Dynamically stable sets in infinite strategy spaces

Evolutionary game theory has largely focused on finite games. Dynamic stability is harder to attain in infinite strategy spaces; Bomze [Bomze, I., 1990. Dynamical aspects of evolutionary stability. Monatsh. Math. 110, 189–206] and Oechssler and Riedel [Oechssler, J., Riedel, F., 2001. Evolutionary dynamics on infinite strategy spaces. Econ. Theory 17, 141–162] provide conditions for the stability of rest points under the replicator dynamics. Here, conditions are given for the stability of sets of strategies under this process.     

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     

Nearly compact and continuous normal form games: characterizations and equilibrium existence

Normal form games are nearly compact and continuous (NCC) if they can be understood as games played on strategy spaces that are dense subsets of the strategy spaces of larger compact games with jointly continuous payoffs. There are intrinsic algebraic, measure theoretic, functional analysis, and finite approximability characterizations of NCC games. NCC games have finitely additive equilibria, and all their finitely additive equilibria are equivalent to countably additive equilibria on metric compactifications. The equilibrium set of an NCC game depends upper hemicontinuously on the specification of the game and contains only the limits of approximate equilibria of approximate games.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

The equilibrium set of two-player games with complementarities is a sublattice

Summary. I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space. Received: May 31, 2001; revised version: October 4, 2002     

Equilibrium Selection and Consistency

In this paper we show that, for two important classes of strategic games, namely the class of mixed extensions of finite games and the class of games with compact and convex strategy spaces and continuous-concave payoff functions, equilibrium selection is incompatible with One Person Rationality, Consistency and (restricted) Non-Emptiness.     

Nash equilibria in ∞-dimensional spaces: an approximation theorem

Summary. We show, by employing a density result for probability measures, that in games with a finite number of players and ∞-dimensional pure strategy spaces Nash equilibria can be approximated by finite mixed strategies. Given ε>0, each player receives an expected utility payoff ε/2 close to his Nash payoff and no player could change his strategy unilaterally and do better than ε. Received: July 15, 1997; revised version: February 6, 1998     

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.     

Local perfection

We investigate solution concepts for normal-form games with large strategy spaces. “Trembling hand” perfection (THP) is extended to games with compact metric strategy sets. A Nash equilibrium is THP if it is the limit of equilibria in which agents face vanishingly small uncertainty. We refine THP by imposing some structure on the kind of uncertainty that agents face: a THP equilibrium is locally perfect if the uncertainty is “predominantly local” in nature, i.e., if agents' probability assessments assign, in the limit, an overwhelmingly lage proportion of the total probability mass to events “in the vicinity of the truth.”     

On equilibrium existence in payoff secure games

We propose a single framework for studying the existence of approximate and exact pure strategy equilibria in payoff secure games. Central to the framework is the notion of a multivalued mapping with the local intersection property. By means of the Fan-Browder collective fixed point theorem, we first show an approximate equilibrium existence theorem that covers a number of known games. Then a short proof of Reny’s (Econometrica 67:1029–1056, 1999) equilibrium existence theorem is provided for payoff secure games with metrizable strategy spaces. We also give a simple proof of Reny’s theorem in its general form for metric games in an appendix for the sake of completeness.     

Purification, saturation and the exact law of large numbers

Purification results are important in game theory and statistical decision theory. We prove a new purification theorem that generalizes several earlier results. The key idea of our proof is to make use of the exact law of large numbers. As an application, we show that every mixed strategy in games with finite players, general action spaces and diffused, conditionally independent incomplete information has many strong purifications.     

Documenting Territory: Passportisation,Territory, and Exception in Abkhazia and South Ossetia

Despite the crucial role it played in the 2008 war between Russia and Georgia, the phenomenon of passportisation has not received a great deal of scholarly attention. Much of the literature has treated the mass distribution of Russian passports to the residents of Abkhazia and South Ossetia as little more than a strategy to manufacture an excuse to go to war with Georgia. Drawing on recent scholarship on territory and territoriality, as well as with literature addressing Agamben's theories of exceptional spaces, this paper contributes to a more nuanced understanding of passportisation by analysing the territorial effects it produced. It argues that the wholesale conversion of Abkhazians and South Ossetians into Russian citizens did not merely manufacture a casus belli, it also produced exceptional spaces within the territory of the Republic of Georgia, where the norms of international law and the modern state system were effectively suspended.     

Pure-Strategy ε-Nash Equilibrium inn-Person Nonzero-Sum Discontinuous Games

In this paper we considern-person nonzero-sum games where the strategy spaces of players are compact subsets ofR^s. The main result states that if the payoff functions are semicontinuous and strongly quasi-concave then an ε-Nash equilibrium in pure strategies exists for every positive ε.Journal of Economic LiteratureClassification Number: C72.     

On the generic inefficiency of differentiable market games

It is shown that a differentiable market game remains generically inefficient when its strategic outcome function is perturbed smoothly. The proof is based on Thom's transversality theorem and removes any restriction regarding the dimension of the strategy spaces. A converse result is that almost all efficient market games that are competitive are characterized by Bertrand-like non-differentiabilities. Finally a synthesis between the Cournot-and-Bertrand-type approaches to Walrasian equilibrium, as recently developed in the literature, is suggested.     

Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections

We prove that the support of mixed strategy equilibria of two-player, symmetric, zero-sum games lies in the uncovered set, a concept originating in the theory of tournaments, and the spatial theory of politics. We allow for uncountably infinite strategy spaces, and as a special case, we obtain a long-standing claim to the same effect, due to R. McKelvey (Amer. J. Polit. Sci.30 (1986), 283-314), in the political science literature. Further, we prove the nonemptiness of the uncovered set under quite general assumptions, and we establish, under various assumptions, the coanalyticity and measurability of this set. In the concluding section, we indicate how the inclusion result may be extended to multiplayer, non-zero-sum games. Journal of Economic Literature Classification Numbers: C72, D72.     

A Note on Adaptive Function-Based Models: The Case of Mobility

I propose a model for classifying innovations related to a particular function – mobility. My analysis demonstrates that existing typologies related to functional economy (FE) remain imprecise when considering specific functions. I include a systems perspective, bringing collective needs into the discussion. More precisely, I highlight the fact that, due to the complex systems in which the transportation system is embedded, collective consumption, like vehicle-sharing systems and self-service schemes, becomes crucial when mobility issues are addressed. Indeed, the satisfaction of individual needs has an impact on collective needs due to traffic congestion and rivalry for the use of limited parking spaces. As a strategy driving sustainability, the FE calls for institutional change.     

Strong Equilibrium in Congestion Games

[6]introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results.Journal of Economic LiteratureClassification Numbers: C71, C72, D62.     

基于生态系统服务空间流的洪涝调节 服务需求制图与生态空间优先级识别

中国快速城市化地区面临严重洪涝风险和洪涝调节 服务供需空间的分离。为识别在满足洪涝调节服务需求方面重 要性高的生态空间以纳入保护的优先级,提出了一种在生态空 间中绘制洪涝调节服务需求的新技术：基于生态系统服务空间 流,利用流量、流边界、流方向、分配原则及其修正因子4个 多维指标,表征将服务需求从建设空间向生态空间分配与投射 过程的关键环节。结果显示：面积大、人口密度高的建设空间 存在高需求流量;借助水文空间流,服务需求由建设空间被分 配至同一集水区内具有洪涝调节能力的生态空间中;被投射为 高服务需求的生态空间大多规模较大或紧邻高需求的建成空 间。通过控制流量、扩展流区域、增加流方向和提升生态空间 服务供应能力,能有效调控投射于生态空间内的洪涝调节服务 高需求。     

Surplus extraction with rich type spaces

We characterize revenue maximizing mechanisms in auction settings with ‘rich’ type spaces, where bidders obtain information from sources other than their own valuation. By considering a relaxed problem, we provide an upper bound on revenue extraction that explicitly builds on the richness of the information structure. We provide a condition under which this upper bound is achieved and describe an optimal mechanism. Under this condition, we also show that the optimal revenue can be achieved through dominant strategy implementation.     

Interim Bayesian Nash equilibrium on universal type spaces for supermodular games

We prove the existence of a greatest and a least interim Bayesian Nash equilibrium for supermodular games of incomplete information. There are two main differences from the earlier proofs and from general existence results for non-supermodular Bayesian games: (a) we use the interim formulation of a Bayesian game, in which each player's beliefs are part of his or her type rather than being derived from a prior; (b) we use the interim formulation of a Bayesian Nash equilibrium, in which each player and every type (rather than almost every type) chooses a best response to the strategy profile of the other players. There are no restrictions on type spaces and action sets may be any compact metric lattices.