This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black‒Scholes market.

We consider a game with K ≥ 2 players, each having an integer-valued fortune. On each round, a pair (i,j) among the players with nonzero fortunes is chosen to play and the winner is decided by flipping a fair coin (independently of the process up to that time). The winner then receives a unit from the loser. All other players' fortunes remain the same. (Once a player's fortune reaches 0, this player is out of the game.) The game continues until only one player wins all. The choices of pairs represent the control present in the problem. While it is known that the expected time to ruin (i.e. expected duration of the game) is independent of the choices of pairs (i,j) (the strategies), our objective is to find a strategy which maximizes the variance of the time to ruin. We show that the maximum variance is uniquely attained by the (optimal) strategy, which always selects a pair of players who have currently the largest fortunes. An explicit formula for the maximum value function is derived. By constructing a simple martingale, we also provide a short proof of a result of Ross (2009) that the expected time to ruin is independent of the strategies. A brief discussion of the (open) problem of minimizing the variance of the time to ruin is given.

An optimal selection problem for bid and ask quotes subject to a stock inventory constraint is investigated, formulated as a constrained utility maximisation problem over a finite time horizon. The arrivals of buy and sell orders are governed by Poisson processes, and a diffusion approximation is employed on assuming the Poisson arrivals intensity is sufficiently large. Using the dynamic programming principle, we adopt an efficient numerical procedure to solve this constrained utility maximisation problem based on a successive approximation algorithm, and conduct numerical experiments to analyse the impacts of the inventory constraint on a dealer's terminal profit and stock inventory level. It is found that the stock inventory constraint significantly affects the terminal stock inventory level.

For positive integers n and q and a monotone graph property $\mathcal{A}$ , we consider the two-player, perfect information game WC(n, q, $\mathcal{A}$ ), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, q + 1 edges of the complete graph Kn which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining q edges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\mathcal{A}$ , then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter q is close to n and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if q ⩾ (1 + ϵ)n, then Client can avoid components of order cϵ−2 ln n for some absolute constant c > 0, whereas for q ⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for every q ⩽ cn, where c > 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.

We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.

Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

In this paper we study Nash equilibrium payoffs for nonzero-sum stochastic differential games with two reflecting barriers. We obtain an existence and a characterization of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations with two reflecting barriers.

We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d . A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN , showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.

Two players take it in turn to claim edges from a graph $G$. The first player (“Maker”) wins if at any point he has claimed $s$ edges at a vertex without the second player (“Breaker”) having claimed a single edge at that vertex. If, by the end of play, this does not occur we say that Breaker wins. Our main aim is to show that for every $s$ there is a graph $G$ in which Maker has a winning strategy.

We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô-Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time-variable coefficients. Moreover, explicit solution formulae are given, so these results are operational. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer.

Schmidt’s game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt’s game and its variations, as well as more general games played on complete metric spaces (for example, fractals). We show that, except for certain exceptional cases, these games are undetermined on certain sets. Judging by the vast numbers of papers utilising these games, we believe that the results in this paper will be of interest to a large audience of number theorists as well as set theorists and logicians.

In this article we study a number of collisions concerning a simple occupancy problem with unequal probabilities. Using combinatorial arguments and negative associations of random variables, we have several limit theorems, namely, a weak law of large numbers and a Poisson law of small numbers including the Chen-Stein estimate.

The objective of this paper is to give a rigorous analysis of a stochastic spatial model of producer-consumer systems that has been recently introduced by Kang and the author to understand the role of space in ecological communities in which individuals compete for resources. Each point of the square lattice is occupied by an individual which is characterized by one of two possible types, and updates its type in continuous time at rate 1. Each individual being thought of as a producer and consumer of resources, the new type at each update is chosen at random from a certain interaction neighborhood according to probabilities proportional to the ability of the neighbors to consume the resource produced by the individual to be updated. In addition to giving a complete qualitative picture of the phase diagram of the spatial model, our results indicate that the nonspatial deterministic mean-field approximation of the stochastic process fails to describe the behavior of the system in the presence of local interactions. In particular, we prove that, in the parameter region where the nonspatial model displays bistability, there is a dominant type that wins regardless of its initial density in the spatial model, and that the inclusion of space also translates into a significant reduction of the parameter region where both types coexist.

Euclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.

We introduce a precise framework for transferring strategies from simpler to more complex games, and use it to construct strategies in certain finite and infinite combinations of games. In particular, we give a finitary characterization of finite hypergraphs X such that the first player can win the positional game on infinitely many copies of X. This resolves a conjecture of Leader.

Following Baurdoux and Kyprianou (2008) we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Lévy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular, we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and ‘thickens’ to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump diffusion case are provided.

We study a class of optimal allocation problems, including the well-known bomber problem, with the following common probabilistic structure. An aircraft equipped with an amount x of ammunition is intercepted by enemy airplanes arriving according to a homogeneous Poisson process over a fixed time duration t. Upon encountering an enemy, the aircraft has the choice of spending any amount 0 ≤ y ≤ x of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of y. Two different goals have been considered in the literature concerning the optimal amount K(x, t) of ammunition spent: (i) maximizing the probability of surviving for time t, which is the so-called bomber problem; and (ii) maximizing the number of enemy airplanes shot down during time t, which we call the fighter problem. Several authors have attempted to settle the following conjectures about the monotonicity of K(x, t): (A) K(x, t) is decreasing in t; (B) K(x, t) is increasing in x; and (C) the amount x - K(x, t) held back is increasing in x. Conjectures (A) and (C) have been shown for the bomber problem with discrete ammunition, while (B) is still an open question. In this paper we consider both time and ammunition to be continuous, and, for the bomber problem, we prove (A) and (C), while, for the fighter problem, we prove (A) and (C) for one special case and (B) and (C) for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (TP2) in the bomber and fighter problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

The lying oracle problem is a problem of finding the optimal strategies in a two-person game where an oracle predicts the outcomes of coin flips and a player bets on the outcomes. The oracle announces whether the coin will land heads or tails, but may at times lie. We analyze the variant of the game which uses a biased coin, where the probability p that the coin lands heads is common knowledge. We determine optimal strategies for both the oracle and player, and we give an explicit expression for the expected payoff to the player when the coin is flipped n times and the oracle may lie at most k times.

A gambler with an initial bankroll is faced with a finite sequence of identical and independent bets. For each bet, he may wager up to his current bankroll, and will win this amount with probability p or lose it with probability 1-p. His problem is to devise a wagering strategy that will maximize his final expected utility with the side condition that the total amount wagered (i.e. the total ‘action’) be at least his initial bankroll. Our main result is an expression that characterizes when the strategy of placing equal-sized wagers on all bets is optimal. In particular, for a given bankroll B, utility function f (concave, increasing, differentiable), and n bets, we show that it is optimal to wager b/n on each bet if and only if the probability of winning each bet is less than or equal to some value p ⋆∈[½,1] (where p ⋆ is an explicit function of B, f, and n). We prove the result by using a basic nonlinear programming technique.