
on Game Theory 
By:  Francesco Caruso (Università di Napoli Federico II); Maria Carmela Ceparano (Università di Napoli Federico II); Jacqueline Morgan (Università di Napoli Federico II and CSEF) 
Abstract:  Twoplayer Stackelberg games may have multiple Subgame Perfect Nash Equilibria (henceforth SPNEs), especially when the best reply correspondence of the follower is not a singlevalued map. Aim of the paper is to investigate the issue of selection of SPNEs in twoplayer Stackelberg games by exploiting perturbations of the payoff functions of the game. To achieve such a goal, since the limit of “perturbed" SPNEs is not necessarily an SPNE of the initial game even for classic perturbations, first we show how to produce an SPNE starting from a sequence of SPNEs of perturbed games. This result allows to define a general selection method for SPNEs that can accommodate various behaviors of the players. More precisely, under mild assumptions on the data of the game we prove that perturbations relying on a Tikhonov regularization, on an adversetomove behaviour and on an altruistic behaviour fit the general method and we present the specific selection results associated to such perturbations. On the one hand, as regards to the Tikhonov regularization and the adversetomove behaviour, we extend or recover the results showed by Morgan and Patrone [Advances in Dynamic Games, (2006), pp. 209221] and by Caruso, Ceparano and Morgan [Dyn. Games Appl., 9 (2019), pp. 416432]. On the other hand, concerning the altruistic behaviour, we present a new specific selection method for SPNEs based on the slightly altruistic approach introduced by De Marco and Morgan [J. Optim. Theory Appl., 137 (2008), pp. 347362] for simultaneousmove games. Finally, we illustrate by examples that the general method carried out under the three different “behaviours" just mentioned can select different SPNEs. 
Keywords:  Subgame perfect Nash equilibrium; selection; twoplayer Stackelberg game; Tikhonov regularization; adversetomove behaviour; slightly altruistic behaviour. 
Date:  2022–01–24 
URL:  http://d.repec.org/n?u=RePEc:sef:csefwp:636&r= 
By:  Emiliano Cantonini; Antonio Penta 
Abstract:  Backward Induction is a fundamental concept in game theory. As an algorithm, it can only be used to analyze a very narrow class of games, but its logic is also invoked, albeit informally, in several solution concepts for games with imperfect or incomplete information (Subgame Perfect Equilibrium, Sequential Equilibrium, etc.). Yet, the very meaning of "backward induction reasoning" is not clear in these settings, and we lack a way to apply this simple and compelling idea to more general games. We remedy this by introducing a solution concept for games with imperfect and incomplete information, Backwards Rationalizability, that captures precisely the implications of backward induction reasoning. We show that Backwards Rationalizability satisfies several properties that are normally ascribed to backward induction reasoning, such as: (i) an incompleteinformation extension of subgame consistency (continuationgame consistency); (ii) the possibility, in finite horizon games, of being computed via a tractable backwards procedure; (iii) the view of unexpected moves as mistakes; (iv) a characterization of the robust predictions of a "perfect equilibrium" notion that introduces the backward induction logic and nothing more into equilibrium analysis. We also discuss a few applications, including a new version of peerconfirming equilibrium (Lipnowski and Sadler (2019)) that, thanks to the backward induction logic distilled by Backwards Rationalizability, restores in dynamic games the natural comparative statics the original concept only displays in static settings. 
Keywords:  backward induction, backwards procedure, backwards rationalizability, incomplete information, interim perfect equilibrium, rationalizability, robustness 
JEL:  C72 C73 D82 
Date:  2022–02 
URL:  http://d.repec.org/n?u=RePEc:upf:upfgen:1815&r= 
By:  Iwan Bos (Maastricht University); Marco A. Marini (Sapienza University of Rome); Riccardo D. Saulle (University of Padova) 
Abstract:  This paper examines capacityconstrained oligopoly pricing with sellers who seek myopic improvements. We employ the Myopic Stable Set solution concept and establish the existence of a unique purestrategy price solution for any given level of capacity. This solution is shown to coincide with the set of purestrategy Nash equilibria when capacities are large or small. For an intermediate range of capacities, it predicts a price interval that includes the mixedstrategy support. This stability concept thus encompasses all Nash equilibria and oers a purestrategy solution when there is none in Nash terms. It particularly provides a behavioral rationale for di erent pricing patterns, including Edgeworth price cycles and states of hypercompetition with supply shortages. We also analyze the impact of a change in firm size distribution. A merger among the biggest firms may lead to more price dispersion as it increases the maximum and decreases the minimum myopically stable price. 
Keywords:  Bounded Rationality, Capacity Constraints, Mergers, Myopic Stable Set, Oligopoly Pricing, Supply Shortages 
JEL:  C72 D43 L13 
Date:  2021–12 
URL:  http://d.repec.org/n?u=RePEc:fem:femwpa:2021.32&r= 
By:  Daniel Clark; Drew Fudenberg; Kevin He 
Abstract:  Learning models do not in general imply that weakly dominated strategies are irrelevant or justify the related concept of "forward induction," because rational agents may use dominated strategies as experiments to learn how opponents play, and may not have enough data to rule out a strategy that opponents never use. Learning models also do not support the idea that the selected equilibria should only depend on a game's normal form, even though two games with the same normal form present players with the same decision problems given fixed beliefs about how others play. However, playing the extensive form of a game is equivalent to playing the normal form augmented with the appropriate terminal node partitions so that two games are information equivalent, i.e., the players receive the same feedback about others' strategies. 
Date:  2022–01 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2201.00776&r= 
By:  John Higgins (Department of Economics, University of Wisconsin, Madison, WI 53706, USA); Tarun Sabarwal (Department of Economics, University of Kansas, Lawrence, KS 66045, USA) 
Abstract:  We study proliferation of an action in binary action network coordination games that are generalized to include global effects. This captures important aspects of proliferation of a particular action or narrative in online social networks, providing a basis to understand their impact on societal outcomes. Our model naturally captures complementarities among starting sets, network resilience, and global effects, and highlights interdependence in channels through which contagion spreads. We present new, natural, and computationally tractable algorithms to define and compute equilibrium objects that facilitate the general study of contagion in networks and prove their theoretical properties. Our algorithms are easy to implement and help to quantify relationships previously inaccessible due to computational intractability. Using these algorithms, we study the spread of contagion in scalefree networks with 1,000 players using millions of Monte Carlo simulations. Our analysis provides quantitative and qualitative insight into the design of policies to control or spread contagion in networks. The scope of application is enlarged given the many other situations across different fields that may be modeled using this framework. 
Keywords:  Network games, coordination games, contagion, algorithmic computation 
JEL:  C62 C72 
Date:  2021–05 
URL:  http://d.repec.org/n?u=RePEc:kan:wpaper:202201&r= 
By:  Dianetti, Jodi (Center for Mathematical Economics, Bielefeld University); Ferrari, Giorgio (Center for Mathematical Economics, Bielefeld University); Fischer, Markus (Center for Mathematical Economics, Bielefeld University); Nendel, Max (Center for Mathematical Economics, Bielefeld University) 
Abstract:  We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow to prove existence and approximation of strong mean field equilibria in models where data may not be continuous with respect to the measure parameter and common noise is allowed. The setting is general enough to encompass qualitatively different problems, such as mean field games for discrete time finite space Markov chains, singularly controlled and reflected diffusions, and mean field games of optimal timing. Our analysis hinges on Tarski's fixed point theorem, along with technical results on lattices of flows of probabiltiy and subprobability measures. 
Keywords:  Mean field games, submodularity, complete lattice of measures, Tarki's fixed point theorem, Markov chain, singular stochastic control, refelcted diffusion, optimal stopping. 
Date:  2022–01–26 
URL:  http://d.repec.org/n?u=RePEc:bie:wpaper:661&r= 
By:  Julien Fixary (Un iversité Paris 1 PanthéonSorbonne, Centre d'Economie de la Sorbonne) 
Abstract:  We extend BichFixary's theorem about the topological structure of the graph of pairwise stable networks. Namely, we show that the graph of pairwise stable networks is not only homeomorphic to the space of societies, but that it is ambient isotopic to a trivial copy of this space (a result in the line of DemichelisGermano's unknottedness theorem. Furthermore, we introduce the notion of (extended) network dynamics which refers to families of vector fields on the set of weighted networks whose zeros correspond to pairwise stable networks. We use our version of the unknottedness theorem to show that most of network dynamics can be continuously connected to each other, without adding additional zeros. Finally, we prove that this result has an important consequence on the indices of these network dynamics at any pairwise stable network, a concept that we link to genericity using BichFixary's oddness theorem 
Keywords:  Pairwise Stability; Unknottedness Theorem; Network Dynamics; Genericity 
JEL:  C61 C62 D85 
Date:  2022–01 
URL:  http://d.repec.org/n?u=RePEc:mse:cesdoc:22002&r= 
By:  Kshitija Taywade; Brent Harrison; Judy Goldsmith 
Abstract:  Many past attempts at modeling repeated Cournot games assume that demand is stationary. This does not align with realworld scenarios in which market demands can evolve over a product's lifetime for a myriad of reasons. In this paper, we model repeated Cournot games with nonstationary demand such that firms/agents face separate instances of nonstationary multiarmed bandit problem. The set of arms/actions that an agent can choose from represents discrete production quantities; here, the action space is ordered. Agents are independent and autonomous, and cannot observe anything from the environment; they can only see their own rewards after taking an action, and only work towards maximizing these rewards. We propose a novel algorithm 'Adaptive with Weighted Exploration (AWE) $\epsilon$greedy' which is remotely based on the wellknown $\epsilon$greedy approach. This algorithm detects and quantifies changes in rewards due to varying market demand and varies learning rate and exploration rate in proportion to the degree of changes in demand, thus enabling agents to better identify new optimal actions. For efficient exploration, it also deploys a mechanism for weighing actions that takes advantage of the ordered action space. We use simulations to study the emergence of various equilibria in the market. In addition, we study the scalability of our approach in terms number of total agents in the system and the size of action space. We consider both symmetric and asymmetric firms in our models. We found that using our proposed method, agents are able to swiftly change their course of action according to the changes in demand, and they also engage in collusive behavior in many simulations. 
Date:  2022–01 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2201.00486&r= 
By:  de Callatay, Pierre (Université catholique de Louvain, LIDAM/CORE, Belgium); Mauleon, Ana (Université catholique de Louvain, LIDAM/CORE, Belgium); Vannetelbosch, Vincent (Université catholique de Louvain, LIDAM/CORE, Belgium) 
Abstract:  We propose the notion of minimal instability to determine the networks that are more likely to emerge in the long run when agents are farsighted. A network is minimally farsighted unstable if there is no other network which is more farsightedly stable. To formulate what it means to be more farsightedly stable, we compare networks by comparing (in the set inclusion or cardinal sense) their sets of farsighted defeating networks. We next compare networks in terms of their absorbtiveness by comparing both their sets of farsighted defeating networks (i.e. in terms of their stability) and their sets of farsighted defeated networks (i.e. in terms of their reachability). A network is maximally farsighted absorbing if there is no other network which is more farsightedly absorbing. We provide general results for characterizing minimally farsighted unstable networks and maximally farsighted absorbing networks, and we study their relationships with alternative notions of farsightedness. Finally, we use experimental data to show the relevance of the new solution concepts. 
Keywords:  networks; stability comparisons; farsighted players 
JEL:  A14 C70 D20 
Date:  2021–09–15 
URL:  http://d.repec.org/n?u=RePEc:cor:louvco:2021012&r= 
By:  Nicolas Eschenbaum; Filip Melgren; Philipp Zahn 
Abstract:  This paper develops a formal framework to assess policies of learning algorithms in economic games. We investigate whether reinforcementlearning agents with collusive pricing policies can successfully extrapolate collusive behavior from training to the market. We find that in testing environments collusion consistently breaks down. Instead, we observe static Nash play. We then show that restricting algorithms' strategy space can make algorithmic collusion robust, because it limits overfitting to rival strategies. Our findings suggest that policymakers should focus on firm behavior aimed at coordinating algorithm design in order to make collusive policies robust. 
Date:  2022–01 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2201.00345&r= 
By:  Taojun Xie (Asia Competitiveness Institute, Lee Kuan Yew School of Public Policy, National University of Singapore); Jiao Wang (Melbourne Institute: Applied Economic & Social Research, The University of Melbourne); Shiqi Liu (Asia Competitiveness Institute, Lee Kuan Yew School of Public Policy, National University of Singapore) 
Abstract:  We develop a tworegion SusceptibleInfectedRecoveredMacroeconomic model to evaluate the cooperative and noncooperative crossborder travel arrangements during a pandemic. In a symmetric setting, the Pareto optimal is a cooperative travel arrangement that emphasizes exclusively on domestic containment. A noncooperative game between social planners results in high travel restrictions with little benefit in economics or health. With asymmetric pandemic dynamics, a border closure on average results in less welfare loss than a noncooperative game, compared to cooperation. A border control allowing minimal essential travel only delays the outbreak in a region initially not infected. This can however be remedied by a timely vaccination plan. Under cooperation, predeparture tests further enhance the welfare. Applying our model, we estimate that the Singapore { Hong Kong travel bubble is valued at US$635.1 per capita, and the Australia { New Zealand travel bubble is valued at US$307.8 per capita. 
Keywords:  Travel Bubbles, Optimal Policy, Cooperation, Noncooperation, COVID19 Pandemic 
JEL:  F41 F42 I10 
Date:  2021–07 
URL:  http://d.repec.org/n?u=RePEc:iae:iaewps:wp2021n10&r= 
By:  Denis Bouyssou (LAMSADE  Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision  Université Paris DauphinePSL  PSL  Université Paris sciences et lettres  CNRS  Centre National de la Recherche Scientifique); Thierry Marchant (UGENT  Universiteit Gent = Ghent University [Belgium]); Marc Pirlot (UMons  Université de Mons) 
Abstract:  The size of maximum antichains in the product of n linear orders is known when the n linear orders have the same length. We present an exact expression for the size of maximum antichains when the linear orders have (possibly) different lengths. From this, we derive an exact expression for the size of maximum antichains in the product of n linear orders with the same length. This expression is equivalent to but different from the existing expression. It allows us to present an asymptotic result for the size of maximum antichains of n linear orders with the same length m going to infinity. 
Keywords:  linear orders Mathematics Subject Classification (2010) 91B06,multichoice cooperative game,Sperner,maximal antichain 
Date:  2021–10 
URL:  http://d.repec.org/n?u=RePEc:hal:journl:hal03484128&r= 
By:  David Martimort (PSE  Paris School of Economics  ENPC  École des Ponts ParisTech  ENS Paris  École normale supérieure  Paris  PSL  Université Paris sciences et lettres  UP1  Université Paris 1 PanthéonSorbonne  CNRS  Centre National de la Recherche Scientifique  EHESS  École des hautes études en sciences sociales  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, EHESS  École des hautes études en sciences sociales); Jérôme Pouyet (THEMA  Théorie économique, modélisation et applications  CNRS  Centre National de la Recherche Scientifique  CY  CY Cergy Paris Université, ESSEC Business School  Essec Business School); Thomas Trégouët (THEMA  Théorie économique, modélisation et applications  CNRS  Centre National de la Recherche Scientifique  CY  CY Cergy Paris Université) 
Abstract:  An incumbent seller contracts with a buyer and faces the threat of entry. The contract stipulates a price and a penalty for breach if the buyer later switches to the entrant. Sellers are heterogenous in terms of the gross surplus they provide to the buyer. The buyer is privately informed on her valuation for the incumbent's service. Asymmetric information makes the incumbent favor entry as it helps screening buyers. When the entrant has some bargaining power visàvis the buyer and keeps a share of the gains from entry, the incumbent instead wants to reduce entry. The compounding effect of these two forces may lead to either excessive entry or foreclosure, and possibly to a fixed rebate for exclusivity given to all buyers. 
Keywords:  excessive entry,foreclosure,exclusionary behavior,incomplete information 
Date:  2021–12 
URL:  http://d.repec.org/n?u=RePEc:hal:pseptp:hal03328387&r= 
By:  Mohamed Belhaj (AMSE  AixMarseille Sciences Economiques  EHESS  École des hautes études en sciences sociales  ECM  École Centrale de Marseille  CNRS  Centre National de la Recherche Scientifique  AMU  Aix Marseille Université); Frédéric Deroïan (AMSE  AixMarseille Sciences Economiques  EHESS  École des hautes études en sciences sociales  ECM  École Centrale de Marseille  CNRS  Centre National de la Recherche Scientifique  AMU  Aix Marseille Université) 
Abstract:  A monopoly sells a network good to a large population of consumers. We explore how the monopoly's profit and the consumer surplus vary with the arrival of public information about the network structure. The analysis reveals that, under homogeneous preferences for the good, degree assortativity ensures that information arrival increases both profit and consumer surplus. In contrast, heterogeneous preferences for the good can create a tension between consumer surplus and profit. 
Keywords:  monopoly,network effects,network information,Bonacich centrality,degree assortativity,assortative mixing 
Date:  2021–03 
URL:  http://d.repec.org/n?u=RePEc:hal:journl:hal03160602&r= 
By:  Kemper, Fynn; Wichardt, Philipp C. 
Abstract:  This paper proposes a simple framework to model contextual influences on procedural decision making. In terms of utility, we differentiate between monetary payoffs and contextual psychological ones, e.g. deriving from the subjects' normative frame of reference. Monetary payoffs are treated as common knowledge while psychological payoffs are treated as partly unforeseeable. Regarding behaviour, we assume that players act optimal given their local perception of the game. As perceptions may be incorrect, we do not consider common equilibrium conditions but instead require strategies to be procedurally justifiable. As we will argue, various common inconsistencies considered in behavioural economics can be understood as procedurally justifiable behaviour. With the present framework, we add an abstract tool to the discussion which allows to consider also the behavioural implications of players foreseeing the corresponding behavioural effects̶ which is often not considered in the respective original models. 
Keywords:  behavioural inconsistencies,context effects,limited foresight,procedural decision making,utility 
JEL:  C70 D01 D91 
Date:  2021 
URL:  http://d.repec.org/n?u=RePEc:zbw:ifwkwp:2202&r= 
By:  David Bounie (i3, CNRS, Télécom Paris, Institut Polytechnique de Paris); Antoine Dubus (Department of Management, Technology and Economics, ETH Zurich Leonhardstrasse 21 Switzerland â€“ 8092 Zurich, Switzerland); Patrick Waelbroeck (i3, CNRS, Télécom Paris, Institut Polytechnique de Paris) 
Abstract:  A monopolist data intermediary collects consumer information that it strategically sells to competing firms in a product market for price discrimination purposes. The intermediary charges a price of information and chooses the optimal partition that maximizes the willingness to pay of firms for information. Different selling mechanisms are compared: list prices, sequential bargaining, and auctions. The intermediary optimally sells information through auctions, whereas consumer surplus is maximized with sequential bargaining and list prices. We discuss the regulatory implications of our results. 
Keywords:  Selling mechanisms; Market for information; Data intermediaries; Competition policy; Regulation of digital markets 
Date:  2022–02 
URL:  http://d.repec.org/n?u=RePEc:eth:wpswif:22367&r= 