promoting cooperation.

Corporate Health Care Enforcement at a Crossroads: Newly Available Data and the Need for Comprehensive False Claims Act Reform

The Department of Justice (“DOJ”) has a long-stated goal of encouraging companies to engage in what the author refers to as “compliant behaviors” – maintenance of an effective pre-existing compliance program, post-enforcement adoption of an effective compliance program, cooperation with a government investigation, and self-disclosure of misconduct. Substantial DOJ guidance over the past two decades, along with the concrete incentive structure of the United States Sentencing Guidelines, have increasingly made clear to organizations when and how such behaviors will be rewarded in criminal matters. Recently, DOJ has made transparency and clarity regarding the benefit of compliant behaviors a priority in calculating and announcing criminal resolutions. With respect to civil False Claims Act (“FCA”) settlements, however, meaningful formal DOJ guidance on the effect of compliant behavior only arrived in 2019. More significantly, DOJ’s treatment of compliant behavior in civil cases, in contrast to that in the case of criminal resolutions, has appeared inconsistent and certainly opaque. This lack of clear implementation philosophy is particularly problematic in the area of health care, an industry for which the FCA is the primary tool for government action in response to misconduct. While much has been written on the systemic efficiencies – or lack thereof – that DOJ’s enforcement practices bring to the activities of regulated industries, this is the first article to use data to ask whether DOJ has a governing practice with respect to civil settlements, or whether instead its settlement practices fail to match its stated principles. For decades, even as resources devoted by market participants to health care compliance have skyrocketed and DOJ has pressed corporate entities to engage in compliant behaviors, the health care industry and the defense bar have expressed skepticism regarding the actual payoff they might realize by engaging in those behaviors. DOJ’s response has been a series of public statements amounting to, “trust us, they matter.” Until now, the health care industry has been without any mechanism to test those assurances. In response to changes in the tax code, however, DOJ made adjustments in 2018 to its practice of disclosing information regarding FCA settlements – changes which have shaken loose data that provides an opportunity to test DOJ’s claims of rewarding compliant behaviors in civil cases. The author is the first and thus far the only person to have identified and analyzed this newly available data. Examination of this data demonstrates that wide-ranging, structural changes are necessary. The data raises substantial questions about: the quantum of credit given for cooperation; conduct DOJ values in resolving FCA cases; and the degree of consistency in cases settled by U.S. Attorney’s Offices across the country. For example, analysis of the data reveals inconsistent benefits for cooperation. Cases where defendants self-disclosed misconduct or cooperated were often not treated more leniently than cases where defendants did not self-disclose or cooperate, and a review of more than 100 settlements did not find a single instance in which DOJ purported to give an entity a reduction based on its pre-existing compliance program. At the same time, DOJ appears to be greatly rewarding defendants for agreeing to settle – highlighting concerns both with regard to whether DOJ is achieving adequate deterrence and with regard to whether the FCA’s potential penalties are coercing settlements. And the data appears to show significant variation in settlement positions depending on the identity of enforcing DOJ component, with cases handled by DOJ’s Civil Division in Washington, D.C. treated more leniently than cases handled solely by U.S. Attorney’s Offices across the country. With the data now public – and at a moment in time when DOJ is emphasizing transparency and clarity in rewarding compliant behaviors in the resolution of criminal cases – DOJ’s corporate health care enforcement regime is at a crossroads. Without change in this area, DOJ risks undercutting its efforts at encouraging compliant behaviors in one of DOJ’s primary enforcement areas. No longer left in the dark about the impact of compliant behaviors in calculating FCA resolutions, the health care industry may be less likely to continue investing in compliance programs at the same rate, to cooperate with government investigations, and especially to self-disclose misconduct to the government. The analysis reveals that a detailed structure of DOJ’s calculations in FCA settlements, with calculations transparent in each FCA resolution, is needed to accomplish DOJ’s goal of encouraging cooperation and investment in compliance programs, as well as to provide an assurance that like cases are treated alike.

Bayesian Graphical Games for Synchronization in Networks of Dynamical Systems

In this paper, differential games with incomplete information, or Bayesian games, are formulated for multiagent continuous-time dynamical systems in a communication graph. These new Bayesian graphical games model the situation where the agents are uncertain about their actual payoff functions and must employ the evidence observed from the behavior of other agents to improve their estimation of the real setting of their environment. Two different solutions are proposed for the game. First, the agents play their best response strategies with respect to the policies of their neighbors; then, the agents prepare themselves for the worst-case neighbor policies. A tight relationship between the beliefs of an agent and its distributed control policy is established. Two belief update methodologies are proposed for the agents to review their strategies using information obtained from the graph topology.

Aversion to risk of regret and preference for positively skewed risks

We assume that the ex-post utility of an agent facing a menu of lotteries depends upon the actual payoff together with its forgone best alternative, thereby allowing for the expost emotion of regret. An increase in the risk of regret is obtained when the actual payoff and its forgone best alternative are statistically less concordant in the sense of Tchen (1980). The aversion to any such risk of regret is thus equivalent to the supermodularity of the bivariate utility function. We show that more regret-risk-averse agents are more willing to choose the risky act in a one-risky-one-safe menu, in particular when the payoff of the risky choice is highly skewed. This is compatible with the possibility effect that is well documented in prospect theory. Symmetrically, we define the aversion to elationrisk that can prevail when the ex-post utility is alternatively sensitive to the forgone worst payoff. We show that elation-risk-seeking is compatible with the certainty effect. We finally show that regret-risk-averse and elation-risk-seeking people behave as if they had rank-dependent utility preferences with an inverse-S shaped probability weighting function that reproduces estimates existing in the literature.

Overconfidence and disappointment in venture capital decision making: An empirical examination

We examine the effects of overconfidence in venture capital investing. Overconfidence in financial decision making is a robust, well‐established finding, and its consequences for decisions by equity market investors, startup entrepreneurs, and CEO's of large firms have been comprehensively examined and documented. This paper considers the behavioral consequences of overconfidence by venture capitalists, comparing their anticipated returns to actual average returns. Our primary metric is Bell's disappointment, a measure of the difference between anticipated and actual payoff. We construct both deterministic and stochastic models, based on empirical data. We find that the average venture capitalist will experience substantial disappointment.

Complexity in risk elicitation may affect the conclusions: A demonstration using gender differences

The Holt and Laury (American Economic Review, 92(5), 1644–1655, 2002) mechanism (HL) is the most widely-used method for eliciting risk preferences in economics. Participants typically make ten decisions with different variance options, with one of these choices randomly chosen for actual payoff. For this mechanism to provide an accurate measure of risk aversion, participants need to understand the choices and give consistent responses. Unfortunately, inconsistent and even dominated choices are often made. Can these mistakes lead to a misrepresentation of economic phenomena? We use gender differences in risk taking to test this question. In contrast to many findings in the literature, HL results typically do not find significant gender differences. We compare the HL approach, where we replicate the lack of significant gender differences, with a simpler presentation of the same choices in which participants make only one of the ten HL decisions; this simpler presentation yields strong gender differences indicating that women are more risk averse than men. We also find gender differences in the consistency of decisions. We believe that the results found in the simpler case are more reflective of underlying preferences, since the task is considerably easier to understand. Our results suggest that the complexity and structure of the risk elicitation mechanism can affect measured risk preferences. The issue of complexity and comprehension is also likely to be present with elicitation mechanisms in other realms of economic preferences.

Economic Harmony: An Epistemic Theory of Economic Interactions

We propose an epistemic theory of micro-economic interactions, termed Economic Harmony. In the theory, we modify the standard utility, by changing its argument from the player’s actual payoff, to the ratio between the player’s actual payoff and his or her aspired payoff. We show that the aforementioned minor epistemic modification of the concept of utility is quite powerful in generating plausible and successful predictions of experimental results, obtained in the standard ultimatum game, and the sequential common pool resource dilemma (CPR) game. Notably, the cooperation and fairness observed in the studied games are accounted for without adding an other-regarding component in the players’ utility functions. For the standard ultimatum game, the theory predicts a division of φ and 1 − φ, for the proposer and responder, respectively, where φ is the famous Golden Ratio (≈0.618), most known for its aesthetically pleasing properties. We discuss possible extensions of the proposed theory to repeated and evolutionary ultimatum games.

Complexity in Risk Elicitation May Affect the Conclusions: A Demonstration Using Gender Differences

The Holt and Laury (2002) mechanism (HL) is the most widely-used method for eliciting risk preferences in economics. Participants typically make ten decisions with different variance options, with one of these choices randomly-chosen for actual payoff. For this mechanism to provide an accurate measure of risk aversion, participants need to understand the choices and give consistent responses. Unfortunately, inconsistent and even dominated choices are often made. Can these mistakes lead to a misrepresentation of economic phenomena? We use gender differences in risk taking to test this question. In contrast to many findings in the literature, HL results typically do not find significant gender differences. We compare the HL approach, where we replicate the lack of significant gender differences, with a simpler presentation of the same choices, in which participants make only one of the ten HL decisions; this simpler presentation yields strong gender differences indicating that women are more risk averse than men. We also find gender differences in the consistency of decisions. We believe that the results found in the simpler case are more reflective of underlying preferences, since the task is considerably easier to understand. Our results suggest that the complexity and structure of the risk-elicitation mechanism can affect measured risk preferences. The issue of complexity and comprehension is also likely to be present with elicitation mechanisms in other realms of economic preferences.

Reactive synthesis without regret

Two-player zero-sum games of infinite duration and their quantitative versions are used in verification to model the interaction between a controller (Eve) and its environment (Adam). The question usually addressed is that of the existence (and computability) of a strategy for Eve that can maximize her payoff against any strategy of Adam. In this work, we are interested in strategies of Eve that minimize her regret, i.e. strategies that minimize the difference between her actual payoff and the payoff she could have achieved if she had known the strategy of Adam in advance. We give algorithms to compute the strategies of Eve that ensure minimal regret against an adversary whose choice of strategy is (1) unrestricted, (2) limited to positional strategies, or (3) limited to word strategies, and show that the two last cases have natural modelling applications. These results apply for quantitative games defined with the classical payoff functions $$\mathsf {Inf}$$Inf, $$\mathsf {Sup}$$Sup, $${\mathsf {LimInf}}$$LimInf, $$\mathsf {LimSup}$$LimSup, and mean-payoff. We also show that our notion of regret minimization in which Adam is limited to word strategies generalizes the notion of good for games introduced by Henzinger and Piterman, and is related to the notion of determinization by pruning due to Aminof, Kupferman and Lampert.

Commitments in Extensive Form Games

We consider extensive-form games in which players have the option to commit to actions before the game is played. We focus on commitment procedures where players make voluntary irreversible commitments in a prescribed order over the decision nodes. We study whether such commitment procedures may lead to Pareto-efficient outcomes. Our main positive result is surprisingly positive:For two-player games, we introduce an order over the decision nodes for which the unique sub-game perfect equilibrium outcome is Pareto efficient regardless of the actual payoffs. We show an impossibility result for games with four players.

Approachability with delayed information

We consider a sequential decision problem where the decision maker is informed of the actual payoff with delay. We introduce a new condition, which generalizes the condition given by Blackwell and ensures that the decision maker can approach a fixed closed and convex set under delay. We show how the convergence rate to the approachable set is sensitive to changes in the information lag and apply our approachability strategy to games with one-sided incomplete information and to regret-free strategies.

Persistence of contrasting traditions in cultural evolution: unpredictable payoffs generate slower rates of cultural change.

We report an experimental test of the hypothesis that contrasting traditions will persist for longer, maintaining cultural differences between otherwise similar groups, under conditions of uncertainty about payoffs from individual learning. We studied the persistence of two alternative, experimentally-introduced, task solutions in chains of human participants. In some chains, participants were led to believe that final payoffs would be difficult to predict for an innovative solution, and in others, participants were aware that their final payoff would be directly linked to their immediate solution. Although the difference between the conditions was illusory (only participants’ impressions were manipulated, not actual payoffs) clear differences were found between the conditions. Consistent with predictions, in the chains that were less certain about final payoffs, the distinctive variants endured over several replacement “generations” of participants. In contrast, in the other chains, the influence of the experimentally-introduced solutions was rapidly diluted by participants’ exploration of alternative approaches. The finding provides support for the notion that rates of cultural change are likely to be slower for behaviors for which the relationship between performance and payoff may be hard to predict.

Double Reference Points in Risky Decision Making

According to prospect theory, decision makers' gain or loss determined by their current states has a profound effect on decisions under risk. In the current paper, the status quo of decision makers is defined as individual reference point, which determines the context as individual gain or loss. Individual reference point is direct, actual, and absolute given that it affects the actual payoffs of decision makers. However, as social comparison theory goes, decision making is also a matter of social comparison. Thus, the status of others is defined as social reference point, which determines social gain or loss. Social reference point is indirect, hypothetical, and relative because it does not influence the actual payoffs of decision makers directly. Risky decision making is affected by social reference point through self-concept, emotion, and cognition. More importantly, both individual and social reference points exist in risky decision making. They have similar impacts on decision makers, both psychologically and behaviorally, thus affecting risky choice jointly. Therefore, the paper proposes the effects of double reference points on risky decision making. However, further research is needed on the mechanisms of double reference points.

Promotion of cooperation induced by a self-questioning update rule in the spatial traveler’s dilemma game

In sociology and economics, evolutionary game theory has provided a powerful framework to illustrate the social dilemma’s problems, and many evolutionary game models are presented, such as prisoner’s dilemma game, snowdrift game, public goods game, and so on. In this paper, however, we focus on another typical pair-wise game model: Traveler’s Dilemma Game (TDG), which has been deeply investigated in economics, but less attention has been paid to this topic within the physics community. We mainly discuss the influence of strategy update rules on the evolution of cooperation in the spatial TDG, and in detail explore the role of a novel self-questioning or self-learning update mechanism in the evolution of cooperation of the TDG model on the square lattice. In our self-questioning rule, each player does not imitate the strategy state of his or her nearest neighbors and simply plays the traveler’s dilemma games twice with nearest neighbors: one is to calculate the actual payoff in the current game round; the other is to perform a virtual game which is used to obtain an intangible payoff if he or she adopts another random strategy. Then, the focal player decides to keep the current strategy or to change into that virtual strategy according to the Fermi-like dynamics. A great number of Monte Carlo simulations indicate that our self-questioning rule is a low information game decision-making mechanism which can greatly promote the evolution of cooperation for some specific conditions in the spatial TDG model. Furthermore, this novel rule can also be applied into the prisoner’s dilemma game, and likewise the behavior of cooperation can be largely enhanced. Our results are of high importance to analyze and understand the emergence of cooperation within many real social and economical systems.

Dynamics of the full information minority game due to the agent’s impact

We propose the full information minority game (FIMG) in which agents perfectly recognize their impact on the market. For reflecting the impact of agents, we introduce an actual total action and an actual payoff which depend on the agent’s strategies. Unlike the perpetual fluctuating dynamics of the standard minority game, the dynamics of the FIMG settles into one of four steady states after a transient time. We classify steady states of the FIMG and analyze their characteristics. In most cases, the total payoff is maximized, and the total action settles into a periodic state. For measuring the degree of cooperation on the FIMG, we calculate a fraction of total payoff maximization and a mean relaxation time for the dynamics to settle into a steady state whose total payoff is maximized.