Antoon van den Elzen

Computing Normal Form Perfect Equilibria for Extensive Two‐Person Games

This paper presents an algorithm for computing an equilibrium of an extensive two-person game with perfect recall. The method is computationally efficient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector. If the starting vector represents a pair of completely mixed strategies, then the equilibrium is normal form perfect. Computational experiments compare the sequence form and the reduced normal form, and show that only the sequence form is tractable for larger games.

An adjustment process for an economy with linear production technologies

In this paper an adjustment process for a general equilibrium model with linear production technologies is given. The process reaches an equilibrium in the economy by simultaneous adaptations of prices and activity levels. Initially the prices of the commodities in excess demand are relatively increased and the prices of the commodities in excess supply are relatively decreased. Firms have only a positive activity level when they make zero profit.An important feature of the process is that it keeps track of the location of the starting price vector throughout its operation. Furthermore, the choice of the starting price vector is only limited by economic considerations. Besides, the process converges under rather weak conditions and possesses an appealing economic interpretation.

Finding a Nash equilibrium in noncooperative N-person games by solving a sequence of linear stationary point problems

In this paper we present an algorithm for finding a Nash equilibrium in a noncooperative normal formN-person game. More generally, the algorithm can be applied for solving a nonlinear stationary point problem on a simplotope, being the Cartesian product of several simplices. The algorithm solves the problem by solving a sequence of linear stationary point problems. Each problem in the sequence is solved in a finite number of iterations. Although the overall convergence cannot be proved, the method performs rather well. Computational results suggest that this algorithm performs at least as good as simplicial algorithms do.For the special case of a bi-matrix game (N=2), the algorithm has an appealing game-theoretic interpretation. In that case, the problem is linear and the algorithm always finds a solution. Furthermore, the equilibrium found in a bi-matrix game is perfect whenever the algorithm starts from a strategy vector at which all actions are played with positive probability.

An adjustment process for the standard Arrow-Debreu model with production

Abstract In this paper we present a taˆtonnement process describing adaptations of prices and quantities in a convex general equilibrium model with production. The adaptations are determined by the state of the market and the starting price vector. The main result of the paper shows that for any economy out of a broad class of so-called, semi-algebraic convex economies and for any starting tuple consisting of a price vector and a related demand and supply, the process defines at least one path connecting the starting tuple and an equilibrium. The second results concerns the uniqueness of the path, given an arbitraty starting tuple, for a general economy out of most standard classes of convex production economies, with consumer demand being unique at all price vectors. In that way, we generalize existing results on converging processes.

Existence of adjustment processes

In this chapter we investigate under what conditions the processes to be considered in this monograph exist and converge. For that we consider one specific process. All processes treated in this monograph, except the ones in Chapter 6, have the same properties and their existence can be studied along the same lines.

Finding Nash equilibria in noncooperative games

In this chapter we consider the problem of finding Nash equilibria in mixed strategies for noncooperative games, in which the payoffs are listed in matrices or more generally in tensors. The main part of this chapter deals with bi-matrix games. Such a game is a noncooperative two-person game with a finite set of actions for each player. The Nash equilibrium is the standard equilibrium concept for a noncooperative game. It states that a strategy is an equilibrium when no player can improve upon his situation by deviating from his strategy while all other players keep on playing their strategies.

An adjustment process for an international trade model

In this chapter we consider price adjustments resulting from shocks in a two country model. For that purpose we present a price adjustment process that reaches an equilibrium price vector when starting from an arbitrarily chosen price vector. The latter vector can be interpreted as the equilibrium price prevailing in the economy before the shock took place.