Uncertain Payoffs Research Articles

Heterogeneous risk preferences in community-based electricity markets

Organization in electricity markets is evolving from centralized pool-based to decentralized peer-to-peer structures. Within this decentralized framework, agents are expected to individually procure their energy, while directly negotiating with other market participants. Since distributed power generation is mostly based on non-dispatchable energy resources with zero marginal cost, any proposed decentralized negotiation mechanism needs to account for uncertainties. When operating uncertain assets, decision makers are affected by subjective attitudes towards uncertain payoffs, impacting not only their energy procurement but also the whole market equilibrium. We propose a new definition of fairness in risky environments and show that, in decentralized electricity markets, heterogeneous risk aversion of participants compromises fairness of the resulting market payments. Consequently, we introduce financial contracts as risk hedging mechanisms and evaluate their impact on market equilibrium and payments. We show that by trading financial products, fairness is restored.

A robust optimization approach for solving two-person games under interval uncertainty

In this paper, robust optimization methodologies for solving incomplete-information two-person zero-sum and nonzero-sum games are developed that consider single or multiple interval inputs (i.e., interval-valued payoffs). Unlike complete-information games, where all parameters of the game such as individual players’ payoffs are assumed common knowledge, incomplete-information games deal with uncertain payoffs. In some cases, the payoffs may be estimated from imprecise data, such as interval data. In such situations, conventional methods that use deterministic payoffs are not appropriate. Also, in many cases, payoffs are available as multiple intervals. This paper proposes robust optimization models for non-cooperative, simultaneous-move, one-shot, two-person games with incomplete information. The proposed approaches are able to aggregate information from multiple sources and thereby result in more realistic outcomes. The robust optimization methods developed in this paper can be used to solve two-person games with interval-valued (single or multiple intervals) payoffs as well as with single-valued (i.e., deterministic) and aleatory (i.e., precise probabilistic information) payoffs or a combination of them. The proposed methodologies are illustrated with several example problems including an investment decision problem and a capacity expansion decision problem. The proposed decoupled approach is compared with some previously developed approaches and it is demonstrated that the proposed formulations generate conservative solutions in the presence of interval uncertainty.

Z-equilibria in Bi-matrix games with uncertain payoffs

The concept of Z-equilibrium has been introduced by Zhuk-ovskii (Mathematical Methods in Operations Research. Bulgarian Academy of Sciences, Sofia (1985) 103–195) for games in normal form. This concept is always Pareto optimal and individually rational for the players. Moreover, Pareto optimal Nash equilibria are Z-equilibria. We consider a bi-matrix game whose payoffs are uncertain variables. By appropriate ranking criteria of Liu uncertainty theory, we introduce some concepts of equilibrium based on Z-equilibrium for such games. We provide sufficient conditions for the existence of the introduced concepts. Moreover, using mathematical programming, we present a procedure for their computation. A numerical example is provided for illustration.

Cognitive Data Offloading in Mobile Edge Computing for Internet of Things

Data offloading to Mobile Edge Computing (MEC) servers is an attractive choice for resource-constrained Internet of Things (IoT) devices, towards reducing their computational effort. In this paper, we investigate the potential of partial data offloading to MEC servers, under the perspective of users’ cognitive IoT devices presenting loss averse and gain seeking behavior. Due to the sharing nature of the access environment and the MEC server’s computational characteristics, we treat the MEC server option as a common pool of resources with uncertain payoff returned to the users, while the local computation capability is treated as a safe option for each user. Following the properties of Prospect Theory, users’ prospect-theoretic utilities are formulated exploiting the local computing and offloading overhead options under probabilistic uncertainty. Such a modeling allows for the infusion of human awareness, inherent cognitive biases and behavioral characteristics into the devices’ operation, their data offloading decisions and the edge computing environment that the devices are interacting with. Accordingly, each user’s optimal offloaded data to the MEC server is obtained as the outcome of a non-cooperative game, with users attempting to maximize their own utilities. The existence and uniqueness of a Pure Nash Equilibrium (PNE) are proven under the probabilistic nature of the respective payoff functions, while a distributed algorithm that convergences to the PNE is designed. Numerical results are provided that demonstrate the operation and superiority of the proposed framework under different IoT scenarios and behaviors, considering both homogeneous and heterogeneous users.

Dynamic Game for Strategy Selection in Hardware Trojan Attack and Defense

The offshore outsourcing introduces serious threats to semiconductor suppliers and integrated circuit(IC) users for the possibility of hardware trojans (HTs). To alleviate this threat, IC designers use the active defenses against HTs which are implanted by the malicious manufacturer. In this paper, a game-theoretic framework based on fuzzy theory is proposed to obtain the optimal strategy. It analyzes the interactions between the active defense designer and the malicious manufacturer. The attack and defense on IC is formulated as a noncooperative dynamic game. The active defense strategy is decided in IC design. And the overall payoff including design costs and losses avoided is optimized. Subsequently, the HT is implanted considering the implantation cost as well as the damage caused by it. To solve the problem of uncertain payoff caused by insufficient information, fuzzy variable is used to represent the influence of the defense coverage rate on the payoff. In order to verify the applicability of fuzzy variable in dynamic game framework, the existence of pure strategy Nash equilibrium in the game is proved. A solution algorithm for pure strategy Nash Equilibrium is proposed to obtain the optimal strategy of attacker and defender. Thanks to the case study of Field Programmable Gate Array (FPGA), the proposed framework is feasible for HT attack and defense game.

Decision-making with measure modeled uncertain payoffs and multiple goals

Our concern is with selecting the best alternative in the face of uncertain satisfactions by the alternatives. Here, we use a measure to model the uncertain outcomes. We look at various approaches for formulating the concept of best in this case of uncertain outcomes. We consider the case where we have one criterion of interest as well as the case where we have multiple criteria of interest.

Public resources allocation using an uncertain cooperative game among vulnerable groups

PurposeThis paper aims to solve the problem of public resource allocation among vulnerable groups by proposing a new method called uncertainα-coordination value based on uncertain cooperative game.Design/methodology/approachFirst, explicit forms of uncertain Shapley value with Chouqet integral form and uncertain centre-of-gravity of imputation-set (CIS) value are defined separately on the basis of uncertainty theory and cooperative game. Then, a convex combination of the two values above called the uncertainα-coordination value is used as the best solution. This study proves that the proposed methods meet the basic properties of cooperative game.FindingsThe uncertainα-coordination value is used to solve a public medical resource allocation problem in fuzzy coalitions and uncertain payoffs. Compared with other methods, theα-coordination value can solve such problem effectively because it balances the worries of vulnerable group’s further development and group fairness.Originality/valueIn this paper, an extension of classical cooperative game called uncertain cooperative game is proposed, in which players choose any level of participation in a game and relate uncertainty with the value of the game. A new function called uncertainα-Coordination value is proposed to allocate public resources amongst vulnerable groups in an uncertain environment, a topic that has not been explored yet. The definitions of uncertain Shapley value with Choquet integral form and uncertainCISvalue are proposed separately to establish uncertainα-Coordination value.

The role of uncertainty in attentional and choice exploration.

The exploitation-exploration (EE) trade-off describes how, when making a decision, an organism must often choose between a safe alternative with a known pay-off, and one or more riskier alternatives with uncertain pay-offs. Recently, the concept of the EE trade-off has been extended to the examination of how organisms distribute limited attentional resources between several stimuli. This work suggests that when the rules governing the environment are certain, participants learn to "exploit" by attending preferentially to cues that provide the most information about upcoming events. However, when the rules are uncertain, people "explore" by increasing their attention to all cues that may provide information to help in predicting upcoming events. In the current study, we examine how uncertainty affects the EE trade-off in attention using a contextual two-armed bandit task, where participants explore with both their attention and their choice behavior. We find evidence for an influence of uncertainty on the EE trade-off in both choice and attention. These findings provide support for the idea of an EE trade-off in attention, and that uncertainty is a primary motivator for exploration in both choice and attentional allocation.

Gender Differences in Job Entry Decisions: A University-Wide Field Experiment

The gender difference in competitiveness has been cited as an important factor driving the gender gap in labor market outcomes. Using a field experiment with more than 35,000 university students, I explore the impact of compensation scheme on willingness to apply for a job. I find that competitive compensation schemes that depend on relative performance disproportionately deter women from applying. I do not find the same deterrence effect under a compensation scheme in which students face uncertain payoffs independent of performance. I also test the impact of describing the job as helping a charity, and this increases the willingness to apply but does not affect the gender gap. My findings are important for managers, who need to understand the impact of different compensation schemes on the share of women recruited for a job posting. The online appendix is available at https://doi.org/10.1287/mnsc.2018.3107 . This paper was accepted by Yan Chen, behavioral economics.

Asymmetrical expectations of future interaction and cooperation in the iterated prisoner's dilemma game

This paper addresses how players’ asymmetrical expectations of future interaction affect their cooperation in the two-player iterated prisoner's dilemma (IPD) game. A player’ expectation of future interaction reflects his/her willingness to continue the interaction with his/her opponent. With the application of a co-evolutionary learning model with a niching technique that is used to search optimal strategies for players, simulation experiments are conducted to present the cooperative or uncooperative outcomes between the players with distinct expectations. Results indicate that, if one player has higher expectation of future interaction than the other, the former may be exploited by the latter. Such exploitation is mainly due to the former player's higher tendency toward mutual cooperation, which triggers the latter player's unilateral defection. Considering that real-world games always involve uncertainty or external disturbance, the games with uncertain payoffs or noise are also performed. Results indicate that the exploitation also exists in the game with uncertain payoffs. However, it may be absent in the presence of high levels of noise. The reason is that the cooperative tendency of the high-expectation player declines with the increasing level of noise. As a result, the two players are engaged in only low levels of cooperation in the IPD game. Nevertheless, when asymmetrical noise is taken into account, it is the player disturbed with higher levels of noise that is exploited by the other. In other words, the player whose strategy is interfered with higher levels of noise is more afraid of endless mutual defection caused by noise, which makes he/she cooperate more than his/her opponent, even though he/she may have a lower expectation on their future interaction.

Games with incomplete information and uncertain payoff: from the perspective of uncertainty theory

In a game with incomplete information, players do not possess full information about their opponents. For such a game, Bayesian approaches which are based on probability theory were employed to find the equilibria. Sometimes, players are lack of data for probabilistic reasoning in which case Bayesian methods cannot be used. In this paper, we adopt a new mathematical framework—uncertainty theory to solve such a game. The player’s type (incomplete information) is modeled as an uncertain variable. In addition to previous studies, the payoff for each player is uncertain in our research, which is a realistic assumption in practice. We first define the games with incomplete information with uncertain payoff (Iu-game). Then, we create a new game (U-game) as a method to solve Iu-game. A theorem is provided to show that the equilibria are the same for both games. We use an example to illustrate the application of the proposed theorem. Finally, we generalize the form of U-game and show the equilibria are independent of the incomplete information.

Information Design: A Unified Perspective

Given a game with uncertain payoffs, information design analyzes the extent to which the provision of information alone can influence the behavior of the players. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used by the analyst to characterize play in the game under many different information structures. We provide an introduction to the basic issues and insights of a rapidly growing literature in information design. We show how the literal and metaphorical interpretations of information design unify a large body of existing work, including that on communication in games (Myerson 1991), Bayesian persuasion (Kamenica and Gentzkow 2011), and some of our own recent work on robust predictions in games of incomplete information. ( JEL C70, D82, D83)

Optimal learning before choice

A Bayesian decision maker is choosing among two alternatives with uncertain payoffs and an outside option with known payoff. Before deciding which alternative to adopt, the decision maker can purchase sequentially multiple informative signals on each of the two alternatives. To maximize the expected payoff, the decision maker solves the problem of optimal dynamic allocation of learning efforts as well as optimal stopping of the learning process. We show that the decision maker considers an alternative for learning or adoption if and only if the expected payoff of the alternative is above a threshold. Given both alternatives in the decision maker's consideration set, we find that if the outside option is weak and the decision maker's beliefs about both alternatives are relatively low, it is optimal for the decision maker to learn information from the alternative that has a lower expected payoff and less uncertainty, given all other characteristics of the two alternatives being the same. If the decision maker subsequently receives enough positive informative signals, the decision maker will switch to learning the better alternative; otherwise the decision maker will rule out this alternative from consideration and adopt the currently more preferred alternative. We find that this strategy works because it minimizes the decision maker's learning efforts. We also characterize the optimal learning policy when the outside option is relatively high, and discuss several extensions.

A new bi-matrix game model with fuzzy payoffs in credibility space

Uncertain theory based on expert's evaluation and non-additive measure is introduced to explore the bi-matrix game with the uncertain payoffs. The uncertainty space based on the axiom of the uncertain measures is presented. Some basic characteristics of uncertain events are described and the expected value of the uncertain variables is given in uncertainty space. A new model of bi-matrix game with uncertain payoffs is established and its equivalent strategy is given. Then, we develop an expected model of uncertain bi-matrix games and define the uncertain equilibrium strategy of uncertain bi-matrix games. By using the expected value of uncertain variable, we transform the model into a linear programming, the expected equilibrium strategy of the uncertain bi-matrix games is identified through solving linear equations.

The Risk-Based Core for Cooperative Games With Uncertainty

In coalitional games with uncertain payoffs, a deviating coalition can only form expectations regarding its post-deviation payoff. Classical approaches address the problem from the side of conservatism, expecting the worst, or by explicit assumptions of the emerging state of the world. We borrow the idea of risk from the finance literature and compare the payoff of staying with the original outcome with the risk of deviating. Employing this idea to the core leads to a new concept that we call the risk-based core. We introduce this concept and discuss its properties. We find an inclusion relation between cores of games with increasingly conservative players. The model is also suitable to study cooperative games in partition function form where the value of a coalition depends on the entire partition. For the cores of such games our main result yields many of the familiar inclusion relations as corollaries, while the inclusion of the optimistic core in the optimistic recursive core turns out to be non-robust.

Risk Assessment Uncertainties in Cybersecurity Investments

When undertaking cybersecurity risk assessments, it is important to be able to assign numeric values to metrics to compute the final expected loss that represents the risk that an organization is exposed to due to cyber threats. Even if risk assessment is motivated by real-world observations and data, there is always a high chance of assigning inaccurate values due to different uncertainties involved (e.g., evolving threat landscape, human errors) and the natural difficulty of quantifying risk. Existing models empower organizations to compute optimal cybersecurity strategies given their financial constraints, i.e., available cybersecurity budget. Further, a general game-theoretic model with uncertain payoffs (probability-distribution-valued payoffs) shows that such uncertainty can be incorporated in the game-theoretic model by allowing payoffs to be random. This paper extends previous work in the field to tackle uncertainties in risk assessment that affect cybersecurity investments. The findings from simulated examples indicate that although uncertainties in cybersecurity risk assessment lead, on average, to different cybersecurity strategies, they do not play a significant role in the final expected loss of the organization when utilising a game-theoretic model and methodology to derive these strategies. The model determines robust defending strategies even when knowledge regarding risk assessment values is not accurate. As a result, it is possible to show that the cybersecurity investments’ tool is capable of providing effective decision support.

Corporate Diversification and Innovation: Managerial Myopia or Inefficient Internal Capital Market

Which is more innovative: the decentralized, diversified corporation or the smaller, more narrowly focused entrepreneurial firm? The finance literature argues that larger, diversified firms have a natural innovation advantage, as their operating units have access to an internal capital market. In contrast, the strategy and entrepreneurship literatures argue that managers of large, decentralized firms suffer from “managerial myopia” which prevents them from investing in projects with long-term, uncertain payoffs. This paper takes a fresh look at the relationship between innovation and diversification using a comprehensive sample of diversified and non-diversified firms along with a novel analytical approach that enables us to tease out the mechanisms influencing the relationship between diversification and innovation. Consistent with the conceptual and empirical work in strategy, we find a robust negative correlation with diversification strategy and research and development intensity, and we also find support for the hypothesis that diversification reduces innovation by discouraging investment. However, our analysis suggests that internal capital market inefficiencies, rather than managerial myopia, is responsible for this negative relationship.

Capacity Planning Under Uncertainty and the Cost of Capital

ABSTRACT We explore how risk aversion affects optimal capacity and pricing decisions within the economic setting of Banker and Hughes (1994). A risk-averse firm invests in fixed capacity and sets a product price, but can also purchase spot capacity at higher unit cost. Initial capacity and price are set by maximizing the firm's mean-variance certainty equivalent. We find that, contrary to common intuition, optimal capacity or list prices can increase under greater risk aversion depending on exogenous fundamentals. We show how the firm's capacity and price choices affect the economic trade-off between the mean and the risk of the firm's uncertain payoffs. We also show that the cost of capital is affected not only by the firm's covariance with other assets, but also by its payoff mean. The objective of minimizing the cost of capital is, therefore, fundamentally inconsistent with maximizing project value.

Response functions

Imagine that John must choose between two uncertain payoff distributions, knowing that the set of possible payoffs is the same for both, but nothing about the shapes of the distributions. In the first period he chooses either alternative and experiences a payoff as a result of his choice. Given this experienced payoff, in the second period he decides whether to choose the same alternative again, or switch. We model John’s second period choice with a response function, i.e., a mapping from obtained payoffs to the probability of choosing the same alternative in the second period. We first provide results on (i) how the shape of the response function affects both expected payoffs and exposure to risk, and (ii) what standard models of choice under uncertainty would predict about the shape of the response function. We then run an experiment to elicit subjects’ response functions, empirically characterize the heterogeneity across subjects with a mixture model, and illustrate how payoffs vary across response function types. Finally, we use our theoretical results, along with additional information that we collected from subjects, to interpret their response functions.

Corporate Diversification and Innovation: Managerial Myopia or Inefficient Internal Capital Markets?

Which is more innovative: the decentralized, diversified corporation or the smaller, more narrowly focused entrepreneurial firm? The economics and finance literatures argue that larger, diversified firms have natural innovation advantages, as their operating units have access to an internal capital market. In contrast, the strategy and entrepreneurship literatures argue that managers of large, decentralized firms suffer from “managerial myopia,” which prevents them from investing in projects with long-term, uncertain payoffs. This paper takes a fresh look at the relationship between innovation and diversification using a comprehensive sample of diversified and non-diversified firms along with a novel analytical approach that teases out the mechanisms influencing the relationship between diversification and innovation. Consistent with both conceptual and empirical work in strategy, we find a robust negative correlation with diversification strategy and research and development intensity. We also find support for the hypothesis that diversification reduces innovation by discouraging investment. However, our analysis suggests that internal capital market inefficiencies, not managerial myopia, is primarily responsible for this observed negative relationship.