A Versatile Stochastic Duel Game

This paper deals with a time dependent game model under the duel type setup. The new hybrid model of the game theory and the fluctuation process could be applied for various practical decision making situations. The unique theoretical stochastic game model is proposed to analyze the two person duel type game in the time domain. The parameters for the strategic decision including the moments of crossings, prior crossings and the optimal number of iterations to get the highest chance for successful shooting are obtained by the compact closed joint functional. In addition, the paper demonstrates to use the new time based stochastic game model for analyzing a conventional duel game model in the distance domain. It also briefly explains to build the strategies into atypical business case and how it actually works.


INTRODUCTION
A game theory has been applied for various strategic situations and developed to solve real-world issues more innovatively. A typical duel is an arranged engagement in combat between two persons, with matched weapons in accordance with agreed upon the rules under different conditions. The duel game in the paper is such that players are shooting a target rather than shooting each other. So, each player could choose either to "shoot" or "wait" for one step closer to the target on his turn (or iteration). In general, a conventional duel game model accurately describes the conditions in the distance domain and finds when and who could win the battle even at the beginning. The backward induction provides a simple solution: regardless of whether you have a better or a worse shoot, the shooting moment when the sum of the success probabilities passes the threshold is the most critical . [18] On the other hand, a variant antagonistic stochastic game in real time between two players A and B has been adapted on the top of a duel game. Unlike conventional antagonistic stochastic games [4][5][7] [12], the players are in a duel game situation at random times with random impacts. Either player can take the best shoot after passing a fixed threshold. According to the backward induction, a player has the chance to take a shoot when its underlying threshold is crossed. Upon that time (referred to as the first passage time), the player can make the strategic decision [6].
In the stochastic duel model, the actions of players are formalized by marked point processes to identify the chances for the successful shootings at the certain points in the time domain. The processes evolve until one of the processes crosses its fixed threshold of the probability of the success. Once the threshold is reached at some point of time, the associated player has the highest chance to succeed his shoot. This basic model could be applied to various business decision making situations even though the rules and conditions of a basic duel game model are relatively restricted. Various duel type games have been studied since 1970s [14][15][18] [20][21] and one of lessons from them is the decision that matter is when to do rather than what to do [18]. Unfortunately, duel type games only consider the deterministic turn around (iteration) for shooting by the success probabilities based on the distance between players.
Atypical marketing strategic decision making based on an antagonistic game could be applied in the smartphone business. Although there are many smartphone manufacturers, Apple and Samsung are the most dominant manufacturers in the market. This game is based on these two companies. Because smartphone technologies are changing rapidly, devices need to be upgraded, even after they are launched. Strategically, the manufacturers may have a greater chance of success if they implement more technologies (or features), but this tactic requires more time spent on research and development. More importantly, a firm could lose its market share if it launches a product that fails to satisfy customers. Therefore, its initial product should be appealing enough to dominate the market. However, at the same time, the firm could also fail if it launches a product too late compared to its rival. This kind of research has been massively studied in the marketing area [1][9-10] [13][19] [23][24] but none of research has studied mathematical approaches. Although the new mathematical game model in this paper is relatively restricted, this newly proposed model could apply various business decision making issues, especially marketing strategic decision for new product launch.
The article presents a versatile stochastic duel game with the complete information which means both players know the success probabilities in the time domain. Unlike a conventional (in the distance domain) duel game, each player might not have the same iteration periods and the iteration periods for each round might be different even within the same player (i.e., stochastic). Lastly, the paper demonstrates the special case of the stochastic duel game which is based on the deterministic same iteration times for both players. This special case shows how the time domain duel game model is related with the distance based duel games.
The paper is organized as follows: Section 2 presents a model of a process where the decision making occurs according to a marked point process in time, with two dimensional marks presenting the cumulative success probabilities up to the dominant point of shooting. We derive a joint functional of each component as the process at the first passing the dominant point and at one step prior. This section also contains practical implications how to understand this new model properly. In Section 3, the special case of a versatile stochastic duel game is covered. It deals with the deterministic iteration times of both players and demonstrates how the stochastic duel game could be applied into a wellknown conventional duel game. It is a relatively simple case but it gives more clear understanding how this new type of stochastic duel games works. This section also contains the way of the time domain transform from the distance domain. Section 4 presents the real-world application which this new model applies in the marketing strategy.
The setup is in the smartphone market which has mainly two dominant players. This section demonstrates how the versatile stochastic duel game could be applied into the realworld situation before the conclusion in Section 5.

A S D G NTAGONISTIC TOCHASTIC UEL AME
The antagonistic duel game of two players (called "A" and "B") are introduced and both players know the full information regarding the success probabilities based on the time. Each player has two strategies either "shoot" or "wait" and choose one strategy at the certain points of time. Let be a payoff function of player A based on the continues E = a b time and be a payoff function of player B at the time .
A payoff function represents the reward value at the time of each player such as the benefits of a player. Both functions are assigned as follows: and are the end of the time which gives the maximum payoff of each player. The values could be implied as the end of the product life cycle when this game is applied in the marketing strategy decision making problem. The probabilities regarding hitting an opponent player at (or ) are considered as follows: Both hitting probabilities are arbitrary incremental continuous functions which reaches 1 when the time (or ) goes to the allowed maximum .
It is noted that the probability of hitting an opponent player becomes 100 % when player A takes a shoot at = > max max ( for player B) which is equivalant with the maximum payoff of player A. The strategic decision in a duel game means to find 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 the shoot (i.e., success probability) and this optimal point becomes the moment of the success in the continuous time domain. This moment is defined as follows: Each player can make the decision at the certain points of the time. Let ( , ( ), ) be a S Y S T probability time space and let be independent -subalgebras. Suppose are -measurable and -measurable renewal point processes with the following Y Y 2.8 7 7 ³ oe X  X ß 5 oe "ß #ß á ß !ß 5 Ÿ !Þ oe a b The game in the paper is a stochastic process describing the evolution of conflicting between players A and B based on the perfectly known information (i.e., the success probabilities of players are known) . Only on the -th epoch , player A could make [22] j W 4 the decision either for taking a shoot or for waiting until another turn (iteration) . He W 4" will have the best chance to hit the player B exceeds its respective threshold (or U V for player B). To further formalize the game, the exit indices are introduced as follows: indicates that player A is starting the game first. In the case of the duel games in 5 ! <! the time domain, the threshold of each player could be converged into one value which > ‡ will be introduced later. Player A will have the best chance to succeed for shooting compare to the failure chance of player B ( and respectively). Hence, . / player A has the highest success probability of shooting at time , unless player B does W . not reach his best shooting chance at time . Thus, the game is ended at . / mine f However, we are targeting the confined duel game for player A on trace -algebra / . / (i. e., player A in the game obtains the best chance for shooting first). The first passage time W . is the associated time from the confined game. The functional of the game will represent the status of both players upon the and the exit time . The latter is of particular interest, because player A wants to predict W ." [2-3] [11] not only his time of the highest chance, but also the moment of the next highest chance prior to this. The establishes an explicit formula for and we abbreviate Theorem 1 Q ./ with (2.12)-(2.19): The Laplace-Carson transform is applied as follows: where is the inverse of the bivariate Laplace transform . _ " [2] Theorem 1. The functional of the game on trace / .
/ satisfies the following formula: Proof: Introduce the families: .a b : oe infe f 7 À W  : Application for to will bypass all terms except for _ Q s :; . Thus, applying operator to random set To prove formula (2.25), we first notice that " " then by Fubini's Theorem and due to (2.25) and (2.28)ß Then, we have can be revived when the inverse of the operator of (2.21) to s Ra The functional Q ./ contains all decision making parameters regarding this game. The information includes the best moments of shooting ( ) and the one step W ß X à . / exit time before the best moment ( ) and the optimal number of iterations = W ß X à . / " " pre-exit time for both players. The information for both players from the closed functional are as follows: It is noted that the best shooting moments of both players may be revised because the optimal index of each player should be an integer. The versatile stochastic duel game allows to analyze the game of two conflicted factors in the continues time domain. Although atypical duel game is limited to the success probabilities of two players based on the distance between players , the continuous time domain duel game in this research [18] could be flexible by using any incremental cost functions based on the continuous time domain.
There are couple of practical implications when we are dealing with this versatile stochastic duel game: First, the backward induction which is the core of a duel game does not be changed even in the time based stochastic games. Therefore, the best strategy for each player is "shoot as soon as that the player cross the point when sum of the success probability of hitting equals one" . The functional [18] Q ./ gives the full analytical information to build up the winning strategies for both players. From (2.32)-(2.36), player A should take a shoot at his -th turn (which means he should wait until the . .  " iterations). The average duration until player A has the best chance to win becomes . "c d W . Similarly, player B should shoot at his -th iteration and the average duration is . / "c d X / Basically, each player will continue their iterations until the accumulated iteration passes the threshold which is from (2.4). Second, > ‡ because of the backward induction, a player who has higher chance to win is determined at the beginning of the game. Again, it does not matter players are better or worse shooting but it is the cumulative success probabilities that matter.

S C D I T PECIAL ASE: ETERMINISTIC TERATION IMES
This section demonstrates the practical case of a versatile stochastic duel game which considers the deterministic iteration time for each player. As it is mentioned before, the continuous time domain duel game in this research could be flexible by using any incremental cost functions based on the continuous time domain. Let us assume that the optimal moment of a time based duel game is already known as from (2.1)-(2. ). Since, > % ‡ we deals with the deterministic iteration process of each player, the durations of one iteration for each player are constant values. From (2.7)-(2.8) and (2. ):

3.1
where and are the fixed iteration duration time of players ( for player A and for .

Lemma 1.
The functional based on the fixed iteration duration (3.1) is as follows: Proof: from (2.12), let us assign (2.13)-(2.19) as follows: Recall the from (2.22), we have Theorem 1 è From (2.32) and (2.34), of the information for both players to get the optimal moment shooting are as follows: The actual moment of best shoot may not same as (2.32) and (2.35) because the exit index of each player should be an integer and this fact also impacts on the of each pre-exit time player: .

Best Strategies for Player A
The best response is the strategies which produce the most favorable outcome for a player [3]. The best response of player A is the of shooting and it responses based on exit index . the iteration time of player B. Let us assign the function of best response correspondences for player A as follows:

3.20
As an illustration, we take 35 5 and As it is mentioned above, > oe ß . oe . − !Þ!"ß > Þ ‡ ‡ + , c d the iteration process of each player is deterministic and the analysis for the best response of player A is demonstrated in Fig. 1. ). It means that player A should . oe "( , take the shoot at 17th iteration to get the highest chance to hit player B. From (3.18), the exit time of player A gives the best hitting chance and the hitting probability of player A is T "! + a b . + from (2.1) and (2.3).

Reconstructing A Conventional Duel Game
The distance based conventional duel game is a special case of the versatile stochastic duel game under the deterministic iteration time condition. This section demonstrates how the duel game in the time domain could be adapted to analyze a conventional duel game. As it has been mentioned in the previous section, the conventional duel game is adapted from Polak . Let us consider the duel game which has the following rules: [18] 1. Each player (player A, B) has a gun with single bullet and player A starts the gameà 2 and they are facing each other with the distance of ; Þ P oe $! Ò7Ó a b 3. let be the probability of player A hitting player B if player A shoots at distance ; T 6 6 + a b 4. let be the player B's probability of hitting player A if player B shoots at distance ; T 6 6 , a b 5. the probability of hitting for each player is give as follows: T 6 oe P  6 ß T 6 oe "  † 6ß ! Ÿ 6 Ÿ Pß " " P P 6. players alternatively have the chance to make the decision; 7. either "shoot" or "one step forward"; a b 6 oe # Ò7Ó ! 8. Every turn makes every closer each other if both players are moving #6 oe % Ò7Ó ! a b forward instead of shooting; 9. the hitting probabilities of both players are known (i.e., perfect information).
According to the above rules, the illustration of the hitting probabilities of both players is shown in Fig. 2. The first shoot should occur after which means that no one should before but there is 6 6 ‡ ‡ a dominance at . As it is mentioned on the rule, player A knows the next move of player 6 ‡ B [7]. Whoever gets the first turn after will be the winner of the game and The value 6 6 ‡ ‡ is determined by their join ability as follows: 6 oe ! Ÿ 6 Ÿ P À T 6  T 6 " ‡ + , From (3.22) and the above given game rules, 6 ""Þ%&* 7 ‡ yields and player A have the c d more chance to win this game if player A shoots at his 5-th turn. It is noted that who shoots first is not necessarily better or worse shooter . This typical duel problem could be [18] solvable by transforming the distance domain to the time domain. Let us assume that the speed of "moving forward" is which is a constant value. All variables in the @ " 7Î= c d distance domain are easily transformed in the time domain (see Table I).
The time domain transforms of the hitting probabilities are illustrated in Fig. 3. As it is illustrated in Fig. 3, the hitting probabilities in both domains are same but the input variable has been changed to time (seconds) from distance (meters). Once the distance domain is mapping in the time domain, we could find the best response from . / a b a b . + (3.21) which could be graphically demonstrated in Fig. 4. The best responses function of player A shows all iteration time range of player B and the best strategy of player A in this duel game is shooting at 5-th iteration moment where . oe . , + (i.e., the downward-pointing triangle in Fig. 4).

C P N P L S ASE RACTICE: EW RODUCT AUNCHING TRATEGY
This section demonstrates the practical application of this research for developing marketing strategy. The case practice shows how stochastic duel games could be applied. The numbers and names are fictional but realistic for this demonstration. In this case practice, there are two major smartphone companies that share the market. Let us make some stories regarding the above situation: Samsung just finished developing its flagship smartphone for this year. Samsung should decide to either "Launch" the smartphone, or "Wait" for running an additional development cycle to add new features. One development cycle takes approximately 6 months. Once the development cycle is started, they cannot launch the product until it is completed. In other words, Samsung must wait 6 months for his next opportunity to make another decision once he chooses to "Wait." Samsung knows that Apple will complete its flagship smartphone 5 months later, and its development cycle to add new features will finish 4 months after it launches the first product. According to the simulated results, after passing the moment , Samsung has the > ‡ opportunity to launch its product before Apple launches their product. But, Samsung should run at least 3 development cycles to create new features before launching (i.e., . oe $ oe W ) after their first commercially ready product is to be launched . If both a b ! companies launch their flagship smartphone before 18 months , customers would a b  > ‡ not buy their products because they lack attractive features. In this game, the process of backward induction shows that Samsung is mostly likely to win because the moment where Samsung shoots is closer to the threshold than what Apple has (i. e., a b oe W > . ‡ >  W  X ‡ . / ). From the perspective of Apple, in this particular scenario, releasing the smartphone on is no longer the optimal strategy because Apple knows that Samsung X / will win this game when it launches its device at . Therefore, it might be better that W . Apple takes the risk to launch the product at ( 17 months) instead of ( 21 X oe X oe / / " months). Probabilistically, Apple will likely lose this game, but the solution to the game is not deterministic. A player with a higher probability of success is not guaranteed to win the game.

CONCLUSION
The new type of an antagonistic stochastic duel game has been studied. In this versatile stochastic duel game, both players could have random iterations in the time domain. A joint functional of the process has been constructed to analyze the information of decision making parameters which give the best chance to win the time domain game of each player. Compact closed forms for Laplace-Carson transforms of named functional have been obtained. Basically, the paper provides the hybrid stochastic model of the game theory and the fluctuation process which is more flexible to apply duel type game problems more effectively. The analytical approach is fully supported to understand the core of a versatile stochastic duel game by applying this model to a conventional duel game as a special case. Furthermore, the actual application of this new model has been demonstrated to support the direct implementation of a versatile stochastic duel game into real-world decision making situations in the smartphone market.