Re ̄ning Indeterministic Choice: Imprecise Probabilities and Strategic Thinking

Often, uncertainty is present in processes that are part of our routines. Having tools to understand the consequences of unpredictability is convenient. We introduce a general framework to deal with uncertainty in the realm of distribution sets that are descriptions of imprecise probabilities. We propose several non-biased re ̄nement strategies to obtain sensible forecasts about results of uncertain processes. Initially, uncertainty on a system is modeled as the nondeterministic choice of its possible behaviors. Our re ̄nement hypothesis translates non-determinism into imprecise probabilistic choices. Imprecise probabilities allow us to propose a notion of uncertainty re ̄nement in terms of set inclusions. Later on, unpredictability is tackled through a strategic approach using uncertainty pro ̄les and angel/daemon games (a=d-games). Here, imprecise probabilities form the set of mixed strategies and Nash equilibria corresponds to natural uncertainty re ̄nements. We use this approach to study the performance of Web applications in terms of response times under stress conditions.


Introduction
Words like uncertainty, imprecision or non-determinism express several facets of a lack of knowledge. They are commonly used in many areas of human reasoning. In an economical context an approach to uncertainty, as a lack of probabilistic information, was studied by Knight. 1 In a mathematical setting, the term imprecision, reshaped as imprecise probabilities is considered by Cooman et al. 2 Non-deterministic choices were introduced by Tony Hoare 3 to describe the semantics of concurrent process when the agents have no full control on the decision making.
In this paper, we provide a general framework to deal with uncertainty. We model full uncertainty as a non-deterministic choice on the possible behaviors. Inspired by McIver and Morgan, 4 we state the re¯nement hypothesis that relates non-determinism to imprecise probabilistic choices. The re¯nement hypothesis claims that when non-determinism is replaced by a totally arbitrary probabilistic choice the system uncertainty does not increase. Once this¯rst re¯nement step is done and depending on our system knowledge, additional steps to lower the unpredictability can be considered by restricting arbitrariness in probabilistic elections.
In the area of social sciences, strategic approaches are a commonly used tools in the¯ght against uncertainty, see for instance Porter work. 5 Gabarr o et al. 6,7 extend strategic thinking to deal with Web apps and short time macroeconomics models. They introduce the uncertainty pro¯le framework that sets up the components, the potential modi¯cations, and the intensity of the actions of two agents, the angel and the daemon (a=d), perturbing a system. Both agents represent the animal spirits 8 modeling antagonistic interests rather than \a weighted average of bene¯ts". 9 In the context of Web apps, a¯rst link with imprecise probabilities was provided in Ref. 10. We extend here this approach to general systems. Uncertainty pro¯les allow us to consider strategic settings through angel/daemon-games. 6 In this strategic condition, it is assumed that the angel modi¯es the systems in such a way that the damage to the system is minimized, while the daemon tries to make that the system works at its worst. Nash equilibria of the a=d-games de¯ne imprecise probabilities that can drastically shrink the system uncertainty.
We apply the framework to Web applications de¯ned as orchestrations of basic Web services or sites. An orchestration compounds the results of site calls to perform some computation. We assume that orchestrations are de¯ned using the Orc language. 11,12 We note that the language itself has operators providing uncertain results. For instance, the asymmetric parallelism operation in Orc, that waits until a result arrives and then triggers the execution of another part of the orchestration is inherently uncertain. In an unreliable Web additional uncertainty sources may emerge and orchestrations have to manage both inherent uncertainty and uncertainty on site calls. In this scenario, semantic characterizations through imprecise probabilities were proposed in Ref. 10. We ground on these semantic representations to analyze the response delay time of an orchestration. We provide information about orchestration response time assuming di®erent knowledge levels about perturbations.
The paper is structured as follows. In Sec. 2, we present our general approach to deal with uncertainty in a step by step presentation. In Sec. 3, we consider Web apps. Finally, in Sec. 4, we post our conclusions and raise some open question.

Re¯ning Non-determinism by Imprecise Probabilities
We introduce a framework to deal with uncertainty consisting of di®erent unsureness levels. In this framework, non-deterministic systems (or processes) are considered at the highest unpredictability level. Random processes described by imprecise probabilities are placed at lower uncertainty levels. Finally, probabilistic systems, de¯ned by distributions with known weights, are at the lowest unpredictability levels. We also introduce several unbiased criteria to improve the predictability of a process. The re¯nement hypothesis and the strategic thinking tools explained below will play a role when reshaping a system into a more predictable one.
First, we introduce some notation.
. u is the non-deterministic choice operator introduced by Tony Hoare. 3 If P and Q are processes, the expression P u Q denotes a process which arbitrarily behaves either like P or Q. . Á k , for k > 1, denotes the probability set . ½ is the choice operator. We use it to represent a probabilistic choice. The notation 4 S 1 @p 1 ½S 2 @p 2 , for ðp 1 ; p 2 Þ 2 Á 2 denotes a system S which behaves as system S 1 with probability p 1 and as S 2 with probability p 2 . . Imprecise probabilistic systems. We extend the previous notation to represent imprecise probabilistic choice systems. For instance, fS 1 @p 1 ½S 2 @p 2 jðp 1 ; p 2 Þ 2 Á 2 g denotes an imprecise probabilistic choice system, containing a probabilistic choice for each ðp 1 ; p 2 Þ 2 Á 2 . . v denotes re¯nement. 4 In our setting S v S 0 means that all the outputs of system S 0 are valid results of S, but S 0 is at least as predictable as S. For instance, a system consisting of throwing n times an unbiased coin is a re¯nement of throwing n times a coin with an imprecise bias in a small range around zero.

Re¯nement hypothesis
Let us develop the interplay between non-deterministic choice and imprecise probabilities through an algebraic approach. Let us start with the simplest system S consisting of the non-deterministic choice among two subsystems S 1 and S 2 . We express this as In words, system S behaves arbitrarily either as system S 1 or S 2 . Following McIver and Morgan, 4 we associate to such a pure indeterministic system a¯rst re¯nement in terms of imprecise probabilities. This re¯nement is a set of probabilistic behaviors, one behavior for each possible probability distribution. We use the symbol È IP to denote the corresponding universe of imprecise probabilities.
Re¯ning Indeterministic Choice: Imprecise Probabilities and Strategic Thinking 455 Formally, we write S 1 È IP S 2 ¼ fS 1 @p 1 ½S 2 @p 2 jðp 1 ; p 2 Þ 2 Á 2 g and we claim that S 1 È IP S 2 provides a re¯nement on the actual process, so we write The intuitive explanation is the following: as S performs an arbitrary choice between S 1 and S 2 , the uncertainty does not increase when the choice is done according to an unknown probability distribution.
In the most general case of a system de¯ned as a non-deterministic choice among n possible systems, arguing similarly, we extend the former re¯nement to where S 1 È IP Á Á Á È IP S n denotes the set of all probabilistic choices between S 1 ; . . . ; S n , that is fS 1 @p 1 ½ Á Á Á ½S n @p n jðp 1 ; . . . ; p n Þ 2 Á n g: Our re¯nement hypothesis states that encoding non-determinism as imprecise probabilities does not increase the uncertainty.

Irreversibility of the re¯nement hypothesis
We show in this section the e®ects of re¯nements on a system analysis. When studying a system behavior an its predictability, applying the re¯nement hypothesis to non-deterministic choices is a non-reversible step. For instance, assume that X 1 ; X 2 ; Y 1 ; Y 2 are four processes each one computing an integer and system S computes the maximum of two non-deterministic choices X 1 u X 2 and Y 1 u Y 2 . We write S ¼ maxðX 1 u X 2 ; Y 1 u Y 2 Þ. First, we analyze the system S allowing non-deterministic choices to evolve according to the CSP 3 distributive law that states the equality fðX u Y Þ ¼ fðXÞ u fðY Þ for a function f. In our case Finally, re¯ning non-deterministic choices, it holds In contrast, if the re¯nement hypothesis is applied on both non-deterministic choices, X 1 u X 2 and Y 1 u Y 2 , before applying distributivity, we get In such a case, writing X 1 È IP X 2 ¼ fX 1 @p 1 ½ X 2 @p 2 jðp 1 ; p 2 Þ 2 Á 2 g and Y 1 È IP Y 2 ¼ fY 1 @q 1 ½Y 2 @q 2 jðq 1 ; q 2 Þ 2 Á 2 g. As the pair ðX 1 ; Y 1 Þ is chosen with probability p 1 q 1 , the maxðX 1 ; Y 1 Þ is also chosen with this probability. Similarly in the other cases. Therefore, maxðX 1 È IP X 2 ; Y 1 È IP Y 2 Þ can be rewritten as, where ðp 1 ; p 2 Þ and ðq 1 ; q 2 Þ are arbitrary distributions in Á 2 . The point is that this re¯nement of system S is di®erent from the previous one. In fact, probability distributions in the former expression are a subset of the total probability space Á 4 involved in re¯nement (1). It can be proved that distributions ðw 1 ; w 2 ; w 3 ; w 4 Þ that can be expressed as a pair in Á 2 Â Á 2 must hold w 1 w 4 ¼ w 2 w 3 , an equality that puts a strong constrain on Á 4 . For instance, distribution ð1=2; 0; 0; 1=2Þ cannot be written as ðp 1 q 1 ; p 1 q 2 ; p 2 q 1 ; p 2 q 2 Þ.
The former paragraph discussion shows that re¯nement steps are non-return points of the predictability analysis. It seems to reinforce the following claim: dealing with uncertainty as soon as possible, is the best way to try to decrease the global unpredictability.
Let us consider a detailed example of re¯nement steps motivated by Web apps. We use an Orc-inspired notation. 11 In Orc, given two Web sites S and T, the symmetric parallelism SjT calls both sites at the same time and returns an interleaving of the results delivered by both sites.
Example 2.1. Consider CrazyChannel, a TV channel o®ering two consecutivē lms, one of Action and another one which is Monarchy related. This can be expressed in Orc as

CrazyChannel ¼ ðActionjMonarchyÞ:
The channel has the inconvenience of being non-deterministic, i.e. we do not known in advance which¯lms will be broadcast. Assume that we know that the action¯lm is drawn either from the Rambo or the Superman series and that the monarchy one is taken either from the Sissi or The Crown series. Formally, As the parallel composition produces an interleaving of the two subsystems the channel o®ers two¯lms, one in each category. As a consequence of the u-distributive law, what couple of¯lms will be broadcast is the result of a non-deterministic choice Finally, we can re¯ne the channel in terms of imprecise probabilities CrazyChannel v fðRambojSissiÞ@r 1 ½ðRambojTheCrownÞ@r 2 ½ðSupermanjSissiÞ@r 3 ½ðSupermanjTheCrownÞ@r 4 jðr 1 ; r 2 ; r 3 ; r 4 Þ 2 Á 4 g: Re¯ning Indeterministic Choice: Imprecise Probabilities and Strategic Thinking 457 A better re¯nement À À À or a less imprecise À À À can be inferred if we know that both non-deterministic choices have been re¯ned into imprecise probabilities internally at, respectively, Action and Monarchy processes. In this case,

Uncertainty pro¯les
Another way to ascertain information about a system is through uncertainty proles. 10 Given a system SðX 1 ; . . . ; X n Þ ¼ S 1 u Á Á Á u S k , let us study its behavior under angel-daemon stress. We use below symbol ½n to denote the set of integers f1; . . . ; ng.
De¯nition 2.1. Given a system SðX 1 ; . . . ; X n Þ, an a=d stress S is formed by a collection of processes ðX i;a ; X i;d ; X i;a;d Þ i2½n . The behavior of component X i under angelic stress is denoted by X i;a . Respectively, X i;d and X i;a;d denote the behavior under daemonic stress or both angelic and daemonic stress, respectively.
For a set a ½n of parameters that su®er angelic stress and a set b ½n of parameters under daemonic stress, we de¯ne the processes X i ½a; d as follows: The system S under stress action ða; dÞ is S½a; d ¼ SðX 1 ½a; d; . . . ; X n ½a; dÞ.
Observe that a stress action describes a global change on the system behavior. Let us consider a¯rst example of a system under a a=d-stress.
Example 2.2. Consider the system SðX 1 ; X 2 Þ ¼ X 1 X 2 where X 1 and X 2 are processes that read integer data from an associated device. Both components truly report the readings under normal working conditions. Assume a stress S that on component X 1 behaves as X 1;a ¼ 2X 1 , X 1;d ¼ 1 Suppose that we know that S will su®er stress, but the extent and location of the perturbations are uncertain. Following Castro et al., 10 we can describe this knowledge by an uncertainty pro¯le.
where S is a stress model, A ½n points out the parameters that can su®er angelic stress, D ½n the ones that can su®er daemonic stress, b a ; b d 2 N, and u : 2 A Â 2 D ! N is the utility À À À or value À À À function.
Agents a and d have the capability to act, respectively, on the given subsets of parameters. The integers b a and b d measure the intensity of the real a and d actions, giving the number of parameters that a or d can really stress. Finally, u is a performance measure on the system Sð½a; dÞ. The possible actions to be undertaken by a and d are given by the sets With no further information about agents a and d, the system can behave as S½a; d, for any ða; dÞ 2 A a Â A d , and it can be described as the non-deterministic choice among all the possible ways to perturb S according to the pro¯le, i.e.
Let us reconsider the system SðX 1 ; Example 2.3. Let S be as in Example 2.2. Assume a high lack of knowledge about which components will be perturbed. In such case A ¼ D ¼ f1; 2g. Moreover, suppose we know that when perturbation arrives it will be focused on a single component, so b a ¼ b d ¼ 1, meaning that both a and d can perturb just one component. Thus, A a Â A d ¼ fðf1g; f1gÞ; ðf1g; f2gÞ; ðf2g; f1gÞ; ðf2g; f2gÞg, and the system under a=d perturbation is SðUÞ ¼ Sðf1g; f1gÞ u Sðf1g; f2gÞ u Sðf2g; f1gÞ u Sðf2g; f2gÞ ¼ SðX 1;a;d ; X 2 Þ u SðX 1;a ; X 2;d Þ u SðX 1;d ; X 2;a Þ u SðX 1 ; X 2;a;d Þ: Tuple U ¼ hSðX 1 ; X 2 Þ; S; f1; 2g; f1; 2g; 1; 1; ui where the utility function can be chosen to be any performance measure of interest.
Choices for a and d can be de¯ned probabilistically. Mixed strategies for a and d are, respectively probability distributions : A a ! ½0; 1 and : A d ! ½0; 1. The utility of the mixed strategy ð; Þ is de¯ned as uð; Þ ¼ P ða;dÞ2A a ÂA d ðaÞuða; dÞðdÞ. To simplify notation, let Á a ¼ Á b a and Á d ¼ Á b d denote the set of mixed strategies for a and d, respectively. We associate to U the probabilistic multiset This set describes, in terms of imprecise probabilities, all the possible behaviors of the system under the stressed environment described by the joint actions of the a=d agents. By the re¯nement hypothesis in Sec. 2.1, SðUÞ v SðUÞ Mixed . We can associate the set of expected values for mixed strategies Informally, EðSðUÞ Mixed Þ describes the set of possible average measurements corresponding to the a=d choices.

Strategic re¯nement using angel-daemon games
In many¯elds and specially in business, strategic approaches are considered to model and restrain unsureness. 5 We incorporate this strategic view into our model through angel-daemon games based on uncertainty pro¯les. In order to model the strategic interaction between a and d, we transform a static description provided by an uncertainty pro¯le U, into a strategic situation called a=d-game. De¯nition 2.3. To each uncertainty pro¯le U, we associate an a=d-game ÀðUÞ ¼ hA a ; A d ; uða; dÞi. The players are a and d and the set of strategy pro¯les is A a Â A d . With no more information, playing the game consists on choosing a tuple ða; dÞ 2 A a Â A d . Players a and d are antagonistic; a tries to maximize the utility and d tries to minimize the value. Formally, ÀðUÞ is a zero-sum game where the utility of a is u a ða; dÞ ¼ uða; dÞ and the utility of d is u d ða; dÞ ¼ Àuða; dÞ.
A pure strategy pro¯le ða; dÞ is a special case of mixed strategy ð; Þ in which ðaÞ ¼ 1 and ðdÞ ¼ 1. A mixed strategy pro¯le ð; Þ is a Nash equilibrium if, for any 0 2 Á a , uð; Þ ! uð 0 ; Þ and, for any 0 2 Á d , uð; Þ uð; 0 Þ. Let Nash denote the set of Nash equilibria. We associate to ÀðUÞ the following multiset: SðUÞ Nash ¼ f½ ða;dÞ2A a ÂA d S½a; d@ðaÞðdÞjð; Þ 2 Nashg: As Nash Á a Â Á d , we have the re¯nement SðUÞ v SðUÞ Mixed v SðUÞ Nash . Therefore, the strategic approach trough a=d-games can decrease uncertainty in S. A pure Nash equilibrium, PNE, is a Nash equilibrium ða; dÞ with pure strategies. It holds that all Nash equilibrium of a zero-sum game À have the same value uðUÞ corresponding to the utility of the row player. 13 For an a=d game ÀðUÞ we write the value of the game as Nash equilibria, and therefore, the value of a zero-sum game, can be characterized as follows (from Osborne's book 14 ).
Lemma 2.1. Given an uncertainty pro¯le U, let ð; Þ be a Nash equilibrium of ÀðUÞ. It holds that, for any a 2 A a such that ðaÞ > 0, we have uða; Þ ¼ uðUÞ and for any d 2 A d such that ðdÞ > 0, we have uð; dÞ ¼ uðUÞ.
Note that from above EðSðUÞ Nash Þ ¼ fuðUÞg. This fact is important because even if SðUÞ Nash contains many imprecise systems, the expected behavior of all of them is the same. This fact opens a door to dramatically remove uncertainty on the expected behavior.
Let us consider an example of application of these ideas. Let U ¼ hSðX 1 ; X 2 Þ; S; f1; 2g; f1; 2g; 1; 1; ui be the uncertainty pro¯le introduced in Example 2.3. The a utilities of the corresponding a=d game ÀðUÞ are the following: About Nash equilibria of this game, we can show the following.

Web Applications
The content of this section is based on the preliminary work, 15 where we introduce uncertainty re¯nements in the¯eld of Web services. In this setting, calls to sites that trigger service executions are the basic resources. Additional operators are o®ered to describe complex computations À À À so called orchestrations À À À that involve interaction of processes in di®erent ways. In any real scenario, uncertainty is inherent in this type of Web computations. We show how to deal with orchestration unpredictability introducing semantic characterizations described in terms of imprecise probabilistic choices.

Web under stress: Imprecision
We introduce now the basic concepts concerning orchestrations and Web uncertainty. An orchestration is a user-de¯ned program that uses services on the Web.

Re¯ning Indeterministic Choice: Imprecise Probabilities and Strategic Thinking 461
A basic service is called a site. A site is silent if it does not publish any result. A site call publish at most one response. An orchestration which composes a number of site calls into a complex computation can be represented by an Orc 11,12 expression. a An orchestration publishes a data stream. We only deal here with orchestrations generating a¯nite number of results. Two Orc expressions (our systems) E and F can be combined using the following operators. 11,12 The symmetric parallelism EjF: E and F are evaluated in parallel. EjF publishes some interleaving of the streams published by E and F. The asymmetric parallelism EðxÞ < x < F: E and F are evaluated in parallel. Some sub-expressions in E may become blocked by a dependency on x. The¯rst result published by F is bound to x, the remainder of F's evaluation is terminated and evaluation of the blocked residue of E is resumed. Finally, the sequence E > x > FðxÞ: E is evaluated and, for each value v published by E, an instance FðvÞ is executed. Given an orchestration E, we denote by sitesðEÞ the set of sites in the de¯nition of E (the parameters). Information on delays is given by an evaluation function providing the return time of each orchestration site. In order to characterize ex-ante the execution of an orchestration E, we introduced in the work 16 the meaning or semantics of E, denoted by [E]. When there is no information about return times, but we know that orchestration results are m 1 ; . . . ; m k , the semantics [E] ¼ bbm 1 ; . . . ; m k cc where, abstracting away any time order, bbm 1 ; . . . ; m k cc is the multiset of results. As we will see later on, a non-deterministic choice of multisets M i may be necessary to express the semantics of an orchestration. We write, in this case, Assume that both delays are unknown, encoded as ðCNNÞ ¼? and ðBBCÞ ¼?. Parameter x in MaryNews ¼ MaryðxÞ < x < TwoNews will get either cnn or bbc, in fact the¯rst one to arrive. As we do not have any prior knowledge of the¯rst a Although it seems that Orc is unnecessary to present the problem under investigation, Orc is an useful tool in order to properly deal with Web uncertainty. In particular, Orc allows us to develop the interplay between non-determinism and imprecise probabilities in a clean mathematical way. b A call to CNN or BBC can be interpreted as a call to https://edition.cnn.com/ or https://www.bbc. com/news. Having full knowledge means that the delay function is known and is de¯ned on each orchestration site. We borrow from Hoare 17 the notation P / Q . R. It should be read: P if Q else R. When (consistent) information increases, the imprecision reduces. However, it is impossible to avoid completely the non-determinism. In the case of Web apps, the proposed knowledge framework is the equivalent to our stress model. De¯nition 3.1. A knowledge framework is a tuple K ¼ hE; ; a ; d i where is the delay function and a and d provide the delay bounds under stress Let ða; dÞ sitesðEÞ Â sitesðEÞ be a pair of site subsets under, respectively, a stress (subset a) and d stress (subset d). We evaluate S½a; d as follows: We denote E½a; d the orchestration under stress where each S 2 sitesðEÞ has been replaced by S½a; d.
When we want to emphasize K, we write S K ½a; d and E K ½a; d. The delay cost function tðE K ½a; dÞ is the delay of the¯rst return based on ðS K ½a; dÞ.
Our next example borrows many ideas from a typical fuzzy approach.

Imprecise probability and re¯nement
A semantics for orchestrations where non-determinism is re¯ned to imprecise probabilities was proposed in Castro et al., 10 following our imprecise re¯nement hypothesis. We adapt here the semantics to deal with delays.
Example 3.6. Let us revisit Example 3.2 where ðCNNÞ ¼ ðBBCÞ ¼?. By the imprecise re¯nement hypothesis, Therefore, [MaryNews] ip is represented by fbbmary cnncc@ 1 ½bbmary bbccc@ 2 j 1 ; 2 ! 0 and 1 þ 2 ¼ 1g: Given ' multisets M 1 ; . . . ; M ' and the Cartesian product of probability spaces Á ¼ Á m 1 Â Á Á Á Â Á m k , we introduce the imprecise probabilistic choice of multisets M ¼ ½ 1 i ' M i @P i ðÞ where multiset M i is chosen with probability P i ðÞ. Here, is any element of Á and P i 's are arithmetic expressions on adding up one. As before, we isolate the probabilities, prbsðMÞ ¼ fðP 1 ðÞ; . . . ; P ' ðÞÞj 2 Ág. This allow us to compare (as before) multiset probabilistic choice. Given ' multisets M 1 ; . . . ; M ' and two multiset choices with imprecise weights on them, M is more imprecise than M 0 (or M 0 is more precise than M), M v M 0 , if and only if prbsðM 0 Þ prbsðMÞ. The next result is an adaptation of Theorem 3 in Ref. 10 to our setting. Proof. We proceed by structural induction showing that re¯nement is monotone through Orc operators. As an illustration, let us consider the parallel composition case. Take E ¼ FjG where

Uncertainty pro¯les and re¯nement
Stressed orchestrations can deliver di®erent results depending on the location of the stress, see for instance the Example 3.5. If we bound the spread of the stress but it is not possible to locate it, what can be said about the delay? To answer this question, we use uncertainty pro¯les. Under knowledge framework K ¼ hE; ; a ; d i the e®ects of the joint interaction of a and d are measured by the cost function tða; dÞ ¼ tðE K ½a; dÞ. The uncertainty pro¯le is then a tuple U ¼ hK; A; D; b a ; b d ; ti. As we are dealing with a cost function, for ÀðUÞ, we extend t in the usual way to deal with mixed strategies. Thus we have As we have seen, in this way we associate a delay value to an uncertain situation looking at it through the associated zero-sum game. When ð; Þ is a Nash equilibrium it holds tðUÞ ¼ tð; Þ.
Example 3.7. We consider for orchestration TwoNews ¼ CNNjBBC the knowledge pro¯le K in Example 3.5. We examine the uncertainty pro¯le U de¯ned as hK; sitesðTwoNewsÞ; sitesðTwoNewsÞ; 1; 1; ti; where both sites can be stressed but, the angel (respectively, the daemon) a®ects only one site. Actions of a in ÀðUÞ are given by A a ¼ fa Re¯ning Indeterministic Choice: Imprecise Probabilities and Strategic Thinking 465 We are interested in modeling how the a=d-games are able to re¯ne the imprecise knowledge on asymmetric parallelism. Consider EðxÞ < x < F where F is a parallel composition of sites S 1 j Á Á Á jS k . Assume [S i ] ¼ bbs i cc, then [F] ¼ bbs 1 ; . . . ; s k cc, and with no delay time information ( ¼?), we can only infer that parameter x will hold any of the values in [F]. So, in this case [x] ¼ bbs 1 cc u Á Á Á u bbs k cc. By the imprecise re¯nement hypothesis Therefore, In order to provide an expression for the re¯nement of [x] ip through a=d-games, we introduce some additional concepts.
De¯nition 3.2. Let U ¼ hK; A; D; b a ; b d ; ti, where K ¼ hF; ; a ; d i and F is a parallel composition of sites S 1 j Á Á Á jS k . For each strategy pro¯le ða; dÞ we consider F½a; d ¼ T 1 j Á Á Á jT k where T ' ¼ S ' ½a; d, for 1 ' k. The indicator function of strategy pro¯le ða; dÞ in ÀðUÞ is the set consisting of all S ' sites in which ðT ' Þ is minimum among fðT 1 Þ; . . . ; ðT k Þg. Formally, Note that in this case tðUÞ is 18=5.
The U imprecise set that¯xes the stress exerted by a and d over the sites, according to weighted strategy pro¯les in ÀðUÞ in a Nash equilibrium, is de¯ned as follows. Observe that we get a chain of improving re¯nements.
Moreover, if we assume that the environment behaves as predicted by U, a more precise behavior can be announced. Assuming that F behaves as predicted by ÀðUÞ, the arrival times to x of the different possible values follow the [x] U;ip distribution. Moreover, assuming that the execution of E is triggered by x, EðtðEðxÞ < x < FÞÞ ¼ tðEÞ þ tðUÞ.
In this way, using the imprecise re¯nement hypothesis, induction and additional rules for more complex composition, we can associate a meaning to any E under uncertainty pro¯le U such that

Conclusions
We have extended previous works 10,15 studying relationships between uncertainty, non-determinism and imprecision. In order to obtain sensible forecasts of uncertain systems, we have proposed to replace non-determinism by an arbitrary probabilistic choice, through the re¯nement hypothesis. Depending on our system knowledge, additional re¯nements shrinking arbitrariness can be considered. Our¯rst conclusion is that addressing uncertainty when it emerges is no worse than, in terms of predictability, postponing the challenge.
Uncertainty pro¯les are formal resources to set up the knowledge about working conditions of a system. To each pro¯le, we associated a zero sum game that allows a strategic analysis. We have proposed the distributions in the Nash equilibrium as an unbiased set of probabilities where to limit arbitrariness. This approach allows us to consider more realistic scenarios and provides an additional analysis tool to support the decisions of the system managers. As a proof of the viability of the proposal, we have analyzed some families of Web orchestrations.
Nash equilibria and the value of the a=d-game is a natural way for re¯nements, however,¯nding Nash equilibria in a=d-games can be computationally di±cult, in fact it is an EXP-complete problem. 18 The approach through mixed strategies, appearing in the Nash equilibria, seems also to suggest the possibility to develop algorithms-based Monte Carlo techniques, that could be more e±cient. As a=d-games are zero-sum games, it is also possible to¯nd the game values through iterative methods. 19 These computational aspects need also to be explored.
It will be of interest to analyze whether the uncertainty analysis performed here, in particular the a=d-re¯nement can be applies to other settings. One area of interest is short-term economic systems, like the IS-LM or IS-MP models, 7 or other decision support models. We can also apply the approach to study perturbations in systems coming from real politics 20 or climate change. 21