Asymmetric Equilibrium Research Articles

Asymmetric equilibria of two nested elastic rings

The packing of soft elastic structures is an important and challenging problem due to the possibility of multiple discrete and continuous zones of contact between different parts of the material. To address this problem, we consider the simplest possible packing problem of a thin elastic ring confined within another shorter flexible ring. The elastic properties as well as the dimensionality of both structures, combined with the contact condition yield a wide a variety of possible equilibrium shapes. When the rings are assumed to be inextensible and unshearable, the equilibrium shapes depend only on their relative bending stiffness κ, and on their relative length μ. Whereas the symmetric equilibria for such a problem have been completely determined, the possibility of asymmetric equilibria with lower energy has not yet been considered. For a fixed value of the relative bending stiffness, we explore these symmetry-breaking equilibria as the length of the inner ring increases. We show that, for μ ≃ 1.9 there is a symmetry-breaking bifurcation and asymmetric equilibria are preferred in order to relax the elastic energy.

Equilibrium selection of a homogenous duopoly with extrapolative foresight

This paper conducts a further analysis concerning equilibrium selection in a widely-studied nonlinear Cournot duopoly game, where each firm is capable to forecast its rival's output in the near future through a straightforward extrapolative foresight technology. Compared with the prior related work in which just one firm can implement such technology, this newly developed adjustment mechanism is endowed with more rationality in the sense that there appear more intelligent firms. It turns out this more rational adjustment mechanism does lead to more convergence to asymmetric equilibrium behavior for a weak foresight ability. Our finding shows that some undesirable complex behavior such as fluctuations and divergent trajectories can be inhibited by increasing the number of such rational firms.

A generalized theory of the figure of the Earth: on the global dynamical flattening

A generalized theory of the figures of the Earth’s interior to a third-order precision of ellipticity is proposed in accompanying paper in which all the odd degree and nonzero order spherical harmonic terms are included. As both the direct and indirect contributions of the asymmetric crust are included, this theory makes a significant improvement for calculating the asymmetric equilibrium figures of the real Earth comparing with the traditional theories which can only deal with the ideal symmetric Earth. The principal moments of inertia (PMOI: A, B, C) and global dynamical flattening (H) are important quantities in studying the rotating Earth. Precession and gravity observations give observation value of H ( $$H_{\mathrm{obs}} \approx 1/305.4559$$ ) with very high precision, while its theoretical calculated value ( $$H_{\mathrm{theory}} \approx 1/308.5$$ ) from traditional theories and a starting symmetric Earth model (like PREM model) is about $$1\%$$ less than $$H_{\mathrm{obs}}$$ . Using the new theory in accompanying paper and replacing the homogeneous outermost crust and oceanic layers in PREM with CRUST1.0 model, we recalculate the equilibrium figures of the Earth’s interior and finally get new values of PMOI and $$H_{\mathrm{theory}}$$ ( $${\approx } \,1/304.7167$$ ) whose consistency with $$H_{\mathrm{obs}}$$ are significantly improved to 0.24%. Furthermore, the asymmetric figures of some interesting boundaries, like inner core boundary, core-mantle boundary, are also given as by-products of this work as these boundaries’ figures are key input for studies of their topographic effect on global rotation and geodynamics, like nutation, normal modes, especially like free core nutation.

EFFECTS OF SWITCHING COSTS ON INNOVATIVE INVESTMENT

Switching costs and innovation are two major issues in economics. Prior research demonstrates the effects of switching costs on competition, but ignores the influence of switching costs to firm innovation. So the purpose of this study is to reveal the relationships between switching costs and cost-reducing innovation by considering brand loyalty. All our theoretical conclusions are captured by game theory based on a two-stage duopoly model. The conclusions of this study show that under moderate conditions, switching costs improve competition. Strong firms implement lower price when switching costs are present than when they are not present. Second, at the asymmetric equilibrium, lower-efficiency firms with switching costs launch less innovative investments than do those without switching costs, while higher-efficiency firms with switching costs launch more innovation. But under symmetric equilibrium, switching costs have no effect on innovative investment. The novel contributions of this paper are that we find switching costs and loyalty have vertical impacts on firms’ cost-reducing innovation, which extends the theory of switching costs.

Medical service provider networks.

In many countries, health insurers or health plans choose to contract either with any willing providers or with preferred providers. We compare these mechanisms when two medical services are imperfect substitutes in demand and are supplied by two different firms. In both cases, the reimbursement is higher when patients select the in-network provider(s). We show that these mechanisms yield lower prices, lower providers' and insurer's profits, and lower expense than in the uniform-reimbursement case. Whatever the degree of product differentiation, a not-for-profit insurer should prefer selective contracting and select a reimbursement such that the out-of-pocket expense is null. Although all providers join the network under any-willing-provider contracting in the absence of third-party payment, an asymmetric equilibrium may exist when this billing arrangement is implemented.

Voltage-Induced Snap-Through of an Asymmetrically Laminated, Piezoelectric, Thin-Film Diaphragm Micro-Actuator—Part 2: Numerical and Analytical Results

Abstract In Part 2 of the two-paper series, the asymmetrically laminated piezoelectric shell subjected to distributed bias voltage as modeled in Part 1 is analytically and numerically investigated. Three out-of-plane degrees-of-freedom (DOFs) and a number of in-plane DOFs are retained to study the shell's snap-through phenomenon. A convergence study first confirms that the number of the in-plane DOFs retained affects not only the number of predicted equilibrium states when the bias voltage is absent but also the prediction of the critical bias voltage for snap-through to occur and the types of snap-through mechanisms. Equilibrium states can be symmetric or asymmetric, involving only a symmetric out-of-plane DOF, and additional asymmetric out-of-plane DOFs, respectively. For symmetric equilibrium states, the snap-through mechanism can evolve from the classical bidirectional snap-through and latching to a new type of snap-through that only allows snap-through in one direction (i.e., unidirectional snap-through), depending on the distribution of the bias voltage. For asymmetric equilibrium states, degeneration can occur to the asymmetric bifurcation points when the radii of curvature are equal. Finally, the unidirectional snap-through renders an explanation to the experimental findings in Part 1.

Asymmetric equilibria in spatial competition under weakly concave scoring rules

The Hotelling–Downs model of spatial competition is used to investigate the strategic position-taking behavior of firms or political parties under scoring rules. Previous studies of non-convergent Nash equilibria – equilibria in which divergent positions are chosen – found that they often do not exist and, when they do, they are fairly symmetric. In particular, this is true for convex scoring rules (Cahan and Slinko, 2017). Here, we investigate non-convergent equilibria for the broad class of weakly concave scoring rules. Surprisingly, we find that only asymmetric equilibria can exist, and we present several examples.

Pricing and Benefit of Decentralization for Competing Supply Chains With Fixed Costs

This paper develops two duopoly game models to explore price decisions and the channel benefit of decentralization for two supply chains, and investigates the channel structure decisions in the presence of fixed marketing and manufacturing costs. We give the price decisions and discuss price distortion. Under the downward decentralization model, our analysis reveals that first, symmetric decentralization improves supply chain profit only when the fixed marketing costs are high or product substitutability is high; second, decentralization improves channel profit, and symmetric decentralization (i.e., both manufacturers use decentralization) is an equilibrium if the fixed marketing (or unit production) costs are sufficiently high and the market scales are sufficiently small; and third, the presence of fixed marketing costs and asymmetry of supply chains support the existence of asymmetric equilibrium. Under the upward decentralization model, we find that symmetric outsourcing emerges when the fixed manufacturing cost or the unit production cost is low while both market scale and product substitutability are large in the supplier-led setting, the channel profit under downward decentralization is higher than that under symmetric outsourcing only if product substitutability is high, and the sourcing strategy largely depends on the specific game sequence.

A game-theoretic analysis of international trade and political conflict over external territories

For analyzing conflict between two large open countries over external territories rich in natural resources, we develop a game-theoretic model of trade under resource appropriation possibilities. We show that greater trade openness (by lowering trade costs) reduces the overall intensity of arming when the adversary countries are symmetric in all dimensions. This finding is consistent with the liberal peace hypothesis that trade reduces conflict. We further analyze how equilibrium is affected by differences in national resource endowments. The resulting asymmetric equilibrium reveals that arming by the more endowed country exceeds that of the less endowed country and the two adversaries respond to lower trade costs differently: the more endowed country cuts back on its arming, whereas the less endowed country may increase it. Under resource endowment asymmetry, the aggregate arming allocations of the adversaries could increase despite greater trade openness.

Asymmetries in the revenue–expenditure nexus: new evidence from South Africa

In this study, we relax the conventional assumption of a linear cointegration relationship in the revenue–expenditure nexus by examining asymmetric equilibrium effects in the South African fiscal budget using quarterly data collected between 1960:Q1 and 2016:Q2. Our mode of empirical investigation is the MTAR model supplemented with a TEC component. Our estimation results primarily point to a weakly sustainable budget in which there exists bidirectional causality between revenues and expenditures, a result which offers support in favour of the fiscal synchronization hypothesis. Collectively, our empirical results have important implications for South African fiscal policymakers.

Agency Model and Wholesale Pricing: Apple versus Amazon in the E-Book Market

Apple’s choice of the agency model (i.e., Apple demands a share from the retail price set by the publishers) when entering the e-book market was surprising because: (i) the upstream firms can accrue all rents in a simultaneous move game if it determines the retail price; and (ii) the incumbent, Amazon, used wholesale pricing arrangements. This paper compares the two different contract types, pure and mixed: one retailer opts for wholesale, the other for the agency model. Departing from a standard and symmetric oligopolistic setup of Bertrand competing retailers and producers, the model accounts for retailers having (a) a significant contribution to the final value and (b) a strategic first-mover advantage. Both conditions combined are necessary (but not sufficient) in order to explain Apple’s choice and the possibility of an asymmetric equilibrium.

Scrape-off-layer current loops and floating potential in limited tokamak plasmas

We investigate the question of how plasma currents circulate and close in the scrape-off-layer (SOL) of convection-limited tokamak plasmas. A simplified two-fluid model describes how currents must evacuate charge at the sheaths due to cross-field currents that are not divergence-free. These include turbulence-driven polarization currents and poloidally asymmetric equilibrium diamagnetic currents. The theory provides an estimate for the radial profile of the floating potential, which reveals a dipolar structure like the one observed experimentally. Simulations with a fluid turbulence code provide evidence for the predicted behaviour of currents and floating potential.

Equal treatment and socially optimal R&D in duopoly with one‐way spillovers

AbstractThis paper examines the standard symmetric two‐period R&D model with a deterministic one‐way spillover structure: know‐how flows only from the high R&D firm to the low R&D firm (but not vice versa). Though firms are ex ante identical, one obtains a unique asymmetric equilibrium (pair) in R&D investments, leading to interfirm heterogeneity in the industry. R&D cooperation by means of a joint lab is considered and compared to the non cooperative solution. The main part of the paper provides a second‐best welfare analysis in which we show that the joint lab yields a socially optimal R&D level subject to an equal treatment (of firms) constraint, which also coincides with the noncooperative solution in the absence of spillovers. We also investigate the welfare costs of this equal treatment constraint and find that they can be quite significant.

Uniqueness, stability and comparative statics for two-person Bayesian games with strategic substitutes

This paper considers a class of two-player symmetric games of incomplete information with strategic substitutes. First, we provide sufficient conditions under which there is either a unique equilibrium which is stable (in the sense of best-reply dynamics) and symmetric or a unique (up to permutations) asymmetric equilibrium that is stable (together with an unstable symmetric equilibrium). Thus, (i) there is always a unique stable equilibrium, (ii) it is either symmetric or asymmetric, and hence, (iii) a very simple local condition—stability of the symmetric equilibrium (i.e., the slope of the best-response function at the symmetric equilibrium)—identifies which case applies. Using this, we provide a very simple sufficient condition on primitives for when the unique stable equilibrium is asymmetric (and similarly for when it is symmetric). Finally, we show that the conditions guaranteeing the uniqueness described above also yield novel comparative statics results for this class of games.

Pay-What-You-Want Pricing Under Competition: Breaking the Bertrand Trap

This paper investigates the viability of Pay-What-You-Want (PWYW) pricing when firms compete without restrictions of a minimum payment requirement. We show that the equilibrium outcomes are different when underpayers, consumers paying less than marginal cost, are present as opposed to when they are absent. In particular, when PWYW pricing is practiced without restricting the presence of underpayers or any minimum payment requirement, then the only two equilibrium structures are: either both firms use the posted price and earn zero profits, or one firm adopts PWYW pricing and the other uses the posted price. The asymmetric pricing equilibrium leads to a softening of price competition where both firms earn positive profits and the Bertrand Trap is broken.

Endogenous Market Structures in the Presence of a Socially Responsible Firm

We consider the endogenous selection of strategic contracts in an asymmetric duopoly with substitutable goods. the duopoly comprises a typical managerial firm with a sales delegation and a socially responsible firm (CSR firm) with a linear combination of social welfare and quantity as its managerial delegation contract. In particular, we examine how the equilibrium market structure changes from the case wthere both firms adopt sales delegation contracts to the case wthere one of the firms becomes a CSR firm, after the owners of the firms select their strategic contracts. We show that two market structures that are asymmetric with respect to their strategic contracts can become equilibrium market structures under the pure strategic contract class. Furthermore, we consider a unique mixed strategy equilibrium to examine how the risk domination between the two asymmetric equilibrium market structures affects equilibrium selection. there, we find that the competition wthere the firm with the sales delegation and the CSR firm have a price contract and a quantity contract, respectively, risk-dominates the competition wthere the firms have a quantity contract and a price contract, respectively. Finally, by deriving the order of social welfare among the four subgames, we show that the social incentive does not coincide with the private incentive in the robust equilibrium with respect to risk domination in the endogenous selection game of the strategic contracts of the asymmetric duopoly with the firm with a sales delegation and the CSR firm.

Duopoly Competition with Network Effects in Discrete Choice Models

We consider two firms selling products to a market of network-connected customers. Each firm is selling one product and the two products are substitutable. The customers make purchases based on the multinomial logit model and the firms compete for their purchasing probabilities. We characterize possible Nash equilibria for homogeneous network interactions and identical firms: when the network effects are weak, there is a symmetric equilibrium that the two firms evenly split the market; when the network effects are strong, there exist two asymmetric equilibria additionally, in which one firm dominates the market; interestingly, when the product quality is low and the network effects are neither too weak nor too strong, the resulting market equilibrium is never symmetric although the firms are ex ante symmetric. We extend these results along multiple directions. First, when the products have heterogeneous qualities, the firm selling inferior product can still retain market dominance in equilibrium due to the strong network effects. Second, when the network effects are heterogeneous, customers with higher social influences or larger price sensitivities are more likely to purchase either product in the symmetric equilibrium. Third, when the network consists of two communities, market segmentation may arise. Fourth, we extend to the dynamic game when the network effects build up over time to explain the first-mover advantage.

Strategic investments under competition for access provision

We examine competition for access provision when symmetric vertically integrated firms invest in infrastructure upgrades. Spillovers through access have two effects (a wholesale-profit effect and a retail-production effect) on infrastructure investment made by vertically integrated firms. When the vertically integrated firms freely set access charges, due to the dominance of the wholesale-profit effect, quality differentials endogenously occur between these firms (asymmetric equilibria). When access charges are regulated, symmetric equilibria occur with multiple equilibrium investments due to the retail-production effect. Because competition for access provision induces a strong incentive for infrastructure investment, it also achieves a higher social welfare than does access regulation.

Endogenous comparative advantage, gains from trade and symmetry-breaking

Similar countries often choose very different policies and specialize in very distinct industries. This paper proposes a novel mechanism to explain policy diversity between similar countries from an open economy perspective. I study optimal policies in a two country model where policies affect the determinants of trade patterns. Symmetric countries choose asymmetric policies in equilibrium if aggregate income is sufficiently convex in own policy. Such high social increasing return to policy, despite constant returns to scale in production, generate enough gains from trade to sustain policy diversity in equilibrium. Any asymmetric equilibrium is welfare-improving compared to the common autarky optimum. Similarly, a more asymmetric Nash equilibrium Pareto dominates a less asymmetric one. As an application, I consider a model in which the government's choice of education policy affects skill distribution and comparative advantage. In this application, the existence of an asymmetric Nash equilibrium in education policy requires that the elasticity of the education cost of agents with respect to skill is relatively low.

Equilibria in symmetric games: Theory and applications

This article presents a new approach to analyze the equilibrium set of symmetric, differentiable games by separating multiple symmetric equilibria and asymmetric equilibria. This separation allows the investigation of, for example, how various parameter constellations affect the scope for multiple symmetric or asymmetric equilibria, or how the equilibrium set depends on the nature of the strategies. The approach is particularly helpful in applications because (i) it allows the complexity of the uniqueness problem to be reduced to a two-player game, (ii) boundary conditions are less critical compared to standard procedures, and (iii) best replies need not be everywhere differentiable. The usefulness of the separation approach is illustrated with several examples, including an application to asymmetric games and to a two-dimensional price-information game.