One challenge facing humans (and nonhuman animal) is that some options that appear attractive locally may not turn out best in the long run. To analyse this human learning problem, we explore human performance in a dynamic decision-making task that places local and global rewards in conflict. We found that experiences that included previous choices and rewards are not easily incorporated into people’s strategy to enhance their performance. Our results suggest that humans are easily driven by concerns about recent feedback, and that choice of a suboptimal behaviour option may be overcome by providing informative cues that indicate a clear immediate outcome for a better option.

This paper obtains an optimal strategy in a finite horizon time for a portfolio of a defined contribution (DC) pension fund for an investor with the CRRA utility function. It employs the optimal stochastic control method in a financial market with two different asset markets, one risk-free and another one risky asset in which its jump follows either by a finite or infinite activity Lévy process. Sensitivity of jump parameters in an uncertainty financial market has been studied.

Hedging is one of the most important topics in finance. When a financial market is complete, every contingent claim can be hedged perfectly to eliminate any potential future obligations. When the financial market is incomplete, the investor may eliminate his risk exposure by superhedging. In practice, both hedging strategies are not satisfactory due to their high implementation costs, which erode the chance of making any profit. A more practical and desirable strategy is to resort to the partial hedging, which hedges the future obligation only partially. The quantile hedging of Föllmer and Leukert (Finance and Stochastics, vol. 3, 1999, pp. 251–273), which maximizes the probability of a successful hedge for a given budget constraint, is an example of the partial hedging. Inspired by the principle underlying the partial hedging, this paper proposes a general partial hedging model by minimizing any desirable risk measure of the total risk exposure of an investor. By confining to the value-at-risk (VaR) measure, analytic optimal partial hedging strategies are derived. The optimal partial hedging strategy is either a knock-out call strategy or a bull call spread strategy, depending on the admissible classes of hedging strategies. Our proposed VaR-based partial hedging model has the advantage of its simplicity and robustness. The optimal hedging strategy is easy to determine. Furthermore, the structure of the optimal hedging strategy is independent of the assumed market model. This is in contrast to the quantile hedging, which is sensitive to the assumed model as well as the parameter values. Extensive numerical examples are provided to compare and contrast our proposed partial hedging to the quantile hedging.

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒt,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e, t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

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.

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with suitably chosen stakes. This problem has recently been extended in a unifying framework to account for limited playing time as well as future discounting, under which bold play is known to be optimal provided that w ≤ ½ ≤ r. This paper is concerned with a further extension of the Dubins-Savage gambling problem involving time-dependent parameters, and shows that bold play not only maximizes the probability of reaching the goal, but also stochastically minimizes the number of plays needed to reach the goal. As a result, bold play also maximizes the expected utility, where the utility at the goal is only required to be monotone decreasing with respect to the number of plays needed to reach the goal. It is further noted that bold play remains optimal even when the time-dependent parameters are random.

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ ½ ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.

For two-person zero-sum games, where the probability of each player winning is a continuous function of time and is known to both players, the mutually optimal strategy for proposing and accepting a doubling of the game value is known. We present an algorithm for deriving the optimal doubling strategy of a player who is aware of the suboptimal strategy followed by the opponent. We also present numerical results about the magnitude of the benefits; the results support the claim that repeated application of the algorithm by both players leads to the mutually optimal strategy.

Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing the stake with probability 1 - w (r > 0, 0 < w < 1). Mr. G is permitted to gamble at the casino, but each time he places a bet, the amount of his debt is increased by a factor of 1 + α (α ≥ 0). How should Mr. G gamble to maximize his chance of reaching his (moving) target and thereby surviving? Dubins and Savage showed that an optimal strategy is to stake boldly if the primitive casino is subfair or fair (i.e. w(1 + r) ≤ 1) and the inflation rate α is 0. Intuitively, a positive inflation rate would motivate Mr. G to try to reach his goal as quickly as possible, so it seems plausible that the bold strategy is optimal. However, Chen, Shepp, and Zame found that, surprisingly, the bold strategy is no longer optimal for subfair primitive casinos with inflation if both r > 1 and α satisfies 1/r ≤ α < r. They also conjectured that the bold strategy is optimal for subfair primitive casinos with inflation if r < 1. It is shown in the present paper that this conjecture is true provided that w ≤ ½. Furthermore, by introducing an interesting notion of sharp strategy, additional results are obtained on optimality of the bold strategy.

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + α, where α is a nonnegative constant, and will continue to go up at this rate, becoming (1 + α) n on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.

We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager II's winning probability when both managers play optimally. Finally, we show that manager II's winning probability is strictly increasing in n, is always less than c, and converges to c as n →∞, where c = 0.3275624139 · ·· is a solution of the equation ln(2) + x ln(x) = x.

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let Fn∗ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that Fn/Fn∗ is a supermartingale with E(Fn/Fn∗) ≤ 1 and, consequently, E(lim Fn/Fn∗) ≤ 1. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that E(Fn/Fn∗) = 1 for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that Fn∗/Fn is a submartingale with E(Fn∗/Fn) ≤ 1 and E(lim Fn∗/Fn) ≤ 1; and E(Fn∗/Fn) = 1 if and only if at each time period j, 1 ≤ j ≤ n, the strategies leading to Fn and Fn∗ are ‘the same’.