1 Introduction
Twoplayer zerosum games played on directed graphs form an adequate framework for the synthesis of reactive systems facing an uncontrollable environment [PR89]. To model properties to be enforced by the reactive system within its environment, games with Boolean objectives and games with quantitative objectives have been studied, for example games with regular objectives [2001automata] and meanpayoff games [ZP96].
Recently, games with multiple objectives have received a lot of attention since in practice, a system must usually satisfy several properties. In this framework, the system wins if it can ensure that all objectives are satisfied no matter how the environment behaves. For instance, generalized parity games are studied in [ChatterjeeHP07], multimeanpayoff games in [VelnerC0HRR15], and multidimensional games with heterogeneous regular objectives in [BruyereHR16].
When multiple objectives are conflicting or if there does not exist a strategy that can enforce all of them at the same time, it is natural to consider tradeoffs. A general framework for defining tradeoffs between (Boolean) objectives consists in assigning to each infinite path of the game a payoff such that iff satisfies , and then to equip with a preorder to define a preference between pairs of payoffs: whenever payoff is preferred to payoff . Because the ideal situation would be to satisfy all the objectives together, it is natural to assume that the preorder has the following monotonicity property: if is such that whenever then , then it should be the case that is preferred to .
As an illustration, let us consider a game in which Player 1 strives to enforce three objectives: , , and . Assume also that Player has no strategy ensuring all three objectives at the same time, that is, Player 1 cannot ensure the objective . Then several options can be considered, see e.g. [BouyerBMU12]. First, we could be interested in a strategy of Player 1 ensuring a maximal subset of the three objectives. Indeed, a strategy that enforces both and should be preferred to a strategy that enforces only. This preference is usually called the subset preorder. Now, if is considered more important than itself considered more important than , then a strategy that ensures the most important possible objective should be considered as the most desirable. This preference is called the maximize preorder. Finally, we could also translate the relative importance of the different objectives into a lexicographic preorder on the payoffs: satisfying and would be considered as more desirable than satisfying and but not . Those three examples are all monotonic preorders.
In this paper, we consider the following threshold problem: given a game graph , a set of regular objectives^{1}^{1}1We cover all classical regular objectives: reachability, safety, Büchi, coBüchi, parity, Rabin, Streett, explicit Muller, or Muller. , a monotonic preorder on the set of payoffs, and a threshold , decide whether Player 1 has a strategy such that for all strategies of Player 2, the outcome of the game has payoff greater than or equal to (for the specified preorder), i.e. . As the number of objectives is typically much smaller than the size of the game graph , it is natural to consider a parametric analysis of the complexity of the threshold problem in which the number of objectives and their size are considered to be fixed parameters of the problem. Our main results are as follows.
Contributions. First, we provide fixed parameter tractable solutions to the threshold problem for all monotonic preorders and for all classical types of regular objectives. Our solutions rely on the following ingredients:

We show that solving the threshold problem is equivalent to solve a game with a single objective that is a union of intersections of objectives taken among (Theorem 3.2). This is possible by embedding the monotonic preorder in the subset preorder and by translating the threshold in preorder into an antichain of thresholds in the subset preorder. A threshold in the subset preorder is naturally associated with a conjunction of objectives, and an antichain of thresholds leads to a union of such conjunctions.

We provide a fixed parameter tractable algorithm to solve games with a single objective as described previously for all types of regular objectives , leading to a fixed parameter algorithm for the threshold problem (Theorem 3.1). Those results build on the recent breakthrough of Calude et al. that provides a quasipolynomial time algorithm for parity games as well as their fixed parameter tractability [Calude], and on the fixed parameter tractability of games with an objective defined by a Boolean combination of Büchi objectives (Proposition 2).
Second, we consider games with a preorder having a compact embedding, with the main condition that the antichain of thresholds resulting from the embedding in the subset preorder is of polynomial size. The maximize preorder, the subset preorder, and the lexicographic preorder, given as examples above, all possess this property. For games with a compact embedding, we go beyond fixed parameter tractability as we are able to provide deterministic polynomial time solutions for Büchi, coBüchi, and explicit Muller objectives (Theorem 4.1). Polynomial time solutions are not possible for the other types of regular objectives as we show that the threshold problem for the lexicographic preorder with reachability, safety, parity, Rabin, Streett, and Muller objectives cannot be solved in polynomial time unless (Theorem 4.2). Finally, we present a full picture of the study of the lexicographic preorder for each studied objective. We give the exact complexity class of the threshold problem, show that we can obtain the values from the threshold problem (which thus yields a polynomial algorithm for Büchi, coBüchi and Explicit Muller objectives, and an algorithm for the other objectives) and provide tight memory requirements for the optimal and winning strategies (Table 3).
Related work. In [BouyerBMU12], Bouyer et al. investigate concurrent games with multiple objectives leading to payoffs in which are ordered using Boolean circuits. While their threshold problem is slightly more general than ours, their games being concurrent and their preorders being not necessarily monotonic, the algorithms that they provide are nondeterministic and guess witnesses whose size depends polynomially not only in the number of objectives but also in the size of the game graph. Their algorithms are sufficient to establish membership to for all classical types of regular objectives but they do not provide a basis for the parametric complexity analysis of the threshold problem. In stark contrast, we provide deterministic algorithms whose complexity only depends polynomially in the size of the game graph. Our new deterministic algorithms are thus instrumental to a finer complexity analysis that leads to fixed parameter tractability for all monotonic preorders and all regular objectives. We also provide tighter lowerbounds for the important special case of lexicographic preorder, in particular for parity objectives.
The particular class of games with multiple Büchi objectives ordered with the maximize preorder has been considered in [AlurKW08]. The interested reader will find in that paper clear practical motivations for considering multiple objectives and ordering them. The lexicographic ordering of objectives has also been considered in the context of quantitative games: lexicographic meanpayoff games in [BloemCHJ09], some special cases of lexicographic quantitative games in [BruyereMR14, 0001MPRW17], and lexicographically ordered energy objectives in [ColcombetJLS17].
In [AlmagorK17] and [KupfermanPV14], the authors investigate partially (or totally) ordered specifications expressed in LTL. None of their complexity results leads to the results of this paper since the complexity is de facto much higher with objectives expressed in LTL. Moreover no result is provided in those references.
Structure of the paper. In Section 2, we present all the useful notions about games with monotonically ordered regular objectives. In Section 3, we show that solving the threshold problem is equivalent to solve a game with a single objective that is a union of intersections of objectives (Theorem 3.2), and we establish the main result of this paper: the fixed parameter complexity of the threshold problem (Theorem 3.1). Section 4 is devoted to games with a compact embedding and in particular to the threshold problem for lexicographic games. The last section is dedicated to the study of computing the values and memory requirements of optimal strategies in the case of lexicographic games (Table 3).
2 Preliminaries
We consider zerosum turnbased games played by two players, and , on a finite directed graph. Given several objectives
, we associate with each play of this game a vector of bits called
payoff, the components of which indicate the objectives that are satisfied. The set of all payoffs being equipped with a preorder, wants to ensure a payoff greater than or equal to a given threshold against any behavior of . In this section we give all the useful notions and the studied problem.Preorders.
Given some nonempty set , a preorder over is a binary relation that is reflexive and transitive. The equivalence relation associated with is defined such that if and only if and . The strict partial order associated with is then defined such that if and only if and . A preorder is total if or for all . A set is upperclosed if for all , , if , then . An antichain is a set of pairwise incomparable elements, that is, for all , if , then and .
Game structures.
We give below the definition of a game structure and notations on plays.
Definition 1
A game structure is a tuple where

is a finite directed graph, with the set of vertices and the set of edges such that^{2}^{2}2This condition guarantees that there is no deadlock. It can be assumed w.l.o.g. for all the problems considered in this article. for each , there exists for some ,

forms a partition of such that is the set of vertices controlled by player with .
A play of is an infinite sequence of vertices such that for all . We denote by the set of plays in . Histories of are finite sequences defined in the same way. Given a play , the set denotes the set of vertices that occur in , and the set denotes the set of vertices visited infinitely often along , i.e., and . Given a set and a set , we denote by the set and by the set .
Strategies.
A strategy for is a function assigning to each history a vertex such that . It is memoryless if for all histories ending with the same vertex , that is, if is a function . It is finitememory if it can be encoded by a deterministic Moore machine where is a finite set of states (the memory of the strategy), is the initial memory state, is the update function, and is the nextaction function. The Moore machine defines a strategy such that for all histories , where extends to histories as expected. The size of the strategy is the size of its machine . Note that is memoryless when .
The set of all strategies of is denoted by . Given a strategy of , a play of is consistent with if for all such that . Consistency is naturally extended to histories in a similar fashion. Given an initial vertex , and a strategy of each player , we have a unique play consistent with both strategies , called outcome and denoted by .
Single objectives and ordered objectives.
An objective for is a set of plays . A game is composed of a game structure and an objective . A play is winning for if , and losing otherwise. As the studied games are zerosum, has the opposite objective , meaning that a play is winning for if and only if it is losing for . Given a game and an initial vertex , a strategy for is winning from if for all strategies of . Vertex is thus called winning for . We also say that is winning from or that he can ensure from . Similarly the winning vertices of are those from which can ensure his objective .
A game is determined if each of its vertices is either winning for or winning for . Martin’s theorem [Martin75] states that all games with Borel objectives are determined. The problem of solving a game means to decide, given an initial vertex , whether is winning from (or dually whether is winning from when the game is determined).
Instead of a single objective , one can consider several objectives that are ordered with respect to a preorder over in the following way. We first define the payoff of a play as a vector^{3}^{3}3Note that in the sequel, we often manipulate equivalently vectors in and sequences of bits. of bits the components of which indicate the objectives that are satisfied.
Definition 2
Given a game structure , and objectives , the payoff function assigns a vector of bits to each play , where for all , if and otherwise.
Given the preorder over , prefers a play to a play whenever . We call ordered game the tuple , the payoff function of which is defined w.r.t. the objectives and its values are ordered with . In this context, we are interested in the following problem.
Problem 1
The threshold problem for ordered games asks, given a threshold and an initial vertex , to decide whether (resp. ) has a strategy to ensure the objective from (resp. ).^{4}^{4}4Note that when and is the usual order over , we recover the notion of single objective with the threshold .
In case (resp. ) has such a winning strategy, we also say that he can ensure (resp. avoid) a payoff .
Classical examples of preorders are the following ones [BouyerBMU12]. Let .

Counting: if and only if . The aim of is to maximize the number of satisfied objectives.

Subset: if and only if . The aim of is to maximize the subset of satisfied objectives with respect to the inclusion.

Maximise: if and only if . The aim of is to maximize the higher index of the satisfied objectives.

Lexicographic: if and only if either or such that and , . The objectives are ranked according to their importance. The aim of is to maximise the payoff with respect to the induced lexicographic order.
In this article, we focus on monotonic preorders. A preorder is monotonic if it is compatible with the subset preorder, i.e. if implies . Hence a preorder is monotonic if satisfying more objectives never results in a lower payoff value. This is a natural property shared by all the examples of preorders given previously.
Example 1
Consider the game structure depicted on Figure 1, where circle vertices belong to and square vertices belong to . We consider the ordered game with for and the lexicographic preorder . Therefore the function assigns value to each play on the first (resp. second) bit if and only if visits infinitely often vertex (resp. ). In this ordered game, has a strategy to ensure a payoff from . Indeed, consider the memoryless strategy that loops in and in . Then, from , decides to go either to leading to the payoff , or to leading to the payoff . As , this shows that any play consistent with satisfies . Notice that while can ensure a payoff from , he has no strategy to enforce the single objective and similarly no strategy to enforce .
Homogeneous regular objectives.
In the sequel of this article, given a monotonically ordered game , we want to study the threshold problem described in Problem 1 for homogeneous regular objectives, in the sense that all the objectives are of the same type, and taken in the following list of wellknown regular objectives.
Given a game structure and a subset of called target set:

The reachability objective asks to visit a vertex of at least once, i.e. .

The safety objective asks to always stay in the set , i.e. .

The Büchi objective asks to visit infinitely often a vertex of , i.e. .

The coBüchi objective asks to eventually always stay in the set , i.e. .
Given a family of sets , and a family of pairs , with :

The explicit Muller objective asks that the set of vertices seen infinitely often is exactly one among the sets of , i.e.

The Rabin objective asks that there exists a pair such that a vertex of is visited infinitely often while no vertex of is visited infinitely often, i.e. .

The Streett objective asks that for each pair , a vertex of is visited infinitely often or no vertex of is visited infinitely often, i.e. .
Given a coloring function that associates with each vertex a color, and a family of subsets of :

The parity objective asks that the minimum color seen infinitely often is even, i.e. .

The Muller objective asks that the set of colors seen infinitely often is exactly one among the sets of , i.e.
In the sequel, we make the assumption that the considered preorders are monotonic, and by ordered game, we always mean monotonically ordered games. When the objectives of an ordered game are of kind , we speak of an ordered game, or of a game if we want to specify the used preorder . As already mentioned, when , an ordered game (with equal to ) resumes to a game with a single objective , that is traditionally called an game. For instance, an ordered game where are reachability objectives and is the lexicographic preorder is called a lexicographic reachability game, and when is called a reachability game.
Note that given an ordered game with nonhomogeneous regular objectives , we can always construct a new equivalent ordered parity game, since each objective can be translated into a parity objective [2001automata].
Useful results on games with a single objective.
Let us end this section by providing some results on games with a single regular objective taken among those defined previously or among the additional ones given herafter. All these results will be useful in the proofs.
Let be a game structure and be target sets and be a Boolean formula over variables . We say that a play satisfies if the truth assignment ( if and only if , and otherwise ) satisfies .

Boolean combination of Büchi objectives, or shortly Boolean Büchi objective:
All operators , , are allowed in Boolean Büchi objectives. However we denote by the size of equal to the number of disjunctions and conjunctions inside , and we say that the Boolean Büchi objective is of size and with variables. The definition of is not the classical one that usually counts the number of operators and variables. This is not a restriction since one can transform any Boolean formula into one such that negations only apply on variables.
We need to introduce some other kinds of regular objectives with Boolean combinations of objectives that are limited to

intersections of objectives: like a generalized reachability objective or a generalized Büchi objective ,

unions of intersections (UI) of objectives: like a UI reachability objective
a UI safety objective , or a UI Büchi objective .
Games with regular objectives are determined by Martin’s theorem [Martin75]. We recall the complexity class of solving those games, as well as the kind (memoryless, finitememory) of winning strategies for both players. See Theorem 2.1 and Table 1 below. For each type of objective, the complexity of the algorithms is expressed in terms of the sizes and of the game structure , the number of colors (for and ), the number of pairs (for and ), the size of the family (for and ), the size of the formula (for ), the number of intersections of objectives (for and ), and the number (resp. ) of intersections (resp. unions) in UI objectives (for , and ).
Theorem 2.1
For games with regular objectives, we have:

Solving reachability or safety games is complete (with an algorithm in time) and both players have memoryless winning strategies [Beeri80, 2001automata, Immerman81].

Solving Büchi or coBüchi games is complete (with an algorithm in time) and both players have memoryless winning strategies [ChatterjeeH14, EmersonJ91, Immerman81].

Solving explicit Muller games with a family is complete (with an algorithm in time) and exponential memory strategies are necessary and sufficient for both players [DziembowskiJW97, Horn08].

Solving Rabin (resp. Streett) games with pairs is complete (resp. cocomplete) [EmersonJ88] (with an algorithm in time [PitermanP06]). In Rabin games (resp. Streett games) memoryless strategies are sufficient for (resp. for ) [Emerson85] and exponential memory strategies are necessary and sufficient for (resp. ) [DziembowskiJW97]

Solving parity games with colors is in  (with an algorithm in time [Calude]) and both players have memoryless winning strategies [Jurdzinski98].

Solving Muller games is complete (with an algorithm in time [McNaughton93]) and exponential memory strategy are necessary and sufficient for both players [DziembowskiJW97, HunterD05].

Solving Boolean Büchi games is complete (with an algorithm in time and exponential memory strategies are necessary and sufficient for both players [AlurTM03].^{5}^{5}5The algorithm complexity and the memory requirements do not appear explicitly in [AlurTM03] but can be deduced straightforwardly thanks to the proposed algorithm.

Solving generalized reachability games with target sets is complete (with an algorithm in time) and exponential memory strategies are necessary and sufficient for both players [FijalkowH13].

Solving generalized Büchi games with target sets is complete (with an algorithm in time) and linear memory (resp. memoryless) strategies are necessary and sufficient for (resp. ) [ChatterjeeDHL16].

Solving UI reachability and UI safety objectives is complete (with an algorithm in time) and exponential memory strategies are necessary and sufficient for both players, where denotes the number of distinct target sets.

Solving UI Büchi games with an objective is complete (with an algorithm in time), and exponential memory (resp. memoryless) strategies are necessary and sufficient for (resp. ) [BloemCGHJ10].
Proof
All the statements follow from the literature except for the case of UI reachability and UI safety games for which we provide a proof. We only consider the reachability case, since the proof is similar for the safety case. First, as solving UI reachability games is harder than solving generalized reachability games (when there is no union), we immediately obtain the lower bounds for the complexity and the memory requirements. Indeed, solving generalized reachability games is complete, and exponential memory strategies are necessary for both players [FijalkowH13].
Let us now prove the upper bounds by following the same approach as proposed in [FijalkowH13] to solve generalized reachability games. Let be a UI reachability game where . We define the function that enumerates all the distinct sets . From , we construct the function such that if . If , we abusively write .
We construct from a new game structure in a way to remember which sets have been visited so far, for . Formally, for , and if and only if and for all , if or , and otherwise. With the initial vertex in , we associate the initial vertex in where if and otherwise. We then have that is winning in the original UI reachability game from if and only if is winning in from for the objective where .
Note that solving this reachability game can be done in time linear in the size of the game with memoryless winning strategies for both players by [2001automata]. Coming back to the initial UI reachability game, this leads to an algorithm working in time, and to exponential memory winning strategies for both players.
Now, as done for generalized reachability games [FijalkowH13], one can notice that if is winning for , then he has a strategy to do so within steps. Moreover, given a path of this size, one can check in polynomial time if there exists some such that the path visits all for
. Thus, we can use an alternating Turing machine that simulates the game for up to
steps and checks whether is winning. As the alternating Turing machine works in polynomial time and , this yields the algorithm. ∎Objectives  Complexity class  memory  memory 

Reachability, safety  complete  memoryless  
Büchi, coBüchi  
Explicit Muller  exponential  
Generalized Büchi  linear  memoryless  
Generalized reachability  complete  exponential  
UI reachability, UI safe  
Parity  memoryless  
Rabin  complete  memoryless  exponential 
Streett  complete  exponential  memoryless 
UI Büchi  
Muller  complete  exponential  
Boolean Büchi 
In the sequel, we need some classical properties on regular objectives that we summarize in the following proposition.
Proposition 1

A safety (resp. coBüchi, Streett) objective is the complement of a reachability (resp. Büchi, Rabin) objective.

A parity objective is both a Rabin and a Streett objective.

Rabin and Streett objectives with one pair are parity objectives with colors. Thus, a Rabin (resp. Streett) objective is the union (resp. intersection) of parity objectives with colors.

The intersection of (resp. union of ) explicit Muller objectives is an explicit Muller objective where (resp. ).

A parity objective with colors (resp. Streett objective with pairs, Rabin objective with pairs, Muller objective with colors and a family ) is a Boolean Büchi objective of size at most (resp. , ) and with (resp. , ) variables.
Proof
First, Item 1 immediately follows from the definitions and Items 2 and 3 are detailed in [ChatterjeeHP07].
Let us consider Item 4. For the intersection we have where , and thus . For the union we have with with .
Let us prove the last item by beginning with Muller objectives. It suffices to note that a play belongs to if and only if there exists an element of such that all colors of are seen infinitely often along the play while no other color is seen infinitely often. This is obviously a Boolean Büchi objective , where each corresponds to a color, that is, is the set of vertices labeled by this color. Note that, in this case, the size of the related formula is at most . The arguments are similar for parity, Streett and Rabin objectives (for instance, a play belongs to if and only there exists an even color seen infinitely often along the play and no lower color seen infinitely often). ∎
3 Fixed parameter complexity of ordered regular games
In this section, we study the fixed parameter tractability of the threshold problem.
Parameterized complexity.
A parameterized language is a subset of , where is a finite alphabet, the second component being the parameter of the language. It is called fixed parameter tractable (FPT) if there is an algorithm that determines whether in time time, where is a constant independent of the parameter and is a computable function depending on only. We also say that belongs to (the class) . Intuitively, a language is FPT if there is an algorithm running in polynomial time w.r.t the input size times some computable function on the parameter. In this framework, we do not rely on classical polynomial reductions but rather use so called reductions. An reduction between two parameterized languages and is a function such that

if and only if ,

is computable by an algorithm that takes time where is a constant, and

for some computable function .
Moreover, if is in , then is also in . We refer the interested reader to [DowneyF99] for more details on parameterized complexity.
Our main result states that the threshold problem is in for all the ordered games of this article. Parameterized complexities are given in Table 2.
Theorem 3.1
The threshold problem is in for ordered reachability, safety, Büchi, coBüchi, explicit Muller, Rabin, Streett, parity, and Muller games.
Objectives  Parameters  Threshold problem 

Reachability, Safety  
Büchi  
coBüchi  
Explicit Muller  
Rabin, Streett  ,  
Parity  ,  
Muller  , 
The proof of this theorem needs to show that solving the threshold problem for an ordered game is equivalent to solving a game with a single objective equal to the union of intersections of objectives taken in . It also needs to show that solving Boolean Büchi games is in .
Monotonic preorders embedded in the subset preorder.
We here present a key tool of this paper: solving the threshold problem for an ordered game is equivalent to solving a game with a single objective equal to the union of intersections of objectives taken in . The arguments are the following ones. (1) We consider the set of payoffs ordered with as well as ordered with the subset preorder (see the example of Figure 2 where is the lexicographic preorder). To any payoff , we associate the set containing all indices such that objective is satisfied. (2) Consider the set of payoffs embedded in the set ordered with . By monotonicity of , we obtain an upperclosed set that can be represented by the antichain of its minimal elements (with respect to ), that we denote by . (3) can ensure a payoff if and only if he has a strategy such that any consistent outcome has a payoff for some , equivalently such that satisfies (at least) the conjunction of the objectives such that . (4) The objective of is thus a disjunction (over ) of conjunctions (over ) of objectives . This statement is formulated in the next theorem (see again Figure 2).
Theorem 3.2
Let be an ordered game, be some threshold, and be an initial vertex. Then, can ensure a payoff from in if and only if has a winning strategy from in the game with the objective . ∎
Note that we obtain the following corollary as a direct consequence of Theorem 3.2 and Martin’s theorem [Martin75].
Corollary 1
Let be an ordered game. If are Borel sets, then has a strategy to ensure a payoff from if and only if it is not the case that has a strategy to avoid a payoff from . ∎
Parameterized complexity of Boolean Büchi games. In order to show that solving the threshold problem for ordered games is in , we need to recall some known results of parameterized complexity for games with a single objective and to prove that solving Boolean Büchi games belongs to .
It is proved in [FijalkowH13] that generalized reachability games belong to . Parity, Rabin, Streett, and Muller games are shown to be interreducible in
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