Two-Person Stochastic Duel with Energy Fuel Constraint Ammo

This paper deals a novel variation of the versatile stochastic duel game, which incorporates an energy fuel constraint in a two-player duel game. The energy fuel not only measures the vitality of players but also determines the power of the shooting projectile. The game requires players to carefully balance their energy usage while trying to outmaneuver their opponent. This unique theoretical framework of the stochastic game model provides a valuable method for understanding strategic behavior in competitive environments, particularly in decision-making scenarios with fluctuation processes. The proposed game provides players with the challenge of optimizing their energy fuel usage while managing the risk of losing the game. The unique rules and constraints of the game in this research are expected for contributing insights into the decision-making strategies and behaviors of players in a wide range of practical applications.


INTRODUCTION
Game theory is a powerful tool for modeling and analyzing strategic decision-making in a wide range of fields, from economics and political science to computer science and engineering [1][2][3][4][5][6].Modern game theory has seen significant developments in recent years, particularly in the areas of repeated games, evolutionary game theory, and network games.One important recent development is the use of machine learning techniques in game theory, which has led to advances in predicting and optimizing outcomes in complex, multi-player games [7].Another area of active research is the analysis of games in which players have incomplete information, which has applications in contract theory, mechanism design, and more [8].Network games have also become an important area of research, as they model interactions among agents in a social or economic network, with implications for contagion and diffusion dynamics [9].Another growing area of interest is the use of game theory in analyzing cybersecurity, particularly in the context of defending against cyber-attacks [3,5,10].Overall, modern game theory has seen exciting developments and applications in a wide range of fields.As researchers continue to develop new methods and models, game theory is likely to remain a valuable tool for understanding strategic decision-making in complex, dynamic environments.Duel games have been the focus of study in game theory due to their applicability in modeling competitive situations, ranging from military conflicts to economic decisions [11].Recent research has explored variations of the duel game, such as the generalized stochastic duel game [12] and the duel game with multiple asymmetric players [13,14] and penalties.Other studies have examined the impact of different strategies on the outcome of duel games, including the use of mixed strategies [15] and time dependent strategies [16,17].Several studies have also explored the application of duel games in specific contexts, such as decision-making in the presence of incomplete information [18], analysis of political competitions [19], and modeling of predator-prey interactions in ecological systems [20].

S D G F C A TOCHASTIC UEL AME WITH UEL ONSTRAINT MMO
The antagonistic duel game of two players (called "A" and "B") in the time domain is considered and both players know the full information regarding the success probabilities based on the time domain [1,2].Let us assigned that      is the monotone increasing CPFs (Probability Density Functions) for player A regarding hitting the opponent player (player B) at the time .Similarly, hitting the       is the probability of player B for opponent player (player A) at the time .
 Each hitting probability (CDF) could be arbitrary chosen and it reaches 1 (i.e., 100 %) when the time goes to the infinite .The      strategic decision for the shooting moment means finding the moment when a player will have the best chance to hit the other.There is a certain point that maximizes the chance for succeeding in the shoot, and this optimal point becomes the moment of success in the continuous time domain.This moment is defined as follows: Additionally, each player has the energy fuel level which randomly drooped and the spent (or usage) of the energy fuel for each player is monotone and non-decreasing.Let ( , are related with the energy drain of each player.The energy level of each player becomes the shooting power to hit a opponent player.The energy fuel of player A drains at times and the magnitudes of energy drains are formalized by process .The energy  drain of player B are described by process similarly.The processes and are    specified their transforms: The game is observed at random times in accordance with the point process which is assumed to be delayed renewal process.
forms an observation process upon embedded over with respective increments and The observation process could be formalized as and it is with position dependent marking and with and being dependent with the By using the double expectation, we have where Let us consider the maximum energy levels of players and .The energy level of     player A after draining fuel is and for player B from (2.9).The stochastic process the energy level of each player are as follows:

2.15
 and the game is over when the -th observation epoch , the shooting power of player A    which is equivalent with the remained energy level at the moment of the shooting is greater than the energy fuel level of player B: To further formalization of the game, the exit index could be defined as follows: 18 where Since player A is assumed to win the game at time , we shall be targeting the confined   game in the view point of player A. The passage tme is associated exit time from the   confined game and the formula (2.15) will be modified as which the path of the game from , which gives an exact definition of the model observed until The joint functional of the    stochastic duel with the fuel limited ammunition is as follows: This model will represent the status of both players upon the and the pre-exit exit time   time .The pre-exit time is a particular interest because player A wants to predict not   only her time for the highest chance, but also the moment for the next highest chance prior to this.The establishes an explicit formula for and we abbreviate Theorem-1


The linear operators are defined as follows:

2.29
where is a sequence, with the inverse otherwise. (2.30) Theorem-1.The functional of the game on trace  satisfies the following formula: We find the explicit formula of the joint function  and applying the operator to  random family , we have

2.36
The functional       contains all decision making parameters regarding this standard stopping game.The information includes the optimal number of iterations of players (i.e.,     and ), the best moments of shooting ( , ; ) and the one step prior to the   exit time best shooting times ( , ; ).The information for player A from the       pre-exit time closed functional are as follows:

.38)
Additionally, there are some special case could be considered.First case is the assets of both players are the same (i.e.,    ).From (2.36), the formula is changed as follows: This implication means that both players have the same energy fuel level and the winning strategy for this game should take the shot as soon as passing the threshold from (2.1).
The other case might be the initial energy level of player A is smaller than what player B has (i.e., ).The best strategy of player A should wait until player B shoots (and      fail) because player A has no chance to win even he hits the target correctly.

CONCLUSION
A new successor to the antagonistic stochastic duel game was studied.This research primarily focused on deriving compact closed-form solutions utilizing transformation and flexible analysis techniques, which were adapted by varying the concept of the energy fuel level.In this innovative duel game, a player can win the game only if their bullets hit the target player and if their shooting power exceeds the remaining energy fuel level of the target player at the moment of shooting.A joint functional of the standard stopping game was constructed to analyze the strategic decision parameters, which indicated the best moment for shooting in the time domain stochastic game.This study provides a thorough account of the innovative analytic approach, shedding light on the fundamental principles that underlie the stochastic duel game model with energy fuel constraints.