No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Valuing inputs under supply uncertainty: The Bayesian Shapley value
Abstract
We consider the problem of valuing inputs in a production environment in which input supply is uncertain. Inputs can be workers in a firm, risk factors for a disease, securities in a financial market, or nodes in a networked economy. Each input takes its values from a finite set and uncertainty is modeled as a probability distribution over this set. First, we provide an axiomatic solution to this problem, uniquely characterizing a valuation scheme called the a priori Shapley value. Second, we solve the problem of valuing inputs a posteriori - that is, after observing output -, obtaining the Bayesian Shapley value. Third, we address the question of rationalizing uncertainty in labor supply in a non-cooperative production game where payoffs are given by the Shapley wage function. We also provide an intuitive condition for the existence of a pure strategy Nash equilibrium. Illustrations of the theory include an application to fidelity networks.
... Since different strategies entail different kinds of grand coalitions, rationalizable strategies provide information about the nature of the grand coalition that may form. The notion of rationalizibility in (Pongou and Tondji, 2018) is different from ours. In (Pongou and Tondji, 2018) a player is an input to a production function, and a pure strategy of an input is quality. ...
... The notion of rationalizibility in (Pongou and Tondji, 2018) is different from ours. In (Pongou and Tondji, 2018) a player is an input to a production function, and a pure strategy of an input is quality. A mixed strategy of an input is defined to be a probability distribution on the set of pure strategies. ...
... A mixed strategy of an input is defined to be a probability distribution on the set of pure strategies. (Pongou and Tondji, 2018) call a vector, i.e., a vector in which each entry refers to a mixed strategy of an input, of mixed strategies rationalizable if the vector constitutes a Nash equilibrium of the associated complete information game. In the associated game the pay-off from a vector of pure strategy is the Shapley value from the characteristic form game in which each player is identified with a quality of an input. ...
The distribution of efficient individuals in the economy and the efforts that they will put in if they are hired are two important concerns for a technologically advanced firm when the firm wants to open a new branch. The firm does not have the information about the exact level of efficiency of an individual when she is hired. We call this situation 'incomplete information'. The standard principal-agent models assume that employees know their efficiency levels. Hence these models design incentive compatible mechanisms. An incentive compatible mechanism ensures that a participant does not have the incentive to misreport her efficiency level. This paper does not assume that employees know how efficient they are. This paper assumes that the production technology of the firm is intelligent, i.e., the output of the machine reveals the efficiency levels of employees. Employees' marginal contributions to the total output of the intelligent machine, the probability distribution of the levels of efficiency, and employees' costs of efforts together define a game of incomplete information. A characterization of ex-ante Nash Equilibrium is established. The characterization results formalize the relationship between the distribution of efficiency levels and the distribution of output.
... Dans notre quatrième et dernier chapitre, nous nous intéressons au problème de l'évaluation de la productivité d'un travailleur et de l'organisation des travailleurs dans un environnement de production incertain. Pongou and Tondji [2018] ont proposé quelques solutions sur l'évalua- ...
... A N uncertain production environment (Pongou and Tondji [2018]) for a finite set of inputs is given by a production function that assigns to each input profile a measurable output. ...
... The Shapley [1953] value is probably the best-known efficient one-point solution concept for TU games. Pongou and Tondji [2018] consider the problem of valuing inputs in a production environment in which input supply is uncertain. They suggest an extension of the Shapley value as a solution to the valuation problem and called this solution "the a priori Shapley value". ...
Un jeu coopératif à utilité transférable (jeu TU) représente toute situation où des agents en interaction ont la possibilité de communiquer librement et de passer des accords contraignants de coopération pour un gain collectif à redistribuer. Il existe une littérature très riche sur les jeux TU et leurs extensions. Dans cette thèse, nous proposons plusieurs généralisations de différents résultats de cette littérature. L'un des problèmes fondamentaux des jeux TU est celui de la répartition des gains issus de la coopération entre les agents. La contribution marginale pure (l'effet marginal en terme de gain lorsque chaque agent rejoint l’ensemble des autres agents) semble être la méthode naturelle permettant de repartir les gains entre les agents. Casajus et Huettner (Casajus and Huettner, Games and Economic Behavior, 108, 37-48, 2018) critiquent cette approche et illustrent leurs propos via une décomposition de la contribution marginale pure à partir de la valeur de Shapley (Shapley, Annals of Mathematics Studies, 28, 307-317,1953.). De manière similaire Chantreuil et al. (Chantreuil et al., Social Choice and Welfare, 54, 1-16, 2019) proposent une décomposition de la valeur de Shapley centrée sur la contribution marginale pure dans le cadre de la théorie des inégalités. Ces deux résultats sont les points de départ de mes travaux de thèse.Dans le chapitre 1, nous généralisons ces différentes approches en utilisant les indices d'interaction (mesures permettant de capter l'interaction qui existe entre les agents). Par l'approche axiomatique, on définit une famille d'indices d'interaction décomposables. Un exemple illustratif en aide à la décision multi-critère est proposé.Dans le chapitre 2, nous étendons l'approche utilisée par Chantreuil et al. (2019) sur des jeux TU avec structure de coalitions. On obtient ainsi trois décompositions de la valeur de Shapley-Owen (Owen, Mathematical Economics and Game Theory, 1977) ; ce qui permet par la suite d'en déduire trois décompositions des indices d'inégalités. Un exemple d'application en théorie des inégalités est proposé.Notre troisième chapitre est consacré à l'axiomatisation de la valeur de Shapley-Owen. Casajus (Casajus, Discrete Applied Mathematics, 304, 212–219, 2021) définit les axiomes du premier ordre comme étant les axiomes énoncés à partir de la contribution marginale des agents. Il définit un axiome du second ordre comme étant un axiome énoncé à partir d'une double contribution marginale (contribution marginale d'un agent i à la contribution marginale d'un autre agent j quand i quitte le jeu). Suivant cette approche, les axiomes utilisés par Young (Young, International Journal of Game Theory, 14, 65–72, 1985) pour l'axiomatisation de la valeur de Shapley sont des axiomes du premier ordre. Casajus (2021) propose donc un résultat similaire avec les axiomes du second ordre. Nous montrons qu'il existe une relation entre les axiomes de Casajus (2021) et ceux de Young (1985). On déduit que l'axiomatisation de Casajus (2021) est une seconde version de celle de Young (1985). Nous proposons par la suite une seconde version de l'axiomatisation de la valeur de Shapley-Owen proposée par Khmelnitskaya et Yanovskaya (Khmelnitskaya and Yanovskaya, Mathematical Methods of Operations Research, 66, 255–261, 2007).Dans notre quatrième et dernier chapitre, nous nous intéressons au problème de l'évaluation de la productivité d'un travailleur et de l'organisation des travailleurs dans un environnement de production incertain. Pongou et Tondji (Pongou and Tondji, Games and Economic Behavior, 108, 206–224, 2018) ont proposé quelques solutions sur l'évaluation de la productivité d’un travailleur dans un tel environnement. Nous ajoutons à ce modèle une structure de communication, c'est-à-dire que les travailleurs peuvent s'organiser suivant un réseau, et nous évaluons la productivité de chaque travailleur.
... Owing to its desirable properties, the classical Shapley value, defined for transferableutility games under the assumption of full information, has been applied to various classes of problems including expected utility theory (Roth, 1977), fair division (Moulin, 1992), political power measurement (Shapley and Shubik (1954), Freixas (2005), Freixas, Marciniak, and Pons (2012), Pongou and Tchantcho (2021)), network centrality measurement (Grofman and Owen (1982), operations management and supply chains (Gopalakrishnan, Granot, Granot, Sošić, and Cui (2021)), cost allocations (Dubey (1982)), contract design (Winter, 2002), Pongou and Tondji (2018)), queueing problems (Maniquet, 2003), input valuation in discrete settings , unfairness and income inequality (Aguiar, Pongou, and Tondji (2018), Aguiar, Pongou, Serrano, and Tondji (2019)), among others. 5 However, the extension of the Shapley value to study problems involving imperfect information has received less attention ). ...
... 13 In fact, in a highly concentrated sexual network in which a few men share all the women, an infected man who is not single will infect several women, whereas an infected woman will infect fewer men; one can show, using the notion of the contagion potential of a network (see Serrano (2013, 2016) and Pongou and Tondji (2018)), that this creates an infection bias against women. As infidelity decreases more for men than for women following an increase in the sex ratio, infected men who are not single infect fewer women, reducing the infection bias against women. ...
How does the sex ratio affect sexual network formation, relationship stability, and the spread of sexually transmitted diseases, and how do these effects differ by gender? We address these questions by developing a dynamic theory of sexual network formation in a two-sided economy, where agents enjoy having many sexual partners but do not want their partners to have others. We fully characterize steady state sexual networks. This characterization implies that a larger male-to-female ratio decreases sexual infidelity and the spread of sexually transmitted infections (STIs). The effect on sexual infidelity is larger for men compared to women, whereas the effect on STIs is larger for women compared to men. We test these predictions of the model using a unique individual-level dataset, in combination with census data from England and Wales. Exploiting variation in cohort/ethnicity/region-specific sex ratios as a quasi-natural experiment, we find that a larger sex ratio decreases sexual infidelity and the number of serial partners, increases the likelihood of safe sex, and reduces the likelihood of acquiring STIs. In line with the model, these effects vary by gender. These findings are robust to using the instrumental variables approach and various robustness checks. We also conduct a placebo test, showing that the sex ratio has no impact on several "atheoretical" health conditions, such as Parkinson's disease, chronic lung disease, heart disease, stroke, hypertension, arthritis, and diabetes, which do not arise from sexual interactions.
... This can be due to a variety of factors, including the distinct social and economic characteristics of each individual. It is argued that individuals who occupy more central positions in networks are more likely to be infected and to spread an infection, e.g., Anderson and May (1992), Pastor-Satorras and Vespignani (2001), Newman (2002), Hethcote and Yorke (2014), Pongou and Tondji (2018), and Rodrigues (2019). This suggests that an optimal lockdown policy should be targeted at more central agents in a network. ...
We develop a model of optimal lockdown policy for a social planner who balances population health with short-term wealth accumulation. The unique solution depends on tolerable infection incidence and social network structure. We then use unique data on nursing home networks in the US to calibrate the model and quantify state-level preference for prioritizing health over wealth. We also empirically validate simulation results derived from comparative statics analyses. Our findings suggest that policies that tolerate more virus spread (laissez-faire) increase state GDP growth and COVID-19 deaths in nursing homes. The detrimental effects of laissez-faire policies are more potent for nursing homes that are more peripheral in networks, nursing homes in poorer counties, and nursing homes that operate on a for-profit basis. We also find that US states with Republican governors have a higher tolerable incidence level, but these policies tend to converge with a high death count.
Supplementary information:
The online version contains supplementary material available at 10.1007/s00148-022-00916-y.
... It is argued that individuals who occupy more central positions in networks are more likely to be infected and to spread an infection (Anderson & May, 1992;Pastor-Satorras & Vespignani, 2001;Newman, 2002;Hethcote & Yorke, 2014;Pongou & Tondji, 2018;Rodrigues, 2019). This suggests that an optimal lockdown policy should be targeted at more central agents in a network. ...
This study develops an economic model for a social planner who prioritizes health over short-term wealth accumulation during a pandemic. Agents are connected through a weighted undirected network of contacts, and the planner's objective is to determine the policy that contains the spread of infection below a tolerable incidence level, and that maximizes the present discounted value of real income, in that order of priority. The optimal unique policy depends both on the configuration of the contact network and the tolerable infection incidence. Comparative statics analyses are conducted: (i) they reveal the tradeoff between the economic cost of the pandemic and the infection incidence allowed; and (ii) they suggest a correlation between different measures of network centrality and individual lockdown probability with the correlation increasing with the tolerable infection incidence level. Using unique data on the networks of nursing and long-term homes in the U.S., we calibrate our model at the state level and estimate the tolerable COVID-19 infection incidence level. We find that laissez-faire (more tolerance to the virus spread) pandemic policy is associated with an increased number of deaths in nursing homes and higher state GDP growth. In terms of the death count, laissez-faire is more harmful to nursing homes than more peripheral in the networks, those located in deprived counties, and those who work for a profit. We also find that U.S. states with a Republican governor have a higher level of tolerable incidence, but policies tend to converge with high death count.
... 25,34) is a winning tripartition) and D(R 45 ) = x * y. It follows that d(R 5 , D(R) = α, d(R 5 , D(R 45 ) = λ = α = d(R 5 , D(R), which implies that 4 ∼ P α,β,γ 5 at the profile R. ...
A round-robin political tournament is an election format where multiple candidates contest in pairs, and votes are aggregated using a general rule to form a social ranking. We formalize this tournament as a strategic form game and provide a necessary and sufficient condition under which truthful voting is a Nash equilibrium. Building on this analysis, we study the concept of effective power, defined as a voter's ability to bring about a social ranking that maximizes his preferences. We show that the classical theories of political power do not translate into effective power in general. We then provide a full characterization of the classes of political tournaments and utility metrics for which these theories capture effective power. We provide both structural and behavioral interpretations of the findings, and derive practical implications for the design of political tournaments that are compatible with truth-telling.
... The case j = 2 leads to simple games and has been already discussed in Sects. 2 and 3. Therefore, the cases we refer to in the following sections are for j ≥ 3. The case j = 3 corresponds to ternary voting games, which have been considerably studied in the last two decades, see e.g, Felsenthal and Machover (1997), Tchantcho et al. (2008), Freixas (2012. The case j > 3 has hardly been studied, it corresponds to four or more levels of approval, and can be considered a special case of the multichoice games studied by Hsiao and Raghavan (1992), Hsiao and Raghavan (1993) and also a special case of the more general framework proposed in Pongou and Tondji (2018). This approach considers an environment in which each player may have a 'different' set of actions to choose from, and the input supply is uncertain so that a probability distribution is defined over the set of actions of each agent. ...
In this paper we discuss on several ways to extend power indices defined on simple games to the context of (j, 2)-games, i.e., games with 2 possible outputs and j possible input levels. The purpose of this paper is more conceptual than technical, and shows that different power indices can be obtained in (j, 2)-games depending on how the criticality of a player is measured. Some power indices for simple games, regarded as measures of power as payment, can be obtained from the selection of two items: (1) a probabilistic model, and (2) a collection of winning coalitions. The combination of two particular probabilistic models and three collections of winning coalitions lead to six well-known power indices for simple games. The Banzhaf and the Johnston power indices are among them. The main goal of the paper is to discuss the generalizations of these six power indices from (2, 2)-games to (j, 2)-games for \(j\ge 3\). The probabilistic model and the collection of winning coalitions are trivially extendable to (j, 2)-games. However, at least four possible ways of measuring the criticality of a player can be considered in this new context, and lead to different possible extensions of the six power indices to (j, 2)-games. We highlight two of them as being the most interesting under our point of view. The first one would be suitable for assessing sensitivity, because it measures the player’s ability to modify the output by a small change in his input level. The second one would be adequate to evaluate strength, because it measures the voter’s ability to modify the output by any change in his voting. The main conclusion is that different extensions of the considered power indices, as the Banzhaf and the Jonnston indices, are possible for \(j\ge 3\) and that they should be taken into account when working in a multiple input context.
... 34 The Shapley value has been applied for solving incentives/cost sharing and contribution distribution problems in operations research (OR) 61,62 and economic situations. 63,64 In this section, we apply the concept of Shapley value to probability transformations. ...
In Dempster–Shafer evidence theory, how to use the basic probability assignment (BPA) in decision‐making is a significant issue. The transformation of BPA into a probability distribution function is one of the common and feasible schemes. To overcome the problems of the existing methods, we propose a marginal probability transformation method based on the Shapley value approach. The proposed method allocates BPA values in terms of how much an element contributes to a set, which is an equitable and effective distribution mechanism. Furthermore, we use probabilistic information content to evaluate the effect of each transformation method. Moreover, some numerical examples are used to demonstrate the efficiency and feasibility of the proposed method. Further, two applications, target recognition, fault diagnosis are used to verify the superiority and effectiveness of the proposed method in practice.
... This can be due to a variety of reasons, including the distinct social and economic roles played by each individual. It is argued that individuals who occupy more central positions in networks are more likely to be infected and to spread an infection (see, for example,Pongou & Tondji (2018) andRodrigues ...
... This can be due to a variety of reasons, including the distinct social and economic roles played by each individual. It is argued that individuals who occupy more central positions in networks are more likely to be infected and to spread an infection (see, for example,Pongou & Tondji (2018) andRodrigues ...
... This can be due to a variety of reasons, including the distinct social and economic roles played by each individual. It is argued that individuals who occupy more central positions in networks are more likely to be infected and to spread an infection (see, for example,Pongou & Tondji (2018) andRodrigues ...
We address the problem of finding the optimal lockdown and reopening policy during a pandemic like COVID-19, for a social planner who prioritizes health over short-term wealth accumulation. Agents are connected through a fuzzy network of contacts, and the planner's objective is to determine the policy that contains the spread of infection below a tolerable incidence level, and that maximizes the present discounted value of real income, in that order of priority. We show theoretically that the planner's problem has a unique solution. The optimal policy depends both on the configuration of the contact network and the tolerated infection incidence. Using simulations, we apply these theoretical findings to: (i) quantify the tradeoff between the economic cost of the pandemic and the infection incidence allowed by the social planner, and show how this tradeoff depends on network configuration; (ii) understand the correlation between different measures of network centrality and individual lockdown probability, and derive implications for the optimal design of surveys on social distancing behavior and network structure; and (iii) analyze how segregation induces differential health and economic dynamics in minority and majority populations, also illustrating the crucial role of patient zero in these dynamics.
... Aguiar, Pongou, and Tondji (2018) propose the Shapley distance as a measure of the extent to which output sharing among the stakeholders of an organization can be considered unfair. It measures the distance between an arbitrary pay profile and the Shapley pay profile under a given technology, the latter profile defining the fair distribution. ...
Aguiar, Pongou, and Tondji (2018) propose the Shapley distance as a measure of the extent to which output sharing among the stakeholders of an organization can be considered unfair. It measures the distance between an arbitrary pay pro�file and the Shapley pay pro�file under a given technology, the latter profile defining the fair distribution. We provide an axiomatic characterization of the Shapley distance, and show that it can be used to determine the outcome of an underlying bargaining process. We also present applications highlighting how favoritism in income distribution, egalitarianism, and taxation violate the different ideals of justice that defi�ne the Shapley value. The analysis has implications that can be tested using real-world data sets.
... We begin by generalizing the framework of a transferable-utility environment to an environment where agents have more than two options. 13 By generalizing the Shapley value to this class of environments, our work is related to recent studies including Freixas (2005), Hsiao and Raghavan (1993), Courtin et al. (2016), and Pongou and Tondji (2017). However, we have a different scope, which is to test the axioms of the Shapley value. ...
... We begin by generalizing the framework of a transferable-utility environment to an environment where agents have more than two options. 13 By generalizing the Shapley value to this class of environments, our work is related to recent studies including Freixas (2005), Hsiao and Raghavan (1993), Courtin et al. (2016), and Pongou and Tondji (2017). However, we have a different scope, which is to test the axioms of the Shapley value. ...
The unique properties of the Shapley value-efficiency, equal treatment of identical input factors, and marginality-have made it an appealing solution concept in various classes of problems. It is however recognized that the pay schemes utilized in many real-life situations generally depart from this value. We propose a nonparametric approach to testing the empirical content of this concept with limited datasets. We introduce the Shapley distance, which, for a fixed monotone transferable-utility game, measures the distance of an arbitrary pay profile to the Shapley pay profile, and show that it is additively decomposable into the violations of the classical Shapley axioms. The analysis has several applications. In particular, it can be used to assess the extent to which an income distribution or a cost allocation can be considered fair or unfair, and whether any particular case of unfairness is due to the violation of one or a combination of the Shapley axioms.
... ,Hsiao and Raghavan (1993),Courtin et al. (2016), andPongou and Tondji (2017). However, we have a different scope, which is to test the axioms of the Shapley value. ...
We study efficiency and maximality in anonymous two-sided economies, where each agent only cares about trading with a desired number of partners from the opposite side. For both homogeneous and heterogeneous economies, we provide a necessary and sufficient condition that ensures the efficiency of all pairwise stable matchings. In addition, we provide a sufficient condition for maximality in heterogeneous economies. These conditions have implications for the design of anonymous markets that function optimally without the continuous intervention of a social planner.
How does justice affect individual incentives and efficiency in a political economy? We show that elementary principles of distributive justice guarantee the existence of a self-enforcing contract whereby agents non-cooperatively choose their inputs and derive utility from their pay. Chief among these principles is that your pay should not depend on your name, and a more productive individual should not earn less. We generalize our analysis to incorporate inclusivity, ensuring basic pay to unproductive agents, implemented through progressive taxation and redistribution. Our findings show that without redistribution, any self-enforcing agreement may be inefficient, but a minimal level of redistribution guarantees the existence of an efficient agreement. Our model has several applications and interpretations. In addition to highlighting the structure of economies and organizations in which fairness and efficiency are compatible, we develop an application to the formation of rent-seeking political alliances under the threat of fake news.
We study a class of probabilistic cooperative games which can be treated as an extension of the classical cooperative games with transferable utilities. The coalitions have an exogenous probability of being realized. This probability distribution is known beforehand and the distribution of the expected worth needs to be done before the realization of the state. We obtain a value for this class of games and present three characterizations of this value using natural extensions of the axioms used in the seminal axiomatization of the Shapley value. The value, which we call the Expected Shapley value, allocates the players their expected worth with respect to a probability distribution.
Do the effects of government policy response to health crises differ for for-profit and not-for-profit organizations? We address this question through the lens of a two-sector continuous-time individual-based mean-field theoretical model that incorporates a non-random social network. Using unique data on nursing home networks in the United States, we calibrate the model and jointly quantify state-level lockdown effectiveness and preference for enforcing stringent containment strategies during the COVID-19 pandemic. We validate our estimated policy measures using external data. Simulations and regression-based analyses show significant interactions between lockdown effectiveness and the ownership status of a nursing home. In particular, differences in COVID-19 death rates between for-profit and not-for-profit nursing homes are driven by lockdown effectiveness.
Pongou and Tondji (2018) describe an uncertain production environment as a situation where input supply is uncertain. Each input has a finite set of actions, and uncertainty is formalized as a probability distribution over this set. These inputs can be workers in a firm, vertices in a networked economy, securities in a financial market, etc. Then, the authors examined the problem of valuing inputs in that environment. By using axiomatic methods, they provided a solution called the a priori Shapley value. Knowing the output level enabled them to bring forth a solution named the Bayesian Shapley value. In this paper, we examine some applications of the Myerson value (Myerson, 1977) in an uncertain production environment. By defining some intuitive axioms, we solve the problem of valuing inputs in Pongou and Tondji’s (2018) environment, and improve it with a communication structure. Depending on the information structure, this leads to the a priori Myerson value and its individual rational revision called the a priori Myerson R value , or the Bayesian Myerson value, and its individual rational revision called the Bayesian Myerson R value. Furthermore, we generalize the core to this environment.
Background
SHapley Additive exPlanations (SHAP) based on tree-based machine learning methods have been proposed to interpret interactions between exposures in observational studies, but their performance in realistic simulations is seldom evaluated.
Methods
Data from population-based cohorts in Sweden of 47,770 men and women with complete baseline information on diet and lifestyles were used to inform a realistic simulation in 3 scenarios of small (ORM = 0.75 vs. ORW = 0.70), moderate (ORM = 0.75 vs. ORW = 0.65), and large (ORM = 0.75 vs. ORW = 0.60) discrepancies in the adjusted mortality odds ratios conferred by a healthy diet among men and among women. Estimates were obtained with logistic regression (L-ORM; L-ORW) and derived from SHAP values (S-ORM; S-ORW).
Results
The sensitivities of detecting small, moderate, and large discrepancies were 28, 83, and 100%, respectively. The sensitivities of a positive sign (L-ORW > L-ORM) in the 3 scenarios were 93, 100, and 100%, respectively. Similarly, the sensitivities of a positive discrepancy based on SHAP values (S-ORW > S-ORM) were 86, 99, and 100%, respectively.
Conclusions
In a realistic simulation study, the ability of the SHAP values to detect an interaction effect was proportional to its magnitude. In contrast, the ability to identify the sign or direction of such interaction effect was very high in all the simulated scenarios.
Frequent violations of fair principles in real-life settings raise the fundamental question of whether such principles can guarantee the existence of a self-enforcing equilibrium in a free economy. We show that elementary principles of distributive justice guarantee that a pure-strategy Nash equilibrium exists in a finite economy where agents freely (and non-cooperatively) choose their inputs and derive utility from their pay. Chief among these principles is that: 1) your pay should not depend on your name, and 2) a more productive agent should not earn less. When these principles are violated, an equilibrium may not exist. Moreover, we uncover an intuitive condition---technological monotonicity---that guarantees equilibrium uniqueness and efficiency. We generalize our findings to economies with social justice and inclusion, implemented in the form of progressive taxation and redistribution, and guaranteeing a basic income to unproductive agents. Our analysis uncovers a new class of strategic form games by incorporating normative principles into non-cooperative game theory. Our results rely on no particular assumptions, and our setup is entirely non-parametric. Illustrations of the theory include applications to exchange economies, surplus distribution in a firm, contagion and self-enforcing lockdown in a networked economy, and bias in the academic peer-review system.
Aguiar et al. (2018) propose the Shapley distance as a measure of the extent to which output sharing among the stakeholders of an organization can be considered unfair. It measures the distance between an arbitrary pay profile and the Shapley pay profile under a given technology, the latter profile defining the fair distribution. We provide an axiomatic characterization of the Shapley distance, and show that it can be used to determine the outcome of an underlying bargaining process. We also present applications highlighting how favoritism in income distribution, egalitarianism, and taxation violate the different ideals of justice that define the Shapley value. The analysis has implications that can be tested using real-world data sets.
We study the long-run stability of trade networks in a two-sided economy of agents labelled men and women. Each agent desires relationships with the other type, but having multiple partners is costly. This cost-benefit trade-off results in each agent having a single-peaked utility over the number of partners, - the volume of trade - , peak being greater for men than for women. We propose a stochastic matching process in which self-interested agents form and sever links over time. Links can be added or deleted, sometimes simultaneously by a single agent. While the unperturbed process yields each pairwise stable network as an absorbing state, stochastic stability in two perturbed processes provides a significant refinement, leading respectively to egalitarian and anti-egalitarian pairwise stable networks. This has implications for the concentration on each side of the market of a random information shock. The analysis captures stylized facts, related to market fragmentation, concentration and contagion asymmetry, in several two-sided economies.
More than half of the HIV/AIDS-infected population today are women. We study a dynamic model of (in)fidelity, which explains the HIV/AIDS gender gap by the configuration of sexual networks. Each individual desires sexual relationships with opposite sex individuals. Two Markov matching processes are defined, each corresponding to a different culture of gender relations. The first process leads to egalitarian pairwise stable networks in the long run, and HIV/AIDS is equally prevalent among men and women. The second process leads to anti-egalitarian pairwise stable networks reflecting male domination, and women bear a greater burden. The results are consistent with empirical observations.
We study the stability and welfare properties of …delity networks. These are networks that form in a mating economy with two types of agents (e.g., men and women). Each enjoys having relationships with the opposite type. However, having multiple partners is viewed as in…delity, which is punished if detected. There is female discrimination in that in…delity is punished more severely for women than for men. We obtain a complete characterization of pairwise stable networks, which is sensitive to the size of the market. In most networks, women obtain their optimal number of partners, whereas men obtain at most their optimum. Furthermore, we show that a pairwise stable network is Pareto e¢ cient if and only if it is female-optimal. Subsequently, we study how a random, unexpected information shock a¤ects women di¤erently from men. In particular, we introduce and completely characterize female-information-biased economies, which are economies in which the di¤usion of such a shock a¤ects more women than men in all pairwise stable networks. We show that female-information-biased economies are segmented (or strati…ed), with the size of each segment being determined by a correspondence. This result generalizes to a probabilistic framework. We extend the analysis to: (1) economies characterized by female-to-male subjugation, a generalization to many-to-many matching markets of the normative principle that underlies relationship formation in one-to-one and many-to-one matching markets; and (2) hierarchical mating economies, wherein each agent has a distinct social rank and higher-ranked agents are more desired as partners. We …nd that any economy of female-to-male subjugation is female-information-biased. Any hierarchical mating economy admits a unique pairwise stable network, and is female-information-biased. The model has testable implications within and beyond the …delity context. Applied to sexual mar-kets, our analysis uncovers the social mechanism underlying the gender gap in HIV/AIDS prevalence, and demonstrates the distinct and complementary role of female discrimination, market strati…cation and social inequality in creating an HIV infection bias against women. Applied to academic markets, our …ndings highlight a factor in the persistence of inequality between academic institutions, and have implications for how new ideas a¤ect students and faculty di¤erently. Applied to the country-citizen market for highly skilled workers, the model predicts a country's brain drain as a function of its development, and derives implications for the gap between countries in the concentration of new technologies.
The authors analyze the Shapley value allocation of an economy with differential information. They address the following questions: How do coalitions of agents share their private information? How can one measure the information advantage or superiority of an agent? Is each agent's private information verifiable by other members of a coalition? Do coalitions of agents pool their private information? Do agents have an incentive to report their true private information? Do value allocations exist in an economy with differential information? The authors provide answers to each of these questions. Copyright 1994 by The Econometric Society.
We extend cooperative games in characteristic function form to multichoice cooperative games. In such multichoice games, Shapley value is extended. The explicit formula is given to compute the value. American Mathematical Society Classification Numbers: 90D12, 90D40; Journal of Economic Literature Classification Numbers: C71, D46, D61, D72.
A macro-prudential approach to financial regulation should reasonably evaluate the systemic risk of individual institutions. We introduce a measure of the systemic risk of interconnected banks using the weighted Shapley value, which characterizes banks' systemic risk by capturing two characteristics: (i) a bank's risk exposure of the primitive asset structure; and (ii) its position in the interbank network. An empirical implementation reveals that the primitive asset structure and interbank connections are main drivers of each bank's systemic risk.
This is a survey of Lloyd Shapley's contributions to matching theory and game theory in general, starting with the work that inspired the Swedish Academy to award the 2012 Nobel Memorial Prize in Economic Sciences to Lloyd Shapley and Alvin Roth.
We characterize the evolution over time of a network of credit relations among financial agents as a system of coupled stochastic processes. Each process describes the dynamics of individual financial robustness, while the coupling results from a network of liabilities among agents. The average level of risk diversification of the agents coincides with the density of links in the network. In addition to a process of diffusion of financial distress, we also consider a discrete process of default cascade, due to the re-evaluation of agents’ assets. In this framework we investigate the probability of individual defaults as well as the probability of systemic default as a function of the network density. While it is usually thought that diversification of risk always leads to a more stable financial system, in our model a tension emerges between individual risk and systemic risk. As the number of counterparties in the credit network increases beyond a certain value, the default probability, both individual and systemic, starts to increase. This tension originates from the fact that agents are subject to a financial accelerator mechanism. In other words, individual financial fragility feeding back on itself may amplify the effect of an initial shock and lead to a full fledged systemic crisis. The results offer a simple possible explanation for the endogenous emergence of systemic risk in a credit network.
Financial contagion is modeled as an equilibrium phenomenon. Because liquidity preference shocks are imperfectly correlated across regions, banks hold interregional claims on other banks to provide insurance against liquidity preference shocks. When there is no aggregate uncertainty, the first-best allocation of risk sharing can be achieved. However, this arrangement is financially fragile. A small liquidity preference shock in one region can spread by contagion throughout the economy. The possibility of contagion depends strongly on the completeness of the structure of interregional claims. Complete claims structures are shown to be more robust than incomplete structures.
Composed in honour of the sixty-fifth birthday of Lloyd Shapley, this volume makes accessible the large body of work that has grown out of Shapley's seminal 1953 paper. Each of the twenty essays concerns some aspect of the Shapley value. Three of the chapters are reprints of the 'ancestral' papers: Chapter 2 is Shapley's original 1953 paper defining the value; Chapter 3 is the 1954 paper by Shapley and Shubik applying the value to voting models; and chapter 19 is Shapley's 1969 paper defining a value for games without transferable utility. The other seventeen chapters were contributed especially for this volume. The first chapter introduces the subject and the other essays in the volume, and contains a brief account of a few of Shapley's other major contributions to game theory. The other chapters cover the reformulations, interpretations and generalizations that have been inspired by the Shapley value, and its applications to the study of coalition formulation, to the organization of large markets, to problems of cost allocation, and to the study of games in which utility is not transferable.
The principle of monotonicity for cooperative games states that if a game changes so that some player's contribution to all coalitions increases or stays the same then the player's allocation should not decrease. There is a unique symmetric and efficient solution concept that is monotonic in this most general sense — the Shapley value. Monotonicity thus provides a simple characterization of the value without resorting to the usual additivity and dummy assumptions, and lends support to the use of the value in applications where the underlying game is changing, e.g. in cost allocation problems.
We analyze the Shapley value allocation of an economy with differential information. Since the intent of the Shapley value
is to measure the sum of the expected marginal contributions made by an agent to any coalition that he/she belongs to, the
value allocation of an economy with differential information provides an interesting way to measure the information advantage
of an agent. This feature of the Shapley value allocation is not necessarily shared by the rational expectation equilibrium.
Thus, we analyze the informational structure of an economy with differential information from a different and new viewpoint.
In particular we address the following questions: How do coalitions of agents share their private information? How can one
measure the information advantage or superiority of an agent? Is each agent’s private information verifiable by other members
of a coalition? Do coalitions of agents pool their private information? Do agents have an incentive to report their true private
information? What is the correct concept of a value allocation in an economy with differential information? Do value allocations
exist in an economy with differential information? We provide answers to each of these questions.
Voting systems with several levels of approval in the input and output are considered in this paper. That means games with n≥2 players, j≥2 ordered qualitative alternatives in the input level and k≥2 possible ordered quantitative alternatives in the output. We introduce the Shapley–Shubik power index notion when passing from ordinary simple games or ternary voting games with abstention to this wider class of voting systems. The pivotal role of players is analysed by means of several examples and an axiomatization in the spirit of Shapley and Dubey is given for the proposed power index.
I. Introduction, 88. — II. Mathematical-institutional economics and agrarian reform, 89. — II. A simple production function,
91. — IV. The feudal system, 93. — V. The capitalist and landless peasants, 94. — VI. A small landowner capitalist society,
97. — VII. The village commune, 100. — VIII. Corporate ownership, 102. — IX. Threats and point product, 104. — X. Summary,
104. — Appendix, 105.
Measuring and decomposing the distance to the Shapley wage function with limited data
- V H Aguiar
- R Pongou
- J B Tondji
- F Allen
- D Gale
Aguiar, V.H., Pongou, R., Tondji, J.B., 2016. Measuring and decomposing the distance to the Shapley wage function with limited data. Mimeo.
Allen, F., Gale, D., 2000. Financial contagion. J. Polit. Economy 108 (1), 1-33.