nep-gth New Economics Papers
on Game Theory
Issue of 2012‒03‒08
27 papers chosen by
Laszlo A. Koczy
Hungarian Academy of Sciences and Obuda University

  1. On the Existence of Markov Perfect Equilibria in Perfect Information Games By Hannu Salonen; Hannu Vartiainen
  2. Characterizing Belief-Free Review-Strategy Equilibrium Payoffs under ConditionalIndependence By Yuichi Yamamoto
  3. On the Likelihood of Dummy players in Weighted Majority Games By Fabrice Barthelemy; Dominique Lepelley; Mathieu Martin
  4. The Bounded Core for Games with Precedence Constraints By Michel Grabisch; Peter Sudhölter
  5. Equilibrium Uniqueness in Network Games with Strategic Substitutes By Yann Rébillé; Lionel Richefort
  6. Replicator Dynamics and Evolutionary Stable Strategies in Heterogeneous Games By Zuazo Garín, Peio; Rocha, André Barreira da Silva; Laruelle, Annick
  7. Double route to chaos in an heterogeneous triopoly game By Ahmad Naimzada; Fabio Tramontana
  8. The Asymmetric Leximin Solution By Driesen, Bram W.
  9. Endougenous Timing in a Mixed Duopoly By Rabah Amir; Giuseppe De Feo
  10. Robust mechanism design and dominant strategy voting rules By Borgers, Tilman; Smith, Doug
  11. Implementing quotas in university admissions: An experimental analysis By Sebastian Braun; Nadja Dwenger; Dorothea Kübler; Alexander Westkamp
  12. Stability of Coalitional Equilibria within Repeated Tax Competition By Sonja Brangewitz; Sarah Brockhoff
  13. Natural implementation with partially hones agents By Lombardi Michele; Yoshihara Naoki
  14. On the Private Provision of Public Goods on Networks By Nizar Allouch
  15. A noncooperative model of network formation with decreasing productivity By Charoensook, Banchongsan
  16. Public Good Provision with Convex Costs By Senatore, L
  17. Multidimensional Political Competition with Non-Common Beliefs By Kazuya Kikuchi
  18. Pairwise interactive knowledge and Nash equilibrium By Bach Christian W.; Tsakas Elias
  19. Global green economy and environmental sustainability: a coopetitive model By Carfì, David; Schilirò, Daniele
  20. Bilateral Trading in Networks By Daniele Condorelli; Andrea Galeotti
  21. Cognitive hierarchies in adaptive play By Khan Abhimanyu; Peeters Ronald
  22. From the lab to the field: envelopes, dictators and manners By Stoop, Jan
  23. Information acquisition during a Dutch auction By Miettinen, Paavo
  24. Endogenous Trading Networks By Daniele Condorelli; Andrea Galeotti
  25. Non-anonymous ballot aggregation: an axiomatic generalization of Approval Voting. By Jorge Alcalde-Unzu; Marc Vorsat
  26. Fixed Water Sharing Agreements Sustainable to Drought By Ambec, Stefan; Dinar, Ariel; McKinney, Daene
  27. Fair Apportionment in the Italian Senate : Which Reform Should Be Implemented? By Fabrice Barthelemy; Gabriele Esposito; Mathieu Martin; Vincent Merlin

  1. By: Hannu Salonen (Department of Economics and PCRC, University of Turku, 20014 Turku, Finland); Hannu Vartiainen (HECER, P.O. Box 17 (Arkadiankatu 7), FI-00014 University of Helsinki)
    Abstract: We study the existence of pure strategy Markov perfect equilibria in two-person perfect information games. There is a state space X and each period player's possible actions are a subset of X. This set of feasible actions depends on the current state, which is determined by the choice of the other player in the previous period. We assume that X is a compact Hausdorff space and that the action correspondence has an acyclic and asymmetric graph. For some states there may be no feasible actions and then the game ends. Payoffs are either discounted sums of utilities of the states visited, or the utility of the state where the game ends. We give sufficient conditions for the existence of equilibrium e.g. in case when either feasible action sets are finite or when players' payoffs are continuously dependent on each other. The latter class of games includes zero-sum games and pure coordination games.
    Keywords: dynamic games, Markov perfect equilibrium
    JEL: C72 C73
    Date: 2011–10
  2. By: Yuichi Yamamoto (Department of Economics, University of Pennsylvania)
    Abstract: This paper proposes and studies a tractable subset of Nash equilibria, belief-free review-strategy equilibria, in repeated games with private monitoring. The payoff set of this class of equilibria is characterized in the limit as the discount factor converges to one for games where players observe statistically independent signals. As an application, we develop a simple sufficient condition for the existence of asymptotically efficient equilibria, and establish a folk theorem for N-player prisoner’s dilemma. All these results are robust to a perturbation of the signal distribution, and hence remain true even under almost-independent monitoring.
    Keywords: repeated game, private monitoring, conditional independence, belief-free review-strategy equilibrium, prisoner’s dilemma
    JEL: C72 C73 D82
    Date: 2012–02–22
  3. By: Fabrice Barthelemy; Dominique Lepelley; Mathieu Martin (THEMA, Universite de Cergy-Pontoise; CEMOI,universite de la réunion; THEMA, Universite de Cergy-Pontoise)
    Abstract: When the number of players is small in a weighted majority voting game, it can occur that one of the players has no influence on the result of the vote, in spite of a strictly positive weight. Such a player is called a “dummy” player in game theory. The purpose of this paper is to investigate the conditions that give rise to such a phenomenon and to compute its likelihood. It is shown that the probability of having a dummy player is surprisingly high and some paradoxical results are observed.
    Keywords: Cooperative game theory, weighted voting games, dummy player, likelihood of voting paradoxes.
    JEL: C7 D7
    Date: 2011
  4. By: Michel Grabisch (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon Sorbonne); Peter Sudhölter (University of Southern Denmark - Department of Business and Economics and COHERE)
    Abstract: An element of the possibly unbounded core of a cooperative game with precedence constraints belongs to its bounded core if any transfer to a player from any of her subordinates results in payoffs outside the core. The bounded core is the union of all bounded faces of the core, it is nonempty if the core is nonempty, and it is a continuous correspondence on games with coinciding precedence constraints. If the precedence constraints generate a connected hierarchy, then the core is always nonempty. It is shown that the bounded core is axiomatized similarly to the core for classical cooperative games, namely by boundedness (BOUND), nonemptiness for zero-inessential two-person games (ZIG), anonymity, covariance under strategic equivalence (COV), and certain variants of the reduced game property (RGP), the converse reduced game property (CRGP), and the reconfirmation property. The core is the maximum solution that satisfies a suitably weakened version of BOUND together with the remaining axioms. For games with connected hierarchies, the bounded core is axiomatized by BOUND, ZIG, COV, and some variants of RGP and CRGP, whereas the core is the maximum solution that satisfies the weakened version of BOUND, COV, and the variants of RGP and CRGP.
    Keywords: TU game, core, restricted cooperation.
    Date: 2012–01
  5. By: Yann Rébillé (LEMNA - Laboratoire d'économie et de management de Nantes Atlantique - Université de Nantes : EA4272); Lionel Richefort (LEMNA - Laboratoire d'économie et de management de Nantes Atlantique - Université de Nantes : EA4272)
    Abstract: A local public goods game in weighted and directed networks is analyzed. Individual efforts are imperfect substitutes, players' preferences are heterogeneous and local externalities are non-uniform and asymmetric. Sufficient conditions under which the game admits a unique equilibrium are established in terms of the number of links between agents in the original network. It appears that these latter conditions for uniqueness are met if, and only if, the structure of relationships is \emph{productive}. That is, a parallel can be established between network games with strategic substitutes and the input-output theory pioneered by Wassily Leontief.
    Keywords: local public goods ; Nash equilibrium ; generalized degree ; productive matrix ; Leontief model
    Date: 2012–02–17
  6. By: Zuazo Garín, Peio; Rocha, André Barreira da Silva; Laruelle, Annick
    Abstract: We generalise and extend the work of Iñarra and Laruelle (2011) by studying two person symmetric evolutionary games with two strategies, a heterogenous population with two possible types of individuals and incomplete information. Comparing such games with their classic homogeneous version vith complete information found in the literature, we show that for the class of anti-coordination games the only evolutionarily stable strategy vanishes. Instead, we find infinite neutrally stable strategies. We also model the evolutionary process using two different replicator dynamics setups, each with a different inheritance rule, and we show that both lead to the same results with respect to stability.
    Date: 2011–12
  7. By: Ahmad Naimzada (University of Milano-Bicocca); Fabio Tramontana (Department of Economics and Quantitative Methods, University of Pavia)
    Abstract: We move from a triopoly game with heterogeneous players (E.M.Elabassy et al., 2009. Analysis of nonlinear triopoly game with heterogeneous players. Computers and Mathematics with Applications 57, 488-499). We remove the nonlinearity from the cost function and introduce it in the demand function. We also introduce a different decisional mechanism for one of the three competitors. A double route to complex dynamics is shown to exist, together with the possibility of multistability of different attractors, requiring a global analysis of the dynamical system.
    Keywords: Triopoly game; Heterogeneous players; Global analysis
    Date: 2011–06
  8. By: Driesen, Bram W.
    Abstract: In this article we define and characterize a class of asymmetric leximin solutions, that contains both the symmetric leximin solution of Imai[5] and the two-person asymmetric Kalai-Smorodinsky solution of Dubra [3] as special cases. Solutions in this class combine three attractive features: they are defined on the entire domain of convex n-person bargaining problems, they generally yield Pareto efficient solution outcomes, and asymmetries among bargainers are captured by a single parameter vector. The characterization is based on a strengthening of Dubra’s [3] property Restricted Independence of Irrelevant Alternatives (RIIA). RIIA imposes Nash’s IIA (Nash [9]), under the added condition that the contraction of the feasible set preserves the mutual proportions of players’ utopia values. Our axiom, entitled RIIA for Independent Players (RIP), says RIIA holds for a group of players, given that the contraction of the feasible set does not affect players outside that group.
    Keywords: Bargaining; asymmetric bargaining solution; leximin solution
    JEL: C78
    Date: 2012–02–17
  9. By: Rabah Amir (Department of Economics, University of Arizona); Giuseppe De Feo (Department of Economics, University of Pavia)
    Abstract: This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private – domestic or foreign – firm competes with a public, welfare maximizing firm. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. From a methodological viewpoint we make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.
    Keywords: Mixed markets, endogenous timing, Cournot equilibrium, Stackelberg equilibrium, privatization.
    JEL: C72 D43 H42 L13
    Date: 2012–02
  10. By: Borgers, Tilman; Smith, Doug
    Abstract: We develop an analysis of voting rules that is robust in the sense that we do not make any assumption regarding voters’ knowledge about each other. In dominant strategy voting rules, voters’ behavior can be predicted uniquely without making any such assumption. However, on full domains, the only dominant strategy voting rules are random dictatorships. We show that the designer of a voting rule can achieve Pareto improvements over random dictatorship by choosing rules in which voters’ behavior can depend on their beliefs. The Pareto improvement is achieved for all possible beliefs. The mechanism that we use to demonstrate this result is simple and intuitive, and the Pareto improvement result extends to all equilibria of the mechanism that satisfy a mild refinement. We also show that the result only holds for voters’ interim expected utilities, not for their ex post expected utilities.
    Keywords: robust mechanism design; dominant strategies; voting; Gibbard-Satterthwaite theorem
    JEL: D7 C7
    Date: 2011–11–03
  11. By: Sebastian Braun; Nadja Dwenger; Dorothea Kübler; Alexander Westkamp
    Abstract: Quotas for special groups of students often apply in school or university admission procedures. This paper studies the performance of two mechanisms to implement such quotas in a lab experiment. The first mechanism is a simplified version of the mechanism currently employed by the German central clearinghouse for university admissions, which first allocates seats in the quota for top-grade students before allocating all other seats among remaining applicants. The second is a modied version of the student-proposing deferred acceptance (SDA) algorithm, which simultaneously allocates seats in all quotas. Our main result is that the current procedure, designed to give top-grade students an advantage, actually harms them, as students often fail to grasp the strategic issues involved. The modified SDA algorithm significantly improves the matching for top-grade students and could thus be a valuable tool for redesigning university admissions in Germany.
    Keywords: College admissions, experiment, quotas, matching; Gale-Shapley mechanism, Boston mechanism
    JEL: C78 C92 D78 I20
    Date: 2012–01
  12. By: Sonja Brangewitz (University of Paderborn); Sarah Brockhoff (University of Freiburg)
    Abstract: This paper analyzes the stability of capital tax harmonization agree- ments in a stylized model where countries have formed coalitions which set a common tax rate in order to avoid the inefficient fully non- cooperative Nash equilibrium. In particular, for a given coalition struc- ture we study to what extend the stability of tax agreements is affected by the coalitions that have formed. In our set-up, countries are sym- metric, but coalitions can be of arbitrary size. We analyze stability by means of a repeated game setting employing simple trigger strategies and we allow a sub-coalition to deviate from the coalitional equilib- rium. For a given form of punishment we are able to rank the stability of different coalition structures as long as the size of the largest coali- tion does not change. Our main results are: (1) singleton regions have the largest incentives to deviate, (2) the stability of cooperation de- pends on the degree of cooperative behavior ex-ante.
    Keywords: capital tax competition, tax coordination, coalitional equilibria, repeated game
    JEL: C71 C72 H71 H77
    Date: 2012–02
  13. By: Lombardi Michele; Yoshihara Naoki (METEOR)
    Abstract: The paper proposes necessary and sufficient conditions for the natural implementation of(efficient) social choice correspondences (SCCs) in pure finite exchange economies when some ofthe agents are partially honest. A partially honest agent is an agent who strictly prefers to tellthe truth when lying has no better material consequences for her. Firstly, it is shown that ifthere is even one partially honest agent in the economy (and the planner does not know heridentity), then any SCC is Nash implementable by a natural price-allocation mechanism. Secondly,and in sharp contrast with the results of conventional models of natural implementation, it isshown that the equivalence relationship between natural price-allocation mechanisms and naturalprice-quantity² mechanisms no longer holds. Finally, and even more strikingly, the paper reportsthat the class of implementable SCCs by natural price-quantity mechanisms is significantly enlarged.
    Keywords: mathematical economics;
    Date: 2012
  14. By: Nizar Allouch (Queen Mary, University of London)
    Abstract: This paper analyzes the private provision of public goods where consumers interact within a fixed network structure and may benefit only from their direct neighbors' provisions. We present a proof for existence and uniqueness of a Nash equilibrium with general best-reply functions. Our uniqueness result simultaneously extends similar results in Bergstrom, Blume, and Varian (1986) on the private provision of public goods to networks and Bramoullé, Kranton, and D'Amours (2011) on games of strategic substitutes to nonlinear best-reply functions. In addition, we investigate the neutrality result of Warr (1983) and Bergstrom, Blume, and Varian (1986) whereby consumers are able to offset income redistributions and tax-financed government contributions. To this effect, we establish that the neutrality result has a limited scope of application beyond regular networks.
    Keywords: Public goods, Uniqueness of Nash equilibrium, Network games, Neutrality, Bonacich centrality, Main eigenvalue
    JEL: C72 D31 H41
    Date: 2012–02
  15. By: Charoensook, Banchongsan
    Abstract: This paper develops a model of noncooperative network formation. Link formation is two-sided. Information flow is two-way. The paper is based on Bala and Goyal (2000) with the following difference in assumption: the value of information decays as it flows through each agent, and the decay is increasing and concave in the number of his links. Thus, an agent may choose to avoid accessing an agent who possess many links since he is aware of the decay incurred through this agent. This avoidance leads to two particular results in the analysis of Nash networks: (1) Nash networks are not always connected; (2) Nash networks do not exist under some parameters. Since disconnectedness is reminiscent of a common feature of real-world network, the model may explain why real-world networks may exhibit this feature even when there is no heterogeneity among agents. Discussion on this insight is provided.
    Keywords: Social Networks; Game Theory; Network Formation
    JEL: Z13 D85 C72
    Date: 2012–02–10
  16. By: Senatore, L
    Abstract: This paper considers a model of voluntary public good provision with two players and convex costs. I demonstrate that the provision of public good is higher in the sequential framework under fairly general conditions. This outcome shows that introducing convex costs may reverse under some condition the results of Varian ( 1994).
    Keywords: Public Goods; Contribution Games; Private Provision of Public Goods
    JEL: D0 H40 H41 C72
    Date: 2011–12
  17. By: Kazuya Kikuchi
    Abstract: This paper extends a probabilistic voting model with a multidimensional policy space, allowing candidates to have different prior probability distributions of the distribution of voters' ideal policies. In this model, we show that a platform pair is a Nash equilibrium if and only if both candidates choose a common generalized median of expected ideal policies. Thus, the existence of a Nash equilibrium requires not only that each candidate's belief have an expected generalized median, which is already a knife-edge condition, but also that the two medians coincide. We also study limits of ε-equilibria of Radner (1980) as ε → 0, which we call "limit equilibria." Limit equilibria are policy pairs that approximate choices by the candidates who almost perfectly optimize. We show that a policy pair is a limit equilibrium if and only if both candidates choose the same policy around which they form "opposite expectations" in a certain sense. For a limit equilibrium to exist (equivalently, for ε-equilibria to exist for all ε > 0), it is sufficient, though not necessary, that either candidate has an expected generalized median.
    Date: 2012–02
  18. By: Bach Christian W.; Tsakas Elias (METEOR)
    Abstract: We provide epistemic conditions for Nash equilibrium, which are considerably weaker than thestandard ones by Aumann and Brandenburger (1995). Indeed, we simultaneously replace commonknowledge of conjectures and mutual knowledge of rationality with strictly weaker epistemicconditions of pairwise common knowledge of conjectures and pairwise mutual knowledge ofrationality respectively. It is also shown that, unlike the Aumann and Brandenburger''s conditions,ours do not imply common knowledge of rationality. Surprisingly, they actually do not even implymutual knowledge of rationality.
    Keywords: microeconomics ;
    Date: 2012
  19. By: Carfì, David; Schilirò, Daniele
    Abstract: This paper provides a coopetitive model for a global green economy taking into account the environmental sustainability. In particular we propose a differentiable coopetitive game G (in the sense recently introduced by D. Carfì) to represent a basic green economy interaction among a country c and the rest of the world w. Our game G is a linear parametric (Euclidean) perturbation of the classic Cournot duopoly. In the paper we offer the complete study of the proposed model and in particular a deep examination of its possible coopetitive solutions.
    Keywords: environmental sustainability; global green economy; coopetitive games; infinite game
    JEL: Q42 Q30 C78 C71 Q56 Q20 Q58 C72
    Date: 2012–03
  20. By: Daniele Condorelli; Andrea Galeotti
    Abstract: We study an incomplete-information model of sequential bargaining for a single object, with the novel feature that agents are located in a network. In each round of trade, the current owner of the object either consumes it or makes a take-it-or-leave-it offer to some connected trader. Traders may buy in order to consume or to resale to others. We show that the equilibrium price dynamics is non-monotone and that traders that intermediate the object arise endogenously and attain a pro t. We also provide insights on how traders' equilibrium payoffs depend on their location in the network.
    Date: 2011–12–14
  21. By: Khan Abhimanyu; Peeters Ronald (METEOR)
    Abstract: Inspired by the behavior in repeated guessing game experiments, we study adaptive play bypopulations containing individuals that reason with different levels of cognition. Individualsplay a higher order best response to samples from the empirical data on the history of play, wherethe order of best response is determined by their exogenously given level of cognition. As inYoung''s model of adaptive play, (unperturbed) play still converges to a minimal curb set. However,with the random perturbations of this (higher order) best response dynamic, the stochasticallystable states obtained may now be different, but in a deterministic manner. Perhapscounter-intuitively, higher cognition may actually be bad for both the individual with highercognition and his parent population.
    Keywords: microeconomics ;
    Date: 2012
  22. By: Stoop, Jan
    Abstract: Results are reported of the first natural field experiment on the dictator game, where subjects are unaware that they participate in an experiment. In contrast to predictions of the standard economic model, dictators show a large degree of pro-social behavior. This paper builds a bridge from the laboratory to the field to explore how predictive findings from the laboratory are for the field. External validity is remarkably high. In all experiments, subjects display an equally high amount of pro-social behavior, whether they are students or not, participate in a laboratory or not, or are aware that they participate in an experiment or not.
    Keywords: altruism; natural field experiment; external validity
    JEL: D63 D64 C70 C93 C91
    Date: 2012–03–02
  23. By: Miettinen, Paavo (Bank of Finland Research)
    Abstract: In this paper we consider equilibrium behavior in a Dutch (descending price) auction where the bidders are uninformed of their valuations with probability q and can acquire information about their valuation at a positive cost during the auction. We assume that the information acquisition activity is covert. We characterize the equilibrium behavior in a setting where bidders are ex ante symmetric and have independent private values. We show that, if the number of bidders is large, the Dutch auction produces more revenue than would a first price auction.
    Keywords: auctions; information acquisition
    JEL: D44 D82 D83
    Date: 2012–02–14
  24. By: Daniele Condorelli; Andrea Galeotti
    Abstract: We investigate the effects of a class of trading protocols on the architecture and efficiency properties of endogenously formed trading networks. In our model, the opportunity to sell valuable objects occurs randomly to different individuals. A sale can only be realized if two individuals are connected, directly or indirectly, but forming and maintaining a trading relation is a costly investment. When the outcome of trading is efficient and provides no intermediation rents, a tension between equilibrium and efficient networks emerges when the cost of forming a link is at an intermediate level. There are two types of inefficiencies. Either all equilibrium networks are under- connected when compared to efficient networks, or a multiplicity of equilibriam may exist and agents may fail to coordinate on the efficient equilibrium network
    Date: 2011–12–14
  25. By: Jorge Alcalde-Unzu (Departamento de Economía-UPNA); Marc Vorsat (Fundación de Estudios de Economía Aplicada (FEDEA))
    Abstract: We study axiomatically situations in which the society agrees to treat voters with different characteristics distinctly. In this setting, we propose a set of six intuitive axioms and show that they jointly characterize a new class of voting procedures, called Personalized Approval Voting. According to this family, each voter has a strictly positive and finite weight (the weight is necessarily the same for all voters with the same characteristics) and the alternative with the highest number of weighted votes is elected. Hence, the implemented voting procedure reduces to Approval Voting in case all voters are identical or the procedure assigns the same weight to all types.
    Keywords: approval voting, characterization, anonymity.
    JEL: D71
    Date: 2011
  26. By: Ambec, Stefan; Dinar, Ariel; McKinney, Daene
    Abstract: By signing a fixed water sharing agreement (FWSA), countries voluntarily commit to release a fixed amount of river water in exchange for an agreed compensation. We examine the vulnerability of such commitments to reduced water ows. Among all FWSAs that are acceptable to riparian countries, we find out the one which is sustainable to the most severe drought scenarios. The so-called upstream incremental FWSA assigns to each country its marginal contribution to its followers in the river. Its mirror image, the downstream incremental FWSA, is not sustainable to reduced ow at the source. We apply our analysis to the Aral Sea basin agreement.
    Keywords: international river treaty, water, stability, core, compliance, Aral sea.
    JEL: D74 Q25 Q28 Q54
    Date: 2011–10
  27. By: Fabrice Barthelemy; Gabriele Esposito; Mathieu Martin; Vincent Merlin (THEMA, Universite de Cergy-Pontoise; EHESS and GREQAM; THEMA, Universite de Cergy-Pontoise; CREM, Université de Caen)
    Abstract: In this paper we analyze the fairness of the 2007 reform proposal concerning the apportionment of the seats between the regions for the Italian Senate. Theory of power indices is used to compare the actual case with the proposed one. Two scenarios are proposed, senators belonging to the same region voting in blocks and senators voting according party lines, using both the Impartial Culture and the Impartial Anonymous Culture models. Our objective is to determine which apportionment is closer to the equal distribution of power among the citizens. In addition, we will seek for apportionments that are closer to the ideal representation than the ones proposed by politicians. We will also derive the probability that different apportionments produce a referendum paradox, i.e. exhibit a majority in the Senate different from the national popular majority.
    Keywords: Power index, Banzhaf, Italian Senate, apportionment, voting paradox, Monte Carlo simulation.
    JEL: C7 D7
    Date: 2011

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