Maximin Values Research Articles

On a Simplified Method of Defining Characteristic Function in Stochastic Games

In the paper, we propose a new method of constructing cooperative stochastic game in the form of characteristic function when initially non-cooperative stochastic game is given. The set of states and the set of actions for any player is finite. The construction of the characteristic function is based on a calculation of the maximin values of zero-sum games between a coalition and its anti-coalition for each state of the game. The proposed characteristic function has some advantages in comparison with previously defined characteristic functions for stochastic games. In particular, the advantages include computation simplicity and strong subgame consistency of the core calculated with the values of the new characteristic function.

Sustainability standards, multicriteria maximin, and viability

AbstractThis paper deals with sustainability criteria and standards. The maximin criterion, as the highest performance that can be sustained over time, promotes intergenerational equity, a pivotal issue for sustainability. The viable control approach, by investigating trajectories and actions complying over time with various standards and constraints, provides major insights into strong sustainability. The present paper addresses the links between maximin and viability approaches in a multicriteria context. It first shows how “Pareto maximin” can be characterized through viability kernels. Such a result makes it possible to determine the trade‐offs and/or synergies between nonsubstitutable economic and ecological standards underlying strong sustainability. The second main result of the paper is to propose algorithms derived from the viability version of dynamic programming to approximate numerically Pareto maximin values, controls, and sustainability standards. Two examples relying on renewable resource management illustrate these analytic and numerical findings. In particular, synergies between sustainability standards of resource conservation, production or profitability are identified for overexploited stocks.

Some minimax theorems of set-valued maps and their applications

In the paper, we consider two types of scalarization functions of sets and investigate their properties. Moreover, based on two set-relations, we propose two kinds of minimax and maximin values of set-valued maps, respectively, and show some minimax theorems of set-valued maps with respect to those minimax and maximin values by using several properties of the above two functions. As an application of these results, we give common saddle point theorems of vector-valued functions.

On question about extension of maximin problem with phase constraints

We study the asymptotic behavior of maximin values of a payoff function, when relaxed constraints are tightened. The payoff function depends on the trajectories of controlled systems of the first and second player. An extension in the class of the Radon measures is used. The asymptotic equivalence between two types of the constraints relaxations is shown.

Purely finitely additive measures as generalized elements in a maximin problem

We study the asymptotic behavior of maximin values of a payoff function, when admissible controls tend to infinity. The payoff function is superposition of a continuos function and a function that is uniform limit of step functions. An extension in the class of finitely additive measures is used.

A BARGAINING MODEL FOR FINITE N-PERSON MULTI-CRITERIA GAMES

In this paper a multi-criteria game model, which allows interactions between players, is studied. The problem addressed is considered as a cooperative game in order to achieve consensus solutions which are evaluated with respect to several criteria simultaneously. The main idea consists of analyzing finite multi-criteria n-person games as multi-criteria bargaining games. The notion of Pareto-optimal guaranteed payoffs as a generalization of the maximin values of scalar games is proposed, together with a solution concept which can be characterized as the solutions of multi-criteria linear programming problems. A procedure to incorporate additional information about the players' preferences in order to reach a final consensus is also provided.

Nonlinear eigenvalue problems for even functionals

Let H be a Hilbert space and let g∈C 1(H,ℝ) be an even Fréchet differentiable functional with completely continuous derivative. We introduce maximin values σ k (t) which are critical values of g restricted to the sphere and show that the functions σ k (t) have right and left derivatives and that are eigenvalues of g', i.e. there exist such that Applications of the result are given to semilinear elliptic equations.

A general minimax result for relative entropy

Suppose nature picks a probability measure P/sub /spl theta// on a complete separable metric space X at random from a measurable set P/sub /spl Theta//={P/spl theta/:/spl theta//spl isin//spl Theta/}. Then, without knowing /spl theta/, a statistician picks a measure Q on S. Finally, the statistician suffers a loss D(P/sub 0//spl par/Q), the relative entropy between P/sub /spl theta// and Q. We show that the minimax and maximin values of this game are always equal, and there is always a minimax strategy in the closure of the set of all Bayes strategies. This generalizes previous results of Gallager(1979), and Davisson and Leon-Garcia (1980).

The unprofitability of mixed-strategy equilibria in two-person games: A second folk-theorem

In this paper, it is demonstrated that the payoffs of mixed-strategy equilibria are equal to the maximin values for all (2 X2) matrix games if maximim implies completely mixed strategies. This characterizes a subclass of unprofitable games. Maximin, instead of Nash equilibrium, has been suggested as an appropriate solution concept for these games.

On the existence of the value of a differential game of specified duration: PMM vol. 36, n≗2, 1972, pp.189–200

A natural extension of Isaac's formulation [1] leads to differential games which are described by differential equations with aftereffect and as regards information are determined by information on the whole prehistory of the game. In this paper relating closely to the investigations in [2 – 10], the emphasis is on the proof of the existence theorem for the value of the differential game in such an extended formulation. We examine a differential game of prescribed duration with a pay-off given in the form of a functional on the game's trajectory. The differential game is approximated by a certain family of multistage games. We consider two sequences (minimax and maximin) comprised of the minimax and the maximin values of the payoff in the multistage games. Conditions are obtained under whose fulfillment these sequences converge and their limits are equal. The proof of the convergence of the minimax and maximin sequences is carried out by a scheme suggested by Fleming [2], The proof of the equality of the limits of these sequences is based on the results of Krasovskii and Subbotin [8, 9].