Evolving densities in continuous strategy games through particle simulations

Many cases of strategic interaction between agents involve a continuous set of choices. It is natural to model these problems as continuous space games. Consequently, the population of agents playing the game will be represented with a density function defined over the continuous set of strategy choices. Simulating evolutionary dynamics on continuous strategy spaces is a challenging problem. The classic approach of discretizing the strategy space is ineffective for multidimensional strategy spaces. We present a principled approach to simulation of adaptive dynamics in continuous space games using sequential Monte Carlo methods. Sequential Monte Carlo methods use a set of weighted random samples, also named particles to represent density functions over multidimensional spaces. Sequential Monte Carlo methods provide computationally efficient ways of computing the evolution of probability density functions. We employ resampling and smoothing steps to prevent particle degeneration problem associated with particle estimates. The resulting algorithm can be interpreted as an agent based simulation with elements of natural selection, regression to mean and mutation. We illustrate the performance of the proposed simulation technique using two examples: continuous version of the repeated prisoner dilemma game and evolution of bidding functions in first-price closed-bid auctions.     

Uncertainty,Learning and International Environmental Policy Coordination

In this paper we construct a simple model of global warming which captures a number of key features of the global warming problem: (i) environmental damages are related to the stock of greenhouse gases in the atmosphere; (ii) the global commons nature of the problem means that these are strategic interactions between the emissions policies of the governments of individual nation states; (iii) there is uncertainty about the extent of the future damages that will be incurred by each country from any given level of concentration of greenhouse gases but there is the possibility that at a future date better information about the true extent of environmental damages may become available; an important aspect of the problem is the extent to which damages in different countries may be correlated. In the first part of the paper we consider a simple model with two symmetric countries and show that the value of perfect information is an increasing function of the correlation between damages in the two countries in both the cooperative and non-cooperative equilibria. However, while the value of perfect information is always non-negative in the cooperative equilibrium, in the non-cooperative equilibrium there is a critical value of the correlation coefficient below which the value of perfect information will be negative. In the second part of the paper we construct an empirical model of global warming distinguishing between OECD and non-OECD countries and show that in the non-cooperative equilibrium the value of perfect information for OECD countries is negative when the correlation coefficient between environmental damages for OECD and non-OECD countries is negative. The implications of these results for international agreements are discussed.     

Stackelberg leadership in a dynamic duopoly with stochastic capital accumulation

I propose a dynamic duopoly model where firms enter simultaneously but compete hierarchically á la Stackelberg at each instant over time. They accumulate capacity through costly investment, with capital accumulation dynamics being affected by an additive shock the mean and variance of which are known. The main findings are the following. First, the Stackelberg game is uncontrollable by the leader; hence, it is time consistent. Second, the leader invests more than the follower; as a result, in the steady state, the leader’s capacity and profits are larger than the follower’s. Therefore, the present analysis does not confirm Gibrat’s Law, since the individual growth rate is determined by the timing of moves.JEL Classification: C61, C73, D43, D92, L13Financial support within the project The post-entry performance of firms: technology, growth and survival lead by Enrico Santarelli, co-financed by the University of Bologna and MIUR, is gratefully acknowledged. I thank Uwe Cantner, Roberto Cellini, Roberto Golinelli, Helen Louri, Enrico Santarelli, Antonello Scurcu, Peter Thompson, two anonymous referees and the audience at the final workshop of the project (Bologna, November 22-23, 2002) for useful comments and discussion. The usual disclaimer applies.     

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N→∞, the expected number of Nash equilibria is approximately . Letting M=N→∞, the expected number of Nash equilibria is , where is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.     

Supporting others and the evolution of influence

In this paper we study environments in which agents can transfer influence to others by supporting them. When planning whom to support, they should take into account the future effect of this, since the receiving agent might use this influence to support others in the future. We show that in the presence of a finite horizon there is an essentially unique optimal support behavior which can be characterized in terms of associated marginal value functions. The analysis of these marginal value functions allows us to derive qualitative properties of optimal support strategies under different specific environments and to explicitly compute the optimal support behavior in some numerical examples. We also investigate the case of an infinite horizon. Examples show that multiple equilibria may appear in this setting, some of which sustaining a degree of cooperation that would not be possible under a finite horizon.     

Achieving efficiency with manipulative bargainers

Two agents bargain over the allocation of a bundle of divisible commodities. After strategically reporting utility functions to a neutral arbitrator, the outcome is decided by using a bargaining solution concept chosen from a family that includes the Nash and the Raiffa–Kalai–Smorodinsky solutions. When reports are restricted to be continuous, strictly increasing and concave, it has been shown that this kind of “distortion game” leads to inefficient outcomes. We study the distortion game originated when agents are also allowed to claim non-concave utility functions. Contrasting with the previous literature, any interior equilibrium outcome is efficient and any efficient allocation can be supported as an equilibrium outcome of the distortion game. In a similar fashion to the Nash demand game we consider some uncertainty about the opponent's features to virtually implement the Nash bargaining solution.     

Price Competition Between Teams

This study uses an experimental approach to examine whether markets are sensitive to the internal incentive structure of the competitors. Toward this goal, we modeled the competitors in a price competition duopoly game as three-player teams. Each player simultaneously declares a bid (price) and the team whose total bid was lower won the competition and was paid accordingly. The losing team was paid nothing, and in case of a tie, each team was paid half its price. This duopoly game was studied under two conditions; a cooperative treatment in which the team's profit was divided equally amongst its members and a non-cooperative one in which each individual member was paid her own bid. Whereas the Nash equilibrium is for each player in either treatment to demand the minimal price possible, we predicted that convergence to the competitive price would be much faster in the cooperative treatment than in the non-cooperative one. The experimental results firmly confirmed this prediction.     

INSTITUTIONS AND (SHORT-RUN) MACROECONOMIC PERFORMANCE

Abstract. This paper reviews the growing interest in institutions. It argues that the explanation for this interest is a fresh awareness of how institutions affect economic performance. This connection arises because many economic interaction take the form of one or other of three 'classic' games (prisoners' dilemma, coordination and chicken) and the outcome in these games (and hence performance of the economy) is influenced by the institutional context of these interactions. The paper provides a detailed illustration of this argument with a survey of the literature which relates unemployment performances to the institutions of wage setting in OECD countries.     

东亚货币一体化博弈分析

东亚发展中国家与地区如何选择汇率制度一直是个颇有争议的话题。在“欧元”单一货币区成功实施的背景下,寻求货币一体化便成为东亚各国与地区关注的焦点。实际上,东亚货币一体化就是各参与方根据其参与的成本和收益,权衡利弊得失的决策博弈过程,该过程具有两阶段博弈的特征。