Elon Kohlberg

ON THE STRATEGIC STABILITY OF EQUILIBRIA

A basic problem in the theory of noncooperative games is the following: which Nash equilibria are strategically stable, i.e. self-enforcing, and does every game have a strategically stable equilibrium? We list three conditions which seem necessary for strategic stabilitybackwards induction, iterated dominance, and invariance-and define a set-valued equilibrium concept that satisfies all three of them. We prove that every game has at least one such equilibrium set. Also, we show that the departure from the usual notion of single-valued equilibrium is relatively minor, because the sets reduce to points in all generic games.

The Asymptotic Theory of Stochastic Games

We study two person, zero sum stochastic games. We prove that limn→∞{Vn/n} = limr→0rVr, where Vn is the value of the n-stage game and Vr is the value of the infinite-stage game with payoffs discounted at interest rate r > 0. We also show that Vr may be expanded as a Laurent series in a fractional power of r. This expansion is valid for small positive r. A similar expansion exists for optimal strategies. Our main proof is an application of Tarskis principle for real closed fields.

On Stochastic Games with Stationary Optimal Strategies

We study two-person zero-sum stochastic games in which the state and action spaces are finite. We give both necessary and sufficient conditions for the players to have stationary optimal strategies in the infinite-stage game.

Optimal strategies in repeated games with incomplete information

The paper is concerned with zero-sum two-person repeated games with lack of information on one side. The main result in the construction of an optimal strategy for the uninformed player in the infinitely repeated game.

The Nucleolus as a Solution of a Minimization Problem

The following problem is discussed : Is it possible to represent the nucleolus of an n-person game as a solution of a linear programming problem? An affirmative answer is given for the case that the set of payoff vectors is a polytope. An example is described, showing that the above may not be possible when the set of payoff vectors is just convex and compact.

The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilberts projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...

Asymptotic Behavior of Nonexpansive Mappings in Normed Linear Spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→xf(Tnx/n)=limn→x‖Tnx/n‖=α, where α≡infy∈c‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichTnx/n converges weakly for allx (infz∈fg(Tnx/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT.

Equally distributed correspondences

Abstract The distribution D ϕ of a correspondence ϕ is defined, and its connection with the set DL ϕ of distributions of its integrable selections is explored. The main result is that if ϕ1 and ϕ2 are equally distributed, i.e., if D ϕ1 = and D ϕ2, then DL ϕ1 and DL ϕ2 have the same closure in the weak convergence topology.

Invariant Half-Lines of Nonexpansive Piecewise-Linear Transformations

It is shown that if f is a nonexpansive piecewise-linear mapping of Rm into itself, there exists a unique half-line that f maps into itself and such that restriction of f thereto is a translation. One easy consequence of this result is that there exists a unique m-vector α such that for every m-vector x, the sequence fnx-nα remains bounded. In particular, fnx/n converges to the same limit α, for all x. Also, f has a fixed point if and only if α = 0. These results are applied to give alternative proofs of several known facts concerning the maximum expected n-period reward in a finite Markov decision process.

The Asymptotic Solution of a Recursion Equation Occurring in Stochastic Games

We show that there exists a Laurent series in a fractional power of n which approximates Vn up to log n, where Vn is the value of an n-stage two person zero sum stochastic game. We prove this result by showing that the Laurent series is an approximate solution of the dynamical programming equation for Vn, Vn+1 = fVn. It seems that our methods could be used to find approximate solutions to other difference equations. Our proof makes repeated use of Tarskis principle for real closed fields.