
on Game Theory 
By:  Edith Elkind; Abheek Ghosh; Paul W. Goldberg 
Abstract:  We study a general scenario of simultaneous contests that allocate prizes based on equal sharing: each contest awards its prize to all players who satisfy some contestspecific criterion, and the value of this prize to a winner decreases as the number of winners increases. The players produce outputs for a set of activities, and the winning criteria of the contests are based on these outputs. We consider two variations of the model: (i) players have costs for producing outputs; (ii) players do not have costs but have generalized budget constraints. We observe that these games are exact potential games and hence always have a purestrategy Nash equilibrium. The price of anarchy is $2$ for the budget model, but can be unbounded for the cost model. Our main results are for the computational complexity of these games. We prove that for general versions of the model exactly or approximately computing a best response is NPhard. For natural restricted versions where best response is easy to compute, we show that finding a purestrategy Nash equilibrium is PLScomplete, and finding a mixedstrategy Nash equilibrium is (PPAD$\cap$PLS)complete. On the other hand, an approximate purestrategy Nash equilibrium can be found in pseudopolynomial time. These games are a strict but natural subclass of explicit congestion games, but they still have the same equilibrium hardness results. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.08151&r= 
By:  Kun Zhang 
Abstract:  I study a class of verifiable disclosure games where the sender's payoff is state independent and the receiver's optimal action only depends on the posterior mean of the state. The sender's messages are verifiable in the sense that every message must contain the true state. I identify necessary and sufficient conditions for a particular information design outcome to be achieved by an equilibrium of the verifiable disclosure game, and give sufficient conditions on model primitives under which the sender does not benefit from commitment power. These results help in characterizing the sender's preferred equilibria and her equilibrium payoff set in a class of verifiable disclosure games. I apply these insights to study influencing voters and selling with quality disclosure. 
Date:  2022–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2206.09918&r= 
By:  Frédéric Koessler (PSE  Paris School of Economics  UP1  Université Paris 1 PanthéonSorbonne  ENSPSL  École normale supérieure  Paris  PSL  Université Paris sciences et lettres  EHESS  École des hautes études en sciences sociales  ENPC  École des Ponts ParisTech  CNRS  Centre National de la Recherche Scientifique  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, PJSE  Paris Jourdan Sciences Economiques  UP1  Université Paris 1 PanthéonSorbonne  ENSPSL  École normale supérieure  Paris  PSL  Université Paris sciences et lettres  EHESS  École des hautes études en sciences sociales  ENPC  École des Ponts ParisTech  CNRS  Centre National de la Recherche Scientifique  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement); Marie Laclau (HEC Paris  Ecole des Hautes Etudes Commerciales, GREGHEC  Groupement de Recherche et d'Etudes en Gestion  HEC Paris  Ecole des Hautes Etudes Commerciales  CNRS  Centre National de la Recherche Scientifique); Jérôme Renault (TSE  Toulouse School of Economics  UT1  Université Toulouse 1 Capitole  Université Fédérale Toulouse MidiPyrénées  EHESS  École des hautes études en sciences sociales  CNRS  Centre National de la Recherche Scientifique  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement); Tristan Tomala (HEC Paris  Ecole des Hautes Etudes Commerciales, GREGHEC  Groupement de Recherche et d'Etudes en Gestion  HEC Paris  Ecole des Hautes Etudes Commerciales  CNRS  Centre National de la Recherche Scientifique) 
Abstract:  This paper studies zerosum splitting games with finite sets of states. Players dynamically choose a pair of martingales {pt,qt}t, in order to control a terminal payoff u(p∞,q∞). A first part introduces the notion of "Mertens–Zamir transform" of a realvalued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square [0,1]2. A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian εoptimal strategies and we characterize it as the unique concaveconvex function satisfying two new conditions. 
Keywords:  Splitting games,MertensZamir system,Repeated games with incomplete information,Bayesian persuasion,Information design 
Date:  2022 
URL:  http://d.repec.org/n?u=RePEc:hal:journl:halshs03672222&r= 
By:  Lisa Bruttel (Universität Potsdam, Department for Economics and Social Sciences, Chair for Economics Markets, Competition, and Institutions  AugustBebelStr. 89  14482 Potsdam, Germany); Muhammed Bulutay (Technische Universität Berlin, Chair of Macroeconomics, H 52  Straße des 17. Juni 135  10 623 Berlin, Germany); Camille Cornand (Univ Lyon, CNRS, GATE UMR 5824, F69130 Ecully, France); Frank Heinemann (Technische Universität Berlin, Chair of Macroeconomics, H 52  Straße des 17. Juni 135  10 623 Berlin, Germany); Adam Zylbersztejn (Univ Lyon, Université Lumière Lyon 2, GATE UMR 5824, F69130 Ecully, France; research fellow at Vistula University Warsaw (AFiBV), Warsaw, Poland) 
Abstract:  Strategic uncertainty is the uncertainty that players face with respect to the purposeful behavior of other players in an interactive decision situation. Our paper develops a new method for measuring strategicuncertainty attitudes and distinguishing them from risk and ambiguity attitudes. We vary the source of uncertainty (whether strategic or not) across conditions in a ceteris paribus manner. We elicit certainty equivalents of participating in two strategic 2x2 games (a staghunt and a marketentry game) as well as certainty equivalents of related lotteries that yield the same possible payoffs with exogenously given probabilities (risk) and lotteries with unknown probabilities (ambiguity). We provide a structural model of uncertainty attitudes that allows us to measure a preference for or an aversion against the source of uncertainty, as well as optimism or pessimism regarding the desired outcome. We document systematic attitudes towards strategic uncertainty that vary across contexts. Under strategic complementarity [substitutability], the majority of participants tend to be pessimistic [optimistic] regarding the desired outcome. However, preferences for the source of uncertainty are distributed around zero. 
Keywords:  risk attitudes, ambiguity attitudes, strategicuncertainty attitudes, staghunt game, marketentry game 
JEL:  C72 C91 C92 D81 
Date:  2022 
URL:  http://d.repec.org/n?u=RePEc:gat:wpaper:2211&r= 
By:  Lukasz Balbus; Wojciech Olszewski; Kevin Reffett; Lukasz Wozny 
Abstract:  For an increasing upper order hemicontinuous correspondence F selfmapping sigmacomplete lattice A, we first provide tight fixedpoint bounds for sufficiently large iterations on F starting from any initial point a in A. We use this result for conducting iterative fixedpoint comparative statics, and then apply our results to monotone games and economies. For games of strategic complementarities, we improve the correspondence principle based results of Echenique (2002) by allowing for divergent learning processes, unstable fixed points, equilibrium indeterminacies, and unordered perturbations. We also apply our results to the comparative statics of stationary equilibria in large economies and the set of recursive equilibria in macroeconomic models with indeterminacies. 
Keywords:  comparative statics; comparative dynamics; adaptive learning; monotone iterations; games with strategic complementarities 
JEL:  C62 C65 C72 
Date:  2022–02 
URL:  http://d.repec.org/n?u=RePEc:sgh:kaewps:2022072&r= 
By:  Yunhan Huang; Tao Zhang; Quanyan Zhu 
Abstract:  This paper aims to formulate and study the inverse problem of noncooperative linear quadratic games: Given a profile of control strategies, find cost parameters for which this profile of control strategies is Nash. We formulate the problem as a leaderfollowers problem, where a leader aims to implant a desired profile of control strategies among selfish players. In this paper, we leverage frequencydomain techniques to develop a necessary and sufficient condition on the existence of cost parameters for a given profile of stabilizing control strategies to be Nash under a given linear system. The necessary and sufficient condition includes the circle criterion for each player and a rank condition related to the transfer function of each player. The condition provides an analytical method to check the existence of such cost parameters, while previous studies need to solve a convex feasibility problem numerically to answer the same question. We develop an identity in frequencydomain representation to characterize the cost parameters, which we refer to as the Kalman equation. The Kalman equation reduces redundancy in the timedomain analysis that involves solving a convex feasibility problem. Using the Kalman equation, we also show the leader can enforce the same Nash profile by applying penalties on the shared state instead of penalizing the player for other players' actions to avoid the impression of unfairness. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.05303&r= 
By:  Aditya Bhattacharjea (Department of Economics, Delhi School of Economics); Srishti Gupta (Department of Economics, Delhi School of Economics) 
Abstract:  We derive several variations of a model in which two upstream firms supply a differentiated product to two downstream firms under exclusive contracts of different kinds. We first derive a benchmark model with upstream firstmover pricing. We then compare its outcomes with four other types of vertical arrangements representing different modes of exploiting buyer power: downstream first mover pricing; Nash bargaining, alternatively with linear and twopart tariffs; and vertical integration. In each case, we show how the equilibrium values of wholesale and retail prices as well as downstream firms’ profits are affected by changes in the exogenous parameters (degree of product differentiation, bargaining power, and production costs). We evaluate the various vertical regimes from the perspective of downstream firms’ profits as well as consumer welfare, and show how more powerful downstream firms can benefit consumers by exercising “countervailing power” against upstream firms. Key Words: Buyer power, Bertrand duopoly, Vertical contracts, Nash bargaining, Vertical integration. JEL Codes: D43, L13, L22 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:cde:cdewps:326&r= 
By:  M. Lunkenheimer; A. Kracklauer; G. Klinkova; M. Grabinski 
Abstract:  In many models in economics or business a dominantly selfinterested homo economicus is assumed. Unfortunately (or fortunately), humans are in general not homines economici as e.g. the ultimatum game shows. This leads to the fact that all these models are at least doubtful. Moreover, economists started to set a quantitative value for the feeling of social justice, altruism, or envy and the like to execute utilitarian calculation. Besides being ethically doubtful, it delivers an explanation in hindsight with little predicting power. We use examples from game theory to show its arbitrariness. It is even possible that a stable Nash equilibrium can be calculated while it does not exist at all, due to the wide differences in human values. Finally, we show that assigned numbers for envy or altruism and the like do not build a field (in a mathematical sense). As there is no homomorphism to real numbers or a subset of it, any calculation is generally invalid or arbitrary. There is no (easy) way to fix the problem. One has to go back to ethical concepts like the categorical imperative or use at most semi quantitative approaches like considering knaves and knights. Mathematically one can only speculate whether e.g. surreal numbers can make ethics calculable. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.02902&r= 
By:  Jamie TuckerFoltz; Richard Zeckhauser 
Abstract:  We study the classic divideandchoose method for equitably allocating divisible goods between two players who are rational, selfinterested Bayesian agents. The players have additive private values for the goods. The prior distributions on those values are independent and common knowledge. We characterize the structure of optimal divisions in the divideandchoose game and show how to efficiently compute equilibria. We identify several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, the divider has a compelling "diversification" incentive in creating the chooser's two options. This incentive, hereto unnoticed, leads to multiple goods being divided at equilibrium, quite contrary to the divider's optimal strategy when preferences are known. In many contexts, such as buyandsell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players' uncertainties about each other's values, the number of goods being divided, and whether the divider can offer multiple alternative divisions. We prove that, when values are independently and identically distributed across players and goods, the chooser is strictly better off for a small number of goods, while the divider is strictly better off for a large number of goods. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.03076&r= 
By:  Zhaohua Chen; Chang Wang; Qian Wang; Yuqi Pan; Zhuming Shi; Chuyue Tang; Zheng Cai; Yukun Ren; Zhihua Zhu; Xiaotie Deng 
Abstract:  Throttling is one of the most popular budget control methods in today's online advertising markets. When a budgetconstrained advertiser employs throttling, she can choose whether or not to participate in an auction after the advertising platform recommends a bid. This paper focuses on the dynamic budget throttling process in repeated secondprice auctions from a theoretical view. An essential feature of the underlying problem is that the advertiser does not know the distribution of the highest competing bid upon entering the market. To model the difficulty of eliminating such uncertainty, we consider two different information structures. The advertiser could obtain the highest competing bid in each round with fullinformation feedback. Meanwhile, with partial information feedback, the advertiser could only have access to the highest competing bid in the auctions she participates in. We propose the OGDCB algorithm, which involves simultaneous distribution learning and revenue optimization. In both settings, we demonstrate that this algorithm guarantees an $O(\sqrt{T\log T})$ regret with probability $1  O(1/T)$ relative to the fluid adaptive throttling benchmark. By proving a lower bound of $\Omega(\sqrt{T})$ on the minimal regret for even the hindsight optimum, we establish the near optimality of our algorithm. Finally, we compare the fluid optimum of throttling to that of pacing, another widely adopted budget control method. The numerical relationship of these benchmarks sheds new light on the understanding of different online algorithms for revenue maximization under budget constraints. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.04690&r= 
By:  Eric Langlais; Andreea CosnitaLanglais 
Abstract:  The paper considers how product liability may shape firm size, product specification choices and market structure. We introduce a spatial Cournot duopoly on the linear market, where firms make an initial decision of product differentiation, then invest in precaution, before competing in quantity. Our main results are fourfold; 1) with full coverage of the market by the duopoly, there exist two equilibria (in pure strategies): central agglomeration (which is stable for low liability costs), and dispersion (which is stable for not too large liabiliy costs); 2) for larger liability costs, a mixed market structure duopoly/monopoly sustains a unique equilibrium with product differentiation; 3) this equilibrium enables a scope of differentiation higher (smaller) than the full duopoly (the social optimum); 4) the impact of liability costs on firms size and profits is complex, since it depends on the impact on both product differentiation and market structure. Finally, we show that consumer surplus and social welfare are both higher under the mixed market structure than under the full duopoly in an equilibrium with product differentiation. 
Keywords:  horizontal differentiation, Cournot competition, spatial model, endogenous market structures, product liability, strict liability, negligence 
JEL:  L41 K21 D82 
Date:  2022 
URL:  http://d.repec.org/n?u=RePEc:drm:wpaper:202218&r= 
By:  Menna Hassan; Nourhan Sakr; Arthur Charpentier 
Abstract:  This paper designs a sequential repeated game of a microfounded society with three types of agents: individuals, insurers, and a government. Nascent to economics literature, we use Reinforcement Learning (RL), closely related to multiarmed bandit problems, to learn the welfare impact of a set of proposed policy interventions per $1 spent on them. The paper rigorously discusses the desirability of the proposed interventions by comparing them against each other on a casebycase basis. The paper provides a framework for algorithmic policy evaluation using calibrated theoretical models which can assist in feasibility studies. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.01010&r= 
By:  Kazuya Kikuchi; Yukio Koriyama 
Abstract:  We consider collective decision making when the society consists of groups endowed with voting weights. Each group chooses an internal rule that specifies the allocation of its weight to the alternatives as a function of its members' preferences. Under fairly general conditions, we show that the winnertakeall rule is a dominant strategy, while the equilibrium is Pareto dominated, highlighting the dilemma structure between optimality for each group and for the whole society. We also develop a technique for asymptotic analysis and show Pareto dominance of the proportional rule. Our numerical computation for the US Electoral College verifies its sensibility. 
Date:  2022–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2206.09574&r= 
By:  Zeyi Chen 
Abstract:  The model of Network Coloring Game (NCG) is used to simulate conflict resolving and consensus reaching procedures in social science. In this work, we adopted some Markov Chain Techniques into the investigation of NCG. Firstly, with no less than $\Delta + 2$ colors provided, we proposed and proved that the conflict resolving time has its expectation to be $O(\log n)$ and the variance $O((\log n)^2)$, thus is $O_p(\log n)$, where $n$ is the number of vertices and $\Delta$ is the maximum degree of the network. This was done by introducing an absorbing Markov Chain into NCG. Secondly, we developed algorithms to reduce the network in postconflictresolution adjustments when a Borda rule is applied among players. Markov Chain Monte Carlo methods were employed to estimate both local and global optimal payoffs. Supporting experimental results were given to illustrate the corresponding procedures. 
Date:  2022–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2206.09153&r= 
By:  Lamprirni Zarpala; Dimitris Voliotis 
Abstract:  We introduce the \enquote{localglobal} approach for a divisible portfolio and perform an equilibrium analysis for two variants of coreselecting auctions. Our main novelty is extending the NearestVCG pricing rule in a dynamic tworound setup, mitigating bidders' freeriding incentives and further reducing the sellers' costs. The tworound setup admits an informationrevelation mechanism that may offset the \enquote{winner's curse}, and it is in accord with the existing iterative procedure of combinatorial auctions. With portfolio trading becoming an increasingly important part of investment strategies, our mechanism contributes to increasing interest in portfolio auction protocols. 
Date:  2022–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2206.11516&r= 
By:  Vahid Ashrafimoghari; Jordan W. Suchow 
Abstract:  In a digital age where companies face rapid changes in technology, consumer trends, and business environments, there is a critical need for continual revision of the business model in response to disruptive innovation. A pillar of innovation in business practices is the adoption of novel pricing schemes, such as PayWhatYouWant (PWYW). In this paper, we employed game theory and behavioral economics to model consumers' behavior in response to a PWYW pricing strategy where there is an information asymmetry between the consumer and supplier. In an effort to minimize the information asymmetry, we incorporated the supplier's cost and the consumer's reference prices as two parameters that might influence the consumer's payment decision. Our model shows that consumers' behavior varies depending on the available information. As a result, when an external reference point is provided, the consumer tends to pay higher amounts to follow the social herd or respect her selfimage. However, the external reference price can also decrease her demand when, in the interest of fairness, she forgoes the purchase because the amount she is willing to pay is less that what she recognizes to be an unrecoverable cost to the supplier. 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2207.08923&r= 
By:  Philippe Jehiel (PSE  Paris School of Economics  UP1  Université Paris 1 PanthéonSorbonne  ENSPSL  École normale supérieure  Paris  PSL  Université Paris sciences et lettres  EHESS  École des hautes études en sciences sociales  ENPC  École des Ponts ParisTech  CNRS  Centre National de la Recherche Scientifique  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, PJSE  Paris Jourdan Sciences Economiques  UP1  Université Paris 1 PanthéonSorbonne  ENSPSL  École normale supérieure  Paris  PSL  Université Paris sciences et lettres  EHESS  École des hautes études en sciences sociales  ENPC  École des Ponts ParisTech  CNRS  Centre National de la Recherche Scientifique  INRAE  Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, UCL  University College of London [London]) 
Abstract:  A unified definition of analogybased expectation equilibrium (ABEE) for strategic environments involving multiple stages and private information is presented. Various alternative interpretations of the concept are proposed as well as a discussion of how to use ABEE in practice. A variety of applications including two new ones related to speculative trading and personnel economics is reviewed. A discussion of a number of alternative equilibrium concepts follows emphasizing the links and differences with ABEE. Finally, a discussion of possible next steps in particular related to the endogeneization of analogy partitions is proposed. 
Keywords:  AnalogyBased Expectation,bounded rationality,limited learning 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:hal:psewpa:halshs03735680&r= 
By:  James C. Cox; Vjollca Sadiraj; Susan Xu Tang 
Abstract:  Consequentialist rational choice theory, including models of (unconditional) social preferences, is challenged by decades of robust data from payoffequivalent public good games with provision or appropriation as well as by robust data showing contributions to public goods, funded by lumpsum taxation, do not crowd out voluntary contributions on a oneforone basis. This paper offers an extension of rational choice theory that incorporates observable moral reference points. This morally monotonic choice theory is consistent with robust data in the literature and has idiosyncratic features that motivate new experimental designs that introduce nonbinding quotas on appropriations or floors on provisions. Data, from three previous experiments and our new experiment, favor moral monotonicity over alternative theoretical models including rational choice theory, prominent beliefbased models of kindness, and popular referencedependent models with loss aversion. 
Keywords:  choice theory, public goods, experiment, payoff equivalence, nonbinding contractions, moral reference points, beliefbased psychological models, referencedependent choices 
JEL:  C91 D03 H41 
Date:  2022–07 
URL:  http://d.repec.org/n?u=RePEc:exc:wpaper:202201&r= 
By:  Evans, Alecia; Sesmero, Juan 
Keywords:  Institutional and Behavioral Economics 
Date:  2022–08 
URL:  http://d.repec.org/n?u=RePEc:ags:aaea21:322804&r= 