Adversarial Contention Resolution Games

1 Introduction

Agents using a shared channel concurrently need to resolve contention for access to the shared medium if they want to broadcast on the channel as quickly as possible. There are n agents and the adversary chooses some k ≤ n of them as players, each having a packet to transmit. A packet is transmitted successfully if there are no other simultaneous transmissions. The challenge is to resolve contention while not knowing the selection of the k participants in advance. Performance of algorithms scheduling transmissions is measured by latency defined as the delay of the last transmitted packet. Contention resolution on a shared channel has been defined in context of Ethernet by [Metcalfe and Boggs, 1976], and has won Metcalfe the 2023 Turing Award. It has been extensively studied in cooperative settings, see, e.g., [Chlebus, 2001; De Marco et al., 2019].

[Fiat et al., 2007] initiated the work on contention resolution algorithms for a shared channel that provide the number of contenders as feedback and selfish agents seek to minimize their individual latency costs. They designed randomized algorithms that are Nash equilibria with bounded latency. [Christodoulou et al., 2014] studied games on a shared channel where the cost of each agent depends on the number of attempted transmissions before success. They designed randomized algorithms that are Nash equilibria with bounded expected cost.

1.1 Summary of Contributions

We study contention resolution (CR) games where selfish agents use only deterministic strategies. We assume minimal feedback from the channel in that once a player successfully transmits, it receives an acknowledgment of this fact and may no longer contend for access to the medium. To provide a framework facilitating design of equilibria, we incorporate the adversary who chooses a configuration of k players from among all n agents. We introduce a concept of adversarial equilibrium (AE) for such games. AE models a situation where the players are risk averse so they would not deviate even if there exist one configuration in the game that could be chosen by the adversary and make them worse-off. This concept combines a notion of classical worst-case adversary from distributed computing [Greenberg and Winograd, 1985; Koml´os and Greenberg, 1985], with that of Nash equilibrium and Pareto optimal solution concepts from game theory [Osborne and Rubinstein, 1994]. AE is Pareto optimal in a sense that for any player there is no alternative strategy that is as good for each configuration, and strictly better for at least one configuration.

We design efficient deterministic communication algorithms that are also AE, with sublinear maximum latency. A summary of our communication algorithms along their performance is given in Table 1. The existence of such algorithms allows us to consider the price of anarchy and the price of stability [Nisan et al., 2007; Maschler et al., 2020] in CR games with respect to AE.

Our equilibria with deterministic strategies imply an upper bound of O(1) on the price of stability (PoS) in the CR game for any n and k = 2 and k = 3. To compare, [Fiat et al., 2007] design Nash equilibria in randomized strategies in their (different CR) model, achieving O(n) latency with overwhelming probability.

We characterize the price of anarchy (PoA) by showing it is in the interval [ n Θ(logn) , n+1 Θ(log n) ] when k = 2 and it is unbounded when k ≥ 3. In comparison, [Fiat et al., 2007] prove that PoA is always unbounded in their model of shared channel, and PoS is O(1) with high probability.

A CR game can be modeled as a simultaneous play of an extensive game, see, e.g., [Nisan et al., 2007, Sec. 3.7]. A special kind of an extensive game is a repeated game, where each stage is the same game. Our game is quite different because the games played in each stage (time) might be different. A player who deviates and transmits successfully, no longer participates in the game and cannot be "punished" later for its misbehavior. For this reason "folk theorems" [Maschler et al., 2020; Osborne and Rubinstein, 1994] about cooperation and punishment in repeated games do not apply, see [Fiat et al., 2007]. On the other hand, designing equilibria is impossible without using an appropriate "punishment" mechanism. [Fiat et al., 2007] used deadlines as such a mechanism, and using it showed existence of equilibria. We use a grim trigger punishment mechanism to design our equilibria.

1.2 RelatedWork on Cooperative Contention Resolution

Shared channels model contention occurring in local-area networks, see [Metcalfe and Boggs, 1976; Chlebus, 2001]. [Bender et al., 2005] proposed to study broadcasting in shared channels with queue-free stations in the framework of adversarial queuing. [Chlebus et al., 2012] introduced deterministic distributed broadcast performed by stations with queues in adversarial shared channels. [Anantharamu et al., 2019] studied latency of broadcasting by deterministic algorithms in shared channels with adversarial packet injection. The meaning of "adversarial" in these papers refers to the way of how players are generated and is different to our adversary, who decides which k ≤ n players are present in a configuration of the game. These papers do not study game theoretic settings.

Contention resolution (CR) algorithms have been studied to help multiple users use efficiently a shared channel. Initially it was assumed that agents would respect the given algorithm. [Greenberg and Winograd, 1985] proved a lower bound Ω(k logk n) on the length of CR protocols, later improved in [Clementi et al., 2001], who showed a lower bound Ω(k log(n/k)). [Koml´os and Greenberg, 1985] proved that there exist CR protocols of length O(k log(n/k)), matching the lower bound in [Clementi et al., 2001], but their proof was only existential. Later, [Kowalski, 2005] showed a polynomial time CR protocol of length O(k logc n), for a constant c > 1, which employs partial selectors by [Indyk, 2002; Chlebus and Kowalski, 2005], efficient dispersers ([Ta-Shma et al., 2007]) and superimposed codes ([Kautz and Singleton, 1964] and [Porat and Rothschild, 2011]).

Acknowledgement-based shared-channel algorithms, considered in this work, have been extensively studied in various communication problems, both deterministic and randomized [Abramson, 1970; Chlebus et al., 2012; De Marco and Stachowiak, 2017; Hradovich et al., 2021; Koml´os and Greenberg, 1985; Kowalski, 2005].

2 Technical Preliminaries

There are n selfish agents (players), each having a single packet to be broadcast on a shared channel. We will use the term agent and player interchangeably. Each agent has a unique name (id) which is an integer in the range [n]. The communication is in synchronous time slots (also called rounds or steps, interchangeably).

A shared channel. If only one agent transmits a message in a round then the transmission is successful. If two or more agents transmit their packets at the same time slot then there is a collision on the channel and none of them is successful. Agents attached to the channel receive feedback in each round. In the model of acknowledgements we use, a successful transmission results in the transmitting agent receiving an acknowledgement (ack) while the feedback for other agents is undetermined. If an agent transmits a dummy message to increase contention, then this is a noisy transmission.

2.1 Contention Resolution Games

A distributed algorithm executed by an agent serves as its strategy. We consider only deterministic algorithms. An algorithm determines for each round if the agent transmits, pauses, or possibly halts and exits. An algorithm is oblivious if the sequence of attempts to transmit and pauses for each agent is determined in advance and encoded as a sequence of zeros and ones. An oblivious algorithm can be represented as a binary array with n rows, row i representing agent i's schedule of transmissions. The number of columns in such array is an upper bound on the algorithm's running time. An array representing an oblivious algorithm, with the property that if k agents only participate in an execution and each of them succeeds in its transmission, is an (n, k)-selector.

Each player executes the same deterministic algorithm determined by the parameters n, k. Player's actions are determined by its unique id. In the course of a game, the algorithm executed by a player could use feedback in a round to define its action in the next round(s). In the model with acknowledgements, a player can only hear acknowledgement of its own successful transmission, i.e., it follows a hardwired schedule of transmissions until it hears acknowledgement, and then it switches off.

A game configuration is as a subset of k ≤ n agents designated as players and none of the players knows who other k - 1, players are. Let F k n be the set of all k-element subsets of [n] = {1, . . . , n}, that is, the set of all possible game configurations. Initially an adversary chooses a configuration K ∈ F k n and informs every player of their status with respect to the configuration. We refer to our game as a CR game with acknowledgements. Any algorithm (strategy) for player i, usually denoted si ∈ {0, 1,1×}∗ , could be modelled using two constructs:

• 0/1 sequence for player i, where 1, "transmission one", in position t means that i transmits in step t (unless it has already transmitted successfully and switched off), and 0 means that it is idle in step t. Players can also transmit a "noisy one", denoted 1×. If player i transmits 1× in step t, their packet is not transmitted even if no one else transmits. If any j (j ̸= i), transmits 1 in step t, they are unsuccessful.
• The adaptive switch-off rule: once a player receives an acknowledgement of its successful transmission, it stops following the activities encoded in the sequence (it could also be viewed as the agent swapping all next positions of the sequence to zeros).

Adversarial equilibria. An oblivious algorithm (s1, s2, . . . , sn) for n players is called an (n, k)-adversarial equilibrium, (n, k)-AE, iff for each player i ∈ [n] and any change of his strategy (a.k.a. a deviation) from si to some other strategy s′i ̸= si (and all other players j ̸= i following their strategies sj), if there is a configuration K of k players for which the change strictly decreases the latency of player i, then there is a configuration K′ of k players for which the change strictly increases the latency of player i. K (K′ , resp.) is called an improving (worsening, resp.) configuration for player i under deviation s′i . More formally, s = (s1, s2, . . . , sn) is an (n, k)-AE, if and only if

where lati(s,K) is the latency of player i under s and K, s-i = (s1, . . . , si-1, si+1, . . . , sn) and (s′i , s-i) = (s1, . . . , si-1, s′i , si+1, . . . , sn). Given (s1, s2, . . . , sn), maximum latency is the latest successful transmission time of any player under any configuration without deviation.

Grim-trigger algorithms. Our particular focus is on a natural and broad subclass of protocols, called Grim-trigger algorithms, where a player (before deviation) is allowed to use noisy ones 1×, only after the last transmission one 1. The notion of Grim-trigger strategies were used in a similar context in repeated games, see [Axelrod and Hamilton, 1981; Nisan et al., 2007, Sec. 27.2].

Price of Anarchy and Stability. [Koutsoupias and Papadimitriou, 2009] introduced the notion of the Price of Anarchy (PoA) to measure how far from the socially optimal outcome is the outcome of the worst (Nash) equilibria. More formally, PoA is the ratio of the outcome of the worst case Nash equlibrium and the socially optimal outcome without selfish agents. The other widely studied concept of Price of Stability (PoS) [Anshelevich et al., 2004], is the ratio between the best-case (Nash) equilibrium and the socially optimal outcome.

Algorithm 1: Noisy GTRR(n), player i

1 for t ∈ [n] do 2 if t = i then 3 transmit(packet) (switch-off upon ack) 4 else if t > i then transmit(noise) ;

Notation. Given a sequence ρ, ρ[t1, . . . , ta] denotes a subsequence of ρ consisting of values of a given sequence ρ in positions t1, . . . , ta. Also, [n] = {1, . . . , n}.

3 Algorithmic Equilibria

We first show simple adversarial equilibria (AE's) with maximum latency n, to be used as building blocks to construct more sophisticated AE's with sublinear latency.

3.1 Enhanced Round Robin Equilibria

Consider the following Persistent Round Robin-type algorithm Persistent RR(n): given any deterministic permutation π of (1, 2, . . . , n), player i ∈ [n] transmits in time slot i and if i is unsuccessful in this time slot i, i.e., i has not heard an acknowledgement (ack), then i transmits from that time point until it gets acknowledgement but no later than time slot n (inclusive). Algorithm Persistent RR(n) is obviously an (n, k)-selector with maximum latency n. It can also be shown to be (n, k)-AE for all n, k such that n ≥ k > 2, except n ≥ 3 and k = 2 (these claims are relegated to the full version).

A small modification could turn Persistent RR(n) into an (n, 2)-AE. Namely, if the first transmission of player i in round i is not successful, then the player continues transmitting a dummy message (noise) till the end of round n. The noisy message introduces a permanent channel blocking effect after unsuccessful unique transmission of an active player; hence, belongs to the class of grim-trigger protocols and we call it Noisy GTRR(n). See Figure 1 for illustrations.

(a) Persistent RR(5) for parame-

ter k ≥ 3.

(b) Noisy GTRR(5) for k ≥ 2. Orange are noisy 1×'s.

Figure 1: Round Robin algorithms examples.

Theorem 1 For any n ≥ k ≥ 2, Noisy GTRR(n) is a grimtrigger protocol that is an (n, k)-selector with maximum latency n and an (n, k)-AE for the CR game with acknowledgements.

Proof. Being an (n, k)-selector with latency at most n follows from the fact that without any deviation, there are no

Algorithm 2: Half Size(n), player i

1 si ←Gen(n, i) /* half-size seq. for player i */ 2 tlast ←max{j : si[j] = 1} /* last slot of 1 in si */ 3 for t ∈ {1, 2, . . . , tlast} do 4 if si[t] = 1 then 5 transmit(packet) (switch-off upon ack) 6 else idle ; 7 if t > tlast then transmit(noise) ;

collisions - thus, every player transmits once and successfully in its exclusive time slot. In the remainder, we prove that it is also an (n, k)-AE.

Suppose first k ≥ 3 and an agent i deviates so that his first transmission is in some earlier slot j < i - 1. Then we show existence of a worsening configuration of size k for agent i. Let us consider any configuration K of size k such that j, j + 1, i ∈ K. Both agents i and j will be unsuccessful in round j and, moreover, agent j starts transmitting dummy messages (noise). Then, in time step j +1, agent j +1 will collide with j, hence j + 1 will start transmitting noise. From this time unit on, all time steps will have agents j, j + 1 transmitting dummy noise messages, hence no agent will be successful till the end of the algorithm (time step n inclusive), including agent i. Therefore, the latency of agent i will be larger than n, which is worse than before the deviation (i ≤ n).

The next possible case is when an agent i deviates and his first transmission is in slot j = i - 1. Let us consider any configuration K containing, among other players, i and j = i - 1. Similarly as in the previous case, i and j collide in time slot j and j starts noisy transmissions, which means that agent i cannot successfully transmit in slot i = j +1. Hence, agent i worsens his latency in this configuration when doing this deviation.

Finally, note that if the first transmission of deviating agent i is in slot i or later, he does not have any configuration for which the deviation would improve its latency (originally i), hence in order to satisfy the definition of (n, k)-AE we do not have to show any configuration for which this deviation worsens the latency of agent i.

Let us now assume that n ≥ k = 2. The player i = 1 cannot improve its latency by any deviation, as he has a unique slot 1 to transmit; so, we do not have to show any worsening configuration for him. If i > 1, then any transmission attempt by player i in round j < i worsens his latency in configuration K = {j, i}, as after collision in round j player j triggers its grim of noise messages till the end of step n, blocking all transmissions. □

3.2 Low-Latency Solution for k = 2

We visualize the strategies as n 0/1 sequences of length log n, that is, as a 0/1 matrix M of size n × logn. As the (n, 2)- selector, we choose as the rows of the matrix, all sequences of length log n, each containing (log n)/2 of 1's. The number of such sequences is ( logn (log n)/2) = Θ ( 2log n √log n ) = Θ ( n √log n ) . We need to slightly increase the length of those sequences to say x logn for some x ≥ 1, so that the number of the sequences is exactly n, because we have n agents. We consider the number of players n to be a central binomial coefficient, i.e. n = (2y y ), for some integer y. The first few central binomial coefficients are 2, 6, 20, 70, 252, . . . . There are infinitely many such numbers, and (2(y+1) y+1 ) = (2y y ) · 4 · (1- 1 2(y+1) ) , meaning that the next central binomial coefficient is at least 3 times larger and less than 4 times smaller than the previous one. Hence we choose the appropriate xi = 2i log n , where i ∈ N such that (xi·logn xi ·log n 2 ) = n. After matrix M is defined, in each row of M we change all 0's, if any, into 1×'s, after the right-most 1 in this row. This construction, see Algorithm Half Size(n), is an (n, 2)-AE. Given a player i ∈ [n], procedure Gen(n, i) generates (in any fixed order) all 0/1 sequences of length x log n with exactly (x log n)/2 1's and returns i'th such sequence, called a half-size sequence. Figure 2 shows an example of sequences, deviation and configurations.

(d) {2, 6}-worsening conf. for pl2; pl6 blocks pl2 using 1×'s.

Figure 2: Half size(6), k = 2. Improving and worsening configurations: pl2 deviates by putting 1 in first slot.

Theorem 2 Half Size(n) is a grim-trigger protocol that is an (n, 2)-selector with maximum latency O(logn) and an (n, 2)-AE for the CR game with acknowledgements.

Proof. We first argue that for any k = 2 players, both players successfully transmit. This follows from the fact that given any pair of rows of M, these are two different sequences of length log n, each with exactly (log n)/2 1's. Their symmetric difference has at least two positions that are 0-1 and 1-0, implying that both players will successfully transmit at least in these time units, so this is a (n, 2)-selector with maximum latency O(log n).

Observe that Half size(n) is a grim-trigger protocol, because any player i (without deviations) transmits noisy 1×'s only after its last transmission 1 - see Algorithm 2.

We now argue that Half size(n) is an (n, 2)-AE. For each player i let ti be the time of the left-most (first) 1. Notice that ti ≤ (x log n)/2+1 for each player i. Let us denote by si the strategy Half size(n)i of player i. Observe that there exists a configuration under which player i transmits at time ti. We will denote by s′i the deviation of player i from her strategy si. We consider three cases.

Firstly, consider that player i deviates by modifying some bits in the time interval [1, . . . , ti]. Assume that now there exists an improving configuration in which i transmits strictly earlier than ti. By the construction of these n sequences we observe that there exists another player j such that both i (after deviation) and j have the same subsequence until ti, i.e., s′i [1, . . . , ti] = sj[1, . . . , ti]. Hence under the configuration K = {i, j}, after deviation, i will transmit strictly later than ti. However, before the deviation, i would have transmitted at time ti, since the respective subsequences of players i and j differ at least at one bit and ti is the first 1 of player i. Thus K = {i, j} is a worsening configuration.

Secondly, assume that i deviates by modifying some bits after ti, i.e., in the time interval [ti +1, . . . , x log n]. Assume that this results in an improving configuration. Apparently, in this improving configuration i transmits after ti. We will now construct worsening configurations based on the total number of 1's in s′i , denoted by ones(s′i ):

• ones(s′i ) ≥ (x logn)/2. Let tlast i be the time of the (x logn)/2'th 1 (one) in s′i . By the construction of these n sequences, there exists a player j ̸= i such that s′i [1, . . . , tlast i ] = sj[1, . . . , tlast i ]. Hence, in this configuration, K = {i, j}, they will be both blocked up until tlast i and since j follows the protocol afterwards she will transmit noisy 1×'s and i will not be able to transmit. Thus K = {i, j} is a worsening configuration.
• ones(s′i ) < (x logn)/2. Let tlast i be the time of the last 1 (one) in s′i . We know by the construction of these n sequences that there exists a player j ̸= i such that s′i [1, . . . , tlast i ] = sj[1, . . . , tlast i ]. Before the deviation, under configuration K = {i, j}, since Half Size(n) is an (n, 2)-selector, we know that player i would have transmitted with maximum latency x logn. However, after this deviation she will not transmit at all. Thus K = {i, j} is a worsening configuration.

Lastly, assume that i deviates before and after ti. This falls into the first case as well and therefore we can find a worsening configuration in the same way as above.

Notice that the deviator i may choose to transmit noisy 1×'s, but this has the same effect on the congestion of the channel as transmitting a packet at this given time slot, and it prevents player i from transmit. Thus, the worsening configurations constructed in the same ways as above, remain worsening for player i after such deviations. □

Remark. In case when n is not equal to a binomial central coefficient, our construction can be made to work by using a recent construction in [Griggs et al., 2023] of maximal antichains with an additional chronological property. We will include details in the full version of the paper.

Algorithm 3: Alt Rec(n), player i

1 if n ≤ 5 then Noisy GTRR(n)i ; 2 if i ≤ n/2 then 3 transmit(packet) (switch-off upon ack) 4 idle 5 Alt Rec(n/2)i 6 else 7 idle 8 transmit(packet) (switch-off upon ack) 9 Alt Rec(n/2)i-n/2

3.3 Low-Latency Recursive Solution for k = 3

The "Alternating Recursion" algorithm Alt Rec(n) for player i ≤ n, called Alt Rec(n)i, executes the strategy of player i in Persistent RR(n) for n ≤ 5, otherwise it proceeds recursively as follows. If i ≤ n/2, the player starts to transmit, followed by silence. If the transmission was unsuccessful, it continues by executing Alt Rec(n/2)i. If i > n/2, the player starts with a silence, followed by transmission. If the transmission was not successful, it continues by executing algorithm Alt Rec(n/2)i-n/2. See Table 2 for an illustration.

We first prove some properties of Alt Rec(n). Let n′ ∈ {3, 4, 5} be the last (and smallest) argument used in the recursive process initialized by Alt Rec(n/2). Let σi be the 0-1 transmission sequence of player i generated by Alt Rec(n)i, encoding a transmission of player i in step t as 1 in position t, and no transmission - as 0. We split σi into two, different length, subsequent parts: σ(P) i and σ(R) i . σ(P) i can be viewed as a sequence of x = log(n/n′ ) pairs, σ(P ) i [1, 2], . . . , σ(P) i [2x - 1, 2x], while σ(R) i , as shown next, as a sequence of length n′ .

• Lemma 1 The following holds for any player i ≤ n: (1) Each σ(P ) i [2y-1, 2y] is 1, 0 if the y-th digit of the binary representation of i is 1, and 0, 1 otherwise, for any 1 ≤ y ≤ x. (2) Sequence σ(R) i is equal to Noisy GTRR(n′ )i mod n′ .
• For any 0-1 sequence ρ containing y ≤ x pairs, each being either 10 or 01, there exist n/2y different players j such

that σ(R) j [1, . . . , 2y] = ρ[1, . . . , 2y] and each j has a unique schedule in Alt Rec(n/2y ).

Proof. The proof, by induction on y = 1, . . . , x, is straightforward by Alt Rec(n)'s recursive definition. □

Does Theorem 3 subsume Theorem 2, i.e., is (n, 3)-AE also (n, 2)-AE? This is not true, e.g., if player 1 plays against one other player in Alt Rec (case k = 2), it can deviate and transmit in the second round. This way it improves against other player ≤ n/2 and does not worsen against any player > n/2. Hence, it is not (n, 2)-AE.

Theorem 3 Alt Rec(n) is a grim-trigger protocol that is an (n, 3)-selector with maximum latency O(logn) and an (n, 3)-AE for the CR game with acknowledgements.

Proof. The algorithm's length T(n) is given by the following recursive equations: T(n) = n for n ≤ 5, and T(n) ≤ 2 + T (n/2) otherwise, thus T (n) ≤ O(log n).

Observe that for n ≤ 5, the (n, 3)-selector and (n, 3)-AE follow from Theorem 1. Therefore, in the remainder we focus on the recursive case for n > 5. Assume that the theorem holds for any Alt Rec(n∗ ), where n∗ < n.

We first show that it is an (n, 3)-selector. By inductive assumption, Alt Rec(n/2) is an (n/2, 3)-selector. Consider any three players with ids i1 < i2 < i3 in Alt Rec(n). If all of them are in the same half-range, i.e., all in [1, n/2] or all in [n/2+1, n] respectively, the selection follows from the recursive part of Alt Rec(n/2), applied to ids i1, i2, i3 or to i1-n/2, i2-n/2, i3-n/2, respectively. Otherwise, there is only one player (i1) in the first half-range or only one player (i3) in the second half-rage of ids, and the two other players in the other half-range of ids. The one player transmits successfully during the first two steps (more precisely, in step 1 if it is i1 alone in the half-range [1, n/2] and in the second round if it is i3 alone in half-range [n/2 + 1, n]). The other two players block themselves in the first two rounds (following the same transmission sequences), but then they execute Alt Rec(n/2) with different ids (i2 - n/2, i3 - n/2 in the former case and i1, i2 in the latter, resp.), guaranteeing selection for them by recursive assumption about (n/2, 3)- selection of Alt Rec(n/2). In the latter argument, it is important that the other player transmitted successfully and thus switched off before the two players started Alt Rec(n/2), as otherwise it could have performed Alt Rec(n/2) with same id as one of the other two players and block him (e.g., player i1 ≤ n/2 and player i2 = i1 + n/2 > n/2 would both execute Alt Rec(n/2)i1 , with the same id i1).

It remains to prove (n, 3)-AE. First consider a deviation of a player i which changes only the recursive part of its schedule: Alt Rec(n/2)i if i ≤ n/2 and Alt Rec(n/2)i-n/2 otherwise. So any improvement of the latency of player i could happen only in that part, while the transmissions and their results in the first two steps are unchanged. Thus, i must be blocked in the first two steps by some other player j in the same range of ids as i, and if the third player is in the other half of ids then it would be successful in the first two rounds and switched off in the recursive part. Thus, there will be no players executing the same sequence Alternate Recursive(n/2)i′ in the recursive part, for any i′ ≤ n/2.

Hence, by recursive assumption, if there is any improving configuration in the part Alt Rec(n/2), there will be also a worsening configuration K (for the latency of the deviating agent i) of size 3 in Alt Rec(n/2). Now we have to show such worsening configuration also in the original Alt Rec(n): if i ≤ n/2 then we just take K (all three players in the lower half-range of ids), otherwise we take K′ = {i, j, j ′ } such that K = {i - n/2, j - n/2, j ′ - n/2}.

Now consider a deviation that starts in the first two steps of Alt Rec(n)i. There are three possible flips: both steps flip, only 1 flips to 0, and finally, only 0 flips to 1. The first two deviations lead to a worsening configuration, despite of any further deviations that may happen later. Indeed, we can choose two other players from the other half-range of ids - after the double flip of player i or just flipping 1 into 0, it will have the first two steps identical to those players or full of 0's, resp. Thus they together block one of the steps and keep the other silent, while before the flip (deviation) player i would have transmitted successfully in one of the two first (the unblocked) steps.

It remains to consider the deviation, ρi, with two 1's in the first two steps. Analogously to the original σi obtained from Alt Rec(n)i, we partition ρi into part ρ(P) i containing first x = log(n/n′ ) pairs and the remaining sequence ρ(R) i of the last n′ digits. There are two cases, which we consider below. Case 1: We assume that there is a pair y ≤ x such that ρ(P ) i [y] contains at least one 0 - in this case we apply analogous case analysis as above for the first pair, showing either a direct worsening configuration on steps 2y - 1, 2y or by using a recursive assumption for Alt Rec(n/2y ), see Lemma 1 (3). Case 2: All pairs ρ(P) i [2y - 1, 2y] contain only 1's, for 1 ≤ y ≤ x. We can block all these 1's of deviator i by choosing any players j, j ′ s.t. σ(P) j contains only pairs 10, while σ(P) j ′ contains only pairs 01, see Lemma 1(3). Let z be the location of first 1 in ρ(R) i - it exists, as otherwise i would not have any transmission during the last n′ steps while, as we have argued, it could be blocked along the first 2y steps by some players j, j ′ (so its latency would worsen by not having a successful transmission). As all i, j, j ′ have different first 2x positions, they will be in different copies of Noisy GTRR(n′ ) that are executed in the last n′ steps. It means that each player j, j ′ can execute any of the schedules in Noisy GTRR(n′ ) - it follows by Lemma 1(3). We can choose schedule for j that has first 1 in position z in Noisy GTRR(n′ ), while schedule for j ′ having first 1 on position min{z + 1, n′ }. Together with deviator i, they block position z in ρ(R) i , and j, j ′ block position min{z + 1, n′ } and all the remaining ones in ρ(R) i . So, player i will be blocked by the end of the algorithm, hence worsening its latency. □

Remark. If n is not a power of 2, Alt Rec(n) can be adjusted as follows. We put Alt Rec(⌊n/2⌋)i (line 5), Alt Rec(⌈n/2⌉)i-⌊n/2⌋ (line 9), and additional correction at the end: if some 5 players i, . . . , i+4 of the same prefix σ(P) i execute Noisy GTRR(5) at the end, while other 3 players i′ , . . . , i′+2 of the same prefix σ(P) i′ executeNoisy GTRR(3), player i+4 replaces σ(P) i+4 by σ(P) i′ and "joins" i′ , i′ +1, i′ +2 to execute Noisy GTRR(4) together. Lengths of the resulting schedules differ by at most 1 and the current analysis holds.

4 Price of Anarchy and Price of Stability

We study here the price of anarchy (PoA) and the price of stability (PoS). Theorem 4 follows from the lower bound Ω(k log(n/k)) on the latency of selectors by [Clementi et al., 2001], and its part (1) follows by the properties of Persistent RR; the part (2) is obtained by an application of Theorem 1; and, finally, (3) follows from Theorems 2 and 3.

Theorem 4 The PoS of the CR game with acknowledgements with n players, and k active players is at most:

1. n Θ(k log(n/k)) , when n ≥ k > 2 or n = k = 2,
2. n Θ(k log(n/k)) , when n ≥ k ≥ 2,

(3) O(1), in model with grim-trigger protocols and k = 2, 3.

Here we will provide an almost complete characterisation of the price of anarchy.

Theorem 5 Given the number of players n and active players k ≤ n, the PoA of the CR game with acknowledgements and with grim-trigger protocols is

1. at least n Θ(logn) , and at most n+1 Θ(log n) when k = 2,
2. unbounded when k ≥ 3.

Proof. By Theorem 1, we know that Noisy GTRR(n) is an (n, k)-AE with maximum latency n for k = 2. This implies that the price of anarchy for k = 2 is at least n Θ(log n) , observing that Ω(log n) is the latency of the shortest (n, 2)-selector, see [Clementi et al., 2001].

We will now prove the upper bound of n+1 Θ(log n) on the price of anarchy. Towards this goal we will prove that any (n, 2)- AE has maximum latency at most n + 1. Let (s1, . . . , sn) be any given (n, 2)-selector which is also (n, 2)-AE. Let s1 be the player that has the maximum possible latency in this (n, 2)-AE and let this maximum latency be ℓ. This means that the left-most (i.e., last) transmission 1 in s1 is at time slot ℓ, s1[ℓ] = 1. Because (s1, . . . , sn) is a (n, 2)-selector, there must exist other player, without loss of generality it could be s2, such that player s1 transmits at time ℓ under configuration {1, 2} with no deviations. This means that player 2 blocks player 1 until time slot ℓ - 1. More precisely, if player 1 has the but last 1 in time slot i ≤ ℓ - 1 (the last 1 in time slot ℓ could also be the only 1 in s1), then player 2 must be such that: s1[1, 2, . . . , i] = s2[1, 2, . . . , i], i ≤ ℓ - 2, s1[i + 1, . . . , ℓ - 1] = (0, . . . , 0), and there must be at least one transmission 1 in s2[i + 1, . . . , ℓ - 1]; let i′ ∈ {i + 1, . . . , ℓ - 1} be the time slot of the first 1 in the sequence s2[i + 1, . . . , ℓ - 1]; i′ is the time slot where player 2 transmits under configuration {1, 2} with no deviations, because (s1, . . . , sn) is a (n, 2)-selector. Note that there is no noisy one 1× in sequence s1[1, . . . , ℓ] because the players use grim-trigger protocols. To be precise, s1 has the grim-trigger, i.e., sequence of 1×'s, after time slot ℓ, and the same is true for s2 after the last 1 in the sequence s2[i+1, . . . , ℓ-1]; that is, s2[ℓ] = 1×.

Let us consider the following deviations of player 1: tj = (¬s1[j], s1[-j]) for any j = 1, 2, . . . , ℓ - 1, where ¬0 = 1

and ¬1 = 0, s1[-j] is sequence s1 except position j, and tj is sequence s1 with ¬s1[j] in time slot j. It is easy to check that configuration {1, 2} is improving for player 1 under any deviation tj, except for j ∈ {i, i′ }. But (s1, . . . , sn) is an (n, 2)-AE, for each of these ℓ - 3 deviations tj of player 1, for j ∈ [ℓ - 1] \ {i, i′ }, so there must be a player pj ∈ {3, 4, . . . , n} so that configuration {1, pj} is worsening for player 1 under deviation tj .

We will call these players pj "blockers", and define them as follows. For any j ∈ [ℓ - 1] \ {i, i′ }:

• If s1[j] = 0, then blocker pj has the same prefix as s1 until time slot j-1, it has ¬s1[j] in time slot j, and then follows sequence s1[j + 1, . . . , ℓ].
• If s1[j] = 1, then blocker pj has the same prefix as s1 until time slot j-1, it has ¬s1[j] in time slot j, and then it is arbitrary on time slots j + 1, . . . , ℓ.

It is easy to check that the configuration {1, pj} is worsening for player 1 under deviation tj for any j ∈ [ℓ-1] \{i, i′ }. Players pj have the same prefix as s1 until time slot j -1 and on time slot j they are flipped to ¬s1[j], so they all are pairwise distinct players. They are also all distinct from players 1 and 2. Thus, there are at least ℓ - 3 + 2 = ℓ - 1 distinct players, and ℓ - 1 ≤ n, implying ℓ ≤ n + 1. This concludes the proof for k = 2.

To show that PoA is unbounded if k ≥ 3, consider Noisy GTRR(n), which by Theorem 1 is (n, k)-AE with maximum latency n. Let us add in front of Noisy GTRR(n), ℓ columns containing only 1's, for any integer ℓ > 0. We argue that the resulting n × (ℓ + n) matrix, M ′ , is (n, k)- AE. By Theorem 1, no player can profitably deviate in the Noisy GTRR(n) part of M ′ . If any player i ∈ [n] deviates in the first ℓ time slots, changing any of its 1's to 0's or 1×'s, such deviation has no effect of i's transmission for any chosen other k - 1 ≥ 2 players in the configuration, because these players block player i in the first ℓ time slots. Thus, matrix M ′ is an (n, k)-AE with maximum latency ℓ + n, which is arbitrarily large when ℓ →∞, hence unbounded PoA. □

5 Conclusion and Further Directions

We introduced contention resolution games, with adversarial configurations and deterministic algorithms as agents' strategies. We proposed adversarial equilibrium as a solution concept and efficient algorithms that are such equilibria. There are efficiency gaps for larger k, see Table 1, thus designing more efficient equilibria and/or proving negative results is a natural research direction.

We expect that our methodology of adversarial selection of players in equilibrium applicable to deterministic algorithmic strategies of selfish autonomous agents, could be used to study communication problems beyond transmission of a single packet on a channel. Natural generalizations include shared channels with stronger feedback than acknowledgements, adaptive algorithms, and contention resolution in radio networks with general topologies. Finally, it is intriguing if there are similar, natural definitions of AE for less risk-averse players, which admit efficient deterministic solutions.

Acknowledgements

The work of Dariusz R. Kowalski was partially supported by the NSF grant 2131538. The work of Piotr Krysta has been partially supported by the Network Sciences and Technologies (NeST) initiative of the University of Liverpool.