# Concurrent games and semi-random determinacy

Consider concurrent, infinite duration, two-player win/lose games played on graphs. If the winning condition satisfies some simple requirement, the existence of Player 1 winning (finite-memory) strategies is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games. Several classical winning conditions satisfy this simple requirement. Under an additional requirement on the winning condition, the non-existence of Player 1 winning strategies from all vertices is equivalent to the existence of Player 2 stochastic strategies winning almost surely from all vertices. Only few classical winning conditions satisfy this additional requirement, but a fairness variant of o

## Authors

• 12 publications
• ### Energy mean-payoff games

In this paper, we study one-player and two-player energy mean-payoff gam...
07/02/2019 ∙ by Véronique Bruyère, et al. ∙ 0

• ### Time-aware uniformization of winning strategies

Two-player win/lose games of infinite duration are involved in several d...
07/11/2019 ∙ by Stephane Le Roux, et al. ∙ 0

• ### Optimal Strategies in Weighted Limit Games

We prove the existence and computability of optimal strategies in weight...
09/23/2020 ∙ by Aniello Murano, et al. ∙ 0

• ### Optimal Strategies in Weighted Limit Games (full version)

We prove the existence and computability of optimal strategies in weight...
08/26/2020 ∙ by Aniello Murano, et al. ∙ 0

• ### Optimal strategies in concurrent reachability games

We study two-player reachability games on finite graphs. At each state t...
10/27/2021 ∙ by Benjamin Bordais, et al. ∙ 0

• ### A Formula for Designing Zero-Determinant Strategies

A formula is presented for designing zero-determinant(ZD) strategies of ...
07/07/2021 ∙ by Daizhan Cheng, et al. ∙ 0

• ### The Discontinuity Problem

Matthias Schröder has asked the question whether there is a weakest disc...
12/03/2020 ∙ by Vasco Brattka, et al. ∙ 0

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## 1 Introduction

Computer science models systems interacting concurrently with their environment via infinite duration two-player win/lose games played on graphs: a play starts at a state of the graph, where the players concurrently choose one action each and thus induce the next state, and so on for infinitely many rounds. The winning condition is a given subset of the infinite sequences of states, and Player 1 wins the play iff the sequence of visited states belongs to . A strategy of a player prescribes one action depending on what has been played so far, and a winning strategy is a strategy ensuring victory regardless of the opponent strategy.

There are games where neither of the players has a winning strategy, but Borel determinacy [25] guarantees the existence of a winning strategy in games where the players play alternately and the winning condition is a Borel set. Under Borel condition again, Blackwell determinacy [26] guarantees a weaker conclusion when the players play concurrently: there exists a value such that for all

the players have stochastic strategies guaranteeing victory with probability

and , respectively.

In the special case of concurrent games played on finite graphs with -regular winning conditions, [11] designed algorithms to decide the existence of (stochastic) strategies that are winning, winning with probability one, and winning with probability for all . [11] also mentions a three-state game where only the latter exist, which exemplifies the complexity of the concurrent -regular games on finite graphs. Then [6] studied concurrent prefix independent winning conditions, which is strictly more general than the -regular conditions, and [13] further improved upon some results. Some of these results were extended recently to multi-player multi-outcome games, see e.g. [3], [15].

The new games This article studies slightly different games: when the players concurrently choose one action each, it also produces a color; the winning condition is now a given subset of the infinite sequences of colors; and Player 1 wins the play iff the produced sequence of colors belongs to . There are two differences between the classical games and the new games. First, the winning condition does not involve the visited states but the transitions instead; second it does so indirectly, via colors labeling the transitions. E.g. in the game on the left-hand side of Figure 1, starting at , the action sequence yields the state sequence and the color sequence .

There are several reasons why these new games are interesting.

• The classical games can be encoded easily into the new ones by using state names as colors. Variants such as the games with colored states, or the colorless games with winning condition on the transitions can also be encoded easily into the new games.

• The converse encoding may increase the state space (to infinity for games with infinitely many actions). Note that the transition-versus-state issue was already studied in the turned-based setting in [10]. Likewise, colorless games are encoded easily in games with colors without size increase, and colors usually lead to more succinct winning conditions.

• Colors are widely used in turn-based games, and for all games they help to study the winning conditions independently from the game structure, and thus to approximate or even characterize nice winning conditions for classes of games (usually simple to check) rather than for single games (usually more accurate but harder to check). This is exemplified by the difference between Theorems 5 and 7 in [27].

• Whereas classical one-state games are trivial, the new one-state games are fairly complex and constitute a nice intermediate object towards the understanding of the more complex general games. Likewise, some one-state (aka stateless) objects from the literature are interesting in their own right: [1]

studied one-state multi-objective Markov decision processes; vector addition systems (VAS,

[17]) are still studied despite the vector addition systems with states (VASS, [16]); the Minkowski games [24] defined with finite sets are a special case of the one-state games from this article.

The main results

• If is closed under interleaving and prefix removal, and if states and colors are finitely many, the existence of a Player 1 winning (finite-memory) strategy is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games.

• If, in addition, is factor-prefix complete and there are finitely many actions, either Player 1 has a winning strategy from one state, or every Player 2 constant (stronger than positional!), positive, stochastic strategy is winning almost surely from all vertices. This is semi-random determinacy.

• One-state games enjoy a stronger conclusion under somewhat weaker assumptions: if the winning condition is factor-set complete and closed under interleaving, if Player 2 has finitely many actions, either Player 1 has a winning strategy, or every Player 2 constant, positive stochastic strategy is winning almost surely.

The finitary flavor of the above characterizations yields decidability and memory sufficiency, in the rough range of double exponentials in the number of states times the number of colors.

In the context of semi-random determinacy, a neutral, random Player 2 is therefore as bad for Player 1 as a hostile environment. Also, the victory is clear-cut in the above results: no need for approximate optimal strategies, no need for the notion of value, etc. This is due to the assumptions, and it is legitimate to wonder how restrictive they are.

Several classical winning conditions from computer science are closed under interleaving, see Section 5. The Muller condition is not, but the parity condition is, so the first characterization result extends to the concurrent Muller games via the Last Appearance Record (LAR), as done in [28]. So, closeness under interleaving is not as restrictive as it may seem.

Fewer classical winning conditions are factor-prefix complete (defined in Section 3.2), but the boundedness condition from [24] and a variant of the -regular languages are both closed under interleaving and factor-prefix complete. The variant is as follows: each produced color requests some combinations of colors to occur in the future. In winning plays, the number of currently unsatisfied requests should be uniformly bounded over time. It may be relevant even as a business model: at every time unit the system can pay penalties for every currently unsatisfied request, which may be covered by greater, albeit bounded, instantaneous income.

The above variant relates to the notion of fairness, which requires that co-finitely many requests are eventually satisfied. The finitary fairness [2] additionally requires uniformly bounded response time. This idea was use in [12] to study temporal logic, and in [9] to study finitary parity games. Requiring uniformly bounded response time (or variants thereof) to study games has been further used later, e.g. in [5]. However, these notions of fairness do not enjoy closeness under interleaving and factor-prefix completeness. (Details in Section 5.)

Related works The semi-random determinacy implies the bounded limit-one property from [11] for the new games: if one state has positive value, one state has value one.

Corollary 3.1 generalizes the nice Theorem 4 from [18]. Note that the convexity of winning conditions defined in [18] is essentially the same as the interleaving closeness defined here.

This article also shares similarities with [14]: both use abstract winning conditions, and both characterize the existence of winning strategies in two-player games by the existence of winning strategies in finitely many derived one-player games. Several articles adopted a similar approach: [19] and [20] reduce multi-player multi-outcome Borel games to simpler two-player win/lose Borel games, and characterize the preferences and structures that guarantee the existence of Nash equilibrium in infinite tree-games; [21] does the same to characterize the preferences that guarantee the existence of subgame perfect equilibrium (at low levels of the Borel hierarchy); [23] does the same to almost characterize the existence of finite-memory Nash equilibrium in games on finite graphs; [22] reduces one-shot concurrent two-player multi-outcome games to simpler one-shot concurrent two-player win/lose games, with applications to generalized Muller games and generalized “parity” games.

One of the benefits of abstraction is that it leads to more general results: e.g. [23] noted that the lexicographic product of mean-payoff and reachability objectives cannot be encoded into real-valued payoffs, and [27] proved it.

Structure of the article Section 2 gives basic definitions. Section 3 presents the main results and additional definitions. Section 4 discusses the key elements of the proofs. Section 5 presents applications.

## 2 Definitions

The folklore Observation 2 below will be used extensively to lift properties from finite words to infinite words. It will be first explicitly invoked, and then only implicitly used.

###### Observation .

Let be such that . Then can be uniquely extended to such that for all and .

Games A game (with colors and states) is a tuple such that

• and are non-empty sets (of actions for Player 1 and Player 2),

• is a non-empty set (of states),

• (is the initial state),

• (is the state update function).

• is a non-empty set (of colors),

• (is a color trace),

• (is the winning condition for Player 1)

Histories The full histories (full runs) of such a game are the finite (infinite) words over , the Player 2 histories (Player 2 runs) are the finite (infinite) words over , and the Player 1 histories (Player 1 runs) are the finite (infinite) words over .

Strategies A Player 1 strategy is a function from to . Informally, it requires Player 1 to remember exactly how Player 2 has played so far, and it tells Player 1 how to play.

Induced histories The function is defined inductively below. As arguments it expects a strategy and a Player 2 history in , and it returns a full history: the very full history that, morally, should happen if Player 1 followed the given strategy while Player 2 played the given Player 2 history. and .

By Observation 2 the function is extended to expect opponents runs in and return full runs: is the only action run whose prefixes are the for .

Winning strategies A Player 1 strategy is winning if for all . If there is a Player 1 winning strategy in a game, one says that Player 1 wins the game.

Extending the update and trace functions The state update function is extended to inductively: and . Using , the trace function is naturally lifted to full histories by induction: and . The trace is further extended to full runs by Observation 2. When considering several games, indices may be added to the corresponding and .

Memory A Player 1 strategy is said to use memory , or memory size , if there exist a set and , and two functions and such that , where is defined inductively by and . If is finite, is called a finite-memory strategy. Note that every Player 1 strategy uses memory .

One-player games Intuitively, a one-player game (with colors and states) amounts to a game where Player 2 has only one strategy available, i.e. . Formally, it is a tuple such that , , and are non-empty sets, , , , and . In this context, the full histories (full runs) of such a game are the finite (infinite) words over , and the Player 2 histories of Player 1 are the natural numbers (telling how many rounds have been played). There is only one Player 2 run, namely . Then, a Player 1 strategy is a function from to , and the notation for the induced full histories is overloaded: and . By Observation 2 the function is (again) extended: is the only action run whose prefixes are the for . A Player 1 strategy is winning if .

Prefix removal A set of infinite sequences is closed under prefix removal if the tails of the sequences from the set are again in the set. Formally, is closed under prefix removal if the following holds: . Note that closeness under prefix removal is weaker than the prefix independence assumed in [6], [13], and [18].

Interleaving Interleaving two infinite sequences consists in enumerating sequentially (part of) the two sequences to produce a new infinite sequence. For example, interleaving and can produce the sequences (perfect alternation), , and (by enumerating the first sequence only), but not the sequences or .

Delayed response Consider a game with finite and . For every let be the elements of , where for all . The elements of are called the Player 2 delayed responses. Intuitively, a Player 2 delayed response amounts to a Player 2 positional strategy in (and only in) a sequentialized version of the game. In every round of this version, Player 1 chooses an action first, then Player 2 chooses an action (or more precisely some color and state among the pairs he could induce by choosing an action). E.g. is a delayed response for Figure 1. It means that at state , Player 2 selects for both actions of Player 1, and at state it selects if Player 1 chooses action . Note that delayed responses are not Player 2 (positional) strategies in the concurrent game, e.g. as is not achievable in any column.

Derived one-player games Let be a Player 2 delayed response. The one-player game is defined by , the projection of on the -component such that . Intuitively, is the game obtained by letting Player 2 fix his strategy (to realize) in the sequentialized version of . For example, the game on the left-hand side of Figure 1 applied to the delayed response yields the game on the right-hand side of Figure 1.

## 3 Main results

Section 3.1 characterizes the existence of Player 1 winning strategies and gives a complexity result. Section 3.2 defines additional concepts and uses the above characterization to characterize the existence of Player 2 everywhere-winning stochastic strategies. Section 3.3 studies the special case of one-state games and presents the semi-random determinacy.

### 3.1 Existence of Player 1 winning strategies

Theorem 3.1 below characterizes the existence of Player 1 winning strategies in a game via the existence of winning strategies in finitely many derived one-player games. Theorem 3.1 afterwards drops the assumption on closeness under prefix removal from Theorem 3.1, but at the cost of a universal quantification over the starting state of the game. In Theorems 3.1 and 3.1, the finiteness and the closeness assumptions are used only to prove the 21 implications. Consider a game . If and are finite, and is closed under interleaving and prefix removal, the following are equivalent.

1. Player 1 wins .

2. Player 1 wins for all delayed responses .

If is finite and Player 1 wins, she can do it with memory size , where is a sufficient memory size to win the one-player games using , and . Consider games parametrized by . If and are finite and is interleaving-closed, the following are equivalent.

1. Player 1 wins for all .

2. Player 1 wins for all and delayed responses .

If the above holds, Player 1 wins every with memory size as in Theorem 3.1.

In games that are (or encode) turn-based games, the delayed responses are Player 2 positional strategies. So, restricting Theorems 3.1 and 3.1 to turn-based games yields Corollaries 3.1 and 3.1, respectively. Note that Corollary 3.1 generalizes Theorem 4 from [18] by only assuming closeness under prefix removal instead of prefix independence. This is significant since the safety condition is closed under interleaving and prefix removal, but is not prefix independent. Consider a game encoding a turn-based game. If and are finite, and is closed under interleaving and prefix removal, either Player 1 has a winning strategy or Player 2 has a positional winning strategy. Consider games parametrized by and encoding a turn-based games. If and are finite, and is closed under interleaving, either Player 1 wins all , or Player 2 has a positional winning strategy for some .

The characterizations from Theorems 3.1 and 3.1

yields decidability results and rough algorithmic complexity estimates in Corollary

3.1 below. Note that checking all the possible strategies using memory size given by Theorems 3.1 and 3.1 would be slower than Corollary 3.1. Let , let be closed under interleaving and prefix removal (resp. by interleaving), and let be such that for all finite and all one-player games , it takes at most computation steps to decide the existence of a (finite-memory) winning strategy in the game. Then for all finite games it takes at most

 f(|A1|,|Q|,|C|)⋅(|C×Q|)|Q|2|C×Q|+|Q||A1||A2|

computation steps to decide whether Player 1 wins (with finite memory).

(resp. computation steps to decide whether Player 1 wins (with finite memory) for all .)

### 3.2 Existence of Player 2 almost surely winning random strategies

Consider a game .

Probability distribution

A probability distribution on a finite set

is a function such that . Let us call the set of the probability distributions on .

Stochastic strategies A Player 2 stochastic strategy is a function .

Induced stochastic histories The function is defined inductively below. As arguments it expects a Player 2 stochastic strategy and a Player 1 history , and it returns a probability distribution on . Informally, it tells the probability of a full history if Player 1 plays and Player 2 follows for many rounds. Formally, , and if , and .

Induced probability measure The function is extended to expect Player 1 runs: . Thus, every pair induces a probability measure on .

Almost surely winning stochastic strategies A Player 2 stochastic strategy is said to be winning almost surely if for all .

Factor-prefix completeness Informally, is factor-prefix complete if the following holds: if the prefixes of an infinite sequence occur as factors arbitrarily far in the tail of a second sequence in , the first sequence is also in . (A factor, aka substring, is a subsequence of consecutive elements.) Formally, is factor-prefix complete if the following holds: .

In Theorem 3.2 below, a distribution is said to be positive if it assigns only positive masses. A (stochastic) strategy is said to be constant if it is a constant function, i.e. it returns always the same distribution, which is stronger than being Markovian, memoriless, or positional. [semi-random determinacy] Consider games parametrized by . If and are finite, if is factor-prefix complete and closed under interleaving and prefix removal, the following are equivalent.

1. for all , Player 1 has no winning strategy in .

2. for all , Player 2 has a constant, positive, stochastic strategy winning almost surely.

3. for all every Player 2 stochastic strategy involving probabilities bounded away from (i.e. with positive infimum) wins almost surely.

So in the setting of Theorem 3.2, either Player 1 has a winning strategy for some , or every constant, positive strategy is winning almost surely, hence the determinacy. Also note that semi-random determinacy implies the bounded limit-one property from [11] for the new games: if one state has positive value, one state has value one.

### 3.3 The special case of stateless (i.e. one-state) games

Stateless games Intuitively, a stateless game (with colors) amounts to a game with only one state, i.e. . Formally, it is a tuple such that , , and are non-empty sets, (as opposed to in the general case), and . Histories, runs, strategies, and induced histories are defined as in the general case . It is easier to extend the trace in this context: and .

Restricting Theorem 3.1 to stateless games yields a simpler Corollary 3.3 below. (Note that restricting Theorem 3.1 would yield a weaker variant of Corollary 3.3, i.e. additionally assuming closeness under prefix removal.) Memory size and algorithmic complexity estimates could be obtained essentially by replacing with in Theorem 3.1 and Corollary 3.1. Consider a game with finite and interleaving-closed . Let be the elements of . The following are equivalents.

1. Player 1 has a winning strategy (resp. finite-memory winning strategy).

2. (resp. ),

where are the regular infinite sequences over .

Restricting Theorem 3.2 to stateless games cancels the universal quantification over states, but an even stronger version can be obtained: finiteness of and prefix removal closeness are dropped, and the assumption on factor-prefix completeness is weaken into factor-set completeness, as below.

Factor-set completeness A language of infinite sequences is called factor-set complete if the following holds: if a sequence in the language has factors of unbounded length over some , the language has a sequence over . This is formally defined by contraposition: is factor-set complete if for all and for all , we have .

###### Observation .

Factor-prefix completeness implies factor-set completeness (finite alphabets).

[Stateless semi-random determinacy] Consider a stateless game with finite and . Let us assume that is interleaving-closed and factor-set complete. Then either Player 1 has a winning strategy, or every Player 2 constant, positive, stochastic strategy is winning almost surely.

## 4 The proofs

Theorems 3.1 and 3.1 characterize a concurrent game by finitely many one-player games. A natural idea would be to split their proof into two parts: first, reduce the problem to turn-based games via the well-known observation that a player has a winning strategy in a concurrent game iff she has one in the sequential version of the game where she plays first; second, use similar techniques as in [18]. For this to work, the sequential versions of the concurrent games must allow for colorless transitions, or a fresh color should be used for the transitions where Player 1 plays. This raises three issues: first, true colors should occur infinitely often in every run in these turn-based games, which would require a more complex notion of turn-based game; second, the winning condition should be rephrased to take the fresh color into account, and so should its closeness properties; third, it would be much difficult to obtain stronger results for the one-state concurrent games, since the one-state property may be hard to track through the translation into turn-based games. Instead, this article overcomes the concurrency directly thanks to Lemma 4.

Let be a family of sets. Then

.

###### Proof.

Towards a contradiction, let us assume the negation of the claim, i.e. . By collecting one witness for each , one constructs such that . In particular, taking yields for all , which contradicts the type of . ∎

Consider the one-state game in Figure 2 (to the left), where each cell encloses one vector of the real plane. Player 1’s objective is that the sum of the outcome vectors remains bounded, which is closed under interleaving and prefix removal, so is a concurrent version of the Minkowski games [24]. There are delayed responses, and five of the corresponding one-player games are displayed to the right in Figure 2. Player 1 wins , since for each the vector is in the convex hull of the three vectors defining . The idea is to let Player 1 play as if she were playing in parallel, more specifically in an interleaved way. Then, summing up the eight bounded trajectories yields a bounded trajectory for .

The main difficulty to play the in an interleaved way is that at every stage, Player 1 should pick an action such that whichever action Player 2 chooses, the resulting vector is exactly the expected one by the (fixed) winning strategy for some . Let be the function that tells which action should be played currently in each of the one-player games. By Lemma 4 there exists an action such that the following holds: if Player 2 chooses , there exists expecting the vector in the cell , and likewise if Player 2 chooses , there exists expecting the vector in the cell .

Let us now quickly mention semi-random determinacy. The proof of Theorem 3.2 below uses similar techniques as, e.g., a proof in the submitted journal version of [24].

###### Proof of 1⇒3 from Theorem 3.2.

Let and let be a Player 2 stochastic strategy that always assigns probability at least to every action.

For all , by contraposition of Theorem 3.1 let be a delayed response (in ) such that Player 1 loses the one-player game . For all , anytime a play reaches the state , the probability that from then on Player 2 follows for rounds in a row, as if second-guessing Player 1, is greater than or equal to .

Consider a play where Player 2 follows . Let be a state that is visited infinitely often. (Such a state exists since is finite.) Thanks to the argument above, for all , the probability that, at some point, Player 2 follows for rounds in a row from on is one. Since the countable intersection of measure-one sets has also measure one, the probability that, for all , at some point Player 2 follows for rounds in a row from on is one.

Let be the corresponding full histories. Since and are finite, the tree induced by prefix closure of the is finitely branching, so by Koenig’s Lemma it has an infinite path , which corresponds to Player 2 following infinitely many rounds in a row. So . By factor-prefix closeness the original play is also losing for Player 1, i.e. winning for Player 2. ∎

## 5 Applications

Abstract assumptions need not only be general, they also need to be practical. Section 5.1 shows that the closeness and completeness axioms enjoy nice algebraic properties: individually, w.r.t. Boolean combination, as well as collectively via the derived closure or completion operators. Section 5.2 mentions several classical or recent winning conditions from computer science and tells which of them satisfy the closeness and completeness axioms. Section 5.3 introduces the notion of bounded residual load as an alternative to the finitary fairness [2], and uses it to define a finitary variant of the -regular languages that satisfies the closeness and completeness axioms.

### 5.1 Algebraic properties of the closeness and completeness axioms

Lemma 5.1 below shows how the axioms behave w.r.t. Boolean combination.

1. The set of the factor-set complete languages is closed under union.

2. The set of the interleaving-closed languages is closed under intersection.

3. The set of the factor-prefix complete languages is closed under intersection and union.

The set of the interleaving-closed languages is not closed under union: and are closed under interleaving (and by prefix removal), but is not. The set of the interleaving-closed languages is not closed under complementation: the interleaving of two infinite sequences that are not eventually constant is not eventually constant, but interleaving the eventually constant sequences and may yield . The set of the factor-set complete languages is not closed under intersection: indeed, both two-element sets and are factor-set complete, but their intersection is not. The set of the factor-set (-prefix) complete languages is not closed under complementation: is factor-set (-prefix) complete, but is not.

The closeness under interleaving and prefix removal, and the factor-prefix completeness induce closure operators. If a relevant winning condition fails to satisfy an equaly relevant axiom, such an operator conveniently constructs a (more generous, axiom satisfying) variant of the winning condition. The closure by prefix removal of a set consists in adding the tails of the sequences from the set; the closure by interleaving consists in adding sequences obtained by interleaving the sequences from the set; and the factor-prefix completion consists in adding the sequences whose prefixes occur arbitrarily far in a sequence from the set. Note that factor-set completeness does not induce a canonical closure operator due to the existential quantifier in its definition.

Lemma 5.1

below shows that the operators behave as expected. This is not for granted in general, as one may need to perform the addition operation an ordinal number of times. Here, one step suffices, which is convenient if computation is of concern.

1. Closure by prefix removal yields sets that are closed under prefix removal.

2. Closure by interleaving yields sets that are closed under interleaving

3. Factor-prefix completion yields sets that are factor-prefix complete.

Lemma 5.1 shows that the operators preserve the existing properties.(Lemma 5.1 is invoked as a proof technique.)

1. Closure by prefix removal preserves closeness under interleaving.

2. Closure by prefix removal preserves factor-set and factor-prefix completeness.

3. Closure by interleaving preserves closeness under prefix removal.

4. Closure by interleaving preserves factor-set and factor-prefix completeness.

5. Factor-prefix completion preserves closeness under prefix removal.

### 5.2 Concrete winning conditions

The non-comprehensive list below displays classical or recent winning conditions from computer science. It especially shows that new winning conditions obtained by conjunction of older winning conditions have been recently studied, e.g. in [7] and [4].

Parity for some . A sequence is winning iff the least number occurring infinitely many times in the sequence is even.

Muller for some . Let be a set of subsets of . A sequence is winning iff the numbers occurring infinitely many times in the sequence constitute a set in .

Mean-payoff , and a sequence is winning iff the limit superior of the partial sums is non-negative: is winning iff . (Variants exist with limit inferior or positivity instead of non-negativity.)

Energy , and a sequence is winning iff its partial sums are non-negative: is winning iff .

Boundedness [24] , and a sequence is winning iff its partial sums are uniformly bounded: is winning iff .

Discounted sum is a bounded subset of . Let and . A sequence is winning iff .

Energy-parity [7] for some . The winning condition is the conjunction of the energy (first component) and the parity (second component) conditions.

Average energy [4] . The objective is to maintain a non-negative energy while keeping the average level of energy below a threshold : a sequence is winning iff .

###### Observation .
1. The parity, mean-payoff, energy, boundedness, energy-parity, and average energy conditions are all closed under interleaving. (It uses Lemma 5.1.2 to deals with energy-parity and average energy.)

2. The Muller and discounted sum conditions are not closed under interleaving.

3. The boundedness condition is factor-prefix complete; the others are not.

4. The energy, energy-parity, and discounted sum conditions are not closed under prefix removal; the others are.

The turn-based safety-mean-payoff-parity games are half-positionally determined. (By Corollary 3.1 and Section 5.1.)

It may be disappointing that the Muller condition is not even closed under interleaving, but Proposition 5.2 below extends Theorem 3.1 to the concurrent Muller games. Using results from [11] is likely to yield a better algorithmic complexity, though, but the point here is mainly that Theorem 3.1 can be extended.

###### Proposition .

[Similar to [11]] Consider the finite games where is a Muller condition. Deciding the existence of a Player 1 winning (finite-memory) strategy can be done in big of

 (|A1||A2||C||C|!)2⋅(|Q||C|2|C|!)|Q||C||C|!(2|Q||C|2|C|!)

computation steps.

Unlike Theorems 3.1 and 3.1, Theorems 3.2 and 3.3 are not likely to be extended to include -regular languages. Before defining a variant of the -regular languages that satisfies the closeness and completeness properties from this article, let us consider notions of fairness that can be defined via a predicate on . Intuitivily is supposed to mean that the sequence has satisfied, with delay at most , a request that was formulated in at time .

There are several reasonable ways to express the good behavior of an infinite sequence using the . The classical definition of fairness requires that all problems be eventually solved (see below), or cofinitely many problems (see below), for a usual weakening that ensures prefix independence of the condition. Arguing that this kind of fairness gives no guarantee about response time, [11] strengthened fairness into finitary fairness, which requires the existence of a uniform bound on the waiting time (see below).

Yet another variant, bounded residual load (), is introduced below. It says that satisfies wrt bounded residual load, if the number of problems that have currently not yet been solved is uniformly bounded overtime.

###### Observation .
1. and are incomparable in general.

The finitary fairness and the like may be too strict for some applications: gladly accepting to wait time units, but categorically refusing to wait time units sounds unusual indeed. Instead, the system (which is responsible for solving the problems) could pay a penalty for each problem spending each time unit unsolved. Thanks to the bounded residual load, one has then the guarantee that the amount of money to be paid per time unit is bounded.

It is possible to combine the two ideas, though: by setting an acceptable response time and an acceptable uniform bound on the number of missed deadlines. This however, turns out to be equivalent to the simple , which argues for the robustness of the concept.

###### Observation .

Let , then .

A second justification for the is that it has nice properties that the other notions of fairness lack when is defined to minic -regular languages, as shown below. Consider a non-empty set of colors and a function . A sequence is said to satisfy from position after delay , denoted , if the following holds.

 ∃u∈C(γn),∃(k1,…,k|u|)∈N|u|,n

Intuitively, each color is a problem or a request, and the problem may be solved in several ways, each way consisting in enumerating suitable colors quickly. (This might very well correspond to the positive fragment of some bounded-time temporal logic.) To simulate the parity condition, one can set and and for all . The corresponding is the parity condition with bounded residual load.

Lemma 5.3 below says that however may be instantiated, all Theorems 3.1, 3.1, 3.2, and 3.3 can be applied with the winning condition.

For every non-empty set of colors and every function , the winning condition is closed under prefix removal and interleaving, and factor-prefix complete.

Even when simulates the parity condition as above, none of the corresponding , , or is both closed under interleaving and factor-set complete. is not closed under interleaving: and , but , where can be obtained by interleaving and . is not factor set-complete: , where , but altough factors of ’s occur with arbitrary length in . is neither: first, and , but , altough can be obtained by interleaving and ; second, as above for . Note that the window-parity condition [8],[5] is not closed under interleaving either, as again exemplified by and .

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## Appendix A Existence of Player 1 winning strategies

### a.1 More on interleaving

Interleaving two finite word consists in enumerating sequentially the two words to produce a new word. For example, interleaving and can produce the words and , but neither nor . Formally, interleaving finite words over some alphabet is defined by induction: for all and , set and .

###### Observation .

Interleaving finite words is associative and commutative.

Let us now give a possible formalization of the interleaving of infinite words. Let . Then iff there exist such that

• ,

• ,

• .

Let . Then iff for all there exists such that for all and can be obtained by interleaving the .

###### Proof.

Let us first assume that , and let us prove by induction on that . For the base case, since for all . For the inductive case by definition of the interleaving of infinite words. By I.H . On the one hand, for all , from follows ; on the other hand, since . Therefore by definition of interleaving of finite words.

Conversely, let us assume that for all there exists such that for all and can be obtained by interleaving the . ∎

### a.2 More on Lemma 4

Below is a short story that might help provide useful insight to some readers. Once upon a time, there was a capricious king who loved pastry. There were many bakeries in his kingdom, and each of them could bake a wide range of delicious cakes. Each shop would bake only one type of cake per day, though, and the only way to know which was to visit the shop. One morning, the king summoned his minister to bring him his favorite cake for dinner (among the cakes of the day). Unfortunately, the shops were far apart and one could only visit one of them within a day, and the king’s favorite depended on the cakes of the day in an irrational way. The minister considered buying a cake from some shop and lying about the cakes of the day in the other shops. But the king knew the range of each shop, what if there were no plausible lie? Desperate, the minister sought help from a mathematician: she enquired about the king’s preferences and the range of each shop, bought a cake from one shop, lied about the cakes of the day, and the king ate happily. Lemma 4 shows that the mathematician was bound to succeed: given the ranges of the shops and the king’s preferences, there always exists a safe shop. T

Corollary A.2 below is derived from Lemma 4 by partial Skolemization, i.e. by pulling the before the , and the before the , thus automatically yielding the and the , respectively. Whereas Lemma 4 could be invoked to characterize the existence of winning strategies, Corollary A.2 will be invoked to characterize the existence of winning strategies with (finite) memory, which will be constructed via the functions and . Note that in the statement of Corollary A.2 uses natural numbers as von Neumann ordinals. .

### a.3 Using Lemma 4

Lemma 4 is then used in Lemma A.3 which factors out most of the proof burden of Theorems 3.1 and 3.1. Lemma A.3 involves games that are concurrent at fewer states than in the original game, and then the proofs of Theorems 3.1 and 3.1 proceed by induction on the degree of concurrency. To define these simpler games, consider .

States involving a player A state is said to involve Player 2 if