Maximizing the variance of the time to ruin in a multiplayer game with selection

2016 ◽  
Vol 48 (2) ◽  
pp. 610-630
Author(s):  
Ilie Grigorescu ◽  
Yi-Ching Yao
Keyword(s):  
Explicit Formula ◽  
Open Problem ◽  
Optimal Strategy ◽  
Value Function ◽  
Short Proof ◽  
Fair Coin ◽  
Expected Time ◽  
Maximum Value ◽  
Maximum Variance ◽  
Expected Duration

Abstract 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.

Metric for Disassembly and Reuse Decisions: Formulation and Validation

2008 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Deborah Thurston
Keyword(s):  
Maximum Likelihood ◽  
Value Function ◽  
Choice Theory ◽  
Decision Makers ◽  
Maximum Likelihood Estimates ◽  
Likelihood Method ◽  
Maximum Value ◽  
Long Range Planning ◽  
The Relationship ◽  
The Value Function

Design for disassembly and reuse focuses on developing methods to minimize difficulty in disassembly for maintenance or reuse. These methods can gain substantially if the relationship between component attributes (material mix, ease of disassembly etc.) and their likelihood of reuse or disposal is understood. For products already in the marketplace, a feedback approach that evaluates willingness of manufacturers or customers (decision makers) to reuse a component can reveal how attributes of a component affect reuse decisions. This paper introduces some metrics and combines them with ones proposed in literature into a measure that captures the overall value of a decision made by the decision makers. The premise is that the decision makers would choose a decision that has the maximum value. Four decisions are considered regarding a component’s fate after recovery ranging from direct reuse to disposal. A method on the lines of discrete choice theory is utilized that uses maximum likelihood estimates to determine the parameters that define the value function. The maximum likelihood method can take inputs from actual decisions made by the decision makers to assess the value function. This function can be used to determine the likelihood that the component takes a certain path (one of the four decisions), taking as input its attributes, which can facilitate long range planning and also help determine ways reuse decisions can be influenced.

scholarly journals Optimal capital injections and dividends with tax in a risk model in discrete time

2020 ◽  
Vol 10 (1) ◽  
pp. 235-259
Author(s):  
Katharina Bata ◽  
Hanspeter Schmidli
Keyword(s):  
Discrete Time ◽  
Optimal Strategy ◽  
Bellman Equation ◽  
Value Function ◽  
Risk Model ◽  
Barrier Type ◽  
Optimal Capital ◽  
Capital Injections ◽  
The Value Function ◽  
Special Case

AbstractWe consider a risk model in discrete time with dividends and capital injections. The goal is to maximise the value of a dividend strategy. We show that the optimal strategy is of barrier type. That is, all capital above a certain threshold is paid as dividend. A second problem adds tax to the dividends but an injection leads to an exemption from tax. We show that the value function fulfils a Bellman equation. As a special case, we consider the case of premia of size one. In this case we show that the optimal strategy is a two barrier strategy. That is, there is a barrier if a next dividend of size one can be paid without tax and a barrier if the next dividend of size one will be taxed. In both models, we illustrate the findings by de Finetti’s example.

scholarly journals COMBINATORIAL ASPECTS OF ORTHOGONAL GROUP INTEGRALS

2011 ◽  
Vol 22 (11) ◽  
pp. 1611-1646 ◽  
Author(s):  
TEODOR BANICA ◽  
JEAN-MARC SCHLENKER
Keyword(s):  
Explicit Formula ◽  
Open Problem ◽  
Orthogonal Group ◽  
Exact Computation ◽  
Expansion Formula ◽  
Rodrigues Formula ◽  
Type Formula

We study the integrals of type [Formula: see text], depending on a matrix a ∈ Mp × q(ℕ), whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case a ∈ M2 × q(2ℕ) to the general case a ∈ Mp × q(ℕ), (2) the construction of the "best algebraic normalization" of I(a), in the case a ∈ M2 × q(ℕ), (3) an explicit formula for I(a), for diagonal matrices a ∈ M3 × 3(ℕ), (4) a modeling result in the case a ∈ M1 × 2(ℕ), in relation with the Euler–Rodrigues formula. Most proofs use various combinatorial techniques.

scholarly journals On the Structure of a Swing Contract's Optimal Value and Optimal Strategy

2008 ◽  
Vol 45 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Sheldon M. Ross ◽  
Zegang Zhu
Keyword(s):  
Optimal Strategy ◽  
Value Function ◽  
Market Price ◽  
Time Interval ◽  
Optimal Contract ◽  
Future Time ◽  
Special Cases ◽  
Optimal Value ◽  
Swing Contract ◽  
Time Point

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.

Search in a Maze

1990 ◽  
Vol 4 (3) ◽  
pp. 311-318 ◽  
Author(s):  
S. Gal ◽  
E. J. Anderson
Keyword(s):  
Optimal Strategy ◽  
Exit Point ◽  
Expected Time ◽  
Total Length

Suppose that you find yourself trapped in a maze about which you know nothing except that it has an exit point. We present an optimal strategy that will lead you to the exit point in minimum expected time. This strategy ensures that the expected total length of the arcs you traverse will not exceed the sum of the lengths of the arcs in the maze.

scholarly journals Penney's Game Odds From No-Arbitrage

2019 ◽  
Author(s):  
Joshua Benjamin Miller
Keyword(s):  
Fair Coin ◽  
No Arbitrage ◽  
Zero Sum ◽  
Betting Odds ◽  
Insight Into ◽  
Expected Duration

Penney's game is a two player zero-sum game in which each player chooses a three-flip pattern of heads and tails and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occur first. This paper provides a novel no-arbitrage argument that generates the winning odds corresponding to any pair of distinct patterns. The resulting odds formula is equivalent to that generated by Conway's ``leading number'' algorithm. The accompanying betting odds intuition adds insight into why Conway's algorithm works. The proof is simple and easy to generalize to games involving more than two outcomes, unequal probabilities, and competing patterns of various length. Additional results on the expected duration of Penney's game are presented. Code implementing and cross-validating the algorithms is included.

scholarly journals An explicit formula for derivative polynomials of the tangent function

2017 ◽  
Vol 9 (2) ◽  
pp. 348-359
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Open Problem ◽  
Tangent Function

Abstract In the paper, the authors derive an explicit formula for derivative polynomials of the tangent function, deduce an explicit formula for tangent numbers, pose an open problem about obtaining an alternative and explicit formula for derivative polynomials of the tangent function, and recommend some papers closely related to derivative polynomials of other elementary and applicable functions.

scholarly journals Optimising dividends and consumption under an exponential CIR as a discount factor

2020 ◽  
Vol 92 (2) ◽  
pp. 285-309
Author(s):  
Julia Eisenberg ◽  
Yuliya Mishura
Keyword(s):  
Brownian Motion ◽  
Optimal Strategy ◽  
Value Function ◽  
Insurance Company ◽  
Parameter Choice ◽  
Surplus Process ◽  
Dividend Payments ◽  
Cir Process ◽  
The Value Function ◽  
Special Parameter

AbstractWe consider an economic agent (a household or an insurance company) modelling its surplus process by a deterministic process or by a Brownian motion with drift. The goal is to maximise the expected discounted spending/dividend payments under a discounting factor given by an exponential CIR process. In the deterministic case, we are able to find explicit expressions for the optimal strategy and the value function. For the Brownian motion case, we are able to show that for a special parameter choice the optimal strategy is a constant-barrier strategy.

scholarly journals A note on the optimal dividends paid in a foreign currency

2016 ◽  
Vol 11 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Julia Eisenberg ◽  
Paul Krühner
Keyword(s):  
Brownian Motion ◽  
Optimal Strategy ◽  
Value Function ◽  
Foreign Currency ◽  
Initial Capital ◽  
Brownian Motion With Drift ◽  
Optimal Dividends ◽  
Surplus Process ◽  
Dividend Payments ◽  
The Value Function

AbstractWe consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.

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