An Attrition Problem of Gambler's Ruin

Asymptotics for the time of ruin in the war of attrition

“…We now consider the model in [15], in which each army loses a soldier independently of the size of m and n. Let the success probability be 1 2 , independent of m and n. We refer to such a model as a 'simple random war'. Without further delay, we note that the results of this section up until (14) can also be obtained via the framework of a simple symmetric random walk; however, for the purposes of consistency, we employ the framework from Section 2.…”

“…The jumps J N ± (f ) and their probabilities given by q N s correspond to the dynamics in (15), after the scaling (17). They are…”

“…We thus formulate the following model below, which, like the model in [23], involves a 'war' between units, and like the model in [15], is an attrition ruin problem. The game is played by two armies, which for convenience we call army A and army B, starting with a pair (m, n) of soldiers, where m and n are nonnegative integers, designating the assets of armies A and B, respectively.…”

“…A difficult and essential part of Theorem 1 is completed by noting a surprising connection between p(m, n) and the Eulerian numbers. In Section 3 we take a closer look at the model in [15], where A and B lose a unit with fixed probabilities independent of m and n. The models studied can be viewed as random walks in the first quadrant, for which we refer the reader to [19], [21], and [29]. In Section 4 we consider the continuous-time version of the model presented in Section 2 with X t and Y t denoting the assets of the two players at time t ≥ 0 and T (m, n) denoting the duration of the game, i.e.…”

Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

“…There are many variations of the problem, newer ones are formulated after older ones are solved. For example, the following variations were proposed: infinite amount of money, three or more players [KP02], [RS04], the attrition variation (applies, e.g., to World Series or Stanley Cup finals [Kai79]), some cases of winning probabilities being dependent on current fortune [ES09], [Lef08].…”