Mikhail B. Iskakov

Equilibrium in safety strategies and equilibriums in objections and counterobjections in noncooperative games

The game problem of the sharing of a resource distributed in the section was first stated in the year 1957. The complexity of the model under consideration lies in the fact that for it, in most of the cases, the Nash equilibrium does not exist, but in practice there is an intuitively perceived stable rational behavior of players, which is based on the reflexive accounting of mutual objections. For the description of this behavior, the definition of equilibrium in safety strategies is suggested, which is equivalent to the Nash equilibrium for the cases in which the Nash equilibrium is available and exists for those situations in the stated problem in which the Nash equilibrium is unavailable. This permits investigating the model. The comparison is given of the suggested approach with the concepts used by various authors of equilibrium in objections and counterobjections for noncooperative games.

Application of generalized median voter schemes to designing strategy-proof mechanisms of multicriteria active expertise

Mechanisms of multicriteria active expertise are represented in the form of generalized median voter schemes for collective decision-making in the terms of families of right and left coalition systems. This approach allows for applying the results, obtained in the theory of social choice, to designing strategy-proof mechanisms.

Equilibrium contained by counter-threats and complex equilibrium in secure strategies

We present two generalizations of the concept of equilibrium in secure strategies. In equilibrium contained by counter-threats (ECCT), no player can increase its payoff by a unilateral deviation without creating a threat to lose more than it wins. This condition must be satisfied for any pseudo-equilibrium in the generalized sense and, therefore, any such equilibrium must belong to the set of ECCT. The second generalization is the complex equilibrium in secure strategies. The proposed concept allows identifying a hierarchical structure of mutual threats between players and will be useful for the analysis of problems with asymmetric behavior of players. Search algorithms for the proposed equilibria and their examples in matrix games are provided.

Equilibria in secure strategies in the Bertrand---Edgeworth duopoly

This paper analyzes the Bertrand–Edgeworth duopoly model using a solution concept of Equilibrium in Secure Strategies (EinSS), which describes cautious behavior in noncooperative games. The concept is suitable for studying games where the threats of other players represent an important factor in the decision-making process. We demonstrate that, in some cases where the Bertrand–Edgeworth price duopoly admits no Nash–Cournot equilibria, there exists a unique EinSS with both players choosing an identical equilibrium price lower than the monopoly price. The difference between these prices can be interpreted as an additional reduction in price that allows the players to secure themselves against the mutual threats of undercutting. We formulate and prove a criterion for the EinSS existence.

Games for cautious players: the equilibrium in secure strategies

A non-cooperative solution, the Equilibrium in Secure Strategies (EinSS), is defined as an extension of the Nash equilibrium in pure strategies, and is meant to solve games where players are “cautious,” i.e., looking for secure positions and avoiding threats. This concept abstracts and unifies ad hoc solutions already formulated in various applied economic games that have been discussed extensively in the literature. A general existence theorem is provided and then applied to the price-setting game in the Hotelling location model, to Tullocks rent-seeking contests, and to Bertrand–Edgeworth duopoly. Finally, competition in the insurance market game is re-examined and the Rothschild–Stiglitz–Wilson contract is shown to be an EinSS even when the Nash equilibrium breaks down.

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.

On strategy-proof direct mechanism of active expertise over strictly convex compact set

We investigate the problem of strategy-proofness in the active expertise process where the decision making are based on the messages of experts who can distort the information for their benefit. In our model the expertise result is the arithmetic mean of expert messages and the opinion space is the multi-dimensional strictly convex compact set. We construct the corresponding direct expertise mechanism. We also prove that there is no strategy-proof direct expertise mechanism for this case. The problem of finding mechanism with minimum manipulation equivalent to the arithmetic mean expertise mechanism is formulated.