nep-mic New Economics Papers
on Microeconomics
Issue of 2017‒04‒30
24 papers chosen by
Jing-Yuan Chiou
National Taipei University

  1. Trading under Asymmetric Information: Positive and Normative Implications By ATTAR Andrea; d’ASPREMONT Claude
  2. Long-Term Employment Relations when Agents Are Present Biased By Fahn, Matthias; Schwarz, Marco A.
  3. Auctions versus Negotiations By Herweg, Fabian; Schmidt, Klaus M.
  4. Certification and Market Transparency By Strausz, Roland
  5. Hierarchical competition and heterogeneous behavior in noncooperative oligopoly markets By Ludovic Alexandre Julien
  6. Implementation in undominated strategies with partially honest agents By MUKHERJEE Saptarshi; MUTO Nozomu; RAMAEKERS Eve
  7. Interim Bayesian Persuasion: First Steps By Eduardo Perez
  8. Sender-Receiver Games with Cooperation By Forges, Françoise; Horst, Ulrich
  9. Life Insurance and Life Settlement Markets with Overconfident Policyholders By Hanming Fang; Zenan Wu
  10. Content acquisition by streaming platforms: premium vs. freemium By CARRONI Elias; PAOLINI Dimitri
  11. Formation of coalition structures as a non-cooperative game By Dmitry Levando
  12. A Theory of Crowdfunding By Strausz, Roland
  13. Opinion Dynamics via Search Engines (and other Algorithmic Gatekeepers) By Fabrizio Germano; Francesco Sobbrio
  14. Privacy and Quality By Lefouili, Yassine; Toh, Ying Lei
  15. Optimal R&D investment with learning-by-doing: Multiple steady-states and thresholds By Anton Bondarev; Alfred Greiner
  16. A Dynamic Model of Electoral Competition with Costly Policy Changes By Hans Gersbach; Philippe Muller; Oriol Tejada
  17. Tariffs, Licensing Contracts, and Consumers' Welfare By Tarun Kabiraj
  18. The identification of attitudes towards ambiguity and risk from asset demand By Polemarchakis, Herakles; Selden, Larry; Song, Xinxi
  19. How to license a downstream technology when upstream firms are capacity constrained? By SCHOLZ Eva-Marie
  20. Sequential versus Static Screening: an Equivalence Result By Krähmer, Daniel; Strausz, Roland
  21. Voting and Contributing While the Group is Watching By Emeric Henry; Charles Louis-Sidois
  22. Assisted Self-Persuasion: Advertising with Consumer Adjustment to Choice By Matthew G. Nagler
  23. A Model of Directed Consumer Search By Haan, Marco A.; Moraga-González, José-Luis; Petrikaite, Vaiva
  24. Best reply structure and equilibrium convergence in generic games By Marco Pangallo; Torsten Heinrich; J Doyne Farmer

  1. By: ATTAR Andrea (Università di Roma Tor Vergata, and Toulouse School of Economics); d’ASPREMONT Claude (Université catholique de Louvain, CORE, Belgium)
    Abstract: We study trading situations in which several principals on one side of the market compete to serve privately informed agents on the other side. In such ‘generalized screening’ settings, competitors may post mechanisms instead of prices, and the enforceability and the efficiency of the contractual relationships become difficult to evaluate. We revisit these issues, focusing on three applications: bilateral (or multilateral) trade, where all traders have private information, auctions and insurance, where incomplete information is one-sided. In the first part, as a benchmark, we focus on the standard mechanism design approach with only one principal, the “mechanism designer", and we rely on the revelation principle as a device to characterize equilibrium out- comes. Even then, first-best optimality, combined with Bayesian incentive compatibility and interim individual rationality might be difficult to obtain, as illustrated by Myerson and Satterthwaite (1983) impossibility result, for- mulated for risk-neutral traders with independent beliefs. In auctions, if the buyers types are correlated à la Crémer and McLean (1985,1988), this impos- sibility can be bypassed and the seller can extract the whole surplus. In the more general multilateral trade setting, a simple modification of a condition provided by d’Aspremont and Gérard-Varet (1982) allows to implement any distribution of the surplus (Kosenok and Severinov, 2008). However, under risk-aversion, only second-best outcomes can be implemented, as originally shown by Stiglitz (1977) for the monopolistic case, and by Crocker and Snow (1985) for the competitive one. In the second part, we consider a class of extensive form games in which several principals (with no private information) compete over mechanisms in the presence of privately informed agents. Applying the standard reve- lation principle becomes problematic, as first pointed out by Peck (1997): there exist equilibrium outcomes that can be supported by general commu- nication mechanisms, but not by simple direct ones. We revisit a relevant implication of this impossibility, i.e. the recent folk-theorem-like result of Yamashita (2010): if there are at least three agents, a large set of incentive compatible allocations can be supported at equilibrium. For the result to hold, principals have to rely on message spaces that are larger than the cor- responding agentsÕ type spaces. In the single agent (or common agency) case, the equilibrium analysis can be further simplified using the delegation principle (Peters, 2001, Martimort and Stole, 2002). In this context, we stress the key role played by the possibility to enforce exclusivity clauses. In standard exclusive competition settings (as Rothchild and Stiglitz, 1976), if a pure strategy equilibrium exists, it is second-best efficient (Crocker and Snow, 1985). This is no longer true under nonexclusive competition. In this case, the possibility to complement his rivals’ offers, creates new strategic opportunities for sellers, and crucially modifies equilibrium outcomes: Attar et al. (2014) establish that, in any pure strategy equilibrium, at most one type of agent is actively trading. The impossibility to enforce exclusive trad- ing may further restrict the set of incentive feasible allocations. The recent work of Attar et al. (2016b) characterizes the constraints faced by a planner who does not have access to agents’ private information, and cannot prevent agents’ from engaging in further trades with sellers. They show that this side-trading opportunity dramatically restricts the set of allocations that are available to a planner. As a matter of fact there is only one incentive com- patible allocation that is robust to the possibility of sellers’ side trades. This prevents any redistribution between different types of (privately informed) buyers.
    Keywords: Mechanism design, Bilateral Trade, Competing Mechanism, Constrained Efficiency, Adverse Selection
    JEL: D43 D82 D86
    Date: 2017–03–10
  2. By: Fahn, Matthias (LMU Munich); Schwarz, Marco A. (LMU Munich)
    Abstract: We analyze how agents\' present bias affects optimal contracting in an infinite-horizon employment setting. The principal maximizes profits by offering a menu of contracts to naive agents: a virtual contract - which agents plan to choose in the future - and a real contract which they end up choosing. This virtual contract motivates the agent and allows the principal to keep the agent below his outside option. Moreover, under limited liability, implemented effort can be inefficiently high. With a finite time horizon, the degree of exploitation of agents decreases over the life-cycle. While the baseline model abstracts from moral hazard, we show that the result persists also when allowing for non-contractible effort.
    Keywords: Employment relations; dynamic contracting; present bias;
    JEL: D03 D21 J31 M52
    Date: 2017–03–25
  3. By: Herweg, Fabian (University of Bayreuth); Schmidt, Klaus M. (University of Munich)
    Abstract: For the procurement of complex goods the early exchange of information is important to avoid costly renegotiation. If the buyer can specify the main characteristics of possible design improvements in a complete contingent contract, a scoring auction implements the efficient allocation. If this is not feasible, the buyer must choose between a price-only auction (discouraging early information exchange) and bilateral negotiations with a preselected seller (reducing competition). Bilateral negotiations are superior if potential design improvements are important, if renegotiation is particularly costly, and if the buyer\'s bargaining position is strong. Moreover, negotiations provide stronger incentives for sellers to investigate design improvements.
    Keywords: Adaptation costs; auctions; behavioral contract theory; loss aversion; negotiations; procurement; renegotiation; ;
    JEL: D03 D82 D83 H57
    Date: 2017–03–25
  4. By: Strausz, Roland (Humboldt University Berlin)
    Abstract: In markets with quality unobservable to buyers, third-party certification is often the only instrument to increase transparency. While both sellers and buyers have a demand for certification, its role differs fundamentally: sellers use it for signaling, buyers use it for inspection. Seller induced certification leads to more transparency, because it is informative - even if unused. By contrast, buyer induced certification incentivizes certifiers to limit transparency, as this raises demand for inspection. Whenever transparency is socially beneficial, seller certification is preferable. It also yields certifiers larger profits, so that regulating the mode of certification is redundant.
    Keywords: Market transparency; certification; information and product quality; asymmetric information;
    JEL: D82 G24 L15
    Date: 2017–03–25
  5. By: Ludovic Alexandre Julien
    Abstract: In this paper, we consider a sequential bilateral oligopoly market which embodies a finite number of leaders and followers who compete on quantities. We define a noncooperative equilibrium concept for this two-stage market game with complete and perfect information, namely the Stackelberg-Nash equilibrium (SNE). Then, we study the existence of a SNE with trade. The existence proof requires some steps as this market game displays a rich set of strategic interactions. In particular, to show the existence of a pure strategy subgame perfect Nash equilibrium, we have to determine the conditions under which there exist well defined continuously differentiable best responses. Some examples buttress the approach and discuss the assumptions made on the primitives.
    Keywords: Best responses, Stackelberg-Nash equilibrium, trade, autarky.
    JEL: C72 D51
    Date: 2017
  6. By: MUKHERJEE Saptarshi (Indian Institute of Technology, Delhi, India); MUTO Nozomu (Yokohama National University, Japan); RAMAEKERS Eve (Université catholique de Louvain, CORE, Belgium)
    Abstract: We consider implementation in undominated strategies by bounded mechanisms. We provide a complete characterization of the class of social choice correspondences that are implementable when agents are partially honest, in the sense that they have strict preferences for being sincere when truthfulness does no result in a worse outcome. As an application, we show that the Pareto correspondence is implemented by a finite mechanism.
    Keywords: Implementation in undominated strategies, Partial honesty, bounded mechanism, Pareto correspondence
    JEL: D71 D01
    Date: 2017–03–16
  7. By: Eduardo Perez (Département d'économie)
    Abstract: This paper makes a first attempt at building a theory of interim Bayesian persuasion. I work in a minimalist model where a low or high type sender seeks validation from a receiver who is willing to validate high types exclusively. After learning her type, the sender chooses a complete conditional information structure for the receiver from a possibly restricted feasible set. I suggest a solution to this game that takes into account the signaling potential of the sender's choice.
    JEL: D82 D83
    Date: 2015–05
  8. By: Forges, Françoise (University of Paris-Dauphine); Horst, Ulrich (Humboldt University Berlin)
    Abstract: We consider generalized sender-receiver games in which the sender also has a decision to make, but this decision does not directly affect the receiver. We introduce specific perfect Bayesian equilibria, in which the players agree on a joint decision after that a message has been sent (\"talk and cooperate equilibrium\", TCE). We establish that a TCE exists provided that the receiver has a \"uniform punishment decision\" (UPD) against the sender.
    Keywords: sender-receiver game; commitment; cooperative solution; individual rationality;
    JEL: C72 C65
    Date: 2017–03–25
  9. By: Hanming Fang; Zenan Wu
    Abstract: We analyze how the life settlement market – the secondary market for life insurance – may affect consumer welfare in a dynamic equilibrium model of life insurance with one-sided commitment and overconfident policyholders. As in Daily et al. (2008) and Fang and Kung (2010), policyholders may lapse their life insurance policies when they lose their bequest motives; but in our model the policyholders may underestimate their probability of losing their bequest motive, or be overconfident about their future mortality risks. For the case of overconfidence with respect to bequest motives, we show that in the absence of life settlement overconfident consumers may buy “too much” reclassification risk insurance for later periods in the competitive equilibrium. In contrast, when consumers are overconfident about their future mortality rates in the sense that they put too high a subjective probability on the low-mortality state, the competitive equilibrium contract in the absence of life settlement exploits the consumer bias by offering them very high face amounts only in the low-mortality state. In both cases, life settlement market can impose a discipline on the extent to which overconfident consumers can be exploited by the primary insurers. We show that life settlement may increase the equilibrium consumer welfare of overconfident consumers when they are sufficiently vulnerable in the sense that they have a sufficiently large intertemporal elasticity of substitution of consumption.
    JEL: D03 D86 G22 L11
    Date: 2017–03
  10. By: CARRONI Elias (Universita di Bologna); PAOLINI Dimitri (Universita di Sassari, CRENoS and CORE, UCL)
    Abstract: We analyze the optimal decision of a monopolistic streaming platform. The platform obtains contents from copyright owners (artists) who are paid with a per-user royalty. Advertisers pay a per-user fee to display their commercials. Users value the variety of contents and are heterogeneously bothered by ads. We show that when commercials generate an intermediate nuisance and the size of the potential market is large, the platform finds it optimal to offer only a paying subscription without displaying any ads. In contrast, a small potential market results in the offer of a menu of subscriptions, with ad-intolerant users paying a positive price and moderately-averse users opting for a free-of-charge solution. The second (first) solution is always preferred when commercials generate a strong (weak) nuisance. We also show that there may emerge a misalignment of the platform’s and artists’ interests.
    Date: 2017–03–01
  11. By: Dmitry Levando (National Research University Higher School of Economics)
    Abstract: Traditionally social sciences are interested in structuring people in multiple groups based on their individual preferences. This paper suggests an approach to this problem in the framework of a non-cooperative game theory. Definition of a suggested game includes a family of nested simultaneous non-cooperative finite games with intra- and inter-coalition externalities. In this family, games differ by the size of maximum coalition, partitions and by coalition structure formation rules. A result of every game consists of partition of players into coalitions and a payoff profile for every player. Every game in the family has an equilibrium in mixed strategies with possibly more than one coalition. The results of the game differ from those conventionally discussed in cooperative game theory, e.g. the Shapley value, strong Nash, coalition-proof equilibrium, core, kernel, nucleolus. We discuss the following applications of the new game: cooperation as an allocation in one coalition, Bayesian game, stochastic games and construction of a non-cooperative criterion of coalition structure stability for studying focal points
    Keywords: Non-cooperative Games; Nash equilibrium; Shapley value; strong equilibrium; core
    JEL: C71 C72 C73
    Date: 2017–02
  12. By: Strausz, Roland (Humboldt University Berlin)
    Abstract: Crowdfunding provides innovation in enabling entrepreneurs to contract with consumers before investment. Under aggregate demand uncertainty, this improves screening for valuable projects. Entrepreneurial moral hazard and private cost information threatens this benefit. Crowdfunding\'s after-markets enable consumers to actively implement deferred payments and thereby manage moral hazard. Popular crowdfunding platforms offer schemes that allow consumers to do so through conditional pledging behavior. Efficiency is sustainable only if expected returns exceed an agency cost associated with the entrepreneurial incentive problems. By reducing demand uncertainty, crowdfunding promotes welfare and complements traditional entrepreneurial financing, which focuses on controlling moral hazard.
    Keywords: Crowdfunding; entrepreneurship; moral hazard; demand uncertainty;
    JEL: D82 G32 L11 M31
    Date: 2017–03–25
  13. By: Fabrizio Germano; Francesco Sobbrio
    Abstract: Ranking algorithms are the information gatekeepers of the Internet era. We develop a stylized framework to study the effects of ranking algorithms on opinion dynamics. We consider rankings that depend on popularity and on personalization. We find that popularity driven rankings can enhance asymptotic learning while personalized ones can both inhibit or enhance it, depending on whether individuals have common or private value preferences. We also find that ranking algorithms can contribute towards the diffusion of misinformation (e.g., “fake news”), since lower ex-ante accuracy of content of minority websites can actually increase their overall traffic share.
    Keywords: search engines, ranking algorithm, search behavior, opinion dynamics, information aggregation, asymptotic learning, misinformation, polarization, website traffic, fake news
    JEL: D83 L86
    Date: 2017–04
  14. By: Lefouili, Yassine; Toh, Ying Lei
    Abstract: This paper analyzes the effects of a privacy regulation that caps the level of data disclosure on investment in quality and social welfare. We develop a model in which a monopolist offers a service for free to consumers with heterogeneous privacy preferences, and derives revenues from disclosing consumer data to third parties. We assume that the users of the service choose how much information they provide to the firm. In this setting, regulating the disclosure level can alter both the extensive margin effect and the intensive margin effect of an investment in quality. In the case where the market is fully covered, a welfare-maximizing regulator who can commit ex ante to a cap on disclosure level finds it optimal to do so when the complementarity between quality and information is not too strong. In the case where the market is partially covered, such an an ex ante disclosure cap may be socially desirable even when quality and information are strongly complementary. Finally, we extend our analysis to the case where the regulator maximizes consumer surplus, and to a scenario where the regulator sets a disclosure cap ex post.
    Keywords: Privacy Regulation, Data Disclosure, Quality.
    JEL: L15 L4 M21
    Date: 2017–04
  15. By: Anton Bondarev; Alfred Greiner (University of Basel)
    Abstract: In this paper we present an inter-temporal optimization problem of a representative R&D firm that simultaneously invests in horizontal and vertical innovations. We posit that learning-by-doing makes the process of quality improvements a positive function of the number of existing technologies with the function displaying a convex-concave form. We show that multiple steady-states can arise with two being saddle point stable and one unstable with complex conjugate eigenvalues. Thus, a threshold with respect to the variety of technologies exists that separates the two basins of attractions. From an economic point of view, this implies that a lock-in effect can occur such that it is optimal for the firm to produce only few technologies at a low quality when the initial number of technologies falls short of the threshold. Hence, history matters as concerns the state of development implying that past investments and innovations determine whether the firm produces a large or a small variety of high- or low-quality technologies, respectively.
    Keywords: Optimal control, horizontal and vertical innovations, multiple steadystates, thresholds, lock-in
    JEL: C61 D92 O32
    Date: 2017
  16. By: Hans Gersbach (ETH Zurich, Switzerland); Philippe Muller (ETH Zurich, Switzerland); Oriol Tejada (ETH Zurich, Switzerland)
    Abstract: We consider an infinite-horizon model of elections where policy changes are costly for citizens and parties. The so-called costs of change increase with the extent of the policy shift and make policy history-dependent. First, we provide a detailed description of the equilibrium dynamics and analyze how policies are influenced by history, costs of change, party polarization, and the incumbent's ability. We show that policies converge to a stochastic alternation between two states and that in the long run costs of change have a moderating effect on policies. Second, we analyze welfare as a function of the marginal cost of change. If the initial level of policy polarization is low, welfare is highest for intermediate marginal costs of change. Moreover, any positive level of costs of change will benefit society if the future is sufficiently valuable. If the initial level of policy polarization is high, however, welfare will be highest for low or zero costs of change.
    Keywords: democracy; dynamic elections; political polarization; costs of change; Markov perfect equilibrium
    JEL: C72 C73 D72 D78
    Date: 2017–04
  17. By: Tarun Kabiraj
    Abstract: In a duopolistic trade model we have shown that a tariff can influence the optimal licensing strategy of the foreign firm. A high tariff will induce fee licensing and a low tariff will result in a royalty licensing. From the viewpoint of the consumers both high tariff and high royalty are distortaionary; hence there is a trade-off between a tariff and a royalty. Then the local government can suitably choose a tariff rate that will induce fee licensing, then consumers' welfare is maximized. In the paper we have used the tools and techniques of game theory and industrial organization literature to the issue of technology licensing and consumers' welfare. We have shown that a tariff on foreign products can be strategically chosen so that the foreign firm transfers its superior technology to a domestic firm under a fee licensing contract and consumers' welfare is maximized.
    Keywords: It is a theoretical work, so applicable to any country, Trade issues, Game theoretical models
    Date: 2016–07–04
  18. By: Polemarchakis, Herakles (Department of Economics, University of Warwick and LabEx MME-DII, University of Paris 2); Selden, Larry (Columbia Business School, Columbia University and University of Pennsylvania); Song, Xinxi (International School of Economics and Management, Capital University of Economics and Business)
    Abstract: Individuals behave differently when they know the objective probability of events and when they do not. The smooth ambiguity model accommodates both ambiguity (uncertainty) and risk. For an incomplete, competitive asset market, we develop a revealed preference test for asset demand to be consistent with the maximization of smooth ambiguity preferences ; and we show that ambiguity preferences constructed from finite observations converge to underlying ambiguity preferences as observations become dense. Subsequently, we give sufficient conditions for the asset demand generated by smooth ambiguity preferences to identify the ambiguity and risk indices as well as the ambiguity probability measure. We do not require ambiguity beliefs to be observable : in a generalized specification, they may not even be defined. An ambiguity free asset plays an important role for identification.
    Keywords: risk ; uncertainty ; identification JEL classification numbers: D11 ; D80 ; D81
    Date: 2017
  19. By: SCHOLZ Eva-Marie (Université catholique de Louvain, CORE, Belgium)
    Abstract: In this paper, we study the relationship between capacity constraints and licensing strategies. To doso, we focus on the licensing strategy of an outside innovator who licenses a process innovation to the downstream sector of a vertical Cournot oligopoly. Downstream firms soruce an essential production factor from a capacity constrained upstream sector. In this setting, we show that the innovator optimally licenses large innovations via per-unit royalty contracts and small innovations via fixed fee contracts. Moreover, an increase in the strength of the capacity constraints makes it more likely that hte optimal licensing contract includes a strictly positive per-unit royalty rate. As a final point, we discuss the relationship between capacity constraints and the social optimality of the innovator’s licensing strategy as measured by aggregate welfaore or the diffusion of the innovation on the downstream market
    Keywords: capacity constraints; licensing contracts, vertical Cournot oligopoly
    JEL: D43 L13
    Date: 2017–02–27
  20. By: Krähmer, Daniel (Bonn University); Strausz, Roland (Humboldt University Berlin)
    Abstract: We show that every sequential screening model is equivalent to a standard text book static screening model. We use this result and apply well-established techniques from static screening to obtain solutions for classes of sequential screening models for which standard sequential screening techniques are not applicable. Moreover, we identify the counterparts of well-understood features of the static screening model in the corresponding sequential screening model such as the single-crossing condition and conditions that imply the optimality of deterministic schedules.
    Keywords: Sequential screening; static screening; stochastic mechanisms;
    JEL: D82 H57
    Date: 2017–03–23
  21. By: Emeric Henry (Département d'économie); Charles Louis-Sidois (Département d'économie)
    Abstract: Members of groups and organizations often have to decide on rules that regulate their contributions to common tasks. They typically differ in their propensity to contribute and often care about the image they project, in particular want to be perceived by other group members as being high contributors. In such environments we study the interaction between the way members vote on rules and their subsequent contribution decisions. We show that multiple norms can emerge. We draw surprising policy implications, on the effect of group size, of supermajority rules and of the observability of actions.
    Date: 2015–10
  22. By: Matthew G. Nagler (Ph.D. Program in Economics, Graduate Center, and Department of Economics and Business, City College of New York)
    Abstract: I develop a new theory of persuasive advertising in which consumers rationally adjust to (i.e., improve their attitude toward) the products they choose and advertising facilitates adjustment. Advertisings price effects depend on whether marginal or inframarginal consumers are most heavily targeted, consistent with the literature. But they also depend on advertisings role as an overall adjustment intensifier, whence variation in the cost of adjustment with the strength of the consumers initial product preference determines the equilibrium price level. Whether too much or too little advertising is provided in equilibrium depends on the sign and size of advertisings price effect, the relative density of marginal consumers, and the relative extent to which advertisings adjustment cost reductions benefit marginal consumers.
    Keywords: Persuasive advertising, Hotelling model, consumer decision-making, pricing, welfare
    JEL: D03 D11 L10 M37
    Date: 2017–03–09
  23. By: Haan, Marco A.; Moraga-González, José-Luis; Petrikaite, Vaiva
    Abstract: We present a framework to study directed consumer search. Firms sell products with two attributes. One is readily observable, the other only after visiting a firm. Search is directed as the order of search is influenced by the observable characteristics. Moreover, if prices are readily observable, firms also influence search direction by their choice of price. We show that when consumers observe prices before search, prices and profits are lower than when they do not. A lower price then not only retains more consumers, but is also more likely to attract them; the latter effect makes demand more elastic. When prices are observable before search, prices decrease in search costs. Consumer surplus initially increases in search costs, but may ultimately decrease.
    Keywords: differentiated products; ordered/directed search; price observability
    JEL: D83 L13
    Date: 2017–04
  24. By: Marco Pangallo; Torsten Heinrich; J Doyne Farmer
    Abstract: Game theory often assumes rational players that play equilibrium strategies. But when the players have to learn their strategies by playing the game repeatedly, how often do the strategies converge? We analyze generic two player games using a standard learning algorithm, and also study replicator dynamics, which is closely related. We show that the frequency with which strategies converge to a fixed point can be understood by analyzing the best reply structure of the payoff matrix. A Boolean transformation of the payoff matrix, replacing all best replies by one and all other entries by zero, provides a reasonable approximation of the asymptotic strategic dynamics. We analyze the generic structure of randomly generated payoff matrices using combinatorial methods to compute the frequency of cycles of different lengths under the microcanonical ensemble. For a game with $N$ possible moves the frequency of cycles and non-convergence increases with $N$, becoming dominant when $N > 10$. This is especially the case when the interactions are competitive.
    Date: 2017–04

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