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Journal ArticleDOI

Operations Research Games: a Survey

01 Dec 2001-Top (Springer Verlag)-Vol. 9, Iss: 2, pp 139-199
TL;DR: This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.
Abstract: This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved. Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players. This interplay between optimisation and allocation is the main subject of the area of operations research games. It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.

Summary (4 min read)

1 Introduction

  • In model (1), one might ask why a smooth term for x2 is required whereas a simple linear term seems to be sufficient to capture the impact of x1.
  • These and similar multi-step procedures perform a series of locally optimal fit and selection procedures, however, the quality of the global model fit can only be investigated empirically.
  • The rest of the paper is organized as follows: Structured survival models and penalized likelihood estimation schemes that will be utilized as building blocks in the boosting algorithm are introduced in Section 2.

2 Structured Survival Models

  • To overcome the restrictions of Cox models, as discussed above, the authors allow the inclusion of both, timevarying and smooth effects.
  • In the survival context, P-splines are frequently used to model smooth functions [18].
  • The latter determines the smoothness of the resulting function estimate, where bigger values of κj correspond to smoother functions and κj = 0 corresponds to an unpenalized estimation of the j-th term.

3 Boosting in Survival Models with Time-Varying Effects

  • The authors device an estimation procedure for Cox-type models with additive structure and possibly time-varying effects.
  • Variable selection and model choice play another major role in this setting.
  • To combine all tasks, component-wise boosting methods are applied in the following.

3.1 Basic Considerations

  • Estimation of models can be done with respect to many different criteria.
  • For more details on base-learners in general and on how to choose them the authors refer to Bühlmann and Hothorn [11].
  • Thus, the authors have a variable selection and model choice approach based on component-wise boosting.
  • As classically the authors are using B-splines of degree 3 or higher, the degrees of freedom for difference penalties of order 2 or higher remain always greater than one.

3.2 Likelihood-Based Boosting for Survival Data (CoxFlexBoost)

  • The boosting algorithm, which the authors will present in the following section, is essentially based on the likelihood-based boosting approach as proposed by Tutz and Binder [20].
  • As the authors specially focus on the inclusion of flexible and time-varying terms in Cox-type additive models, they call the new algorithm CoxFlexBoost.
  • In the following, the authors denote the j-th base-learner by gj(x(t);βj), j = 1, . . . , J , where J is the number of base-learners (possibly after decomposing nonlinear effects into several separate base-learners as described in the previous section).
  • The covariates x(t) include classical covariates and possible time-varying effects expressed as artificial time-dependent covariates or the time t itself.
  • Thus, gj(x(t);βj) can correspond to a linear function of x̃, where x̃ is a covariate from x(t), or of time t, with t being the time covariate from x(t), or, more flexible, a smooth function of x̃ or t.

3.2.1 CoxFlexBoost Algorithm

  • Hence, the integrand in (11) is a function depending on the coefficient βj , which the authors try to estimate, and on time t̃.
  • As CV does not involve estimation of the degrees of freedom (which tend to underestimate the true degrees of freedom [24]) this is a more sensible solution.
  • An appropriate stopping iteration is determined as the number of boosting iterations m̂stop,opt that maximizes the log-likelihood on the validation data.

3.2.2 Remarks on Computational Considerations

  • The authors have to integrate over time t̃ for each base-learner, in each boosting iteration and in each step of the optimization method (in their implementation the Broyden-Fletcher-Goldfarb-Shanno [BFGS] method, see, e.g., [25]) used to determine β̂j .
  • The authors believe that specifying the degrees of freedom df to determine the amount of smoothness of each base-learner is much more intuitive.
  • To specify the smoothing parameters via the corresponding degrees of freedom the authors exploit the relation that the latter depend on κj and thus they can solve df(κj)− d̃f j != 0 (13) for κj with a pre-specified value of d̃f j .

4.1.1 Outline of Simulations

  • To gain deeper insights in the properties of the proposed CoxFlexBoost procedure, two simulation studies were performed.
  • Furthermore, the authors investigated the properties of variable selection and wanted to check if other effects, as linear and smooth effects, are detected and modeled “correctly”.
  • The authors will utilize linear or P-spline base-learners in the following.
  • (18) Like in the previous simulation scheme the authors simulated the censoring times Ci i.i.d.∼ Expo(1/t).

4.1.2 Simulation Results 1: Model Choice and Variable Selection

  • In the first scheme the authors simulated 200 randomly drawn replicates of the data set.
  • In the following section, the authors explore the accuracy of model choice (and thus of variable selection) given by the relative frequency of selected base-learners.
  • When the authors look at the noneffective covariates they see that the frequencies of selection are much smaller than those of the effective covariates.
  • The time-varying effect of x6 is always discovered and the (log) baseline hazard is almost always selected.
  • Only in roughly half of the models a flexible time-varying effect is chosen.

4.1.3 Simulation Results 2: Estimated Effects

  • Now, the authors look at the estimated effects and compare them with the real, specified effects.
  • In Figure 1, the authors see a selection of the estimated effects for the six effective covariates.
  • Caused by the sparse data at the boundaries (note that the authors used a standard normal distribution to simulate x3) the boundaries of the sine form of x3 are estimated quite poorly, whereas the middle part is estimated quite sensible (not depicted here).
  • The estimation in the center region is hardly effected and only slight deviations in the areas with less observations can be identified.
  • The time-varying effect of x6 (not depicted here) suffers from the same problem as the baseline hazard, i.e., the estimated function is very unstable in the sparse, right tail.

4.2 Application: Model for Surgical Patients

  • In the following section, the authors aim to build a model for patients with severe sepsis.
  • Baseline characteristics and detailed outcomes of that population were published recently [30, 31, 32].
  • The authors obtained relevant covariates reflecting the state of the patient on admission day, and the 90-day survival time for 462 patients with severe sepsis.
  • To build the model, the authors applied the proposed CoxFlexBoost algorithm to the data.

4.2.1 Application of CoxFlexBoost

  • To asses the stability of the variable selection and model choice process of component-wise boosting, as implemented in CoxFlexBoost [28], the authors used 5 random subsamples, each with 362 observations, of the severe sepsis data from Großhadern.
  • In contrast to the two-stage stepwise procedure, CoxFlexBoost is not able to handle preset covariates.
  • In contrast, in the application of the TSS procedure [3] six mandatory covariates were used.
  • Defining an inclusion rate of 2 or less negligible, only 10 out of 20 covariates can be regarded as influential covariates in the boosting model.
  • For some of the covariates, time-varying effects were also selected.

4.2.2 Comparison of Model Selection Strategies

  • Comparing the results of the application of the two-stage stepwise procedure (see [3]) and CoxFlexBoost to the Großhadern data set of patients with severe sepsis, the authors can conclude that both approaches have advantages with regard to different aspects:.
  • The two-stage stepwise procedure is easily extended in such a way, as showed in Hofner et al. [3].
  • In their application CoxFlexBoost tended to a sparser solution but this could be due to the starting model with mandatory covariates in the two-stage stepwise model.
  • Altogether, the authors see that none of the approaches is superior to the other.
  • Especially in high-dimensional settings with many possible predictors, boosting with its robustness against overfitting and the built in regularization is clearly the preferred method.

5 Summary and Outlook

  • The authors derived boosting methods for flexible survival models with time-varying effects.
  • It is hard to decide if a covariate should enter the model as a linear term, smooth term or as time-varying effect or if the covariate is not required at all.
  • Boosting offers the possibility to estimate the model with inherent model choice and variable selection.
  • One possible alternative to the proposed model choice scheme in CoxFlexBoost could be to fit the model in a similar fashion like that proposed in the MFPT approach by Sauerbrei et al. [8].
  • Including time-varying effects for smooth effects would result in modeling an interaction of two functions:.

Acknowledgments

  • The authors thank W. H. Hartl from the Department of Surgery, Klinikum Großhadern for the data set and stimulating problems and D. Inthorn and H. Schneeberger for initiation and maintenance of the database of the surgical intensive care unit.
  • B. Hofner and T. Hothorn were supported by Deutsche Forschungsgemeinschaft, grant HO 3242/1-3.

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Tilburg University
Operations Research Games
Borm, P.E.M.; Hamers, H.J.M.; Hendrickx, R.L.P.
Publication date:
2001
Link to publication in Tilburg University Research Portal
Citation for published version (APA):
Borm, P. E. M., Hamers, H. J. M., & Hendrickx, R. L. P. (2001).
Operations Research Games: A Survey
.
(CentER Discussion Paper; Vol. 2001-45). Operations research.
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Download date: 10. aug.. 2022

No. 2001-45
OPERATIONS RESEARCH GAMES: A SURVEY
By Peter Borm, Herbert Hamers and Ruud Hendrickx
July 2001
ISSN 0924-7815

Operations Research Games: A Survey
Peter Borm
1;2
Herbert Hamers
1
Ruud Hendrickx
1
Abstract
This paper surveys the research area of cooperative games associated with
several types of operations research problems in which various decision makers
(players) are involved. Cooperating players not only face a joint optimisation
problem in trying, e.g., to minimise total joint costs, but also face an addi-
tional allocation problem in how to distribute these joint costs back to the
individual players. This interplay between optimisation and allocation is the
main subject of the area of operations research games. It is surveyed on
the basis of a distinction between the nature of the underlying optimisation
problem: connection, routing, scheduling, production and inventory.
1 Introduction
Typically, operations research analyses situations in which one decision maker,
guided by an objective function, faces an optimisation problem. The theory con-
centrates on the question of how to act in an optimal way and, in particular, on
the issues of computational complexity and the design of e±cient algorithms. Game
theory on the other hand analyses situations involving at least two interacting de-
cision makers (called players), with possibly diverging interests. Roughly speaking,
it deals with mathematical models of competition and cooperation.
Competitive or noncooperative game theory studies situations in which the play-
ers can negotiate about what to do (i.e., pre-play communication is allowed), but
enforceable binding agreements are assumed not to be possible. Therefore strategic
analysis and individual incentives play a prominent role here. In cooperative game
theory enforceable binding agreements are possible and also side payments may be
allowed. Now the main issue is fair allocation, either of joint costs or joint revenues.
1
CentER and Department of Econometrics and Operations Research, Tilburg University.
2
Corresponding author. P.O. Box 90153, 5000 LE Tilburg, The Netherlands. E-mail address:
pemborm@kub.nl.
1

Since the early developments of operations research and game theory there has
been a strong interplay between the two disciplines. Especially the interrelation
between operations research and noncooperative game theory is well-known: be-
tween duality results in mathematical programming and minimax results for zero-
sum games, between linear complementarity and bimatrix games, between Markov
decision processes and stochastic games, and between optimal control theory and
di®erential games.
The interrelation between operations research and cooperative game theory is of a
more recent date and is summarised under the heading of operations research games.
One can say that an important part of the interplay between cooperative games and
operations research stems from the basic (discrete) structure of a graph, network
or system that underlies various types of combinatorial optimisation problems. If
one assumes that at least two players are located at or control parts (e.g., vertices,
edges, resource bundles, jobs) of the underlying system, then a cooperative game
can be associated with this type of optimisation problem. In working together, the
players can possibly create extra gains or save costs compared to the situation in
which everybody optimises individually. Hence the question arises how to share the
extra revenues or cost savings.
One way to analyse this question is to study the general properties (e.g., bal-
ancedness or convexity) of all games arising from that speci¯c type of operations
research problem and to apply a suitable existing game theoretic solution concept
(e.g., core or Shapley value) to this class. Another way is to create a context speci¯c
allocation rule. Such a rule can be based either on desirable properties in this spe-
ci¯c context or on a kind of decentralised mechanism that prescribes an allocation
on the basis of the algorithmic process along which a jointly optimal combinatorial
structure is established (e.g., following an algorithm to create a minimal cost span-
ning tree). A general reference on cooperative games is Driessen (1988) and various
speci¯c operations research games are treated in Curiel (1997).
The original request of TOP to the authors of this paper was to write a complete
survey on operations research games. Given the abundance of papers on this topic,
starting more or less from the beginning of the seventies, and the vast increase during
the nineties, this constitutes a \mission impossible". Moreover, the de¯nition of
operations research games is not so strict that a unique selection of research streams
is prescribed. Consequently, the choice of topics treated in this survey is somewhat
biased towards our own expertise, knowledge and interests. A ¯rst general aim
2

of this survey is to give an unacquainted reader a °avour of the things that are
going on in this interesting ¯eld of research. Our second aim is to provide a rather
up to date state-of-the-art. Last but not least we hope it stimulates researchers
to enter this ¯eld. There are still many questions to be asked, gaps to be ¯lled
and extensions to be investigated. We have included a brief ¯nal section with our
ideas for possible future research lines; not in a very detailed and elaborate way,
but mainly by making some hints and stating some catchwords to provoke possibly
di®erent individual associations and to enter new research tracks.
To better structure the survey, we have chosen to make a division of operations
research games into ¯ve categories, primarily based on the nature of the underlying
optimisation problem. In our view, this categorisation also allows for a better insight
into the various relationships in methodology, techniques and results across the
di®erent classes of operations research games. We distinguish between:
(i) Connection: ¯xed tree, spanning tree
(ii) Routing: Chinese postman, travelling salesman
(iii) Scheduling: sequencing, permutation, assignment
(iv) Production: linear production, °ow
(v) Inventory
Each category will be treated in a separate section. Within each category we have
chosen one representative class which is discussed in some detail, starting from the
level of a person who is not familiar with the topic. For this reason, relatively
much attention is paid to the modelling phase, i.e., how to go from operations
research to game theory. After discussing the main results from the literature for
the representative class, the other classes within the same category are treated in a
more compact way.
To conclude the introduction we mention some topics which could be considered
as being inside the theory of operations research games, or at least closely related,
but which will not be discussed in this survey. For those interested we have added a
selected list of references. Within a ¯nancial context we mention bankruptcy games,
cf. O'Neill (1982), Aumann and Maschler (1985), Curiel et al. (1987), Young (1988),
Kaminski (2000) and Calleja et al. (2001), deposit games, cf. Izquierdo and Rafels
(1996) and Borm et al. (2001) and shortest path games, cf. Fragnelli et al. (2000),
3

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Cites background from "Operations Research Games: a Survey..."

  • ...(44) However, these limiting assumptions have not prevented game theory from contributing to analysis in numerous related areas.(45) One response that is often offered in defense of the assumption above, that adversary objectives are strictly consequence maximizers, is that it defaults to the worst case and is therefore “erring” in the right direction....

    [...]

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TL;DR: This work develops cost-allocation mechanisms with well-known properties from the cooperative game theory literature, such as budget balance, stability, and cross-monotonicity for a logistics network in which shippers collaborate and bundle their shipment requests to negotiate better rates with a common carrier.
Abstract: We study a logistics network in which shippers collaborate and bundle their shipment requests to negotiate better rates with a common carrier. In this setting, shippers can identify collaborative routes with decreased overall empty truck movements. After the optimal routes that minimize total cost of covering all the shippers' demands are determined, this cost is allocated among the shippers. Our goal is to devise cost-allocation mechanisms that ensure the sustainability of the collaboration. We first develop cost-allocation mechanisms with well-known properties from the cooperative game theory literature, such as budget balance, stability, and cross-monotonicity. Next, we define a set of new properties, such as a guaranteed discount from the standalone cost for each shipper, desirable in our setting, and propose several cost-allocation schemes that could lead to implementable solutions. We also perform a computational study on randomly generated and real-life data to derive insights on the performance of the developed allocation schemes.

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Cites background from "Operations Research Games: a Survey..."

  • ...We refer the reader to Young (1985) that gives a thorough review of basic cost allocation methods and to Borm et al. (2001) for a survey of cooperative games associated with operations research problems....

    [...]

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Frequently Asked Questions (6)
Q1. What are the contributions in "Operations research games: a survey" ?

This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers ( players ) are involved. This interplay between optimisation and allocation is the main subject of the area of operations research games. 

Issues to be considered in the future involve: ² dynamics: changes in the player set and other time-related aspects, ² strategic incentives ( coopetition ), ² minimising private information exchange, ² consistency, monotonicity and continuity arguments for allocation rules, ² stochastic uncertainty, ² asymmetric information between the players with respect to the data of the underlying operations research problem. 

The main objective of inventory management is to minimise average (long term) costs per time unit, while guaranteeing a prespeci¯ed minimal level of service. 

An interesting recent contribution to the theory of °ow situations is the characterisation of the MC solution (as a set valued solution) for so-called simple °ow situations, i.e., situations where each player dictatorially controls exactly one arc, other arcs are publicly available (with control games w with w(S) = 1 for all coalitions S) and all arcs have a capacity of 1. 

Applications that can be analysed using assignment games are, e.g., private markets in used cars, real estate markets and auctions. 

Such a rule can be based either on desirable properties in this speci¯c context or on a kind of decentralised mechanism that prescribes an allocation on the basis of the algorithmic process along which a jointly optimal combinatorial structure is established (e.g., following an algorithm to create a minimal cost spanning tree).