We present the antagonistic duel game of two players (called “A” and “B”) where both players know the full information regarding the success probabilities based on time. Each player has two strategies, either “shoot” or “wait”, and chooses one strategy at certain points of time. Let

$A\left(s\right)$ be a payoff function of Player A based on the continuous time

s and

$B\left(t\right)$ be a payoff function of Player B at the time

t. Both payoff functions are monotone and non-decreasing. A payoff function represents the reward value for the time of each player such as the benefits of a player. Both functions are assigned as follows:

where

${s}_{\mathrm{max}}$ and

${t}_{\mathrm{max}}$ 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

s (or

t) are considered as follows:

Both hitting probabilities are arbitrary incremental continuous functions that reach one when the time

s (or

t) goes to the allowed maximum

$\left({s}_{\mathrm{max}}<\infty \right)$. It is noted that the probability of hitting an opponent player becomes 100 % when Player A takes a shot at

${s}_{\mathrm{max}}$ (

${t}_{\mathrm{max}}$ for Player B), which is equivalent to the maximum payoff of Player A. The strategic decision in a duel game 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 shot (i.e., success probability), and this optimal point becomes the moment of success in the continuous time domain. This moment

${t}^{*}$ is defined as follows:

Each player can make the decision at certain points of time. Let (

$\Omega $,

$\mathcal{F}$(

$\Omega $),

P) be a probability time space, and let

${\mathcal{F}}_{S},$ ${\mathcal{F}}_{T}\subseteq \mathcal{F}\left(\Omega \right)$ be independent

$\sigma $-subalgebras. Suppose:

are

${\mathcal{F}}_{S}$-measurable and

${\mathcal{F}}_{T}$-measurable renewal point processes (

${\epsilon}_{a}$ is a point mass at

a) with the following notation:

The game in the paper is a stochastic process describing the evolution of conflict between Players A and B based on perfectly known information (i.e., the success probabilities of the players are known) [

22]. Only on the j

^{th} epoch

${S}_{j}$, Player A could make the decision either to take a shot or to wait until another turn (iteration)

${S}_{j+1}$. He/she will have the best chance to hit Player B exceeding his/her respective threshold

U (or

V for Player B). To further formalize the game, the exit indices are introduced as follows:

and

${\sigma}_{0}<0$ indicates that Player A is starting the game first. In the case of the duel games in the time domain, the threshold of each player could converge to one value

${t}^{*}$ (i.e.,

$U\to {t}^{*},V\to {t}^{*}$), which is found from (

4). Player A will have the best chance to succeed in shooting compare to the failure chance of Player B (

${P}_{a}\left({S}_{\mu}\right)$ and

$1-{P}_{b}\left({T}_{\nu}\right)$, respectively). Hence, Player A has the highest success probability of shooting at time

${S}_{\mu}$, unless Player B does not reach his/her best shooting chance at time

${T}_{\nu}$. Thus, the game is ended at min

$\left\{{S}_{\mu},\phantom{\rule{4.pt}{0ex}}{T}_{\nu}\right\}$. However, we are targeting the confined duel game for Player A on trace

$\sigma $-algebra

$\mathcal{F}\left(\Omega \right)\cap \left\{{P}_{a}\left({S}_{\mu}\right)+{P}_{b}\left({T}_{\nu}\right)\ge 1\right\}\cap \left\{{S}_{\mu}\le {T}_{\nu}\right\}$ (i.e., Player A in the game obtains the best chance of shooting first). The first passage time

${S}_{\mu}$ is the associated time from the confined game. The functional of the game model:

is the model of a standard stochastic game with a continuum of states and represents the status of both players upon the exit time

${S}_{\mu}$ and the pre-exit time

${S}_{\mu -1}$ [

23,

24,

25]. The pre-exit time is of particular interest because Player A wants to predict not only his/her time for the highest chance, but also the moment for the next highest chance prior to this.

**Proof.** The theorem is abbreviated as (

13)–(

20):

The Laplace–Carson transform is applied as follows:

with the inverse:

where

${\mathcal{L}}^{-1}$ is the inverse of the bivariate Laplace transform [

23]. Let us introduce the families:

Application of

${\widehat{\mathcal{L}}}_{pq}$ to

${\Phi}_{\mu \left(p\right)\nu \left(q\right)}$ will bypass all terms except for

$\left\{{S}_{\mu}<{T}_{\nu}\right\}\cap \left\{{P}_{a}\left({S}_{\mu}\right)+{P}_{b}\left({T}_{\nu}\right)\ge 1\right\}$. Thus, applying operator

${\widehat{\mathcal{L}}}_{pq}$ to random set

$\left\{{\mathbf{1}}_{\left\{\mu \left(p\right)=j,\phantom{\rule{4.pt}{0ex}}\nu \left(q\right)=k\right\}}:p\ge 0,\phantom{\rule{4.pt}{0ex}}q\ge 0\right\}$, we arrive at:

To prove the formula (

25), we first notice that:

Then, iterating the integral of (

21), we have from (

25):

which yields (

25). Denote:

Since,

where:

then by Fubini’s theorem and due to (

25) and (

28),

where:

and:

Now,

${\Phi}_{\mu \left(p\right)\nu \left(q\right)}$ can be revived when the inverse of the operator

${\widehat{\mathcal{L}}}_{pq}$ of (

22) to

$\Psi \left(u,v\right)$:

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