Linear Payoff Functions Research Articles

Intensive care unit/step-down unit queuing game with length of stay decisions

In this paper, length of stay (LOS) competition between two servers in tandem without buffer between them is investigated using queuing games. This system typifies the relationship between the intensive care unit (ICU) and the step-down unit (SDU) of a hospital. We model and analyze the equilibrium LOS decision under four different games (one cooperative and three non-cooperative games) as follows: (i) both servers cooperate; (ii) the servers do not cooperate and make decisions simultaneously; (iii) the servers do not cooperate and the first server, the ICU, is the leader (ICU Stackelberg); (iv) the servers do not cooperate and the second server, the SDU, is the leader (SDU Stackelberg). The payoff of the ICU is expressed as the difference between the service benefit and the waiting in queue penalty, while that of the SDU is the difference between the service benefit and the overstay penalty. The results show that LOS decisions of each server depends critically on the payoff function’s form and the exogenous demand. Secondly, with a linear payoff function, the SDU is only beneficial to the system if the unit cost is greater than the unit reward at the ICU. Our results revealed also that payoffs depend on the substitutability in both ICU Stackelberg and SDU Stackelberg games. When most of the LOS is spent at the ICU unit. Our results suggest that the critical care pathway performs better under coordination and or leadership at the ICU level.

Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions

The contextual combinatorial semi-bandit problem with linear payoff functions is a decision-making problem in which a learner chooses a set of arms with the feature vectors in each round under given constraints so as to maximize the sum of rewards of arms. Several existing algorithms have regret bounds that are optimal with respect to the number of rounds T. However, there is a gap of Õ(max(√d, √k)) between the current best upper and lower bounds, where d is the dimension of the feature vectors, k is the number of the chosen arms in a round, and Õ(·) ignores the logarithmic factors. The dependence of k and d is of practical importance because k may be larger than T in real-world applications such as recommender systems. In this paper, we fill the gap by improving the upper and lower bounds. More precisely, we show that the C2UCB algorithm proposed by Qin, Chen, and Zhu (2014) has the optimal regret bound Õ(d√kT + dk) for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C2UCB algorithm and demonstrate that it enjoys the optimal regret bound for a more general problem that can take into account other objectives simultaneously. We also show that our technique would be applicable to related problems. Numerical experiments support our theoretical results and considerations.

A Characterisation of ‘Phelpsian’ Statistical Discrimination

Abstract We establish that a type of statistical discrimination—that based on informativeness of signals about workers’ skills and the ability appropriately to match workers to tasks—is possible if and only if it is impossible uniquely to identify the signal structure observed by an employer from a realised empirical distribution of skills. The impossibility of statistical discrimination is shown to be equivalent to the existence of a fair, skill-dependent, remuneration for workers. Finally, we connect the statistical discrimination literature to Bayesian persuasion, establishing that if discrimination is absent, then the optimal signalling problem results in a linear pay-off function (as well as a kind of converse).

Analysis of an optimal stopping problem arising from hedge fund investing

We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth-fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.

Applicability of the Analytical Solution to N-Person Social Dilemma Games

The purpose of this study is to present an analysis of the applicability of an analytical solution to the $N-$person social dilemma game. Such solution has been earlier developed for Pavlovian agents in a cellular automaton environment with linear payoff functions and also been verified using agent based simulation. However no discussion has been offered for the applicability of this result in all Prisoners 'Dilemma game scenarios or in other $N-$person social dilemma games such as Chicken or Stag Hunt. In this paper it is shown that the analytical solution works in all social games where the linear payoff functions are such that each agent's cooperating probability fluctuates around the analytical solution without cooperating or defecting with certainty. The social game regions where this determination holds are explored by varying payoff function parameters. It is found by both simulation and a special method that the analytical solution applies best in Chicken when the payoff parameter $S$ is slightly negative and then the analytical solution slowly degrades as $S$ becomes more negative. It turns out that the analytical solution is only a good estimate for Prisoners' Dilemma games and again becomes worse as $S$ becomes more negative. A sensitivity analysis is performed to determine the impact of different initial cooperating probabilities, learning factors, and neighborhood size.

Emergent Behavior of “Rational” Agents in Some Forgotten N-Person Games

This paper investigates N-person games with linear payoff functions, which are defined by four parameters each for the case when some of these parameters are equal to each other. Such cases represent transitions between different games. The participating agents are all greedy simpletons who imitate the behavior of the neighbor in their Moore neighborhood - nine neighbors, including themselves - that received the highest reward in the previous iteration. The results show that the solutions are non-trivial and represent quite irregular emergent behavior when the payoff functions are equal or cross each other.

The evolution of conventions in minimum effort games

The literature on minimum effort game has been concerned with a symmetric game with linear payoff functions. The main aim of the present paper is to study the coordination problem arising in a not necessarily symmetric minimum effort game with two players. The sources of asymmetry can be twofold: the productivity of effort and the distribution of the join output. To select among the Pareto ranked equilibria we use the stochastic stability criterion. We show that, for any configuration parameters, the set of stochastically stable equilibria coincides with the set of potential maximizers. We also show that when the disutility of effort is linear, the Pareto dominant equilibrium is stochastically stable provided that the distributive parameter belongs to a well defined range. When the disutility of effort is nonlinear no distributive arguments can be used to successfully affect the selection process. Lastly we prove that the connection between stochastic stability and maximum potential can fail when more than two agents are considered.

On accuracy, robustness and tolerances in vector Boolean optimization

A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appropriate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples.

Analytical solutions of N-person games

The possibility of analytical solutions of N-person games is presented. A simple formula provides valuable information about the outcomes of such games with linear payoff functions and Pavlovian agents. Experiments performed with our simulation tool for the multiagent stag hunt dilemma game are presented. For the case of Pavlovian agents the game has nontrivial but remarkably regular solutions. If both payoff functions are linear and the real solutions of Eq. (2) are both positive, then the analytical solutions are remarkably accurate. © 2012 Wiley Periodicals, Inc. Complexity, 2012 © 2012 Wiley Periodicals, Inc.

Existence of Nash networks and partner heterogeneity

In this paper, we pursue the line of research initiated by Haller and Sarangi (2005). We examine the existence of equilibrium networks called Nash networks in the non-cooperative two-way flow model by Bala and Goyal (2000a,b) in the presence of partner heterogeneity. First, we show through an example that Nash networks in pure strategies do not always exist in such model. We then impose restrictions on the payoff function to find conditions under which Nash networks always exist. We provide two properties—increasing differences and convexity in the first argument of the payoff function that ensure the existence of Nash networks. Note that the commonly used linear payoff function satisfies these two properties.

Existence of Nash Networks and Partner Heterogeneity

In this paper, we pursue the work of H. Haller and al. (2005, [10]) and examine the existence of equilibrium networks, called Nash networks, in the noncooperative two-way flow model (Bala and Goyal, 2000, [1]) with partner heterogeneous agents. We show through an example that Nash networks do not always exist in such a context. We then restrict the payoff function, in order to find conditions under which Nash networks always exist. We give two properties: increasing differences and convexity in the first argument of the payoff function, that ensure the existence of Nash networks. It is worth noting that linear payoff functions satisfy the previous properties.

A systematic analysis of the N‐person chicken game

AbstractWe report computer simulation experiments based on our agent‐based simulation tool to model the multiperson Chicken dilemma game for the case when the agents are greedy simpletons who imitate the action of that of their neighbors who received the highest payoff for its previous action. The individual agents may cooperate with each other for the collective interest or may defect, i.e., pursue their selfish interests only. After a certain number of iterations the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. The payoff (reward/penalty) functions are given as two straight lines: one for the cooperators and another for the defectors. The payoff curves are functions of the ratio of cooperators to the total number of agents. Even for linear payoff functions, we have four free parameters that determine the payoff functions that have the following properties: (1) Both payoff functions increase with the increasing number of cooperators. (2) In the region of low cooperation the cooperators have a higher reward than the defectors. (3) When the cooperation rate is high, there is a higher payoff for defecting behavior than for cooperating behavior. (4) As a consequence, the slope of the D function is greater than that of the C function and the two payoff functions intersect. (5) All agents receive a lower payoff if all defect than if all cooperate. We have investigated the behavior of the agents systematically. The results show that the solutions have predictable tendencies but they are nontrivial and quite irregular. The solutions show drastic changes in the parameter ranges 0.6 ≤ R ≤ 0.65 for all values of S and 0 ≤ S ≤ 0.2 when R < 0.6 (R is the reward for mutual cooperation and S is the sucker's payoff to a lonely cooperator). © 2010 Wiley Periodicals, Inc. Complexity, 2010

A systematic approach of multi-person games

This paper gives a systematic analysis of a special group of multi-person games. By assuming linear payoff functions, the order of magnitude of four parameters determines the type of the games. In our study we fix three parameters and systematically vary the fourth one. In this way we can obtain the battle of sexes, the benevolent chicken, the chicken and the maximising differences games. Our systematic approach is based on agent based simulation. It presents the varying characteristics of the different games as well as it shows how the transitions are made from one game to another.

Stability and accuracy functions in a coalition game with bans, linear payoffs and antagonistic strategies

A coalition game with a finite number of players in which initial coefficients of linear payoff functions are subject to perturbations is considered. For any efficient solution which may appear in the game, appropriate measures of the quality are introduced. These measures correspond to the so-called stability and accuracy functions defined earlier for efficient solutions of a generic multiobjective combinatorial optimization problem with Pareto and lexicographic optimality principles. Various properties of such functions are studied. Maximum norms of perturbations for which an efficient in sense of equilibrium solution preserves the property of being efficient are calculated.

An [formula omitted]-person battle of sexes game

The n -person Battle of Sexes game is first introduced. The game’s properties are discussed and payoff functions are modeled under the main assumption that players’ payoffs are based on whether they like their choices or not and also on how many other players have the same choices. Linear payoff functions are assumed and the existence of the Nash equilibrium is examined. The dependence of the equilibrium on model parameters is also analyzed.

Finite cooperative games with a parametric concept of equilibrium under uncertainty conditions

The parametric concept of equilibrium (the optimality principle) in a finite cooperative game of several players in a normal form is introduced. This concept is defined by the partitioning of the players into coalitions. In this situation, two extreme cases of this partitioning correspond to the lexicographically optimal situation and the Nash equilibrium situation, respectively. The analysis of the stability of a generalized optimal situation under the perturbations of the coefficients of the linear payoff functions in the l1-metric is performed. The limiting level of such perturbations, conserving the optimality of the situation is found.

Neural nets in a group decision process

In recent years there has been some interest in applying Artificial Adaptive Agents (AAA) to the study of complex adaptive systems, especially economic systems. Neural networks are frequently employed as AAA. Artificial neural nets mimic certain aspects of the physical structure and information processing of the human brain and their most attractive characteristic is their ability to learn a pattern from a given set of examples. In this study, we investigated the ability of neural nets to model human behavior in a group decision process. The context was a market entry game with a linear payoff function and binary decisions. The players had to decide, for each trial, whether or not to enter a market whose capacity is public knowledge. Human behavior in this situation has been modeled and empirically validated by the Nash equilibrium for noncooperative n-person games. A simulation of the game was performed with neural nets instead of human subjects. The nets were trained using the results of the games in which they participated. The simulation with groups of neural nets exhibits phenomena very similar to those observed in groups of human players.

Understanding the Economic Value of Legal Covenants in Investment Contracts: A Real-Options Approach to Venture Equity Contracts

Valuing early-stage high-technology growth-oriented companies is a challenge to current valuation methodologies. This inability to come up with robust point estimates of value should not and does not lead to a breakdown of market liquidity: Instead, efforts are redirected towards the design of investment contracts which materially skew the distribution of payoffs in favor of the venture investors. In effect, limitations in valuation abilities are addressed by designing the investment contracts as baskets of real options instead of linear payoff functions. This paper investigates four common features (covenants) of venture capital investment contracts from a real option perspective, using both analytical solutions and numerical analysis to draw inferences for a better understanding of contract features. The impact of the concept for pricing issues, valuation negotiation and for contract design are considered. It is shown for example how 'contingent pre-contracting' for follow-up rounds is theoretically a better proposition than the simple 'rights of first refusal' commonly found in many contracts. We also provide for results (such as timing of investments, length of rounds, choices of liquidation levels, conversion levels) that take into account full interaction of the different features considered. We document some complex facts, such as the concavity of the VC contract value depending on the amount invested at the different stages, the actual share impact of the most common antidilution feature, some endogenous motivation for early VC exits from otherwise performing companies, and stress overall the importance of a full analysis for efficient contract negociation and understanding.

Tacit Coordination in a Decentralized Market Entry Game with Fixed Capacity

We focus on a class of market entry in which a newly emergent market opportunity may be fruitfully exploited by no more than a commonly known, exogenously determined number of firms. Our results show significant effects of the parameters manipulated in the study, namely, the market capacity, entry fee, and method of subject assignment to groups (fixed vs. random). In contrast to previous market entry games with linear payoff functions, we find no evidence of convergence to equilibrium play on the aggregate level. Shifting the focus of the analysis from the aggregate to the individual level, four clusters of subjects are identified. The patterns are: (1) choice of the same action that is independent of the parameters of the game or the outcome of previous presentations of the same game; (2) random choices with probabilities prescribed by the equilibrium solution for risk-neutral players; (3) random choices with probabilities equal to the individual observed overall proportion of entry; and (4) sequential dependencies that violate any model that assumes randomization. Subjects in the fourth and largest category are shown to adjust their choices in accordance with a simple principle of strategic reasoning.

Best bounds for expected financial payoffs I algorithmic evaluation

A systematic approach to the evaluation of best bounds for expected financial payoffs, in case the mean, variance and range of the distribution are known, is presented. It is based on the majorant/minorant mathematical technique, which consists to bound a payoff by some quadratic polynomial. For the class of piecewise linear payoff functions, a classification of the global triatomic extrema is given, and a general algorithm for evaluation is formulated.