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on Game Theory |
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Issue of 2011‒02‒26
twenty-six papers chosen by Laszlo A. Koczy Hungarian Academy of Sciences and Obuda University |
| By: | Rene van den Brink (VU University Amsterdam); Ilya Katsev (Russian Academy of Sciences, St Petersburg); Gerard van der Laan (VU University Amsterdam) |
| Abstract: | A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions to every TU-game. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. Properties of solutions (the core, the nucleolus, the prekernel and the Shapley value) are given for games on union closed systems. |
| Keywords: | TU-game; restricted cooperation; union closed system; core; prekernel; nucleolus |
| JEL: | C71 |
| Date: | 2011–02–14 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20110036&r=gth |
| By: | Rene van den Brink (VU University Amsterdam); Yukihiko Funaki (School of Political Science and Economics, Waseda University Tokyo) |
| Abstract: | We generalize the null player property (satisfied by the Shapley value) and nullifying player property (satisfied by the equal division solution) to the so-called delta-reducing player property, stating that a delta-reducing player (being a player such that any coalition containing this player earns a fraction delta in [0,1] of the worth of that coalition without that player) earns a zero payoff. This property yields the null player property for delta = 1 and the nullifying player property for delta = 0. We show that efficiency, symmetry, linearity and this delta-reducing player property characterizes the corresponding delta-discounted Shapley value. Moreover, we provide a strategic implementation of these solutions where delta is a discount factor that determines the decrease in value to be distributed in the next round after the proposal is rejected and the remaining players (without the proposer) play a new round of bidding. |
| Keywords: | Cooperative TU-game; Shapley value; equal division solution; delta-discounted Shapley value; Axiomatization; Implementation; Discounting |
| JEL: | C71 C72 |
| Date: | 2010–07–08 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100065&r=gth |
| By: | P. Jean-Jacques Herings (Maastricht University); Harold Houba (VU University Amsterdam) |
| Abstract: | We analyze the simplest Condorcet cycle with three players and three alternatives within a strategic bargaining model with recognition probabilities and costless delay. Mixed consistent subgame perfect equilibria exist whenever the geometric mean of the agents' risk coefficients, ratios of utility differences between alternatives, is at most one. Equilibria are generically unique, Pareto efficient, and ensure agreement within finite expected time. Agents propose best or second-best alternatives. Agents accept best alternatives, may reject second-best alternatives with positive probability, and reject otherwise. For symmetric recognition probabilities and risk coefficients below one, agreement is immediate and each agent proposes his best alternative. |
| Keywords: | Bargaining; Condorcet Paradox; Consistent Subgame Perfect Equilibrium; Risk Aversion; Compromise Prone |
| JEL: | C73 C78 D72 |
| Date: | 2010–03–01 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100026&r=gth |
| By: | Matthijs van Veelen (University of Amsterdam) |
| Abstract: | For games in which there is no evolutionarily stable strategy, it can be useful to look for neutrally stable ones. In extensive form games for instance there is typically no evolutionary stable strategy, while there may very well be a neutrally stable one. Such strategies can however still be relatively stable or unstable, depending on whether or not the neutral mutants it allows for - which by definition do not have a selective advantage themselves - can open doors for other mutants, that do have a selective advantage. This paper defines robustness against indirect invasions in order to be able to discern between those two very different situations. Robustness against indirect invasions turns out to come with a very natural setwise generalisation of evolutionary stability; we prove that if a strategy is robust against indirect invasions, then this strategy and its (indirect) neutral mutants form a set that is asymptotically stable in the replicator dynamics. |
| Keywords: | Robustness against indirect invasions; neutrally stable strategy; evolutionarily stable strategy; extensive form game |
| JEL: | C73 |
| Date: | 2010–03–19 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100033&r=gth |
| By: | Sugato Chakravarty (Purdue University); Yongjin Ma (Purdue University); Sandra Maximiano (Purdue University) |
| Abstract: | The goal of this paper is to investigate the interaction between social ties and deceptive behavior within an experimental setting. To do so, we implement a modified sender-receiver game in which a sender obtains a private signal regarding the value of a state variable and sends a message related to the value of this state variable to the receiver. The sender is allowed to be truthful or to lie about what he has seen. The innovation in our experimental design lies in the fact that, in contrast to the extant sender-receiver games, the receiver can take no action – which eliminates strategic deception. A further innovation lies in the fact that subjects (i.e., senders) are not restricted to choose between truth telling and a unique type of lie but, instead, are allowed to choose from a distinct set of allocations that embodies a multi-dimensional set of potential lies. Our experimental design is, therefore, able to overcome an existing identification problem by allowing us to disentangle lying aversion from social preferences. We implement two treatments: one in which players are anonymous to each other (strangers); and one in which players know each other from outside the experimental laboratory (friends). We find that individuals are less likely to lie to friends than to strangers; that they have different degrees of lying aversion and that they lie according to their social preferences. Pro-social individuals appear to be more lying averse. If they lie, however, they are equally likely to do so with friends and strangers. The deceptive behavior of selfish individuals mimics those of pro social types only when subjects play with friends. Overall, in addition to social preferences, friendship appears to be an important factor in improving our understanding of deceptive behavior. |
| Keywords: | Lying, Friendship, social ties, deceptive behavior, signal, experiment |
| JEL: | G21 D82 O16 |
| Date: | 2011–02 |
| URL: | http://d.repec.org/n?u=RePEc:csr:wpaper:1007&r=gth |
| By: | Matthijs van Veelen (CREED, Universiteit van Amsterdam); Julian Garcia (Vrije Universiteit) |
| Abstract: | Repeated games tend to have large sets of equilibria. We also know that in the repeated prisoners dilemma there is a profusion of neutrally stable strategies, but no strategy that is evolutionarily stable. This paper shows that for all of these neutrally stable strategies there is a stepping stone path out; there is always a neutral mutant that can enter a population and create an actual selective advantage for a second mutant. Such stepping stone paths out of equilibrium generally exist both in the direction of more and in the direction of less cooperation. While the central theorems show that such paths out of equilibrium exist, they could still be rare compared to the size of the strategy space. Simulations however suggest that they are not too rare to be found by a reasonable mutation process, and that typical simulation paths take the population from equilibrium to equilibrium through a series of indirect invasions. Instability does not mean we cannot draw qualitative conclusions though. The very nature of the indirect invasions implies that the population will on average be (somewhat) reciprocal and (reasonably) cooperative. |
| Keywords: | Repeated games; evolution; robust against indirect invasions; simulation |
| JEL: | C73 |
| Date: | 2010–04–08 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100037&r=gth |
| By: | Lombardi, Michele; Yoshihara, Naoki |
| Abstract: | This paper studies implementation problems in the wake of a recent new trend of implementation theory which incorporates a non-consequentialist flavor of the evidence from experimental and behavioral economics into the issues. Specifically, following the seminal works by Matsushima (2008) and Dutta and Sen (2009), the paper considers implementation problems with partially honest agents, which presume that there exists at least one individual in the society who concerns herself with not only outcomes but also honest behavior at least in a limited manner. Given this setting, the paper provides a general characterization of Nash implementation with partially-honest individuals. It also provides the necessary and sufficient condition for Nash implementation with partially-honest individuals by mechanisms with some types of strategy-space reductions. As a consequence, it shows that, in contrast to the case of the standard framework, the equivalence between Nash implementation and Nash implementation with strategy space reduction no longer holds. |
| Keywords: | Nash implementation; canonical-mechanisms; s-mechanisms; self-relevant mechanisms; partial-honesty; permissive results. |
| JEL: | D71 C72 |
| Date: | 2011–02–13 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:28838&r=gth |
| By: | Shiko Maruyama (School of Economics, University of New South Wales) |
| Abstract: | I study the estimation of finite sequential games with perfect information. The major challenge in estimation is computation of high-dimensional truncated integration whose domain is complicated by strategic interaction. I show that this complication resolves when unobserved off-the-equilibrium-path strategies are controlled for. Separately evaluating the likelihood contribution of each subgame perfect strategy profile that rationalizes the observed outcome allows the use of the GHK simulator, the most widely used importance-sampling probit simulator. Monte Carlo experiments demonstrate the performance and robustness of the proposed method, and confirm that misspecification of the decision order leads to underestimation of strategic effect. |
| Keywords: | Inference In Discrete Games; Sequential Games; Monte Carlo Integration; GHK Simulator; Subgame Perfection; Perfect Information |
| JEL: | C35 C63 C72 |
| Date: | 2010–11 |
| URL: | http://d.repec.org/n?u=RePEc:swe:wpaper:2010-22&r=gth |
| By: | Adrian de Groot Ruiz (University of Amsterdam); Theo Offerman (University of Amsterdam); Sander Onderstal (University of Amsterdam) |
| Abstract: | Currently no refinement exists that successfully selects equilibria across a wider range of Cheap Talk games. We propose a generalization of refinements based on credible deviations, such as neologism proofness and announcement proofness. According to our Average Credible Deviation Criterion (ACDC), the stability of an equilibrium is determined by the frequency and the size of credible deviations. In our experiment, we find support for the relevance of credible deviations. In addition, we find support for ACDC in settings where other criteria remain silent. ACDC also explains results from previous experiments. |
| Keywords: | cheap talk; neologism proofness; announcement proofness; credible deviation; ACDC; experiment |
| JEL: | C72 C92 D82 D83 |
| Date: | 2011–02–14 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20110037&r=gth |
| By: | Rene van den Brink (VU University Amsterdam); Youngsub Chun (Seoul National University, Seoul, Korea) |
| Abstract: | We investigate the implications of imposing balanced consistency and balanced cost reduction in the context of sequencing problems. Balanced consistency requires that the effect on the payoff from the departure of one agent to another agent should be equal between any two agents. On the other hand, balanced cost reduction requires that if one agent leaves a problem, then the total payoffs of the remaining agents should be affected by the amount previously assigned to the leaving agent. We show that the minimal transfer rule is the only rule satisfying efficiency and Pareto indifference together with either one of our two main axioms, balanced consistency and balanced cost reduction. |
| Keywords: | Sequencing problem; minimal transfer rule; balanced consistency; balanced cost reduction |
| JEL: | D63 D71 |
| Date: | 2010–03–19 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100035&r=gth |
| By: | Justin Leroux; Walid Marrouch |
| Abstract: | In the context of bilateral bargaining,we deal with issue linkage by developing a two-issue-two-players cooperative bargaining model. The axioms we propose focus on the role of the disagreement points. A family of bargaining rule stands out: the monotonic equal net ratio solutions. These solutions point to Pareto efficient outcomes such that the relative gains for players are equal across issues and reward the players for improving their bargaining power over each issue. We discuss our results in light of international trade and environmental negotiations, which are often put on the bargaining table in a linked fashion. <P>Dans un contexte de négociations bilatérales, nous adressons la question de l’inter-connexion des sujets de négociation dans un modèle coopératif à deux-sujets-deux-agents. Les axiomes que nous proposons insistent sur le rôle des points de menace. Une famille de solutions ressort de l’analyse : les solutions monotones à taux nets identiques. Chacune de ces solutions préconise une issue Pareto efficace de sorte que les gains relatifs de chaque agent sont les mêmes pour les deux sujets de négociation. De plus, ces règles récompensent les agents pour des améliorations de leurs pouvoirs de négociation. Nous discutons nos résultats à la lumière des négociations de commerce international et environnementales, qui sont souvent amenées à la table de négociations de manière liée. |
| Keywords: | Multi-issue bargaining, issue linkages, axiomatic solutions, disagree-ment point. , Négociations Multi-sujets, inter-connexion des sujets, solutions axiomatiques, point de Menace. |
| JEL: | C78 Q56 |
| Date: | 2011–02–01 |
| URL: | http://d.repec.org/n?u=RePEc:cir:cirwor:2011s-30&r=gth |
| By: | Chirantan Ganguly; Indrajit Ray |
| Abstract: | In the Crawford-Sobel (uniform, quadratic utility) cheap-talk model, we consider a simple mediation scheme (a communication device) in which the informed agent reports one of N possible elements of a partition to the mediator and then the mediator suggests one of N actions to the uninformed decision-maker according to the probability distribution of the device. We show that such a simple mediated equilibrium cannot improve upon the unmediated N-partition. Crawford-Sobel equilibrium when the preference divergence parameter (bias) is small. |
| Keywords: | Cheap talk, Mediated Equilibrium |
| JEL: | C72 |
| Date: | 2011–01 |
| URL: | http://d.repec.org/n?u=RePEc:bir:birmec:05-08rr&r=gth |
| By: | Elena Inarra (UPV/EHU); Annick Laruelle (UPV/EHU) |
| Abstract: | We introduce a new variation of the hawk-dove game suggested by an experiment that studies the behavior of a group of domestic fowls when a subgroup has been marked. Speci cally we consider a population formed by two types of individual that fail to recog- nize their own type but do recognize the other type. In this game we nd two evolutionar- ily stable strategies. In each of them, individuals from one type are always attacked more, whatever proportion of the population they represent. Our theoretical results are consis- tent with the conclusions drawn from experimental work, where marked fowls received more pecks than their unmarked counterparts. |
| JEL: | C72 |
| Date: | 2011–02–17 |
| URL: | http://d.repec.org/n?u=RePEc:ehu:ikerla:201150&r=gth |
| By: | Alcalde, Jose; Romero-Medina, Antonio |
| Abstract: | This paper proposes a way to allocate students to schools such that conciliates Pareto efficiency and stability. Taking as a starting point the recent reform proposed by the Boston School Committee, we propose a marginal modification to reach our objective redefine how students are prioritize. Our proposal is to allow schools to prioritize only a small set of students an then use a common priority order for the rest. Under this condition we propose a score based priority ranking that makes the output of the new Boston Mechanism Pareto efficient and stable. |
| Keywords: | School allocation problem; Pareto efficient matching |
| JEL: | D71 C71 C72 |
| Date: | 2011–02–13 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:28831&r=gth |
| By: | Dobbelaere, Sabien (VU University Amsterdam); Luttens, Roland Iwan (Ghent University) |
| Abstract: | We introduce collective bargaining in a static framework where the firm and its risk-neutral employees negotiate over wages in a non-binding contract setting. Our main result is the equivalence between the non-binding collective equilibrium wage-employment contract and the equilibrium contract under binding risk-neutral efficient bargaining. We also demonstrate that our non-cooperative equilibrium wages and profits coincide with the Owen values of the corresponding cooperative game with the coalitional structure that follows from unionization. |
| Keywords: | collective bargaining, union, firm, bargaining power, non-binding contract |
| JEL: | C71 J51 L20 |
| Date: | 2011–02 |
| URL: | http://d.repec.org/n?u=RePEc:iza:izadps:dp5518&r=gth |
| By: | Brams, Steven J.; Kaplan, Todd R; Kilgour, D. Marc |
| Abstract: | We describe a simple 2-stage mechanism that induces two bargainers to be truthful in reporting their reservation prices in a 1st stage. If these prices criss-cross, the referee reports that they overlap, and the bargainers proceed to make offers in a 2nd stage. The average of the 2nd-stage offers becomes the settlement if both offers fall into the overlap interval; if only one offer falls into this interval, it is the settlement, but is implemented with probability 1/2; if neither offer falls into the interval, there is no settlement. Thus, if the bargainers reach the 2nd stage, they know their reservation prices overlap even if they fail to reach a settlement, possibly motivating them to try again. |
| Keywords: | Bargaining; truth-telling mechanisms; probabilistic implementation; incomplete information. |
| JEL: | C78 D74 D02 J52 C72 |
| Date: | 2011–02–18 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:28999&r=gth |
| By: | Marcelo Tyszler (University of Amsterdam); Arthur Schram (University of Amsterdam) |
| Abstract: | We theoretically and experimentally study voter behavior in a setting characterized by plurality rule and mandatory voting, where voters choose from three options. We are interested in the occurrence of strategic voting in an environment where Condorcet cycles may occur. In particular, we focus on how information about the distribution of preferences affects strategic behavior. We also vary the relative importance of the second preferred option to investigate how this affects the strategic vote. Quantal response equilibrium analysis is used to analyze the game and proves to be a good predictor for the experimental data. Our results indeed show that strategic voting arises, the extent of which depends on (i) the availability of information; (ii) the relative importance of the intermediate candidate; (iii) the electorate's relative support for one's preferred candidate; and (iv) the relative position of the plurality-supported candidate in a voter's preference ordering. Our results show that information serves as a coordination device where strategic voting does not harm the plurality-preferred candidate's chances of winning. |
| Keywords: | Voting Behavior; Experimental Economics; Quantal Response Equilibrium |
| JEL: | C92 D72 D83 |
| Date: | 2011–02–10 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20110025&r=gth |
| By: | Jana Vyrastekova (Radboud University Nijmegen); Sander Onderstal (University of Amsterdam) |
| Abstract: | We analyze gender differences in the trust game in a "behind the veil of ignorance" design. This method yields strategies that are consistent with actions observed in the classical trust game experiments. We observe that, on average, men and women do not differ in "trust", and that women are slightly more "trustworthy". However, men's strategies are bimodal, peaking at the subgame perfect Nash equilibrium and the Pareto efficient frontier, while women's strategies are single peaked at moderate transfers. For a given high level of pro-social preferences, men send more than women. This may be linked to men willing to bear more risk than women. |
| Keywords: | trust game; experiment; strategy method behind the veil of ignorance; gender differences |
| JEL: | C72 C91 |
| Date: | 2010–07–05 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100063&r=gth |
| By: | Alaitz Artabe (UPV/EHU); Annick Laruelle (UPV/EHU and Ikerbasque); Federico Valenciano (UPV/EHU) |
| Abstract: | In this paper we address several issues related to collective dichotomous decision-making by means of quaternary voting rules, i.e., when voters may choose between four actions: voting yes, voting no, abstaining and not turning up-which are aggregated by a voting rule into a dichotomous decision: accep- tance or rejection of a proposal. In particular we study the links between the actions and preferences of the actors. We show that quaternary rules (unlike binary rules, where only two actions -yes or no- are possible) leave room for “manipulability” (i.e., strategic behaviour). Thus a preference pro…le does not in general determine an action pro…le. We also deal with the notions of success and decisiveness and their ex ante assessment for quaternary voting rules, and discuss the role of information and coordination in this context. |
| Date: | 2011–02–17 |
| URL: | http://d.repec.org/n?u=RePEc:ehu:ikerla:201148&r=gth |
| By: | K. De Jaegher (Utrecht University); J.J.A. Kamphorst (Erasmus University Rotterdam) |
| Abstract: | This paper characterizes the set of equilibrium networks in the two-way flow model of network formation with small decay, and this for all increasing benefit functions of the players. We show that as long as the population is large enough, this set contains large- as well as small-diameter networks. For all benefit functions, the periphery-sponsored star is the most stable. When the marginal benefits of information are constant, all non-star networks are equally stable. With increasing marginal benefits of information, small-diameter networks in general tend to be more stable. However, with decreasing marginal benefits of information, large-diameter networks tend to be the most robust along with the periphery-sponsored star. |
| Keywords: | Network formation; two-way flow model; decay; non-linear benefits |
| JEL: | C72 D85 |
| Date: | 2010–12–10 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100125&r=gth |
| By: | Lohmann E.; Borm P.; Herings P.J.J. (METEOR) |
| Abstract: | To verify whether a transferable utility game is exact, one has to check a linear inequalityfor each exact balanced collection of coalitions. This paper studies the structure andproperties of the class of exact balanced collections. Comparing the definition of exactbalanced collections with the definition of balanced collections, the weight vector of abalanced collection must be positive whereas the weight vector for an exact balancedcollection may contain one negative weight. We investigate minimal exact balanced collections, and show that only these collections are needed to obtain exactness. The relation between minimality of an exact balanced collection and uniqueness of the corresponding weight vector is analyzed. We show how the class of minimal exact balanced collections can be partitioned into three basic types each of which can be systematically generated. |
| Keywords: | operations research and management science; |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:umamet:2011009&r=gth |
| By: | Michael R. Baye (Indiana University); Dan Kovenock (University of Iowa); Casper G. de Vries (Erasmus University Rotterdam) |
| Abstract: | The Babylonian bridal auction, described by Herodotus, is regarded as one of the earliest uses of an auction in history. Yet, to our knowledge, the literature lacks a formal equilibrium analysis of this auction. We provide such an analysis for the twoplayer case with complete and incompete information, and in so doing identify what we call the "Herodotus Paradox". |
| Keywords: | second price; transfer to loser |
| JEL: | C72 D44 |
| Date: | 2010–09–02 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100090&r=gth |
| By: | Oliver Linton; Sorawoot Srisuma |
| Abstract: | We propose a general two-step estimation method for the structural parameters ofpopular semiparametric Markovian discrete choice models that include a class ofMarkovian Games andallow for continuous observable state space. The estimation procedure is simpleas it directly generalizes the computationally attractive methodology of Pesendorferand Schmidt-Dengler (2008) that assumed finite observable states. This extensionis non-trivial as the value functions, to be estimated nonparametrically in the firststage, are defined recursively in a non-linear functional equation. Utilizingstructural assumptions, we show how to consistently estimate the infinitedimensional parameters as the solution to some type II integral equations, thesolving of which is a well-posed problem. We provide sufficient set of primitives toobtain root-T consistent estimators for the finite dimensional structural parametersand the distribution theory for the value functions in a time series framework. |
| Keywords: | Discrete Markov Decision Models, Kernel Smoothing, Markovian Games, Semi-parametric Estimation, Well-Posed Inverse Problem.D |
| Date: | 2010–08 |
| URL: | http://d.repec.org/n?u=RePEc:cep:stiecm:/2010/550&r=gth |
| By: | V. Filipe Martins-da-Rocha; Nicholas C. Yannelis |
| Date: | 2011 |
| URL: | http://d.repec.org/n?u=RePEc:man:sespap:1105&r=gth |
| By: | Dietmar Fehr; Julia Schmid |
| Abstract: | Contest or auction designers who want to maximize the overall revenue are frequently con- cerned with a trade-off between contest homogeneity and inclusion of contestants with high valuations. In our experimental study, we find that it is not profitable to exclude the most able contestant in favor of greater homogeneity among the remaining contestants, even if the theoretical exclusion principle predicts otherwise. This is because the strongest contestants con- siderably overexert. A possible explanation is that these contestants are afraid they will regret a low but risky bid if they lose and thus prefer a strategy which gives them a low but secure pay-off. |
| Keywords: | experiments, contests, all-pay auction, heterogeneity, regret aversion |
| JEL: | C72 C92 D84 |
| Date: | 2011–02 |
| URL: | http://d.repec.org/n?u=RePEc:hum:wpaper:sfb649dp2011-009&r=gth |
| By: | Erik Ansink (IVM, VU University Amsterdam, and Wageningen University); Harold Houba (VU University Amsterdam) |
| Abstract: | Water markets with market power are analysed as multi-market Cournot competition in which the river structure constrains access to local markets and limited resources impose capacity constraints. Conditions for uniqueness are identified. Lerner indices are larger under binding resource constraints. The number of cases explodes in the number of local markets. Under quadratic benefit functions and symmetric constant marginal extraction costs, closed-form solutions for selected cases are derived, and numerical implementation through a single optimization program is available. Upstream locations face less competition than downstream. Observed price patterns in the Goulburn-Murray Irrigation District are consistent with the theoretical results. |
| Keywords: | Water markets; oligopoly; market power; Cournot-Walras equilibrium |
| JEL: | C72 C73 Q25 |
| Date: | 2010–05–27 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20100054&r=gth |