
on Game Theory 
By:  Ziv Hellman; Yehuda John Levy 
Abstract:  The solution concept of a Bayesian equilibrium of a Bayesian game is inherently an interim concept. The corresponding ex ante solution concept has been termed Harsányi equilibrium; examples have appeared in the literature showing that there are Bayesian games with uncountable state spaces that have no Bayesian approximate equilibria but do admit Harsányi approximate equilibrium, thus exhibiting divergent behaviour in the ex ante and interim stages. Smoothness, a concept from descriptive set theory, has been shown in previous works to guarantee the existence of Bayesian equilibria. We show here that higher rungs in the countable Borel equivalence relation hierarchy can also shed light on equilibrium existence. In particular, hyperfiniteness, the next step above smoothness, is a sufficient condition for the existence of Harsányi approximate equilibria in purely atomic Bayesian games. 
Keywords:  Bayesian games; Equilibrium existence; Borel equivalence relations 
JEL:  C62 C65 C72 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:gla:glaewp:2020_15&r=all 
By:  Leanne Streekstra (Department of Business and Economics, University of Southern Denmark); Christian Trudeau (Department of Economics, University of Windsor) 
Abstract:  We extend the familiar shortest path problem by supposing that agents have demands over multiple periods. This potentially allows agents to combine their paths if their demands are complementary; for instance if one agent only needs a connection to the source in the summer while the other requires it only in the winter. We show that the resulting cost sharing problem always has a nonempty core, regardless of the number of agents and periods, the cost structure or the demand profile. We then exploit the fact that the model encompasses many wellstudied problems to obtain or reobtain nonvacuity results for the cores of sourceconnection problems, (msided) assignment problems and minimum coloring problems. 
Keywords:  shortest path, demand over multiple periods, cooperative game, core, sourceconnection, assignment. 
JEL:  C71 D63 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:wis:wpaper:2003&r=all 
By:  Tomoyuki Ichiba; Tianjiao Yang 
Abstract:  This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrage both in the $N$ investors and mean field regimes. An investor competes with a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the %proportion of initial capital. With market price of risk processes depending on the stock market and investors respectively, the minimal initial capital required is the optimal cost in the $N$player games and mean field games. It can be characterized as the smallest nonnegative continuous solution of a Cauchy problem. The measure flow of wealth appears in the cost, while the joint flow of wealth and strategy is in state processes. We modify the extended mean field game with common noise and its notion of the uniqueness of Nash equilibrium. There is a unique equilibrium in $N$player games and mean field games with mild conditions on the equity market. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.15158&r=all 
By:  Krumer, Alex; Megidish, Reut; Sela, Aner 
Abstract:  We study roundrobin tournaments with four symmetric players and two identical prizes where players compete against each other in games modeled as an allpay contest. We demonstrate that in this common structure players may have an incentive to manipulate the results, namely, depending on the outcomes of the first round, a player may have an incentive to lose in the second round in order to maximize his expected payoff in the tournament. 
Date:  2020–02 
URL:  http://d.repec.org/n?u=RePEc:cpr:ceprdp:14412&r=all 
By:  Andrew Choi; Syngjoo Choi; Yves Gueron; Eungik Lee 
Abstract:  This paper provides experimental evidence on the impacts of irreversibility and imperfect monitoring on the efficiency and the equity of repeated public goods game. We find that irreversibility and imperfect monitoring both lead to inefficient and unequal outcomes through different channels. Irreversibility lowers public goods contribution in earlier periods and makes the initialperiod contribution gap between two players longlasting. Imperfect monitoring hampers conditional cooperation and reduces group contribution persistently. A finite mixture estimation with conditional cooperator provides a coherent account of the treatment effects. 
Keywords:  repeated games; dynamic games; imperfect monitoring; irreversibility; cooperation; 
JEL:  C73 C91 C92 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:snu:ioerwp:no133&r=all 
By:  Giménez Gómez, José M. (José Manuel); Peris, Josep E.; Subiza, Begoña 
Abstract:  A minimum cost spanning tree problem analyzes the way to efficiently connect individuals to a source when they are located at different places; that is, to connect them with the minimum possible cost. This objective requires the cooperation of the involved individuals and, once an efficient network is selected, the question is how to fairly allocate the total cost among these agents. To answer this question the literature proposes several rules providing allocations that, generally, depend on all the possible connection costs, regardless of whether these connections have been used or not in order to build the efficient network. To this regard, our approach defines a simple way to allocate the optimal cost with two main criteria: (1) each individual only pays attention to a few connection costs (the total cost of the optimal network and the cost of connecting by himself to the source); and (2) an egalitarian criteria is used to share costs or benefits. Then, we observe that the spanning tree cost allocation can be turned into a claims problem and, by using claims rules, we define two egalitarian solutions so that the total cost is allocated trying to equalize either the payments in which agents incur, or the benefit that agents obtain throughout cooperation. Finally, by comparing both proposals with other solution concepts proposed in the literature, we select equalizing payments as much as possible and axiomatically analyze it, paying special attention to coalitional stability (core selection), a central property whenever cooperation is needed to carry out the project. As our initial proposal might propose allocations outside the core, we modify it to obtain a core selection and we obtain an alternative interpretation of the Folk solution. Keywords: Minimum cost spanning tree, Egalitarian, Cost sharing, Core. JEL classification: C71, D63, D71. 
Keywords:  Jocs cooperatius, Economia del benestar, 33  Economia, 
Date:  2019 
URL:  http://d.repec.org/n?u=RePEc:urv:wpaper:2072/376029&r=all 
By:  Harry Pei 
Abstract:  I study a reputation model in which a patient player privately observes a persistent state that affects his myopic opponents' payoffs, and can be one of the several commitment types that plays the same (possibly mixed) action over time. The main result is a characterization of the set of environments under which the patient player obtains at least his commitment payoff in all equilibria regardless of his stagegame payoff function. My result implies that small perturbations to a pure commitment action can lead to a discontinuous change in the patient player's equilibrium payoff. The main technical contribution is to use martingale techniques to construct a nonstationary strategy under which the patient player can avoid signaling negative information about the state while at the same time, matching the longrun frequency of his actions to the mixed commitment action and convincing his opponents that his action is close to the commitment action in almost all periods. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.16206&r=all 
By:  Dongshuang Hou (NPU  Northwestern Polytechnical University [Xi'an]); Aymeric Lardon (GATE Lyon SaintÉtienne  Groupe d'analyse et de théorie économique  CNRS  Centre National de la Recherche Scientifique  Université de Lyon  UJM  Université Jean Monnet [SaintÉtienne]  UCBL  Université Claude Bernard Lyon 1  Université de Lyon  UL2  Université Lumière  Lyon 2  ENS Lyon  École normale supérieure  Lyon); Hao Sun (NPU  Northwestern Polytechnical University [Xi'an]) 
Abstract:  Two new notions of stability of coalitions, based on the idea of exclusion or integration of players depending on how they affect allocations, are introduced for cooperative transferable utility games. The first one, called internal stability, requires that no coalition member would find that her departure from the coalition would improve her allocation or those of all her partners. The second one, called external stability, requires that coalitions members do not wish to recruit a new partner willing to join the coalition, since her arrival would hurt some of them. As an application of these two notions, we study the stability of Group Purchasing Organizations using the Shapley value to allocate costs between buyers. Our main results suggest that, when all buyers are initially alone, while small buyers will form internally and externally stable Group Purchasing Organizations to benefit from the best price discount, big buyers will be mutually exclusive and may cooperate with only small buyers. 
Keywords:  Internal and external stability,Group purchasing organization,Cost allocation,Shapley value 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:hal:wpaper:halshs02860639&r=all 
By:  Sela, Aner 
Abstract:  We study two reverse contests, A and B, with two agents, each of whom has both a linear reward function that increases in the agent's effort and an effort constraint. However, since the effort (output) of the agents has a negative effect on society, if the agents' effort constraints are relatively high, the designer in reverse contest A imposes a punishment such that the agent with the highest effort who caused the greatest damage is punished. Conversely, if the agents' effort constraints are relatively low, in reverse contest B, the designer awards a prize to the agent with the lowest effort who caused the smallest damage. We analyze the behavior of both symmetric and asymmetric agents in both contests A and B. In equilibrium, independent of the levels of the agents' effort constraints, both agents are active and they have positive expected payoffs. Furthermore, the agents might have the same expected payoff regardless of their asymmetric values of the prize/punishment or their asymmetric effort constraints. 
Date:  2020–02 
URL:  http://d.repec.org/n?u=RePEc:cpr:ceprdp:14411&r=all 
By:  Anton Kolotilin (School of Economics, UNSW Business School); Hongyi (School of Economics, UNSW Business School) 
Abstract:  We study a communication game between an informed sender and an uninformed receiver with repeated interactions and voluntary transfers. Transfers motivate the receiver’s decisionmaking and signal the sender’s information. Although full separation can always be supported in equilibrium, partial or complete pooling is optimal if the receiver’s decisionmaking is highly responsive to information. In this case, the receiver’s decisionmaking is disciplined by pooling extreme states, where she is most tempted to defect. 
Keywords:  strategic communication, relational contracts 
JEL:  C73 D82 D83 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:swe:wpaper:201812b&r=all 
By:  Sascha Kurz 
Abstract:  Binary yesno decisions in a legislative committee or a shareholder meeting are commonly modeled as a weighted game. However, there are noteworthy exceptions. E.g., the voting rules of the European Council according to the Treaty of Lisbon use a more complicated construction. Here we want to study the question if we loose much from a practical point of view, if we restrict ourselves to weighted games. To this end, we invoke power indices that measure the influence of a member in binary decision committees. More precisely, we compare the achievable power distributions of weighted games with those from a reasonable superset of weighted games 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.05330&r=all 
By:  Jobst Heitzig; Sarah Hiller 
Abstract:  Many realworld situations of ethical relevance, in particular those of largescale social choice such as mitigating climate change, involve not only many agents whose decisions interact in complicated ways, but also various forms of uncertainty, including quantifiable risk and unquantifiable ambiguity. In such problems, an assessment of individual and groupwise moral responsibility for ethically undesired outcomes or their responsibility to avoid such is challenging and prone to the risk of under or overdetermination of responsibility. In contrast to existing approaches based on strict causation or certain deontic logics that focus on a binary classification of `responsible' vs `not responsible', we here present several different quantitative responsibility metrics that assess responsibility degrees in units of probability. For this, we use a framework based on an adapted version of extensiveform game trees and an axiomatic approach that specifies a number of potentially desirable properties of such metrics, and then test the developed candidate metrics by their application to a number of paradigmatic social choice situations. We find that while most properties one might desire of such responsibility metrics can be fulfilled by some variant, an optimal metric that clearly outperforms others has yet to be found. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.07352&r=all 
By:  Pierpaolo Battigalli; Martin Dufwenberg 
Abstract:  The mathematical framework of psychological game theory is useful for describing many forms of motivation where preferences depend directly on own or others’ beliefs. It allows for incorporating, e.g., emotions, reciprocity, image concerns, and selfesteem in economic analysis. We explain how and why, discussing basic theory, experiments, applied work, and methodology. 
Keywords:  psychological game theory, beliefdependent motivation, reciprocity, emotions, image concerns, selfesteem 
JEL:  C72 D91 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:ces:ceswps:_8285&r=all 
By:  Xiang Yu; Yuchong Zhang; Zhou Zhou 
Abstract:  This paper studies competitions with rankbased reward among a large number of teams. Within each sizable team, we consider a meanfield contribution game in which each team member contributes to the jump intensity of a common Poisson project process; across all teams, a mean field competition game is formulated on the rank of the completion time, namely the jump time of Poisson project process, and the reward to each team is paid based on its ranking. On the layer of teamwise competition game, three optimization problems are introduced when the team size is determined by: (i) the team manager; (ii) the central planner; (iii) the team members' voting as partnership. We propose a relative performance criteria for each team member to share the team's reward and formulate some mean field games of mean field games, which are new to the literature. In all problems with homogeneous parameters, the equilibrium control of each worker and the equilibrium or optimal team size can be computed in an explicit manner, allowing us to analytically examine the impacts of some model parameters and discuss their economic implications. Two numerical examples are also presented to illustrate the parameter dependence and comparison between different team size decision making. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.14472&r=all 
By:  Christian Bontemps (ENAC  Ecole Nationale de l'Aviation Civile); Rohit Kumar (Indian Statistical Institute [New Delhi]) 
Abstract:  In this paper, we consider inference procedures for entry games with complete information. Due to the presence of multiple equilibria, we know that such a model may be setidentified without imposing further restrictions. We complete the model with the unknown selection mechanism and characterize geometrically the set of predicted choice probabilities, in our case, a convex polytope with many facets. Testing whether a parameter belongs to the identified set is equivalent to testing whether the true choice probability vector belongs to this convex set. Using tools from the convex analysis, we calculate the support function and the extreme points. The calculation yields a finite number of inequalities, when the explanatory variables are discrete, and we characterized them once for all. We also propose a procedure that selects the moment inequalities without having to evaluate all of them. This procedure is computationally feasible for any number of players and is based on the geometry of the set. Furthermore, we exploit the specific structure of the test statistic used to test whether a point belongs to a convex set to propose the calculation of critical values that are computed once and independent of the value of the parameter tested, which drastically improves the calculation time. Simulations in a separate section suggest that our procedure performs well compared with existing methods. 
Keywords:  setidentification,convex set,entry games,support function 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:hal:journl:hal02137356&r=all 
By:  Benjamin Heymann (CERMICS); Michel de Lara (CERMICS); JeanPhilippe Chancelier (CERMICS) 
Abstract:  We state and prove Kuhn's equivalence theorem for a new representation of games, the intrinsic form. First, we introduce games in intrinsic form where information is represented by $\sigma$fields over a product set. For this purpose, we adapt to games the intrinsic representation that Witsenhausen introduced in control theory. Those intrinsic games do not require an explicit description of the play temporality, as opposed to extensive form games on trees. Second, we prove, for this new and more general representation of games, that behavioral and mixed strategies are equivalent under perfect recall (Kuhn's theorem). As the intrinsic form replaces the tree structure with a product structure, the handling of information is easier. This makes the intrinsic form a new valuable tool for the analysis of games with information. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.14838&r=all 
By:  Victor Aguirregabiria 
Abstract:  Firms make decisions under uncertainty and differ in their ability to collect and process information. As a result, in changing environments, firms have heterogeneous beliefs on the behavior of other firms. This heterogeneity in beliefs can have important implications on market outcomes, efficiency, and welfare. This paper studies the identification of firms' beliefs using their observed actions  a revealed preference and beliefs approach. I consider a general structural model of market competition where firms have incomplete information and their beliefs and profits are nonparametric functions of decisions and state variables. Beliefs may be out of equilibrium. The framework applies both to continuous and discrete choice games and includes as particular cases models of competition in prices or quantities, auction models, entry games, and dynamic investment games. I focus on identification results that exploit a natural exclusion restriction in models of competition: an observable variable that affects a firm's cost (or revenue) but does not have a direct effect on other firms' profits. I present identification results under three scenarios  common in empirical IO  on the data available to the researcher. 
Keywords:  Nonequilibrium beliefs; Structural models of competition; Identification; Revealed beliefs approach 
JEL:  C57 D81 D83 D84 L13 
Date:  2020–06–29 
URL:  http://d.repec.org/n?u=RePEc:tor:tecipa:tecipa670&r=all 
By:  Sela, Aner 
Abstract:  We study bestofthree allpay auctions with two players who compete in three stages with a single match per stage. The first player to win two matches wins the contest. We assume that a prize sum is given, and show that if players are symmetric, the allocation of prizes does not have any effect on the players' expected total effort. On the other hand, if players are asymmetric, in order to maximize the players' expected total effort, independent of the players' types, it is not optimal to allocate a single final prize to the winner. Instead, it is optimal to allocate intermediate prizes in the first stage or/and in the second stage in addition to the final prize. When the asymmetry of the players' types is sufficiently high, it is optimal to allocate intermediate prizes in both two first stages and a final prize to the winner. 
Date:  2020–02 
URL:  http://d.repec.org/n?u=RePEc:cpr:ceprdp:14410&r=all 
By:  Geraldine Bouveret; Antoine Mandel 
Abstract:  We investigate the containment of epidemic spreading in networks from a normative point of view. We consider a susceptible/infected model in which agents can invest in order to reduce the contagiousness of network links. In this setting, we study the relationships between social efficiency, individual behaviours and network structure. First, we exhibit an upper bound on the Price of Anarchy and prove that the level of inefficiency can scale up to linearly with the number of agents. Second, we prove that policies of uniform reduction of interactions satisfy some optimality conditions in a vast range of networks. In setting where no central authority can enforce such stringent policies, we consider as a type of secondbest policy the shift from a local to a global game by allowing agents to subsidise investments in contagiousness reduction in the global rather than in the local network. We then characterise the scope for Pareto improvement opened by such policies through a notion of Price of Autarky, measuring the ratio between social welfare at a global and a local equilibrium. Overall, our results show that individual behaviours can be extremely inefficient in the face of epidemic propagation but that policy can take advantage of the network structure to design efficient containment policies. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.07580&r=all 
By:  Yuval Heller; Erik Mohlin 
Abstract:  We study environments in which agents are randomly matched to play a Prisoner's Dilemma, and each player observes a few of the partner's past actions against previous opponents. We depart from the existing related literature by allowing a small fraction of the population to be commitment types. The presence of committed agents destabilizes previously proposed mechanisms for sustaining cooperation. We present a novel intuitive combination of strategies that sustains cooperation in various environments. Moreover, we show that under an additional assumption of stationarity, this combination of strategies is essentially the unique mechanism to support full cooperation, and it is robust to various perturbations. Finally, we extend the results to a setup in which agents also observe actions played by past opponents against the current partner, and we characterize which observation structure is optimal for sustaining cooperation. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.15310&r=all 
By:  Jihad C. Elnaboulsi (CRESE, Univ. Bourgogne FrancheComté); Wassim Daher (Gulf University for Science and Technology, Department of Mathematics and Natural Sciences); Yigit Saglam (Victoria University of Wellington, School of Economics and Finance) 
Abstract:  We analyze how environmental taxes should be optimally levied when the regulators and firms face costs uncertainties in a StackelbergCournot game. We allow linearquadratic payoffs functions coupled with an affine information structure encompassing common and private information with noisy signals. In the first period, the regulator chooses the intensity of emissions taxes in order to reduce externalities. In the second period, facing industryrelated and firmspecific shocks, firms compete in the marketplace as Cournot rivals and choose outputs. We show that, given costs uncertainties with nonuniform quality of signals across firms, the regulator sets differentiated tax policy. We also examined the social value of information under exante calibrated emissions taxes. We argue that the magnitude of the associated social benefits and costs of more precise private signals hinge largely and fundamentally on the value of the ratio of the slopes of the marginal damage and the marginal consumer surplus. The lack of accurate data clouds the regulatory process by preventing the necessary finetuning of the tax rules towards specific environmental circumstances. Finally, we investigate information sharing between polluters and its impacts on welfare. We stress that, when there are threats of severe environmental damages under deep uncertainties, collusion is welfare reducing and may jeopardize the regulatory process. Numerical simulations illustrate the results that the model delivers. 
Keywords:  Differentiated Taxes, Costs Uncertainties, Signaling, Precision, Information Sharing, Collusion, Energy Markets. 
JEL:  D43 D83 H23 L13 Q58 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:crb:wpaper:202004&r=all 
By:  Ryan Donnelly; Matthew Lorig 
Abstract:  We consider the problem of maximizing portfolio value when an agent has a subjective view on asset value which differs from the traded market price. The agent's trades will have a price impact which affect the price at which the asset is traded. In addition to the agent's trades affecting the market price, the agent may change his view on the asset's value if its difference from the market price persists. We also consider a situation of several agents interacting and trading simultaneously when they have a subjective view on the asset value. Two cases of the subjective views of agents are considered, one in which they all share the same information, and one in which they all have an individual signal correlated with price innovations. To study the large agent problem we take a meanfield game approach which remains tractable. After classifying the meanfield equilibrium we compute the crosssectional distribution of agents' inventories and the dependence of price distribution on the amount of shared information among the agents. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.13585&r=all 
By:  David McAdams 
Abstract:  This paper develops a Nashequilibrium extension of the classic SIR model of infectiousdisease epidemiology ("Nash SIR"), endogenizing people's decisions whether to engage in economic activity during a viral epidemic and allowing for complementarity in socialeconomic activity. An equilibrium epidemic is one in which Nash equilibrium behavior during the epidemic generates the epidemic. There may be multiple equilibrium epidemics, in which case the epidemic trajectory can be shaped through the coordination of expectations, in addition to other sorts of interventions such as stayathome orders and accelerated vaccine development. An algorithm is provided to compute all equilibrium epidemics. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.10109&r=all 