Random Walk Analysis in Antagonistic Stochastic Games

Abstract

This article deals with two "antagonistic random processes" that are intended to model classes of completely noncooperative games occurring in economics, engineering, natural sciences, and warfare. In terms of game theory, these processes can represent two players with opposite interests. The actions of the players are manifested by a series of strikes of random magnitudes imposed onto the opposite side and rendered at random times. Each of the assaults is aimed to inflict damage to vital areas. In contrast with some strictly antagonistic games where a game ends with one single successful hit, in the current setting, each side (player) can endure multiple strikes before perishing. Each player has a fixed cumulative threshold of tolerance which represents how much damage he can endure before succumbing. Each player will try to defeat the adversary at his earliest opportunity, and the time when one of them collapses is referred to as the "ruin time". We predict the ruin time of each player, and the cumulative status of all related components for each player at ruin time. The actions of each player are formalized by a marked point process representing (an economic) status of each opponent at any given moment of time. Their marks are assumed to be weakly monotone, which means that each opposite side accumulates damages, but does not have the ability to recover. We render a time-sensitive analysis of a bivariate continuous time parameter process representing the status of each player at any given time and at the ruin time and obtain explicit formulas for related functionals.

Layers of noncooperative games

“…In a basic model, which is treated in Section 2, we are interested in the ruin time of a selected player upon nearest observation epoch, and the status of the system at this time (along with many other parameters introduced in the upcoming sections). Unlike the recent work by the first author [3,4] with time dependent analysis (and thus more detailed information about the processes at any moment of time), the present modeling and analysis will be concerned with the information for the reference times only. They are: the exit time of player A (considered as a loser, without loss of generality) from the game and the pre-exit time, i.e.…”

“…It is very common and quite reasonable for an observer or user to associate himself with the losing player A (rather than the winner B) to evaluate and prognosticate the worst outcome and to take measures ahead of time. The advantage of this more rudimentary model (by the way, not being a special case of [3,4]) lies in very compact explicit formulas leading to more tame computations. Some of the past work on related techniques was initiated in [2] by the first author and it was merely applied to the stock market.…”

“…In a more advanced model of Section 3, we will emulate some of the time dependence of [3,4] in the following sense. We will embed an ''intermediate threshold'', say M 1 (< M) and see, through this ''layer'', along with the rest of the information, when the overall damage to player A will exceed M 1 before it will exceed M. In other words, this refinement will more closely analyze the behavior of the processes prior to player's A defeat.…”

Multilayers in a modulated stochastic game

“…The closed form enabled one to alter M 1 arbitrarily thereby refining the information on the status of player A w.r.t. various thresholds instead of refining it w.r.t time as in [14]. A further refinement is to lay out the cross-level behavior of the process associated with player B, in addition to that for player A.…”

“…Fluctuation theory has also become a stand-alone area of stochastics, with wide spread applications to physics [23,26,27,37] economics [32][33][34][35], stock market [11,12], biology [26], and queueing theory [21,42]. The use of fluctuation theory in games has not been explored until recently [14][15][16][17].…”