Monotone imitation dynamics in large populations

We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.     

A general structure theorem for the Nash equilibrium correspondence

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003–1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537–545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46–60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.     

On the stability of coalition structures

We resolve a seeming conflict between a non-existence result on solutions to coalition formation in hedonic games [Barberà, S., Gerber, A., 2007. A note on the impossibility of a satisfactory concept of stability for coalition formation games. Economics Letters 95, 85–90] and the universal existence of stable coalition structures in TU games under the χ-value [Casajus, A., 2008. Outside options, component efficiency, and stability, Games and Economic Behavior (forthcoming). doi: 10.1016/j.geb.2007.04.003].     

Comment on “A note on the Hotelling principle of minimum differentiation: Imitation and crowd”

We argue that the minimum differentiation firm location equilibrium and the pure strategy pricing equilibrium in Di Cintio’s [Di Cintio, M., 2007. A note on the Hotelling principle of minimum differentiation: Imitation and crowd. Research in Economics 61 (3), 122–129] “Note” need not exist under the conditions claimed.     

Purification in the infinitely-repeated prisoners' dilemma

This paper investigates the Harsanyi [Harsanyi, J.C., 1973. Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points. International Journal of Game Theory 2 (1), 1–23]-purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of one-period memory mixed strategy equilibria used by Ely and Välimäki [Ely, J.C., Välimäki, J., 2002. A robust folk theorem for the prisoner's dilemma. Journal of Economic Theory 102 (1), 84–105] in their study of the repeated prisoners' dilemma with private monitoring. We find that any such strategy profile is not the limit of one-period memory equilibrium strategy profiles of the perturbed game, for almost all noise distributions. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable.     

Nash equilibrium and generalized integration for infinite normal form games

Infinite normal form games that are mathematically simple have been treated [ Harris, C.J., Stinchcombe, M.B., Zame, W.R., in press. Nearly compact and continuous normal form games: characterizations and equilibrium existence. Games Econ. Behav.]. Under study in this paper are the other infinite normal form games, a class that includes the normal forms of most extensive form games with infinite choice sets.Finitistic equilibria are the limits of approximate equilibria taken along generalized sequences of finite subsets of the strategy spaces. Points must be added to the strategy spaces to represent these limits. There are direct, nonstandard analysis, and indirect, compactification and selection, representations of these points. The compactification and selection approach was introduced [Simon, L.K., Zame, W.R., 1990. Discontinuous games and endogenous sharing rules. Econometrica 58, 861–872]. It allows for profitable deviations and introduces spurious correlation between players' choices. Finitistic equilibria are selection equilibria without these drawbacks. Selection equilibria have drawbacks, but contain a set-valued theory of integration for non-measurable functions tightly linked to, and illuminated by, the integration of correspondences.     

Evolutionary dynamics on infinite strategy spaces

Summary. The study of evolutionary dynamics was so far mainly restricted to finite strategy spaces. In this paper we show that this unsatisfying restriction is unnecessary. We specify a simple condition under which the continuous time replicator dynamics are well defined for the case of infinite strategy spaces. Furthermore, we provide new conditions for the stability of rest points and show that even strict equilibria may be unstable. Finally, we apply this general theory to a number of applications like the Nash demand game, the War of Attrition, linear-quadratic games, the harvest preemption game, and games with mixed strategies. Received: June 25, 1999; revised version: January 31, 2000     

A model of the Italian cut-off system for taxing small businesses

The Studi di Settore are used by the Italian tax administration to calculate reference revenue levels for small businesses and provide a kind of cut-off level for tax audits. Recently new rules have been introduced in order to render the Studi di Settore more efficient in producing realistic estimates, with the aim of reducing the “legalized evasion” that might arise in case of a systematic downward bias. Voices of the involved categories, however, convinced the Government to partially step back. Building upon the standard firm’s tax evasion model of Cowell [Cowell, F.A., 2004. Carrots and sticks in enforcement. In: Aaron, H.J., Slemrod, J. (Eds.), The Crisis in Tax Administration. The Brookings Institution, Washington DC, pp. 230–275] and the approach of Santoro [Santoro, A.C., 2006. Evasione delle società di capitali: evidenze empiriche e proposte di policy. In: Brosio, G., Muraro, M. (Eds.), Il Finanziamento del Settore Pubblico. SIEP, Angeli, Milano, pp. 163–186] we show that, under given conditions, a stringency increase might backfire implying a larger overall tax evasion and a smaller tax revenue.     

Bootstrap methods for autocorrelation test with uncorrelated but not independent errors

By using bootstrap technique we investigate the properties of the Breusch [Breusch, T.S., 1978. Testing for autocorrelation in dynamic linear models. Australian Economic Papers 17, 334–355]–Godfrey [Godfrey, L.G., 1978. Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica 46, 1303–1310] autocorrelation tests in dynamic models with uncorrelated but not independent errors. In this paper we show that, under conditions when the errors are uncorrelated but not independent, even the best likelihood ratio test cannot achieve the asymptotic distribution under the null hypothesis of no autocorrelation. Standard bootstrap methods also fail to produce consistent results. To overcome this problem we applied several bootstrap testing methods for the same purpose and found the stationary bootstrap and Wild bootstrap with static model to perform adequately among the other bootstrap methods.     

Revisiting games of incomplete information with analogy-based expectations

This paper studies the effects of analogy-based expectations in static two-player games of incomplete information. Players are assumed to be boundedly rational in the way they forecast their opponent's state-contingent strategy: they bundle states into analogy classes and play best-responses to their opponent's average strategy in those analogy classes. We provide general properties of analogy-based expectation equilibria and apply the model to a variety of well known games. We characterize conditions on the analogy partitions for successful coordination in coordination games under incomplete information [Rubinstein, A., 1989. The electronic mail game: Strategic behavior under ‘almost common knowledge’. Amer. Econ. Rev. 79, 385–391], we show how analogy grouping of the receiver may facilitate information transmission in Crawford and Sobel's cheap talk games [Crawford, V.P., Sobel, J., 1982. Strategic information transmission. Econometrica 50, 1431–1451], and we show how analogy grouping may give rise to betting in zero-sum betting games such as those studied to illustrate the no trade theorem.     

Consistency of beliefs and epistemic conditions for Nash and correlated equilibria

A consistency condition (action-consistency) on the interim beliefs of players in a game is introduced. Action-consistency is weaker than common priors and, unlike common priors, is characterized by a “no-bets” condition on verifiable events. Using action-consistency, we provide epistemic conditions to Nash and correlated equilibria weakening the common knowledge restrictions in Aumann and Brandenburger [Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180] and Aumann [Aumann, R., 1987. Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1–18].     

Consensus and common knowledge of an aggregate of decisions

McKelvey and Page [McKelvey, R., Page, T., 1986. Common knowledge, consensus and aggregate information. Econometrica 54, 109–127] generalized Aumann's [Aumann, R.J., 1976. Agreeing to disagree. Ann. Statist. 4, 1236–1239] agreement theorem to the case where agents have common knowledge of a statistic of their posterior probabilities of some event. They showed that if individuals have the same prior, and if the statistic satisfies a stochastic regularity condition, then common knowledge of it implies equality of all posteriors. We show a similar result in a more general setting where agents have common knowledge of a statistic of their individual decisions. Decisions can be posteriors as well as discrete actions such as buy or sell. We show that if the decision rule followed by individuals is balanced union consistent, and if the statistic of individual decisions is exhaustive, then common knowledge of it implies equality of all decisions. We give an example showing that neither Cave's [Cave, J., 1983. Learning to agree. Econ. Letters 12, 147–152] union consistency condition nor Parikh and Krasucki's [Parikh, R., Krasucki, P., 1990. Communication, consensus and knowledge. J. Econ. Theory 52, 178–189] convexity condition is sufficient to guarantee the result.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

Capital–skill complementarity and inequality: A sensitivity analysis

Krusell et al. in [Krusell, P., Ohanian, L., Ríos-Rull, J.V., Violante, G.L., 2000. Capital–skill complementarity and inequality: A macroeconomic analysis. Econometrica 68 (5), 1029–1053] analyzed the capital–skill complementarity hypothesis as an explanation for the behavior of the US skill premium. We refit Krusell et al.'s [Krusell, P., Ohanian, L., Ríos-Rull, J.V., Violante, G.L., 2000. Capital–skill complementarity and inequality: A macroeconomic analysis. Econometrica 68 (5), 1029–1053] model with two alternative capital equipment price series: One proposed by Greenwood et al. [Greenwood, J., Hercowitz, Z., Krusell, P., 1997. Long-run implications of investment-specific technological change. Amer. Econ. Rev. 87 (3), 342–362] and the official, revised National Income and Product Accounts (NIPA) data. We find that capital–skill complementarity is preserved, but other results were sensitive to the data used. Specifically, the fit of the model was similar to Krusell et al.'s [Krusell, P., Ohanian, L., Ríos-Rull, J.V., Violante, G.L., 2000. Capital–skill complementarity and inequality: A macroeconomic analysis. Econometrica 68 (5), 1029–1053] using the NIPA data, but not the Greenwood et al. [Greenwood, J., Hercowitz, Z., Krusell, P., 1997. Long-run implications of investment-specific technological change. Amer. Econ. Rev. 87 (3), 342–362] data. Also, both series produce estimates of the elasticity of substitution between unskilled labor and equipment that are substantially larger than Krusell et al.'s [Krusell, P., Ohanian, L., Ríos-Rull, J.V., Violante, G.L., 2000. Capital–skill complementarity and inequality: A macroeconomic analysis. Econometrica 68 (5), 1029–1053] estimates.     

Why evolution does not always lead to an optimal signaling system

This paper gives a complete characterization of neutrally stable strategies for sender–receiver games in the style of Lewis, or Nowak and Krakauer [Lewis, D., 1969. Convention: A Philosophical Study. Harvard Univ. Press, Cambridge, MA; Nowak, M., Krakauer, D., 1999. The evolution of language. Proc. Nat. Acad. Sci. USA 96, 8028–8033]. Due to the dynamic implications of neutral stability, the replicator dynamics of this model does not necessarily lead to the rise of an optimal signaling system, where every state of the world is bijectively linked to one signal and vice versa, but it can be trapped in suboptimum situations where two (or more) signals are used for the same event, or two (or more) events are associated with one and the same signal.     

A re-examination of the technological catching-up hypothesis across OECD industries

This study re-examines the catching-up hypothesis at the industry level across the main OECD countries, using panel data econometric models involving technological gap indicators calculated with a nonparametric distance function suggested by Färe et al. [Färe, R.S., Grosskopf, M.N., Norris, M., Xhang, Z., 1994. Productivity growth, technical progress, and efficiency change in industrialized countries. American Economic Review 84, 66–83]. The results show that there is statistical evidence of a catching-up process at the industry level. Moreover, both tradables and nontradables sectors exhibit catching-up effects and technology adoption from abroad. This result complements the findings by Bernard and Jones [Bernard, A.B., Jones, C.I., 1996a. Comparing apples to oranges: productivity convergence and measurement across industries and countries. American Economic Review 86(5), 1216–1238., Bernard, A.B., Jones, C.I., 1996b. Productivity across industries and countries: Time series theory and evidence. Review Of Economics and Statistics 78, 135–146], Gouyette and Perelman [Gouyette, C., Perelman, S., 1997. Productivity convergence in OECD services industries. Structural Change and Economic Dynamics 8, 279–295] and Hansson and Henrekson [Hansson, P., Henrekson, M., 1997. Catching up, social capability, government size and economic growth, in V. Bergström, eds, Government and Growth, Oxford: Clarendon Press, 61–126] that there is no (or even a slow) catching-up effect in the manufacturing sector. Moreover, social capability indicators evaluated for each country show that “Non-European” and “Central European” tradables sectors have a rather similar degree of inefficiency while “North European” countries are less efficient for both tradables and non-tradables. Lastly, both the cross country and the cross sectors dispersions of inefficiency levels are smaller for tradables sectors than for non-tradables.     

Rapid evolution under inertia

This paper demonstrates that inertia driven by switching costs leads to more rapid evolution in a class of games that includes m×m pure coordination games. Under the best-response dynamic and a fixed rate of mutation, the expected waiting time to reach long-run equilibrium is of lower order in the presence of switching costs, due to the creation of new absorbing states that allow Ellison's [Ellison, G., 2000. Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud. 67, 17–45] “step-by-step” evolution to occur.     

In search of stars: Network formation among heterogeneous agents

This paper reports results from a laboratory experiment on network formation among heterogeneous agents. The experimental design extends the Bala–Goyal [Bala, V., Goyal, S., 2000. A non-cooperative model of network formation, Econometrica 68, 1131–1230] model of network formation with decay and two-way flow of benefits by introducing agents with lower linking costs or higher benefits to others. Furthermore, agents' types may be common knowledge or private information. In all treatments, the (efficient) equilibrium network has a “star” structure. While equilibrium predictions fail completely with homogeneous agents, star networks frequently occur with heterogeneous agents. Stars are not born but rather develop: with a high-value agent, the network's centrality, stability, and efficiency all increase over time. A structural econometric model based on best response dynamics and other-regarding preferences is used to analyze individual linking behavior. Maximum-likelihood estimates of the underlying structural parameters, obtained by pooling data from several treatments, allow us to explain the main treatment effects.     

Cournot–Nash equilibria in continuum games with non-ordered preferences

In the usual framework of continuum games with externalities, we substantially generalize Cournot–Nash existence results [Balder, A unifying approach to existence of Nash equilibria, Int. J.Game Theory 24 (1995) 79–94; On the existence of Cournot–Nash equilibria in continuum games, J. Math. Econ. 32 (1999) 207–223; A unifying pair of Cournot–Nash equilibrium existence results, J. Econ. Theory 102 (2002) 437–470] to games with possibly non-ordered preferences, providing a continuum analogue of the seminal existence results by Mas-Colell [An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ. 1 (1974) 237–246], Gale and Mas-Colell [An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15], Shafer and Sonnenschein [Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348], Borglin and Keiding [Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorems, J. Math. Econ. 3 (1976) 313–316] and Yannelis and Prabhakar [Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12 (1983) 233–245].     

Business cycle dynamics with duration dependence and leading indicators

Durland and McCurdy [Durland, J.M., McCurdy, T.H., 1994. Duration-dependent transitions in a Markov model of US GNP growth. Journal of Business and Economic Statistics 12, 279–288] investigated the issue of duration dependence in US business cycle phases using a Markov regime-switching approach, introduced by Hamilton [Hamilton, J., 1989. A new approach to the analysis of time series and the business cycle. Econometrica 57, 357–384] and extended to the case of variable transition parameters by Filardo [Filardo, A.J., 1994. Business cycle phases and their transitional dynamics. Journal of Business and Economic Statistics 12, 299–308]. In Durland and McCurdy’s model duration alone was used as an explanatory variable of the transition probabilities. They found that recessions were duration dependent whilst expansions were not. In this paper, we explicitly incorporate the widely-accepted US business cycle phase change dates as determined by the NBER, and use a state-dependent multinomial Logit modelling framework. The model incorporates both duration and movements in two leading indexes – one designed to have a short lead (SLI) and the other designed to have a longer lead (LLI) – as potential explanatory variables. We find that doing so suggests that current duration is not only a significant determinant of transition out of recessions, but that there is some evidence that it is also weakly significant in the case of expansions. Furthermore, we find that SLI has more informational content for the termination of recessions whilst LLI does so for expansions.