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An Evolutionary Analysis of Insurance Markets with Adverse Selection
Abstract
Rothschild and Stiglitz (1976) demonstrated that adverse selection may entail nonexistence of equilibrium in competitive insurance markets. We approach this problem in a dynamic model with boundedly rational insurance firms. Firms' behavior is based on imitation of profit making contracts, withdrawal of loss making contracts, and experimentation with random contracts. Consumers choose in each period the best contract available. We show that the candidate competitive equilibrium is the long run prediction if experimentation is rare and every firm experiments with contracts in the vicinity of its current menu.
JEL classication: C70, C72, D82, G22, L1
Keywords: adverse selection, bounded rationality, evolutionary game theory, imitation, insurance markets, learning, local experimentation, stochastic stability.
... about the beliefs is shared by word-of-mouth communication as in Banerjee and Fudenberg [4]. Alternatively, beliefs can be inferred if the menus of contracts are observable as in Ania et al. [1]. ...
... We find three mechanisms to justify why principals infer whether or not they come from the same reference group. Either because of a mechanism based on word-of-mouth communication as in Banerjee and Fudenberg [4], or because the contracts and profits are observed as in Ania et al. [1], or simply because the contracting quantities are observed, because principals with low-cost agents obtain equal quantities. ...
... Hence, their choices about quantities in a time t + 1 are based on the belief about the aggregate quantity, which in our set-up equals the quantity one period beforeq t . However, in t + 1 the payoff is affected by the realization of the aggregate quantityq t+1 which is described by (1). ...
We relax the common assumption of homogeneous beliefs in principal-agent relationships with adverse selection. Principals are competitors in the product market and write contracts also on the base of an expected aggregate. The model is a version of a cobweb model. In an evolutionary learning set-up, which is imitative, principals can have different beliefs about the distribution of agents’ types in the population. The resulting nonlinear dynamic system is studied. Convergence to a uniform belief depends on the relative size of the bias in beliefs.
... (3)(4) Insurance and risk management, Vol. 85 (3)(4) ...
... Insurance and risk management, Vol. 85 (3)(4) ...
... Assurances et gestion des risques/Insurance and risk management Vol. 85(3)(4) ...
Après avoir rappelé les contextes théorique et économique dans lequel s’inscrit cet article, nous présentons un premier modèle économétrique existant d’élasticité-prix. Puis, nous introduisons un modèle micro-économique dynamique en partant de l’exemple concret d’une compagnie d’assurance non-vie qui souhaite changer de stratégie de renouvellement. Nous étudions les effets de son choix de prix (primes) sur son portefeuille et son chiffre d’affaires, dans un contexte de concurrence. La variation du nombre de contrats en portefeuille et du chiffre d’affaires de la compagnie entre deux instants t 0 et t 1 est déterminée. Une application numérique sur trois branches d’assurance d’entreprises précède la conclusion.
... Further, we can suggest how such a pattern of prices might arise in the first place. Ania et al. (2002) describe a model where car insurers have bounded rationality and do not know risk probabilities. In this model, firms offer a menu of contracts, withdrawing unprofitable contracts and occasionally experimenting with new contacts. ...
... As long as consumers are rational and make appropriate choices, the market evolves to a unique equilibrium that is the same as Rothschild and Stiglitz (1976). Ania et al. (2002) do not discuss what would happen if consumers' behaviour were also characterized by bounded rationality, but it is plausible that one potential outcome would be the pattern of prices that we observe in the UK. 16 As we have seen, the pricing anomaly is not confined to the comparison between the TP and CP markets; there are also a small number of providers who charge more for a higher deductible. ...
... 17 Therefore, it is likely that unsophisticated consumers have similar prior beliefs that CP policies are more expensive. As long as aversion to risk is negatively correlated with risk type (propitious selection) and car insurers respond merely by offering menus of contracts that are profitable (as in Ania et al., 2002), this could result in low-risk types buying more insurance and, in the most extreme cases, result in TP policies being more expensive for car insurers due to TP policyholders being so much more risky than CP policyholders. This appears to be what has happened in the UK car insurance market. ...
We document a large and persistent anomaly in the UK car insurance market over the period 2012–13: insurance companies charged
a higher premium for third-party (liability) insurance than comprehensive insurance (which includes third party). Furthermore,
some companies charged higher premiums for comprehensive policies with larger deductibles. In contrast to current theories
of adverse or propitious selection, our evidence suggests both that consumers are too confused or too poorly informed to arbitrage
between policy types and that sellers of car insurance do not implement the incentive-compatibility constraints at the heart
of consumer demand theory. This particular insurance market is much less sophisticated than that characterized by modern microeconomic
theory.
... 1 There are yet methodically different strands in the literature. Ania et al. (2002) take an evolutionary game theory approach, Guerrieri et al. (2010) consider a competitive search model; other directions use cooperative concepts (Lacker and Weinberg, 1999) or a general equilibrium framework (see, for example, Dubey andGeanakoplos 2002 or Bisin andGottardi 2006) and modify equilibrium concepts, e.g., Blandin et al. (2016). For a recent overview see Mimra and Wambach (2014). ...
In competitive common value adverse selection markets, existence of a pure strategy equilibrium is often justified by appealing to Wilson’s (J Econ Theory 16(2):167–207, 1977) concept of ‘anticipatory equilibrium.’ The anticipatory equilibrium is based on the notion that all market participants expect unprofitable contracts to be withdrawn. We present a model of individual contract withdrawals that captures the strategic process underlying the anticipatory equilibrium concept: We introduce an additional—endogenously ending—stage into the Rothschild and Stiglitz (Q J Econ 90(4):629–649, 1976) model in which initial contracts can be withdrawn repeatedly after observation of competitors’ contract offers and withdrawals. Individual contract withdrawal allows for a rich strategic interaction. We show that an equilibrium exists where consumers obtain their respective second-best efficient Miyazaki–Wilson–Spence (MWS) contracts. However, this equilibrium requires latent contracts on offer. Furthermore, any individually rational and incentive-compatible allocation that earns nonnegative profits on aggregate can be sustained as equilibrium allocation. We further allow for contract addition as in Riley’s (Econometrica 47(2):331–359, 1979) ‘reactive equilibrium.’ Allowing for contract addition does not change the set of possible outcomes.
... However, a pooling contract cannot be tendered in equilibrium as insurers would try to cream skim low risks. 1 This equilibrium inexistence result has spurred extensive research, and although several modifications to the RS model have been brought forward, the debate is still ongoing and has gained in interest in recent years. 2 In the literature, mainly two candidates for the equilibrium allocation have emerged in competitive insurance markets with adverse selection. One is the original RS allocation, even if it is not an equilibrium in the RS model (e.g., Riley, 1979;Engers and Fernandez, 1987;Inderst and Wambach, 2001;Ania, Tröger, and Wambach, 2002;Mimra and Wambach, 2016). The second is the Miyazaki-Wilson-Spence (MWS) allocation (e.g., Asheim and Nilssen, 1996;Mimra and Wambach, 2011;Netzer and Scheuer, 2010;Netzer and Scheuer, 2014;Picard, 2014). ...
Even 30 years after Rothschild and Stiglitz's () seminal work on competitive insurance markets with adverse selection, existence and characterization of the equilibrium outcome are still an open issue. We model a basic extension to the Rothschild and Stiglitz () model: we endogenize up-front capital of insurers. Under limited liability, low up-front capital gives rise to an aggregate endogenous insolvency risk, which introduces an externality among customers of an insurer (Faynzilberg, 2006). It is shown that an equilibrium with the second-best efficient Miyazaki-Wilson-Spence allocation always exists.
... Early works by Wilson (1977) and Riley (1979) map the idea of a reaction to competitors by modifying the equilibrium concept to include expectations about competitor behavior. Instead of modifying the equilibrium concept, a second vein of research explicitly models dynamic insurance market interactions (Hellwig 1987;Engers and Fernandez 1987;Jaynes 1978;Asheim and Nilssen 1996;Ania, Tröger, and Wambach 2002). A third strand takes a different approach and does not include dynamics, but changes the contract or insurer characteristics (Inderst and Wambach 2001;Faynzilberg 2006;Picard 2009). ...
... 4 There are yet methodically different strands in the literature. Ania, Tröger, and Wambach (2002) take an evolutionary game theory approach, Guerrieri, Shimer, and Wright (2010) consider a competitive search model; other directions use cooperative concepts (Lacker and Weinberg, 1999) or a general-equilibrium framework (see e.g. Dubey and Geanakoplos (2002) or Bisin and Gottardi (2006)). ...
It has been shown that in competitive insurance markets with adverse selec-tion where the single crossing property does not hold, profit-making contracts can exist in equilibrium. Recently, Snow (2009) has argued that this result is an artefact of the assumption that in the models used firms only offer single contracts. We show that this implication is not correct. Even with multiple contract offers profit making contracts can exist in equilibrium, depending on the specification of the game.
... This approach has led to many important economic applications, such as oligopolies [V97], [RS01], [AFS00], signalling games [NS97], and insurance markets [ATW02], to mention a few. ...
Evolutionary game dynamics is the application of population dynamical methods to game theory. It has been introduced by evolutionary biologists, anticipated in part by classical game theorists. In this survey, we present an overview of the many brands of deterministic dynamical systems motivated by evolutionary game theory, including ordinary differential equations (and, in particular, the replicator equation), differential inclusions (the best response dynamics), difference equations (as, for instance, fictitious play) and reaction-diffusion systems. A recurrent theme (the so-called ‘folk theorem of evolutionary game theory’) is the close connection of the dynamical approach with the Nash equilibrium, but we show that a static, equilibrium-based viewpoint is, on principle, unable to always account for the long-term behaviour of players adjusting their behaviour to maximise their payoff.
... Therefore the asymmetric information problem, as well as the group effect, vanishes in the long-run. This last example provides an evolutionary dynamic for the insurance market, where the contracts are revised in the form described in Section 2. Ania et al. (2002) defined another evolutionary dynamics followed by bounded rational insurance firms; they offer menus of insurance contracts which are periodically revised: profitable competitors' contracts are imitated and loss-making contracts are withdrawn. In the words of Wilson (1977, p. 205), it lacks "an explicitly dynamic model, which describes how firms adjust their policies over time". ...
In asymmetric information problems, agents with less information (principals or contractors) usually take as given the preferences of agents with more information (agents or contractees). Moreover, the distribution of characteristics of contractees is supposed to be invariant. In this article we consider a mixed framework of asymmetric information (adverse selection followed by moral hazard) where those two assumptions are excluded. Specifically, the contractor only knows the current distribution of characteristics and the contractees may change them after signing the contract, if this improves their welfare. Thus, we find that the asymmetric information problem leads to a group effect (changes of characteristics). This feedback defines a sequence of temporary equilibria. We provide conditions for the convergence of that sequence to a stationary long run equilibrium. We also prove that both temporary equilibrium and long-run equilibrium coincide with the equilibrium in classical models of adverse selection and the moral hazard problem vanishes in the long-run..
... Although the AIP products themselves are completely visible, certain aspects such as policy clauses, calculation of premiums, and degrees of coverage do require professional knowledge to understand. This asymmetric information frequently leads to an "adverse selection" phenomenon (from insurers' perspectives), which further complicates the insurance provision (Jee, 1989;Landsberger and Meilijson, 1994;Lewis and Sappington, 1995;Ligon and Thistle, 1996;Inderst and Wambach, 2001;Ania et al., 2002;Theilen, 2003). ...
This paper develops a multinomial logit (MNL) model to identify important factors affecting the selection of automobile insurance policies (AIPs), which are characterized with high similarity among the bundled alternatives. Various nested logit (NL) model specifications are attempted to elucidate the choice behaviors for these highly similar auto insurance products. Some 3,000 voluntary automobile insurance records, randomly drawn from a non-life insurance company in Taiwan, are used to conduct the empirical study. The estimation result of the preferred MNL model shows that vehicle usage, engine capacity, imported/domestic vehicle, and number of claims are significant factors influencing one's selection of AIPs. Owners of newer vehicles, of larger engine capacities, of imported vehicles, and of more insurance claims tend to purchase wider coverage of insurance packages. The NL models confirm that the bundled AIPs are similar and highly correlated products. The empirical results also provide evidence of adverse selection in automobile insurance market --the wider the insurance coverage, the less motivation the insured would reduce the number of claims (or equivalently, prevent the accidents).
... We combine the two latter strands of the literature by extending the dynamic structure of the game to incorporate upfront capital choice and endogenously allow for firm insolvency, similar to Faynzilberg (2006). 7 Despite this multitude of approaches, mainly two possible equilibrium allocations have emerged: The original RS allocation, even if it is not an equilibrium in the RS model (e.g Riley 1979, Inderst and Wambach 2001, Ania et al. 2002), or the WMS allocation (e.g. Asheim and Nilssen 1996, Picard 2009, Mimra and Wambach 2009). ...
Even 30 years after Rothschild and Stiglitz (1976)'s seminal work on compet-itive insurance markets with asymmetric information, (in)existence of a pure-strategy equilibrium is still an open issue. We propose a basic extension to the Rothschild and Stiglitz (1976) model to solve the equilibrium inexistence problem: We endogenize upfront capital of insurers. Under limited liability, low upfront capital gives rise to an endogenous insolvency risk. This intro-duces an externality among customers of an insurer that guarantees equilib-rium existence: an equilibrium in pure strategies with the second-best efficient Wilson-Miyazaki-Spence (WMS) allocation always exists.
... 7 Picard (1996); Viaene et al. (2002); Major and Riedinger (2002); Brockett et al. (2002); Viaene and Dedene (2004). 8 Murray et al. (1994); Cummins and Weiss (1991); Dionne and Doherty (1994); Ania et al. (2002); Janssen and Karamychev (2005). 9 Dellaert et al. (1990Dellaert et al. ( , 1993; Lee and Urrutia (1996); Lee and Ligon (2001); Renn (2004); Caudill et al. (2005). ...
Automobile insurance policies (AIPs) are typically offered by the insurers in different bundled formats and some of which may be highly similar. This study proposes a two-component modelling system, which consists of choice of physical damage coverage and choice of non-physical damage coverage. Both multinomial logit (MNL) and paired combinatorial logit (PCL) models are attempted to explain the choice behaviours of AIP alternatives. The proposed models are tested with a large data set drawn from a Taiwanese non-life insurance company. It is found that the PCL model is structurally superior to the MNL model in analysing the choice of physical damage bundled AIP alternatives. In the context of non-physical damage coverage choice, however, the assumptions from the MNL model hold, suggesting that the use of PCL model is not required. Based on our estimation results, the insurance providers can develop marketing strategies to refine their existing AIPs or to develop new AIPs to better serve their customers.
... However, absent the particular refinement structure used in their model, there is an enormous multiplicity of equilibria. Ania et al. (2002) provide evolutionary analysis of the markets with adverse selection and conditions under which they attain equilibrium. While each model in this short in comparison to the existing literature list provides an appealing alternative to the principles behind the RS notion of equilibrium, assessing which of the principles governs actual markets is ultimately an empirical question. ...
A variation of the Rothschild-Stiglitz' equilibrium is examined in the context of competitive lending under adverse selection.
The predictions of the model are tested in an experimental market setting. If equilibrium exists, the loan contracts offered
and taken should separate projects by quality. When equilibrium exists, the experiments confirm the theory. The entrepreneurs
with high-risk projects take bigger loans and pay higher credit spreads than those with low-risk projects. When equilibrium
does not exist, which happens exactly when the candidate equilibrium does not provide a Pareto-optimal allocation, in half
of the sessions loan trading stabilizes around the candidate equilibrium pair. In the other half, however, markets never settle
down. This finding has important implications. When lenders can offer menus of contracts, as is usually the case in reality,
the outcome may not be the zero-profit separating contracts of the standard model. Worse, fitting the standard model to field
data may lead to serious biases in estimated parameters while falsely accepting the model's main restriction (separation).
... While Nöldeke and Samuelson found some support also for pooling equilibria, Jacobsen et al. found strong support for the Riley equilibrium. Likewise,Ania et al. (2002)applied similar tools to Rothschild's and Stiglitz' (1976) screening model. It would be interesting to investigate the applicability and power of those tools in the present model. ...
We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
... The stream of the learning literature initiated by Kandori, Mailath, and Rob (1993) and Young (1993) has seen a growing number of applications to oligopoly theory in the last years. Starting with the analysis of a Cournot oligopoly presented in Vega-Redondo (1997) (extended in different directions in AlósFerrer, Ania, and Vega-Redondo (1999) , Tanaka (1999) and Hoppé (2000) ), further applications have been studied for Bertrand oligopolies (Alós- Ferrer, Ania, and Schenk-Hoppé (2000) and Hehenkamp (2000) ), differentiatedgoods oligopolies (Tanaka (2000)), signaling (Nöldeke and Samuelson (1997)), and insurance markets (Ania, Tröger, and Wambach (2001)). All these models are based on the ideas pointed out by Alchian (1950). ...
This paper explores the impact of memory in Cournot oligopolies where firms learn through imitation of success as suggested in Alchian [J. Polit. Econ. 57 (1950) 211] and modeled in Vega-Redondo [Econometrica 65 (1997) 375]. As long as memory includes at least one period, the long-run outcome corresponds to any quantity between the Cournot one and the Walras one. The (evolutionary) stability of the walrasian outcome relies on interfirm comparisons of simultaneously observed profits (i.e., whether a firm earns more than others in a given period), whereas the stability of the Cournot-Nash equilibrium is derived from intertemporal comparisons of profits for each given firm (i.e., whether a deviation pays off for that firm).
We propose a model of coupled population games where intra- and intergroup interactions overlap. We analyze the general class of symmetric 2×2 games with coupled replicator dynamics in this framework. Standard one- and two-population predictions extend to a total of ten regions with different sets of attractors, among them novel hybrid points where one population randomizes and the other plays a pure strategy. Building on the theoretical analysis, we run continuous-time laboratory experiments using 48 different variants of coupled games. Observations confirm the theory to a large extent, but we also find a number of systematic deviations. When the attractors' eigenvalues are smaller (in absolute terms), play converges to steady states located further from the prediction.
Classical microeconomics is intended to explain how a price system is able to coordinate the economic agents. But even if it can be extended to incomplete information and externalities, it remains grounded on very heroic assumptions. Agents are endowed with a very strong rationality, equilibrium is stated without a concrete process to achieve it, market is the unique institution considered. Evolutionary microeconomics is aimed at bypassing these limitations by considering a dynamic approach, however not biologically oriented. Agents have local information and bounded rationality, they are involved in explicit processes of interactions through time, various institutions sustain the market or substitute to it. It explains then some phenomena hardly explained by classical microeconomics: dispersion of prices, variety of industrial structures, financial bubbles.
Glossary Definition of the Subject Introduction Normal Form Games Static Notions of Evolutionary Stability Population Games Revision Protocols Deterministic Dynamics Stochastic Dynamics Local Interaction Applications Future Directions Acknowledgments Bibliography
We provide an overview of the paths taken to understand existence and efficiency of equilibrium in competitive insurance markets with adverse selection since the seminal work by Rothschild and Stiglitz (1976). A stream of recent work reconsiders the strategic foundations of competitive equilibrium by carefully modelling the market game.
In this relatively short survey, we present the core elements of the microeconomic analysis of insurance markets at a level suitable for senior undergraduate and graduate economics students. The aim of this analysis is to understand how insurance markets work, what their fundamental economic functions are, and how efficiently they may be expected to carry these out.
With the rapid development of the national real estate industry, we are encountering an unprecedented engineering quality safety challenge in China, and it requires that we should establish the construction quality insurance system to conform to the situation. In order to resolve the moral hazard and adverse selection problems of the contractor in the construction quality insurance system, this article designs a insurance contract with premium differentiation for contractors to be selected where both contractor's types and actions are unobservable, so as to judge their true information in terms of their selecting results and make them work hard. Following the revelation principle, we analyze and solve the model by applying the optimal controlling theory. It could provide further theoretical foundation for the insurer who design the reasonable incentive contract.
When the strategy set of a game is a continuum, its discretization may not conserve local properties even for arbitrarily fine strategy grids. Two technical lemmas are proven to solve the problem, one relying on differentiability properties and another for higher dimensions formulated for open-graph correspondences. Four applications are shown, regarding the discretization of Cournot and Bertrand settings, a consumer optimization problem, and an insurance market.
We analyze the use of information in a repeated oligopolistic insurance market. To sustain collusion, insurance companies might refrain from changing their pricing schedules even if new information about risks becomes available. We therefore provide an explanation for the existence of "unused observables" that is information which
We extend the seminal Rothschild and Stiglitz (1976) model on competitive insurance markets with asymmetric information in the spirit of Wilson (1977)’s ‘anticipatory equilibrium’ by introducing an additional stage in which initial contracts can be withdrawn after observation of competitors’ contract offers. We show that an equilibrium always exists where consumers obtain their respective Wilson-Miyazaki-Spence (WMS) contract. Jointly profit-making contracts can also be sustained as equilibrium contracts. However, the second-best efficient WMS allocation is the unique equilibrium allocation under entry.
Abstract We model a Hotelling market with multidimensional product differentiation in an evolutionary framework. Both evolutionary stability (in the sense of Schaffer, 1989) and stochastic stability (following [Kandori et al., 1993] and [Young, 1993]) are analyzed. It is shown that firms move towards the center in product space, i.e. a "principle of minimum differentiation" on all dimensions of the product space applies.
Mit der Schaffung des europäischen Versicherungsbinnenmarktes wurde 1994 auch der deutsche Versicherungsmarkt liberalisiert. Damit stehen seither den Unternehmen auf diesen Märkten mit der Produktgestaltung, der Prämienkalkulation und der Risikoklassifikation neue Wettbewerbsinstrumente zur Verfügung. Zusätzlich ist der Markteintritt nationaler Unternehmen in andere europäische Märkte erleichtert worden. Die Nutzung der neuen Instrumente im Rahmen von Unternehmensstrategien vor dem Hintergrund eingeschränkter Information und geringer Erfahrung der Marktteilnehmer sowie die Möglichkeiten wohlfahrtssteigernder Markteingriffe sind Inhalt der vorliegenden Untersuchung. Es zeigt sich, daß Informationsbeschränkungen und Marktmacht immanent sind. Die sich daraus ergebenden Marktunvollkommenheiten lassen sich durch einen eingeschränkt informierten Regulierer oder Staat nicht vollständig beheben. Dafür sind Eingriffe möglich, welche die negativen Effekte abzumildern vermögen und dabei einem sehr geringem Informationserfordernis unterliegen. Relevante Informationen beziehen sich in erster Linie auf die Schadenscharakteristika von Versicherungsnehmern. Ein einzelnes Unternehmen profitiert von einem Informationsvorsprung gegenüber seinen Konkurrenten, wenn es diesen für verstärkte Risikoklassifikation einsetzen kann. Aus sozialer Sicht ist es aber sinnvoller, wenn existierende Informationen allen Unternehmen zur Verfügung stehen, so daß eine einheitliche Risikoklassifikation möglich wird. Die hierfür erforderlichen Informationen können nur Beobachtung der Kunden gewonnen werden. Dann hat ein Unternehmen einen Informationsvorsprung gegenüber seinen Konkurrenten in Bezug auf die Charakteristika seiner Kunden. Dieses Effekte begründen, warum Informationsbeschränkungen und Marktmacht auf einem Versicherungsmarkt immanent sind. Versicherungsnehmer, die sich gegen ein Schadensrisiko absichern, stehen Versicherungsschutz und Prävention als substitutive Instrumente zur Verfügung. Die geeignete Wahl der Versicherungsprämie kann effiziente Prävention seitens der Versicherungsnehmer bewirken. Wenn wegen der Informationsbeschränkung verschiedene Risikoklassen nicht identifiziert werden können, kann nur eine einheitliche Prämie für alle Klassen erhoben werden. Dann kann effiziente Prävention nicht bei allen Risikoklassen vorliegen. Ein Eingriff, der diese negativen Wohlfahrtseffekte abmildern kann, besteht in der Vorgabe einer fixen Versicherungsdeckung, so daß die Anreize der verschiedenen Risikoklassen in die richtige Richtung gelenkt werden. Die Marktmacht eines Unternehmens ermöglicht ihm, risikoklassenspezifische Prämien über der fairen Prämie zu wählen. Ein Unternehmen hat stets den Anreiz zu Prämiendifferenzierung, da es ihm einen höheren Gewinn zu erzielen erlaubt. Dagegen ergibt eine Wohlfahrtsuntersuchung, daß eine einheitliche Prämie sozial wünschenswert ist. Ein Verbot der Prämiendifferenzierung ist ein einfaches Mittel, für welches bei einem staatlichen Eingriff nur sehr geringen Informationsanforderungen bestehen. Es zeigt sich, daß die negativen Auswirkungen der Marktmacht dadurch abzumildern sind, indem die Marktmacht eingeschränkt wird, und zwar indem dem Unternehmen nicht die Nutzung aller Wettbewerbsinstrumente gestattet wird. Along with the creation of the European single insurance market in 1994 the German insurance market was liberalized. Since then, the firms are free in their product design, their premium calculation, and their risk classification. Furthermore, market entry within European markets is facilitated. Firm strategies in applying the new instruments and also market interventions aiming at welfare improvements are subject matters of the present investigation. Both are analyzed for an environment with information constraints and sparse experience of the market participants. One result is that information constraints and market power are immanent. The resulting market imperfections cannot be eliminated completely by a regulatory or state authority which also suffers some information constraints. However, there exist some interventions which require a very low amount of information and which can mitigate the negative effects of the market imperfections. In general, relevant information refers to the loss characteristics of consumers. A single firm benefits from an informational advantage compared to its competitor when it intensifies its risk classification activities. From the social perspective, it would be more efficient when existing information is available to all firms on the market to arrange a uniform risk classification. The necessary information can be generated from observing the loss statistics of the customers. Then a firm gains an informational advantage compared to its competitors on the loss characteristics of its customers. These effects explain why informational constraints and market power are immanent on insurance markets. Insurance coverage and loss prevention are two substitute instruments which can be utilized by consumers facing a potential loss. The appropriate insurance premium will induce an efficient use of prevention by the customers. Information constraints may inhibit the identification of distinct risk classes. Then only a uniform premium for different risk classes will be possible and loss prevention cannot be efficient for all risk classes anymore. An intervention which will reduce the resulting welfare loss may consist in a fixed insurance coverage which has to be purchased by all risk classes. Then all risk classes have an incentive to correct the prevention activities towards the efficient level. Market power allows a firm to apply price discrimination. In the context of insurance markets this means that a firm prefers to ask a specific premium from each risk class. Conversely, a welfare analysis shows that a uniform premium may be socially preferable. Imposing a uniform premium is a simple measure with very low informational requirements. Such an intervention may reduce the welfare loss from price discrimination. This shows that the negative effects resulting from market power may be alleviated by restricting the instruments which can be applied by the firm which has market power.
The goal of this essay is to show an insurance market equilibrium defined by an insurance product price and a probability of insolvency for the insurer(s).
This paper considers price discrimination when competing firms do not observe a customer’s type but only some other variable correlated to it. This is a typical situation in many insurance markets—such as motor insurance—where it is also often the case that insurance is compulsory. We characterise the equilibria and their welfare properties under various price regimes. We show that discrimination based on immutable characteristics such as gender is a dominant strategy, either when firms offer policies at a fixed price or when they charge according to some consumption variable that is correlated to costs. In the latter case, gender discrimination can be an outcome of strategic interaction alone in situations where it would not be adopted by a monopolist. Strategic price discrimination may also increase cross subsidies between types, contrary to expectations. Copyright The Geneva Association 2005
The debate about whether insurance companies should be allowed to use results of genetic tests for underwriting purposes is both lively and increasingly relevant as both technology and lawmaking efforts are progressing rapidly. Much concern has been raised about whether allowing firms to use such information to create risk-rated insurance premiums (genetic ratemaking) is unfairly discriminatory. We show that, especially in the context of the manner in which insurance markets operate, arguments about the appropriateness of allowing such information use by insurers which are based on discrimination are very fragile from a conceptual or philosophical perspective. Moreover, a focus on discrimination is not very helpful from a pragmatic or policy oriented perspective. We argue that adopting a Utilitarian welfare framework, as inspired by Harsanyi’s (1953,1955) veil of ignorance interpretation, in order to determine whether risk-rating by use of genetic test results is morally defensible provides a much more promising line of reasoning for settling this controversial issue.
This paper surveys conflicts of interest in the corporate governance arena, with emphasis on auditors, boards of directors, analysts and investment bankers, regulators, management, attorneys and investors. Enron provides a host of examples as well. I stress the multifaceted nature of these conflicts, and the fact most research looks at some conflicts, such as auditor independence, absent the larger setting and potential interactions among various players. I further speculate herding behavior is an important explanatory device in understanding periodic failures.
With the rapid development of the national real estate industry, we are encountering an unprecedented engineering quality safety challenge in China, and it requires that we should establish the construction quality insurance system to conform to the situation. In order to resolve the moral hazard and adverse selection problems of the contractor in the construction quality insurance system, the insurer must design a set of optimal incentive contracts for contractors to be selected, so as to judge their true information in terms of their selecting results and make them work hard. Considering the contractorpsilas intrinsic motivation, this paper sets up an optimal incentive contract model when both contractorpsilas types and actions are unobservable. Following the revelation principle, we analyze and solve the model by applying the optimal controlling theory, and we discuss the importance of intrinsic motivation. The conclusions show that the intrinsic motivation is more efficient than extrinsic incentive. Accordingly the insurer should fully consider the contractorspsila intrinsic motivation in the contract.
This paper gives an overview of a collector-emitter voltage sign-based commutation method. This method is implemented based on the output current sign and commutation voltage polarity signals of matrix converter. All these signals are acquired by a simple unit in the drive and protection boards designed for reverse blocking insulated gate bipolar transistors (RB-IGBTs) of the matrix converter in test. Therefore, no current and/or voltage sensor is required to obtain any other commutation signals. The RB-IGBT samples used are newly developed, and one important feature is that its reverse leakage current is mostly affected by nuge. Under the same temperature condition, when nuce is negative, a positively biased nuge results in a distinct decrease of the reverse leakage current. The proposed commutation method can supply the reverse-blocked RB-IGBTs with positively biased nuge . Hence, the power losses caused by the reverse leakage current can be reduced using the proposed method. Experimental results have proved the feasibility and effectivity of this method
Do boundedly rational agents repeatedly playing a symmetric game with a unique symmetric equilibrium learn over time to play it? In this paper we model the dynamic interaction of two types of such agents, experimenters and imitators, whose behavior is characterized by simple rules of thumb. We find that the stochastic process describing their play isstable in the large: it converges globally and with probability one to a compact neighborhood of the equilibrium. However, its local behavior near the equilibrium depends in interesting ways on the details of the model.Journal of Economic LiteratureClassification Number: D83.
Given falling birth rates, ageing baby boomers approaching retirement age as well as a pension crisis in most advanced economies, understanding the characteristics of the labour supply function of the elderly have taken on a new significance. Even in developing countries, with labour surplus economies, this is a major issue as these poor countries try to build a pension scheme with at least a minimum amount of state provision for the elderly. What motivates retired people to enter or continue in the labour force is the focus of our analysis. We use panel data from Korea which is an interesting country since it transited from developing to developed economy status within the last few decades and therefore exhibits characteristics of both underdevelopment and economic advancement. The econometric methods include probit models of: pooled data; panel data with random effects; and 2SCML, to allow for possible endogeneity bias induced by the self-declared health status of the elderly. We stress the crucial importance of pecuniary and non-pecuniary factors in determining labour supply of the elderly. Contrary to expectations, non-pecuniary factors such as health status are crucial in the decision-making process of whether to work or not to work for the elderly.
The first edition of this book was published in 1979 in Russian. Most of the material presented was related to large-deviation theory for stochastic pro cesses. This theory was developed more or less at the same time by different authors in different countries. This book was the first monograph in which large-deviation theory for stochastic processes was presented. Since then a number of books specially dedicated to large-deviation theory have been pub lished, including S. R. S. Varadhan [4], A. D. Wentzell [9], J. -D. Deuschel and D. W. Stroock [1], A. Dembo and O. Zeitouni [1]. Just a few changes were made for this edition in the part where large deviations are treated. The most essential is the addition of two new sections in the last chapter. Large deviations for infinite-dimensional systems are briefly conside:red in one new section, and the applications of large-deviation theory to wave front prop agation for reaction-diffusion equations are considered in another one. Large-deviation theory is not the only class of limit theorems arising in the context of random perturbations of dynamical systems. We therefore included in the second edition a number of new results related to the aver aging principle. Random perturbations of classical dynamical systems under certain conditions lead to diffusion processes on graphs. Such problems are considered in the new Chapter 8.
We study a market for a homogeneous good in which firms adjust their production decisions on the basis of imitation, learning from own experience, and local experimentation. For any fixed set of firms (more than one), long run behaviour settles on a symmetric marginal-cost pricing equilibrium. When market entry and exit are allowed, we find a sharp effect of technology on long-run market structure. Specifically, we show that, under decreasing returns and some fixed cost, the market grows to "full capacity" at Walrasian equilibrium; on the other hand, if returns are increasing, the unique long run outcome involves a profit-maximising monopolist.
This paper presents an evolutionary model of Bertrand competition in a market for a homogeneous good, where identical firms face a technology with decreasing returns to scale. Only quoted prices and realized profits are observed. The behavior of firms is based on imitation of success and experimentation, and is formally modeled through behavioral principles. We find that, even under simple behavior, the dynamic process selects a strict subset of the Nash equilibria of the underlying game. In the long run all firms make positive profits. Adding more sophistication, we obtain a finer prediction, named "central prices.'' This prediction essentially coincides with the Walrasian equilibrium, if costs are quadratic.
Consider an n-person game that is played repeatedly, but by different agents. In each period, n players are drawn at random from a large finite population. Each player chooses an optimal strategy based on a sample of information about what others players have done in the past. The sampling defines a stochastic process that, for a large class of games that includes coordination games and common interest games, converges almost surely to a pure strategy Nash equilibrium. Such an equilibrium can be interpreted as the "conventional" way of playing the game. If, in addition, the players sometimes experiment or make mistakes, then society occasionally switches from one convention to another. As the likelihood of mistakes goes to zero, only some conventions (equilibria) have positive probability in the limit. These are known as stochastically stable equilibria. They are essentially the same as the risk dominant equilibria in 2 × 2 games, but for general games the two concepts differ. The stochastically stable equilibria are computed by finding a path of least resistance from every equilibrium to every other, and then finding the equilibrium that has lowest overall resistance. This is a special case of a general theorem on perturbed Markov processes that characterizes their stochastically stable states graph-theoretically.
In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise:
where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of .
A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems.
By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit.
In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain.
Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.
This textbook aims to provide a comprehensive overview of the essentials of microeconomics. It offers unprecedented depth of coverage, whilst allowing lecturers to 'tailor-make' their courses to suit personal priorities. Covering topics such as noncooperative game theory, information economics, mechanism design and general equilibrium under uncertainty, it is written in a clear, accessible and engaging style and provides practice exercises and a full appendix of terminology.
This paper reconsiders the proposition, due to Jaynes, that the Rothschild-Stiglitz-Wilson insurence market with adverse selection has a Nash equilibrium once the sharing of information about customers is treated endogenously as part of the game among firms. The proposition is shown to be correct if and only if each firm's communication behaviour is conditioned on the set of contracts that are offered by the other firms. This type of conditioning introduces a reactive element which is akin to the alternative approach of Wilson.
Ever since the seminal work by Rothschild and Stiglitz (Q. J. Econom. 90 (1976) 629) on competitive insurance markets under adverse selection, the problem of non-existence of equilibrium in pure strategies has received much attention in the literature. We extend the original analysis by considering firms which face capacity constraints, which might be due to limited capital. We show that under mild assumptions an equilibrium exists, where every consumer obtains his appropriate Rothschild–Stiglitz contract.
This paper examines an evolutionary model in which the primary source of “noise” that moves the model between equilibria is not arbitrarily improbable mutations but mistakes in learning. We model strategy selection as a birth–death process, allowing us to find a simple, closed-form solution for the stationary distribution. We examine equilibrium selection by considering the limiting case as the population gets large, eliminating aggregate noise. Conditions are established under which the risk-dominant equilibrium in a 2×2 game is selected by the model as well as conditions under which the payoff-dominant equilibrium is selected.Journal of Economic LiteratureClassification Numbers C70, C72.
The paper investigates how a competitive market would allocate insurance policies if firms were not able to determine the riskiness of individual consumers. It is demonstrated that if all firms have static expectations with regard to the policy offers of other firms, no stationary equilibrium may exist. A second equilibrium concept is then introduced which incorporates a different expectation rule. Each firm assumes that any policy will be immediately withdrawn which becomes unprofitable after that firm makes its own policy offer. This equilibrium is shown to exist and some of its welfare properties are investigated.
Economic theorists traditionally banish discussions of information to footnotes. Serious consideration of costs of communication, imperfect knowledge, and the like would, it is believed, complicate without informing. This paper, which analyzes competitive markets in which the characteristics of the commodities exchanged are not fully known to at least one of the parties to the transaction, suggests that this comforting myth is false. Some of the most important conclusions of economic theory are not robust to considerations of imperfect information.
This chapter discusses job market signaling. The term market signaling is not exactly a part of the well-defined, technical vocabulary of the economist. The chapter presents a model in which signaling is implicitly defined and explains its usefulness. In most job markets, the employer is not sure of the productive capabilities of an individual at the time he hires him. The fact that it takes time to learn an individual's productive capabilities means that hiring is an investment decision. On the basis of previous experience in the market, the employer has conditional probability assessments over productive capacity with various combinations of signals and indices. This chapter presents an introduction to Spence's more extensive analysis of market signaling.
The labor market is viewed as a market for labor contracts. A firm is identified as having an internal labor market if the efficient mode of production requires that it employ heterogeneous worker types by offering a wage structure as a set of subsidizing contracts. If the firm is free to offer its choice of wage structures, and if the Wilson notion of equilibrium is considered, then certain irreducible combinations of wage-job contracts will obtain. Depending upon the distribution of worker types, wages in these equilibrium wage structures may not correspond to the marginal productivities of individual workers, but the firm breaks even because the wages of high productivity types subsidize low productivity types within the firm. The theoretical framework of our model is based on recent contributions to the theory of self-selection screening.
Rothschild and Stiglitz have shown than insurance markets and other markets in which an adverse-selection problem exists cannot have Nash-type pooling or subsidized separating equilibria and are unlikely to have Nash-type unsubsidized separating equilibria. Wilson, Miyazaki, and Riley have analyzed anticipatory pooling and subsidized-separating equilibria and reactive unsubsidized separating equilibria. Each of these alternatives to Nash-type equilibria requires strategic behavior on the part of insurance sellers for support. The present paper analyzes as another alternative pooling equilibria that require dissembling behavior on the part of insurance buyers for support. This dissembling model has the attractive feature that it takes explicit account of the convention of requiring insurance buyers to submit applications, a practice that the analysis interprets to be a natural response to the adverse-selection problem.
We show that for many classes of symmetric two-player games, the simple decision rule ""imitate-the-best"" can hardly be beaten by any other decision rule. We provide necessary and sufficient conditions for imitation to be unbeatable and show that it can only be beaten by much in games that are of the rock-scissors-paper variety. Thus, in many interesting examples, like 2x2 games, Cournot duopoly, price competition, rent seeking, public goods games, common pool resource games, minimum effort coordination games, arms race, search, bargaining, etc., imitation cannot be beaten by much even by a very clever opponent.
We study equilibrium selection by evolutionary learning in monotone signalling games. The learning process is a development of that introduced by Young for static games extended to deal with incomplete information and sequential moves; it thus involves stochastic trembles. For vanishing trembles the process gives rise to strong selection among sequential moves equilibria. If the game has separating equilibria, then in the long run only play according to a specific separating equilibrium, the so-called Riley equilibrium, will be observed frequently. This selection, is stronger than, and only partly in accordance with, traditional selection based on restrictions on "out-of-equilibrium" beliefs.
We analyze the long-run outcome of markets in which boundedly rational firms with a decreasingreturns to scale technology compete in prices. The behavior of these firms is based on limitation ofsuccess and experimentation. In this framework, we introduce a new approach to model boundedlyrational behavior, based on the idea of behavioral principles, i.e. formal descriptions. Even with thesimplest ones, the result is that the prices announced are a strict refinement of the set of Nashequilibria. With more sophisticated behavioral principles, the long-run outcome corresponds to theconcept of central prices (wich are also Nash equilibria) introduced here. This is a robust andclear-cut prediction wich, under quadratic costs and arbitrary demand, essentially coincides with theWalrasian equilibrium.
In his work on market signaling, Spence proposed a dynamic model of a signaling market in which a buyer revises prices in light of experience and sellers choose utility-maximizing signals given these prices. Spence also suggested that subjecting the dynamic process to rare perturbations might allow one to choose between multiple equilibria. This paper examines the effect of introducing such perturbations into Spence's dynamic model. We find that refinement results arise naturally from the dynamic analysis. In a broad class of markets, our model selects a separating equilibrium outcome if and only if the equilibrium outcome satisfies a version of the undefeated equilibrium concept, whereas a pooling equilibrium outcome is selected if and only if the equilibrium outcome is both undefeated and satisfies D1.
In this paper we model the dynamic interaction of two types of agents, experimenters and imitators, whose behavior is characterized by simple rules of thumb. The agents repeatedly play a one-shot game in which the agent's actions are strategic substitutes.
We examine the evolutionary foundations of common equilibrium refinement ideas for extensive form games, such as backward and forward induction, by examining the limiting outcome of an evolutionary process driven by stochastic learning and (rare) mutations. We show that the limiting outcome in a class of extensive form games with perfect information always includes the subgame perfect equilibrium outcome, but also contains other outcomes if the subgame perfect outcome fails a (strong) local stability property. The evolutionary system imposes a forward induction requirement that strengthens van Damme's; it selects announcement proof equilibria in a class of cheap talk games; and it yields refinement results in some signaling games. The evolutionary model thus yields results that would often be interpreted as satisfying forward induction, but does not always impose sufficient discipline on actions and conjectures at unreached subgames to yield results consistent with backward induction.
Individuals in a finite population repeatedly choose among actions yielding uncertain payoffs. Between choices, each individual observes the action and realized outcome ofoneother individual. We restrict our search to learning rules with limited memory that increase expected payoffs regardless of the distribution underlying their realizations. It is shown that the rule that outperforms all others is that which imitates the action of an observed individual (whose realized outcome is better than self) with a probability proportional to the difference in these realizations. When each individual uses this best rule, the aggregate population behavior is approximated by the replicator dynamic.Journal of Economic LiteratureClassification Numbers: C72, C79, D83.
An evolutionary model with a finite number of players and with stochastic mutations is analyzed. The expansion and contraction of strategies is linked to their current relative success, but mutuation, perturbing the system from its deterministic evolution, are present as well. The focus is on the long run implications of ongoing mutations, which drastically reduce the set of equilibria. For 2 by 2 symmetric games with two symmetric strict Nash equilibria the risk dominant equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium. Copyright 1993 by The Econometric Society.
Consider an evolutionary context where a given number of quantity-setting oligopolists tend to mimic successful behavior, occasionally experimenting with some small probability. In this context, it is shown that the unique long-run outcome of the process has all firms playing Walrasian, i.e., choosing an output that maximizes profits when taking the market-clearing price as given. (This abstract was borrowed from another version of this item.)
This paper analyses, in a simple two-region model, the undertaking of noxious facilities when the central government has limited prerogatives. The central government decides whether to construct a noxious facility in one of the regions, and how to …nance it. We study this problem under both full and asymmetric information on the damage caused by the noxious facility in the host region. We particularly emphasize the role of the central government prerogatives on the optimal allocations. We …nally discuss our results with respect to the previous literature on NIMBY and argue that taking into account these limited prerogatives is indeed important.
This paper analyses, in a simple two-region model, the undertaking of noxious facilities when the central government has limited prerogatives. The central government decides whether to construct a noxious facility in one of the regions, and how to …nance it. We study this problem under both full and asymmetric information on the damage caused by the noxious facility in the host region. We particularly emphasize the role of the central government prerogatives on the optimal allocations. We …nally discuss our results with respect to the previous literature on NIMBY and argue that taking into account these limited prerogatives is indeed important.
Curb sets [Basu and Weibull, Econ. Letters 36 (1991), 141-146] are product sets of pure strategies containing all individual best-responses against beliefs restricted to the recommendations to the remaining players. The concept of minimal curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium. We introduce the concept of minimal strong curb sets which is a set-theoretic coarsening of the notion of strong Nash equilibrium. Strong curb sets are product sets of pure strategies such that each player's set of recommended strategies must contain all coalitional best-responses of each coalition to whatever belief each coalition member may have that is consistent with the recommendations to the other players. Minimal strong curb sets are shown to exist and are compared with other well known solution concepts. We also provide a dynamic learning process leading the players to playing strategies from a minimal strong curb set.
If buyers are less well-informed about product quality than sellers, market prices will reflect average quality. Sellers of high quality products therefore have an incentive to engage in some distinguishing activity which operates as a signal to potential buyers. This paper explores the viability of such signalling or "informational equilibria." It is established that with a continuum of quality levels there is no Nash equilibrium. An alternative non-cooperative equilibrium concept is then developed in which potential price searching agents take account of possible reactions by other agents. It is shown that there is a unique "reactive" informational equilibrium.
This experiment was designed to test various learning theories in the context of a Cournot oligopoly. The authors derive theoretical predictions for the learning theories and test these predictions by varying the information given to subjects. The results show that some subjects imitate successful behavior if they have the necessary information and, if they imitate, markets are more competitive. Other subjects follow a best reply process. On the aggregate level, the authors find that more information about demand and cost conditions yields less competitive behavior, while more information about the quantities and profits of other firms yields more competitive behavior.
The paper examines the behaviour of “evolutionary” models with ɛ-noise like those which have been used recently to discuss
the evolution of social conventions. The paper is built around two main observations: that the “long run stochastic stability”
of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution
is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states.
The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable
set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful,
the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times.
A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized
to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction
and cycles as long run stochastically stable sets.
In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. In contrast with the principal-agent models that are extended within, here the presence of ties in the agents' best-response correspondences yields discontinuous payoff functions for the agencies. These discontinuities are dealt with via efficient tie-breaking rules. The main results of this paper are a proof of existence of (mixed-strategies) Nash equilibria in the case of profit-maximizing agencies, and of socially efficient allocations when the firms are risk minimizers. It is also shown that in the particular case of the entropic risk measure, there exists an efficient "fix-mix" tie-breaking rule, in which case firms share the whole market over given proportions.
We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
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