Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Foundations Of Statistics

The Theory of Incentives I: The Principal-Agent Model

Preface (R.J. Aumann, S. Hart). Strategic equilibrium (E. van Damme). Foundations of strategic equilibrium (J. Hillas, E. Kohlberg). Incomplete information (R.J. Aumann, A. Heifetz). Non-zero-sum two-person games (T.E.S. Raghavan). Computing equilibria for two-person games (B. von Stengel). Non-cooperative games with many players (M. Ali Khan, Y. Sun). Stochastic games (J-F. Mertens). Stochastic games: recent results (N. Vieille). Game theory and industrial organization (K. Bagwell, A. Wolinsky). Bargaining with incomplete information (L.M. Ausubel, P. Cramton, R.J. Deneckere). Inspection Games (R. Avenhaus, B.V. Stengel, S.Zamir). Economic history and game theory (A. Greif). The shapley value (E. Winter). Variations on the shapley value (D. Monderer, D. Samet). Values of non-transferable utility games (R. McLean). Values of games with infinitely many players (A. Neyman). Values of perfectly competitive economies (S. Hart). Some other economic applications of the value (J-F. Mertens). Strategic aspects of political systems (J. Banks). Game-theoretic analysis of legal rules and institutions (J-P. Benoit, L.A. Kornhauser). Implementation Theory (T. Palfrey). Game Theory and experimental Gaming (M. Shubik).

Handbook of game theory with economic applications

We examine a three-person stochastic game where the only existing equilibria consist of cyclic Markov strategies. Unlike in two-person games of a similar type, stationary e-equilibria (e > 0) do not exist for this game. Besides we characterize the set of feasible equilibrium rewards.

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Cyclic Markov equilibria in stochastic games

We prove that every multiplayer perfect-information game with bounded and lower-semicontinuous payoffs admits a subgame-perfect e-equilibrium in pure strategies. This result complements Example 3 in Solan and Vieille [Solan, E., N. Vieille. 2003. Deterministic multi-player Dynkin games. J. Math. Econom.39 911--929], which shows that a subgame-perfect e-equilibrium in pure strategies need not exist when the payoffs are not lower-semicontinuous. In addition, if the range of payoffs is finite, we characterize in the form of a Folk Theorem the set of all plays and payoffs that are induced by subgame-perfect 0-equilibria in pure strategies.

/pdf/perfect-information-games-with-lower-semicontinuous-payoffs-44rs7pgrjj.pdf

Perfect-Information Games with Lower-Semicontinuous Payoffs

We examine so-called product-games with an aperiodic transition structure, with respect to the average reward, for which we present an approach based on communicating states. For the general n-player case, we establish the existence of 0-equilibria. In addition, for the special case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies.

/pdf/stochastic-games-on-a-product-state-space-1b5yri7ypp.pdf

Stochastic Games on a Product State Space

We show the existence of stationary limiting average e-equilibria e > 0 for two-person recursive repeated games with absorbing states. These are stochastic games where all states but one are absorbing, and in the nonabsorbing state all payoffs are equal to zero. A state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions. For the purpose of our proof, we introduce properness for stationary strategy pairs. Our result is sharp since it extends neither to the case with more nonabsorbing states, nor to the n-person case with n > 2. Moreover, it is well known that the result cannot be strengthened to the existence of 0-equilibria and that repeated games with absorbing states generally do not admit stationary e-equilibria.

Recursive Repeated Games with Absorbing States

https://dke.maastrichtuniversity.nl/f.thuijsman/additive.pdf

János Flesch

Papers

Cyclic Markov equilibria in stochastic games

Perfect-Information Games with Lower-Semicontinuous Payoffs

Stochastic Games on a Product State Space

Recursive Repeated Games with Absorbing States

Stochastic games with additive transitions