Three-Player Gambler’s Ruin Problem: Some Extensions

For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.

Parisian excursion of a Levy process is defined as the excursion of the process below or above a pre-defined barrier continuously exceeding a certain time length. In this thesis, we study classical and Parisian type of ruin problems, as well as Parisian excursions of collective risk processes generalized on the classical Cramer-Lundberg risk model. We consider that claim sizes follow mixed exponential distributions and that the main focus is claim arrival process converging to an inverse Gaussian process. By this convergence, there are infinitely many and arbitrarily small claim sizes over any finite time interval. The results are obtained through Gerber-Shiu penalty function employed in an infinitesimal generator and inverting corresponding Laplace transform applied to the generator. In Chapter 3, the classical collective risk process under the Cram´er-Lundberg risk model framework is introduced, and probabilities of ruin with claim sizes following exponential distribution and a combination of exponential distributions are also studied. In Chapter 4, we focus on a surplus process with the total claim process converging to an inverse Gaussian process. The classical probability of ruin and the joint distribution of ruin time, overshoot and initial capital are given. This joint distribution could provide us with probabilities of ruin given different initial capitals in any finite time horizon. In Chapter 5, the classical ruin problem is extended to Parisian type of ruin, which requires that the length of excursions of the surplus process continuously below zero reach a predetermined time length. The joint law of the first excursion above zero and the first excursion under zero is studied. Based on the result, the Laplace transform of Parisian ruin time and formulae of probability of Parisian type of ruin with different initial capitals are obtained. Considering the asymptotic properties of claim arrival process, we also propose an approximation of the probability of Parisian type of ruin when the initial capital converges to infinity. In Chapter 6, we generalize the surplus process to two cases with total claim process still following an inverse Gaussian process. The first generalization is the case of variable premium income, in which the insurance company invests previous surplus and collects interest. The probability of survival and numerical results are given. The second generalization is the case in which capital inflow is also modelled by a stochastic process, i.e. a compound Poisson process. The explicit formula of the probability of ruin is provided.

In the classic three-player ruin problem, the play continues until at least one of the players is completely ruined. In this research, we present a novel version of the classic three-player game with interest lies in a specific player. We determine the ruin probabilities and expected durations of the game given that our player (i.e. the gambler or the casino's player) wins or loses. The desired game plan is executed for both the cases, i.e. symmetric and asymmetric with all of the players having equal initial stakes. We also obtained the asymptotic results of the ruin probabilities and expected durations of the proposed game plan. Further, the validity of the desired game structure is also verified through a Monte Carlo simulation study.

The gambler's ruin problem with n players and asymmetric play

ABSTRACTThis paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.

The asymmetric n-player gambler's ruin problem with equal initial fortunes

An interest in gambling has greatly increased over the last few decades with the more common use of slot machines and online gambling, especially sports betting. A concern that has been publically raised is addiction and eventual gambler’s ruin (loss of all money). In this paper we provide a solution to the Gambler’s Ruin problem in regards to roulette. We compute the probability of a gamblers ruin with applications to the various betting opportunities playing roulette by determining the W+1 roots of the relevant polynomials and from there determine the probability of a gamblers ruin. We find situations where the payoff becomes higher, the probability of ruin becomes lower. Lower goals of gain are associated with a lower probability of ruin and larger bets and larger odds payoff also increase the probability of ruin.

We introduce new theoretical insights into two-population asymmetric games allowing for an elegant symmetric decomposition into two single population symmetric games. Specifically, we show how an asymmetric bimatrix game (A,B) can be decomposed into its symmetric counterparts by envisioning and investigating the payoff tables (A and B) that constitute the asymmetric game, as two independent, single population, symmetric games. We reveal several surprising formal relationships between an asymmetric two-population game and its symmetric single population counterparts, which facilitate a convenient analysis of the original asymmetric game due to the dimensionality reduction of the decomposition. The main finding reveals that if (x,y) is a Nash equilibrium of an asymmetric game (A,B), this implies that y is a Nash equilibrium of the symmetric counterpart game determined by payoff table A, and x is a Nash equilibrium of the symmetric counterpart game determined by payoff table B. Also the reverse holds and combinations of Nash equilibria of the counterpart games form Nash equilibria of the asymmetric game. We illustrate how these formal relationships aid in identifying and analysing the Nash structure of asymmetric games, by examining the evolutionary dynamics of the simpler counterpart games in several canonical examples.

In this article, we present the general expressions for the variance of the ruin time of the classic two-player gambler's ruin problem with successive and nonoverlapping trials. The rationale of this game plan is motivated by its exhibition in the game of tennis, where a player is required to win two consecutive serves to win the point after achieving deuce. This strategy (i.e., decision is based on successive and nonoverlapping trials) is in favor of the player, who plays with a better skill set and reduces the chances of decision based only on luck. We explicitly derive the general expressions of variance up to m successive and non-overlapping trials for the case of symmetric and asymmetric games. It is proved that the expressions given in literature for the symmetric and asymmetric cases are the sub cases of our proposed expressions. Finally, some special games (i.e., m = 2) are simulated and the results are verified with the proposed formulas.

On the distribution of classic and some exotic ruin times

In the context of the classical two-player gambler's ruin problem, the winning probabilities and initial stakes are pre-decided. If a player (who is in financial crisis) starts with less amount than his/her opponent in the symmetric game, has more chances to be ruined. Besides, a player (based on previous record data) with more winning probability than his/her competitor, has fewer chances to be ruined. We observe that most of the time, usually a weaker player is not fully willing to make a contest with a strong player. To give a fair chance to fight back for a weaker player and to develop the audience's interest, equity-based modeling is required. In this research, we propose some new equity-based models for the game of two players. In this way, we advocate the weaker player (with less winning probability or less amount to start the game) is motivated to participate in the contest because of a fair chance to make a comeback. The working methodology of newly proposed schemes is executed by deriving general expressions of the ruin probabilities for mathematical evaluation along with observing the ruin times, and then findings are compared with the results of a classic two-player game. Hence, the prime objectives related to the study are achieved by taking diverse parametric settings in the favor of equity-based modeling.

We analyze the main dynamical properties of the evolutionarily stable strategy (ℰ𝒮𝒮) for asymmetric two-population games of finite size and its corresponding replicator dynamics. We introduce a definition of ℰ𝒮𝒮 for two-population asymmetric games and a method of symmetrizing such an asymmetric game. We show that every strategy profile of the asymmetric game corresponds to a strategy in the symmetric game, and that every Nash equilibrium (𝒩ℰ) of the asymmetric game corresponds to a (symmetric) 𝒩ℰ of the symmetric version game. We study the (standard) replicator dynamics for the asymmetric game and we define the corresponding (non-standard) dynamics of the symmetric game. We claim that the relationship between 𝒩ℰ, ℰ𝒮𝒮 and the stationary states (𝒮𝒮) of the dynamical system for the asymmetric game can be studied by analyzing the dynamics of the symmetric game.KeywordsNash EquilibriumMixed StrategyPure StrategyEvolutionary GameStable StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article, both classical discrete‐ and continuous‐time risk models are introduced; then the time of ruin and the severity of ruin are defined. The probability of ruin and some quantities concerning the time of ruin, such as the expectation of the present value of the time of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin, and the time of ruin caused by a claim, and the moments of the time of ruin due to oscillation and claim, respectively, are studied.

In this article, both classical discrete‐ and continuous‐time risk models are introduced; then the time of ruin and the severity of ruin are defined. The probability of ruin and some quantities concerning the time of ruin, such as the expectation of the present value of the time of ruin, the (discounted) moment of the severity of ruin caused by a claim, the joint moment of the severity of ruin, and the time of ruin caused by a claim, and the moments of the time of ruin due to oscillation and claim, respectively, are studied.

The asymmetric [formula omitted]-player gambler’s ruin problem with ties allowed and simulation