Equilibrium in safe strategies

Abstract

The interaction of many participants dividing among themselves a resource located on a certain set is investigated. For the problem stated, the Mash equilibrium does not exist in most of the cases, but at the same time there is an intuitively perceived stable rational behavior of participants, which relies on the reflexive accounting of mutual threats. For the description of such a behavior, the definition is set up of an equilibrium in safe strategies (ESS's), which coincides with the strict Nash equilibrium where it is available and exists for those situations in the stated problem where it is unavailable. Using the introduced definition, the initial problem is examined. The introduced equilibrium is also illustrated by examples of bimatrix games, a competitive system of stimulation, and a game in which the ESS's does not exist.

A Competitive Stimulation Mechanism

“…We have a game of agents in normal form [1]: a set N of agents, their aim functions (2), and a set of admissible actions are given; agents make their choice once, concurrently and independently, and the description of the game is common and known to all agents. The problem here is to a solution (equilibrium) of the game of agents.…”

“…On the one hand, our model is close to the design of the optimal competitive stimulation system, which is investigated in many papers [2][3][4]. In a competitive stimulation system, the control organ (the principal) defines a procedure for comparative estimation of the activities of agents, number of classes, number of places in each of the classes, and rate of commission to agents belonging to different classes.…”

“…Equilibrium in safe strategies in this case is a set of strategies, which, when violated by a player in isolation, either he decreases his gain or encounters a threatening state [2].…”

“…There are several agents, each of which takes (concurrently with other agents and independently) decisions on his actions [1,2]. The agent that obtains the best results (called the "winner") is awarded a fixed commission.…”