Contention Resolution under Selfishness

G. Christodoulou∗ K. Ligett† E. Pyrga‡

Abstract

In many communications settings, such as wired and wireless local-areanetworks, when multiple users attempt to access a communication channelat the same time, a conflict results and none of the communications are suc-cessful. Contention resolution is the study of distributed transmission and re-transmission protocols designed to maximize notions of utility such as chan-nel utilization in the face of blocking communications.

An additional issue to be considered in the design of such protocolsis that selfish users may have incentive to deviate from the prescribed be-havior, if another transmission strategy increases their utility. The work ofFiat et al. [8] addresses this issue by constructing an asymptotically optimalincentive-compatible protocol. However, their protocol assumes the cost ofany single transmission is zero, and the protocol completely collapses undernon-zero transmission costs.

In this paper,1 we treat the case of non-zero transmission cost c. Wepresent asymptotically optimal contention resolution protocols that are robustto selfish users, in two different channel feedback models. Our main result isin the Collision Multiplicity Feedback model, where after each time slot, thenumber of attempted transmissions is returned as feedback to the users. Inthis setting, we give a protocol that has expected cost Θ(n+c logn) and is ino(1)-equilibrium, where n is the number of users.

1 Introduction

Consider a set of sources, each with a data packet to be transmitted on a sharedchannel. The channel is time-slotted; that is, the transmissions are synchronized∗University of Liverpool, U.K. gchristo@liv.ac.uk Partially supported by DFG grant Kr

2332/1-3 within Emmy Noether Program and by EPSRC grant EP/F069502/1.†California Institute of Technology, Pasadena, CA. katrina@caltech.edu Supported in

part by an NSF Graduate Research Fellowship, NSF grant 0937060 to the Computing ResearchAssociation for the CIFellows Project, and NSF grant 1004416.

‡Technische Universitat Munchen, Germany. pyrga@in.tum.de1A preliminary version of this work appeared in ICALP 2010.

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and can only take place at discrete steps. The packets are of fixed length and eachfits within one time slot. If only one user transmits in a given slot, the transmissionis successful. If more than one user attempts to transmit messages in the same slot,a collision occurs, and all transmissions for that time step fail; the failed packetswill need to be retransmitted later. Typically, the goal of the system designer isto optimize global notions of performance such as channel utilization or averagethroughput, i.e., the average number of successful transmissions per time unit. If allof the sources were under centralized control, avoiding collisions would be simple:Simply allow one source to transmit at each time step, alternating in a round-robinor other “fair” fashion.

What happens, though, if the transmission protocol must be executed in a dis-tributed fashion, with minimal additional communication? Further, what happensif each source selfishly wishes to minimize the expected time before she transmitssuccessfully, and will only obey a protocol if it is in her best interest? Can we stilldesign good protocols in this distributed, adversarial setting? What informationdoes each user need to receive on each time step in order for such protocols tofunction efficiently?

In this game theoretic contention resolution framework, Fiat et al. [8] designan incentive-compatible transmission protocol which guarantees that (w.h.p.) allplayers will transmit successfully in time linear in the total number of users. Oneof the nice features of their protocol is that it only needs a very simple feedbackstructure to work: They assume only that each player receives feedback of the form0/1/2+ after each time step (ternary feedback), indicating whether zero, one, ormore than one transmission was attempted. This positive result is actually basedon a very negative observation, that the price of anarchy [15] in this model is un-bounded: If transmission costs are assumed to be zero, it is an equilibrium strategyfor all users to transmit on all time steps! Clearly, this is an undesirable equilib-rium. Fiat et al. [8] construct their efficient equilibrium using this bad equilibriumquite cleverly as a “threat”— the players agree that if not all players have exited thegame by a certain time, they will default to the always-transmit threat strategy fora very large time interval. This harsh penalty then incentivizes good pre-deadlinebehavior.

It is natural, however, to assume that players incur some transmission cost c> 0(attributable to energy consumption; see, for example, [17, 24]) every time theyattempt a transmission, in addition to the costs they incur due to the total length ofthe protocol. If the cost of transmitting is even infinitesimally non-zero, though, thethreat strategy used in [8] is no longer at equilibrium, and the protocol breaks down.We address this here, by developing efficient contention resolution protocols thatare ε-incentive-compatible even under strictly positive transmission costs, where ε

is a decreasing function of the number of players.

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To achieve this goal, we need to strengthen the channel feedback structure,since the ternary feedback seems to be too limited to deal with the new require-ments. We begin by considering a perfect information feedback model, wherein allplayers can identify which players transmitted at each step. Using this model willalso make it easier to introduce the main ideas of our protocols. Next, we reducethe amount of information that the feedback provides, moving to a model closerto to the initial ternary model. In particular, we consider the Collision Multiplic-ity Feedback (CMF) model. Here, after each collision, while the identity of thecollided packets is lost, the number of packets that participated in the collision isbroadcast to all users. The CMF model has gained substantial attention in the liter-ature because it admits significantly better throughput results than can be achievedwith ternary feedback. In this setting, we present an efficient contention resolutionprotocol that is robust to selfish players.

Even though our algorithm for the CMF model would also work with a perfectinformation feedback channel, the perfect information model allows us to design asimpler, more efficient algorithm. As such, our perfect information algorithm pro-vides an introduction to the main idea behind our CMF approach— iteratively split-ting the players into subgroups—and helps highlight the role of additional feedbackinformation.

1.1 Our Results

There are three main technical challenges we address:

1. Separating players by developing unique ids. Our protocols differ from pre-vious incentive-compatible contention resolution protocols in that they arehistory dependent; that is, the protocol uses the transmission history in or-der to construct a random ordering of the users. We use algorithmic tech-niques similar to those for “tree”-style protocols, originally developed byCapetanakis [6, 7] and later further developed in the information theory andnetworks literatures. Such protocols resolve collisions by gradually splittingthe colliding players in successively smaller groups.

2. Developing a “threat”. Like Fiat et al. [8], we use a threat strategy to incen-tivize good behavior. However, the threat they use collapses under nonzerotransmission costs.2 On the other hand, we use a weaker threat strategy andneed to develop efficient protocols that are incentive compatible despite ourweaker punishment.

2Recall from the earlier description that the threat of Fiat et al. requires that all remaining playerswill be transmitting in every step with probability 1 for a large number of steps. This is an equilibriumonly when the cost of a transmission is zero.

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3. Detecting deviations. In order for a threat to be effective, the players need tobe able to detect and punish when one of them cheats. Our protocol employsa system of checksums and transmissions whose purpose is to communi-cate information on cheating—despite the fact that a “transmission” givesno other information to the players than its presence or absence.

We address these challenges in two different feedback models, described be-low. In each feedback model we consider, we present asymptotically optimal pro-tocols that guarantee to each user expected average cost that is linear in the totalnumber of users. The protocol for our main result is in ε-equilibrium, where ε

goes to 0 as the number of players grows. This form of approximate equilibrium isa very reasonable equilibrium notion in our setting, as games with a large numberof players are technically the only interesting case—if there were only a constantnumber of players, one could simply play the exact time-independent equilibriumthat appears in [8], although its cost grows exponentially with the number of play-ers.

We also note that we do not assume the existence of global ids for the play-ers. If such ids were available, then a trivial protocol would already be sufficient,i.e., simply transmitting in order of lexicographically increasing ids would be ef-ficient and incentive compatible. Of course, this algorithm would not be “fair”in the sense that certain nodes are more favored than others (the one with lexico-graphically smallest id would always be the first to transmit). Moreover, globallyknown ids are not always available. Given the fact that the only means of com-munication is via the common channel, and given the lack of any further optionsfor coordinating such a communication process, obtaining globally known ids inan incentive compatible manner is a non-trivial task. Our protocols can be seen asimplicitly creating unique ids for each player, despite the limited communicationand decentralized setting.

Perfect Information First, in Section 3, we consider the feedback model of per-fect information, where each user finds out after each time step what subset ofsources attempted a transmission. Perfect information is a very common assump-tion in non-cooperative game theory. Making this strong assumption allows us tohighlight the game theoretic aspects of the problem, and the insight we develop inthe context of this strong feedback model is useful in developing algorithms undermore restrictive feedback. Unlike the trivial protocol under global ids, the ran-domized protocol that we present is fair in the sense that all users have the sameexpected cost. Protocol PERFECT has expected cost (n+ 1)/2+ logn, while themaximum gain that any player has by deviating is no more than 1.

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Collision Multiplicity Next, in Section 4, we present the main result of thiswork. Here, we study a model with Collision Multiplicity Feedback (CMF), (oth-erwise known as M-ary feedback; see, e.g., [25, 20, 23]), in which after each timeslot, users find out the exact number of packets that were transmitted during theslot, but not the identities of the transmitting players. In some practical situations,this feedback can be measured as the total energy on the channel during one slot, bymeans of a number of energy detectors. Our protocol MULTIPLICITY has expectedcost Θ(n+c logn), where c is the transmission cost, and is a o(1)-equilibrium w.r.t.the number of players n, i.e., it is an ε-equilibrium, such that ε = o(1).

1.2 Related Work

Contention resolution for communication networks is a well-studied problem. TheALOHA protocol [1], given by Abramson in 1970 (and modified by Roberts [22]to its slotted version), is one of the most famous multiaccess communication pro-tocols. However, Aloha leads to poor channel utilization due to an unnecessarilylarge number of collisions. Many subsequent papers study the efficiency of mul-tiaccess protocols when packets are generated by some stochastic process (see forexample [12, 11, 21]). Such statistical arrival models are very useful, but cannotcapture worst-case scenarios of bursty inputs, as in [5], where batched arrivals aremodeled by all n packets arriving at the same time. To model this worst-case sce-nario, one needs n nodes, each of which must simultaneously transmit a packet;this is also the model we use in this work.

One class of contention resolution protocols explicitly deals with conflict res-olution; that is, if k ≥ 2 users collide (out of a total of n users), then a resolutionalgorithm is called on to resolve this conflict (it makes sure that all the packets thatcollided are successfully transmitted), before any other source is allowed to use thechannel. For example, [6, 7, 14, 26] describe tree algorithms whose main idea isto iteratively give priority to smaller groups, until all conflicts are resolved, withΘ(k+k log(n/k)) makespan. We use a similar splitting technique in the algorithmswe present here.

A variety of upper and lower bounds for the efficiency of various protocols havebeen shown. For the binary model (transmitting players learn whether they succeedor fail; non-transmitting players receive no feedback) when k is known, [10] pro-vides an O(k+ logk logn) algorithm, while [18] provides a matching lower bound.For the ternary model, [13] provides a bound of Ω(k(logn/ logk)) for all deter-ministic algorithms. In all of the results mentioned so far in this subsection, it isassumed that players will always follow the protocol given to them, even if it is notin their own best interest.

The CMF model we consider in this paper was first considered by Tsybakov [25],

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where he proposed a protocol with throughput 0.533. Later Pippenger [20] showed,using a random-coding existential argument, that the capacity of the channel is 1.Ruszinko and Vanroose later in [23] gave a constructive proof of the same re-sult, by designing a particular protocol reaching the throughput 1. Georgiadis andPapantoni-Kazakos [9] considered the case when the collision multiplicity can beestimated by energy detectors, up to an upper bound.

More recently, a variety of game theoretic models of slotted Aloha have alsobeen proposed and studied in an attempt to understand selfish users; see for ex-ample [2, 16, 3]; also [17, 2] for models that include non-zero transmission costs.Much of the prior game theoretic work only considers transmission protocols thatalways transmit with the same fixed probability (a function of the number of play-ers in the game). By contrast, we consider more complex protocols, where aplayer’s transmission probability is allowed to be an arbitrary function of her playhistory and the sequence of feedback she has received. Other game theoretic ap-proaches have considered pricing schemes [27] and cases in which the channelquality changes with time and players must choose their transmission levels ac-cordingly [19, 28], and [4] for a related model.

As discussed above, Fiat et al. [8] study a model very similar to the one wepresent here; while the feedback mechanism they assume is not as rich as ours,crucially, their model does not incorporate transmission costs. The idea of a threatis used both in [8] and in our work, as a way to incentivize the players to be obe-dient. However, the technical issues involved are completely different in the twopapers. The protocol in Fiat et al. [8] relies for its threat on the existence of an ex-tremely inefficient equilibrium, where all players constantly transmit their packetsfor an exponentially long period. This equilibrium is used as an artificial dead-line; that is, the protocol switches to that inefficient equilibrium after linear time.This threat is history independent, in the sense that it will take place after a linearnumber of time steps, regardless of any transmission history of the players. Thishistory-independent threat relies critically on the absence of transmission costs. Onthe other hand, our threat is history dependent and makes use of a cheat-detectionmechanism: any deviation from the protocol is identified with at least constantprobability, and the existence of a deviation is communicated to all players. In re-sponse, all players switch their transmission strategy according to the exponentialtime-independent equilibrium of Fiat et al. (all players transmit at every slot withprobability Θ(1/

√n)).

The communication allowed by the feedback models we employ is critical inallowing us to perform cheat detection and in allowing us to communicate the pres-ence of a cheater, so that she can be punished. Although we are not aware of lowerbound results that explicitly rule this out, we suspect that the ternary feedbackmechanism used by Fiat et al. [8] is not rich enough to allow such communication.

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2 Definitions

Game structure Let N = 1,2, . . . ,n be the set of players in the game. Ev-ery player holds a single packet of information that she wants to send through acommon channel, and all players (and all packets) are present at the start of theprotocol. We assume that time is discretized, divided into slots t = 1,2, . . .. Atany given slot t, a pending player i (one who has not yet successfully transmitted)has two available pure strategies: She can either try to transmit her packet in thatslot or stay quiet. We represent the action of a player i by an indicator variableXit that takes the value 1 if player i transmitted at time t, and 0 otherwise. Thetransmission vector Xt at time t is represented by Xt = (X1t ,X2t , . . . ,Xnt), whilethe number of attempted transmissions at time t is denoted by Yt = ∑

ni=1 Xit . The

transmission sequence X t is the sequence of transmission vectors of time up to t:X t = (X1,X2, . . . ,Xt). In a (mixed) strategy pi, a player transmits at time t withprobability pit = Pr[Xit = 1]. If exactly one player transmits in a given slot, wesay that the transmission is successful, and the player who transmitted leaves thegame;3 the game then continues with the rest of the players. If two or more playersattempt to transmit at the same slot, then they all fail and remain in the game. Thegame ends when all players have successfully transmitted.

Feedback At a time step t, the actual transmission vector is Xt . However, in-dividual players do not have access to this complete information. Instead, afterplaying at step t, each player i receives some feedback Iit that is a function of Xt .The feedback is symmetric, i.e., every player who receives feedback gets the sameinformation: I jt = Ikt = It . At any time step t, a player’s selected action is a (ran-domized) function of the entire sequence of actions she has taken so far and thehistory Hit = (Ii1, . . . , Ii(t−1)) of the feedback that she has received from the chan-nel. For convenience, we define a player i’s personal history hit = (Xi1,Xi2, . . . ,Xit)as the sequence of her actions.

We distinguish between two different feedback structures: (1) Perfect informa-tion: After each time step t, all players receive as feedback the identities of theplayers who attempted a transmission, i.e., Iit = Xt ,∀i ∈ N. However, there are noshared global ids for the players; each player potentially has a different local or-dering on the player set, so, for example, the concept of the “lexicographically firstplayer” has no common meaning. If, for example, the same player elected to trans-mit in two adjacent slots, all other players would be aware of this, but one of heropponents might internally consider her “player 1”, while another might considerher “player 327”. (2) Collision Multiplicity Feedback (CMF): After each time step

3Alternatively, we can assume that player transmits with pit = 0 on all subsequent rounds.

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t, all players receive as feedback the cardinality of the set of players who attempteda transmission, but do not find out their identities. Here, Iit =Yt = ∑

nj=1 X jt ,∀i∈N.

In this setting, the feedback does not allow one to distinguish two adjacent trans-missions by the same player from two adjacent transmissions by different players.

Transmission protocols We define fit , a decision rule for player i at time t, asa function that maps a pair (hi(t−1),Hi(t−1)) to a probability pit . A protocol fi

for player i is simply a sequence of decision rules fi = fi1, fi2, · · · . A protocolis symmetric or anonymous and is denoted by f = f1 = . . . = fn iff the decisionrule assigns the same probabilities to all players with the same personal history. Inother words, if hit = h jt for two players i 6= j, it holds that fi,t+1(hit ,Hit) = pi(t+1) =

p j(t+1) = f j,t+1(h jt ,H jt).4

Individual utility Given a transmission sequence XT wherein all players even-tually transmit successfully, define the latency or success time Si of agent i asargmint(Xit = 1,X jt = 0, ∀ j 6= i). The cost to player i is the sum of the costsfor the time-to-success and the transmission costs: Ci(XT ) = Si+c∑t≤Si Xit . Givena transmission sequence XT of actions so far, a decision rule f induces a proba-bility distribution over sequences of further transmissions. In that case, we writeE f

i (XT ) for the expected total cost incurred by a sequence of transmissions that

starts with XT and then continues based on f . In particular, for X0 := /0, E fi (X

0) isthe expected cost of the sequence induced by f .

Equilibria The objective of every player is to minimize her expected cost. Wesay that a protocol f is in equilibrium if for any transmission sequence X t theplayers cannot decrease their cost by unilaterally deviating; that is, for all playersi, E f

i (Xt) ≤ E( f ′i , f−i)

i (X t), for all f ′i , t. Similarly, we say that a protocol f is in an

ε-equilibrium if for any transmission history X t , E fi (X

t)≤ (1+ε)E( f ′i , f−i)i (X t), for

all f ′i , t. Notice that this is a very strong equilibrium requirement, in that there mustnot ever be any time step in which any player has significant incentive to deviate,even under low-probability transmission sequences.

Social utility We are interested in providing the players with a symmetric pro-tocol that is in equilibrium (or ε-equilibrium, for some small ε > 0) such that theaverage expected cost E f

i (X0) of any player i is low.5

4Notice that since the feedback is symmetric, hit = h jt implies Hit = H jt .5Since we are interested in symmetric protocols, then the average expected cost is equal to the

expected cost of any player, and hence the social utility coincides with the individual utility.

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3 Perfect Information

In this section we consider the perfect information setting, where in every timeslot t, every user i receives feedback Iit = Xt , the exact vector of transmissions int. It is important to note that although in this setting each player can distinguishbetween her opponents, we do not assume that the players have globally visibleid numbers.6 Instead, we assume that each player has a set of local id numbersto distinguish among her opponents, but one player’s local labeling of all otherplayers may be independent of another’s. Unlike global ids, local ids do not posevery hard implementation constraints; for instance, in sensor networks or ad-hocmobile wireless networks, users may be able to derive local ids for the other playersbased on relative topological position.

The main idea of the protocol we present here is to generate unique, randomids for each player based on her transmission history. Next, with the use of thoserandom ids, the players are synchronized in a fair manner, and can exit the gamewithin short time.

As mentioned earlier, this section is presented with transmission costs c assome negligible, nonzero ε , rather than treating general nonzero costs. This greatlysimplifies the presentation of this section and allows us to focus on the main ideasthat the more complicated protocol of Section 4 is based on. No such assumptionon c will be made when describing the protocol for the more restricted (and thusmore challenging) feedback model. Further, in Section 4.2 we discuss how nonnegligible transmission costs can be incorporated into this protocol as well.

3.1 Protocol PERFECT

Protocol PERFECT works in rounds. In a round k there is exactly one split slotand some `k ≥ 0 leave slots. In a split slot, all pending players (i.e., the playersthat are still in the game) transmit with probability 1/2, independently of eachother and their personal history. The purpose of these split slots is to aid in thegeneration of unique ids for each player. We define the id idi(k) of player i, afterthe kth split slot has taken place, as a binary string of length k that represents thetransmission history of player i only on the split slots. When, after some round k,a player i manages to obtain a unique id, i.e., idi(k) 6= id j(k), for all i 6= j, she willbe assigned a leave slot. During a leave slot a single prescribed player transmitswith probability 1, while all other players stay quiet. Such a player has a successfultransmission and exits the game.

6In fact, if the identities of the players were common knowledge, then simply transmitting inlexicographic order would be incentive-compatible and achievable.

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The protocol is given as Protocol 3.1. We will now describe what happens in around in more detail. Consider the beginning of round k+ 1, for k ≥ 0. The firstslot s is a split slot. Let nT be the total number of players that transmitted in s.Every player observes the actions of all other players in slot s, and updates theirids: The id of player j is updated by appending “1” or “0” at the end of id j(k),depending on whether j transmitted in s or not. If there are players that obtainedunique ids within slot s then they must be assigned leave slots. Let Uk+1 be the setcontaining those players. The order in which those players will be assigned leaveslots depends on the number nT of players that transmitted in s: The players inUk+1 are assigned leave slots in order of increasing id if nT is even, and in order ofdecreasing id otherwise. All players in Uk+1 will transmit successfully and exit thegame.7

The order in which the players in Uk+1 are assigned leave slots is not fixed,so as to avoid giving incentive to a player to strategically create an id, instead ofobtaining a random one. If the players in Uk+1 were always assigned leave slotsin order of increasing id, then players would have incentive to remain quiet in splitslots, as this would result in an id of smaller value (they append a “0” to their id)than if they transmitted (they would append a “1” to their id). To avoid this, theprotocol prescribes the order to be increasing or decreasing with equal probability(since nT is equally likely to be even or odd); this way the players have nothing togain by choosing any specific id.

Fixing a player i, any player j (including i) is equally likely to obtain a uniqueid in any round k, regardless of whether i transmits or stays quiet during the splitslot of round k. This is because in each round every player randomizes uniformlybetween transmitting and staying quiet, independent of the transmission historyand independent of the other players. Therefore, i cannot reduce the expectednumber of rounds she needs until she obtains a unique id, nor the expected numberof players that will be assigned leave slots before she does. Also, when we assumearbitrarily small transmission cost c, player i has no incentive to deterministicallystay quiet during a split so as to save on the transmission costs she incurs. There-fore, if no player could succeed during a split slot (i.e, have a successful transmis-sion due to being the single player transmitting in a split), then the expected costof i would have been exactly the same, whether i transmits or stays quiet in a split.(Player i has no reason to deviate from the protocol in a leave round, as this canonly increase her total cost.) The possibility of a player successfully exiting duringa split by being the only one to transmit creates an imbalance between the expected

7A player i who is currently assigned a leave slot keeps transmitting until she succeeds. Thistechnical detail ensures that the protocol will not collapse if some other player j tries to transmit atthat slot. Such an event would occur only if j deviated from the protocol by transmitting in the leaveslot assigned to i.

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Protocol 3.1 PERFECTRequire: Code for player i, at the beginning of round k+1:

N′ : set of all players still in the game (pending players)id j(k) : id of player j ∈ N′ w.r.t. the first k split slots

1: Transmit in current split slot s with probability 1/22: nT := #players that transmitted in slot s (known due to perfect information

feedback)3: for j ∈ N′ do4: X j(k+1) := 1 if j transmitted in s, 0 otherwise5: id j := id j(k) ·X j(k+1) Update the id’s of all players, appending X j(k+1)6: Uk+1 := j ∈ N′ : id j 6= id j′ , for all j′ ∈ N′, j′ 6= j7: if |Uk+1| > 0 then There are players that obtained a unique id with the last

split8: if nT is even then Sort Uk+1 in increasing or decreasing order according

to the value of nT9: sort Uk+1, in increasing order of id j

10: else11: sort Uk+1, in decreasing order of id j

12: for j ∈Uk+1 do Players in Uk+1 are assigned leave slots13: if i = j then i is the player that is assigned the current leave slot14: Transmit in current slot si with probability 115: else another player is assigned the current leave slot16: Stay quiet in current slot s j with probability 1

[End of round k+1]

costs a player has per step when she transmits and when she stays quiet during thesplit. However, we will prove that the maximum gain over all possible deviationsis very small.

3.2 Analysis of Protocol PERFECT

We begin by proving a bound on the expected total cost of participating in theprotocol.

Theorem 3.1. Assuming negligible transmission costs, the expected total cost ofeach player when all players follow Protocol PERFECT is Ei(X0) ≤ (n+ 1)/2+logn.

Proof. Note that player i is expected to obtain a unique id in logn rounds, sincethe number of players that share the same id is halved (in expectation) after each

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split round. In each of these rounds, she pays one for the split slot. All playersare equally likely to be assigned the k-th leave slot, for all 1 ≤ k ≤ n, since at thebeginning of the game, all players are equally likely to obtain any particular id.A player pays one for each leave slot she experiences. The total expected cost isactually slightly lower, since there is a small (but nonzero) chance that a playermight be the only one to transmit during a split slot.

Next, we wish to bound the expected cost a player can achieve by any sequenceof deviations. First, Lemma 3.2 will bound the amount a player can gain by devi-ating from the protocol in any single slot. Notice that transmitting with probability1/2 in split slots is nearly an equilibrium strategy, due to the way the random idsare created and the fact that it is not known in advance whether the leave slots willbe assigned in increasing or decreasing order. If it were not for the possibility ofplayers exiting during a split slot, the expected costs of transmitting and of stayingquiet would be the same for every player i. However, for each split slot t, thereis some positive (but small) probability that among the n′ pending players, only asingle player transmits in t. That player not only obtains a unique id in t, but alsohas a successful transmission during the split. When this is the case, there is noneed for a leave slot to be assigned to her, as she already exits the game during thesplit, so the rest of the pending players are saved one slot of waiting. We will seethat (even given our assumption that the transmission cost is arbitrarily small) it isactually (slightly) beneficial for a player to stay quiet during split slots, hoping thatsomeone else will succeed during the split, instead of transmitting in the hope thatshe will be the unique player doing so (see Lemma 3.2 for more details). We willsee that a player minimizes her expected cost if she stays quiet with probability 1in all split slots and follows the protocol during leave slots (but the incentive todeviate in split slots is small).

Lemma 3.2. Let t be some split slot, in the beginning of which there are n′ ≥ 2pending players; let i be one of them. i cannot gain more than n′−2

2n′−1 by deviatingfrom the protocol at t.

Proof. In the following we define the group of a player j at the beginning of someslot t ′ to be the set of players—including j—that share the same id as j at thebeginning of slot t ′.

Note that the only deviation we must consider is to change the probability withwhich i will transmit during the split slot t (instead of transmitting with probability1/2). As we have already argued, such a deviation cannot decease the expectednumber of splits that i (or any other player) needs until exiting the game, nor theexpected number of players that will be assigned a leave slot before i. Rather, theonly benefit may come from the fact that such a deviation changes the probability

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that some player will exit during the split slot t. That implies that i’s deviation at thas no impact on her future expected cost, other than reducing the number of leaveslots for the current round.

We will pair up events with identical probabilities, depending on the the num-ber of players that transmit at t. We will have four cases to consider: all of the op-ponents transmit, exactly one opponent transmits, all of the opponents stay quiet,exactly one opponent stays quiet.

We first consider the case when all players other than i transmit, and note:

• if player i also transmits, the current round is essentially a no-op

• if player i stays quiet, she obtains a unique id, and exits in the next step

We can pair these events with the case when all players other than i stay quiet:

• if player i stays quiet, the current round is essentially a no-op

• if player i transmits, she obtains a unique id and exits immediately

Note that each of the two cases has probability 1/2n′−1. Thus, with probability1/2n′−1, staying quiet results in an increase in cost of one (versus transmitting),due to having to wait for a leave slot, as opposed to exiting directly during thesplit.

We now consider the other two cases. When exactly one opponent transmits, ifplayer i also transmits, neither player exits during the split and the future costs canbe paired with the situation where exactly one opponent remains quiet and i alsoremains quiet. Note that it may be the case that i, as well as other players, obtain aunique id in this round, but they will all have to wait for leave slots to be assignedto them. Thus the two situations left to consider are:

• Exactly one opponent transmits and i is quiet: The opponent exits immedi-ately. The size of all groups except the opponent’s, remain the same.

• Exactly one opponent is quiet and i transmits: The opponent obtains a uniqueid, and exits in the next slot. The size of all groups except the opponent’s,remain the same.

Note that the cases when exactly one opponent transmits or exactly one opponentis quiet each have probability (n′−1)/2n′−1. Thus, with probability (n′−1)/2n′−1,staying quiet results in an decrease in cost of one, due to the fact that the singleopponent that transmitted exited during the split slot.

Thus, i maximizes her gain by staying quiet with probability 1. Since the pro-tocol calls for uniform randomization between staying quiet and transmit, this gainis bounded by (n′−1)/2n′−1 +1/2n′−1 = (n′−2)/2n′−1.

13

We are now ready to bound the total amount that any player can gain by devi-ating.

Theorem 3.3. If all other players follow Protocol PERFECT , the total expectedgain any player would have from any sequence of deviations is no more than 1,assuming negligible transmission costs.

Proof. Consider any given transmission sequence X t , for some time slot t, and anyplayer i who is still pending at t. Since a player can only benefit by deviating duringsplit slots and any such deviation has no impact on the expected number of splitsthat any player (including i) will need until obtaining a unique id, we only need toconsider the expected gain for each split slot separately.

Let N′ be the set of pending players at the beginning of t, with |N′| = n′ andlet ni be the size of i’s group at t. (Again we refer to the set of players—includingi—that share the same id as i at the beginning of slot t as i’s group.)

Note that every time that player i gains by deviating from the protocol at asplit slot, there must be a successful transmission by some player i′ 6= i during thatsplit. Therefore, every time this happens the number of pending players reducesby at least one.8 Also, note that the expected gain from deviating is a decreasingfunction in the number n′ of pending players (by Lemma 3.2). Therefore, if ineeds exactly k rounds until she exits the game, and gi(m) is her expected gain bydeviating in a round where there are m≤ n′ remaining players, then her maximumtotal expected gain by deviating is bounded by G(k) = ∑

k+1j=2 gi( j). (Note that it

i can never be the only player remaining at the beginning of a round; she wouldhave obtained a unique id in the previous round, and thus would already haveexited the game.) Using Lemma 3.2, the above quantity becomes G(k) = ∑

k+1j=2( j−

2)2−( j−1) = 2−k(2k− k−1

).

Finally note that the probability that i needs exactly k additional rounds untilshe obtains a unique id (when initially she belongs to a group of ni players) isPi(k) =

(1−2−k

)ni−1−(1−2−(k−1)

)ni−1. Therefore, the total expected gain of i

by deviating in all remaining split slots is bounded byn

∑k=1

Pi(k) ·G(k) =n

∑k=1

2−k((

1−2−k)ni−1

−(

1−2−k+1)ni−1

)(2k− k−1

)≤

n

∑k=1

((1−2−k

)ni−1−(

1−2−k+1)ni−1

)=(1−2−n)ni−1

≤ 1,

8It may reduce by two if the player that succeeded during the split belonged to a group of size 2.The other player in that group would be assigned a leave slot in the same round.

14

which completes the proof.

Remark 3.4. Note that Protocol PERFECT is technically not at o(1)-equilibrium,despite players having incentive < 1 to deviate, as in some low probability events,a player could reach a point where her expected cost is only a constant. Thiscan easily be remedied by enforcing a small minimum cost on all players, thatincreases (arbitrarily slowly) with n. Enforcement of a minimum cost is illustratedin the protocol in the following section.

4 Collision Multiplicity Feedback

In this section, we present the main result of the paper, Protocol MULTIPLICITY.This protocol works under the CMF channel model, in which, after each slot t,all players are informed about the number of attempted transmissions, i.e., Yt =

∑i∈N Xit . This information can be provided by a number of energy detectors, and itis broadcast to all users. CMF is a well-studied and important feedback model ininformation theory, with many nice theoretical results (see e.g. [25, 20, 9, 23]).

Here we give an overview of the main ideas of Protocol MULTIPLICITY and wewill describe it in detail in Section 4.1. Protocol PERFECT presented in Section 3.1crucially relies on users knowing how each player acts at every time step, andthus does not work under this feedback model. Instead, we design a differentprotocol that overcomes this difficulty and has expected average cost Θ(n+c logn).Recall that in Protocol PERFECT the id of a player is a string denoting all therandom choices the player had to make during split slots. For every time t, theplayers are partitioned into groups according to their ids, i.e., all the players in thesame group share the same id. Moreover, all players perform a split at the sametime, regardless of the group they belong to. The protocol MULTIPLICITY that wepresent here, again uses the idea of randomly generated ids, but here, each groupsplits separately, in a sequential manner. Each split is followed by a validation slot,whose aim is to verify that the split has been performed “correctly”.

In the CMF model, the players have an incomplete view of history. This mightgive incentive to some players to “cheat” and pretend to belong to a different group(of smaller size), in order to exit the game sooner. We discourage such a behaviorusing a “threat”: if any deviation from the protocol is observed, then all playersswitch their transmission strategy according to a costly equilibrium strategy. Fi-nally, a cheat-detecting mechanism is needed. The validation slots accomplishexactly this task; the cheating is noted by all players if some validation step fails.In that case, all the players get punished by switching to an exponential-cost time-independent equilibrium protocol (punishment protocol):

15

The punishment protocol Fiat, Mansour and Nadav [8] show that, for trans-mission cost c = 0 and k pending players, the time-independent symmetric proto-col where every player transmits with probability p = Θ(1/

√k) is in equilibrium.

It gives average packet latency eΘ(√

k), and thus switching to this protocol can beused as a high-cost punishment for detected defections. For our punishment pro-tocol, we adapt this protocol to the general case of transmission cost c ≥ 0: thetransmission probability becomes pc = Θ(1/

√k(1+ c)) and the expected cost is

(1+ c)eΘ(√

k/(1+c)).9

4.1 Protocol MULTIPLICITY

The protocol works in rounds. Every round consists of a sequence of split andvalidation slots such that each player participates in exactly one split slot in eachround. We say that a player participates in a split slot s, if the protocol gives theplayer nonzero transmission probability in s.10

Assume that at the end of round k ≥ 0, there are Mk different groups (eachgroup consists of players that have the same transmission history with respect tothe split slots they have participated in).11 Let G j,k be the jth group. The playersdo not know which players belong to group G j,k, but they will be aware of the size|G j,k| of the group.

Consider round k+1. At the beginning of the round, the players decide, using(as in the perfect information setting) the parity of the total number xk of playersthat transmitted during all the split slots of round k, whether groups will split inincreasing or decreasing order of ids. According to this order, every group willperform a split, followed by a validation. Let G j,k be the current group performinga split and validation.

Split slot When it is group G j,k’s turn, all players that belong to this group (andonly these players) will transmit with probability 1/2. All players (regardless ofwhether they belong in G j,k or not) note the number nT, j of the members of G j,kthat transmitted in this slot. These nT, j players will form one of the new subgroupsof G j,k.

Validation slot The immediately next slot is a validation slot. All players of G j,kthat transmitted in the previous split slot must now stay quiet, while those that did

9More details can be found in the full-version of [8], atwww.cs.tau.ac.il/˜urinadav/papers/JournalSelfishMac.pdf

10As opposed to a player that the protocol instructs to stay quiet during s with probability 1.11At round 0 all players belong to the same group, i.e., M0 = 1.

16

not, must now transmit. This second set of players will form the other subgroup ofG j,k. Again all players can see the number nQ, j of players that transmitted duringthis slot.

Right after the validation step has happened (and before the next group’s turncomes), all players check if the members of G j,k were properly split into the twonew subgroups, by checking that the sum of the sizes of the two subgroups equals|G j,k|. If that is true, this group is properly divided, and the next group will split.

If the check failed, then there is some player that deviated from the protocol:Either one of the members of G j,k did not transmit in any of the split or validationslots of group G j,k (or even transmitted in both slots); or, some player that did notbelong in G j,k transmitted in either of these two slots. All players note thereforethe fact that someone has deviated, and they all switch to the punishment protocol.

We note that since now each group splits separately, there is no longer the needfor explicit leave-slots. A user that obtains a unique id in round k will transmitsuccessfully when her group G j,k performs a split if she is the only one to transmit(i.e., if she was the only member of G j,k to append a ’1’ to her id); or she willtransmit successfully when her group performs the validation step if she was theonly from G j,k that stayed quiet in the split (i.e., she was the only member of G j,kto append a ‘0’ to her id).

If all players of a group G j,k transmitted during their corresponding split, thevalidation that normally follows will be skipped. The reason is that if all players ofG j,k transmitted in the split, no one would transmit in the validation slot. All play-ers would know that this slot should be empty, which would provide an incentiveto “attack”, i.e., transmit during that slot, disobeying the protocol.

Finally, for technical reasons we discuss in the analysis, if before log logn timesteps have elapsed, there is some group G j,k that splits in a way such that either ofits subgroups has size less than logn, then that particular split will be canceled. Inthat case, the protocol continues as usual, and group G j,k retains in round k+1 thesize it had in round k.

4.2 Analysis

Theorem 4.1. Each player’s expected cost when all players follow Protocol MUL-TIPLICITY is Θ(n+ c logn).

Proof. We will postpone considering the condition of Line 24. The expected num-ber of split-slots required for a player to obtain a unique id is logn. Consider around k. Every player will transmit exactly once during this round (either in thesplit slot corresponding to her group, or in the validation slot immediately follow-ing that split). If Mk−1 is the number of groups that have formed by the end of round

17

k− 1, then the duration of round k is at most 2Mk−1 slots, where Mk−1 ≤ 2k−1.Therefore the expected cost for i is bounded by 2n + c logn. Noting now alsothat the condition of Line 24 may only introduce a further delay of log logn stepsand half as many transmissions for i, we obtain the bound on the total expectedcost.

For the following, note that the o(1) notation refers to the total number ofplayers, i.e., the number of players n that were present at the beginning of the game,and not the number of pending players at any given time t during the execution ofthe protocol.

Theorem 4.2. Protocol MULTIPLICITY is a o(1)-equilibrium.

Proof. We consider the various ways in which a player might deviate from theprotocol. If a player belonging to group G j,k, for some j,k, did not transmit at thesplit slot of G j,k, nor at the corresponding validation slot, then the sum nT, j +nQ, j

will be found less than |G j,k|. This will make all players switch to the (high cost)punishment protocol. Similarly, if a player transmits during the validation slotof group G j,k when she was not supposed to, then nT, j + nQ, j > |G j,k|. We willconsider the case of a player manipulating her transmission probability away from1/2 during her group’s split slot below.

The remaining case is when a player i /∈ G j,k cheats by transmitting duringthe split slot of group G j,k.12 Assume that i cheats when there are n′ ≤ n pendingplayers in total. She will get caught with probability at least 1/4 (if G j,k is ofminimum size, i.e. 2).13 In that case she will have expected future cost (1 +

c)eΘ(√

n′/(1+c)), worse than the corresponding cost of following the protocol, i.e.,O(n′+ c logn′). Therefore, the expected cost of i becomes larger if she deviatesinstead of following the protocol.

We now observe that a player cannot gain by deterministically choosing to stayquiet during a split for her group. She does not save on the transmission cost c, asshe would have to transmit during the validation slot anyway. Moreover, stayingquiet or transmitting in the split does not affect the number of rounds a player mustwait until she obtains a unique id: Consider a player i belonging to some groupG j,k. All other players from G j,k transmit during the corresponding split slot with

12Validation slots only happen if there is at least one player from G j,k that is supposed to transmitin them. Thus, a player cannot have a successful transmission by attacking a validation slot.

13If none of the members of G j,k transmitted during the split slot, then i has a successful trans-mission. If all members but one transmitted, then the cheating does not get caught (and i only hada failed transmission). In this case, it looks as if all members of G j,k transmitted, and the validationslot is skipped. In all other cases, i gets caught.

18

probability 1/2; the expected sizes of the two new groups to be formed are thesame and i has exactly the same probability of obtaining a unique id, whether shetransmits in the split, or stays quiet.

On the other hand, splits always happen before the corresponding validations.If i obtains a unique id during round k, then she exits the game earlier if she wasthe only player of her group transmitting at the split, than if she were the onlyquiet player. This implies that “always transmit” gives a smaller expected (future)cost. It is therefore the optimal strategy. Nevertheless, if pτ is the probability for aplayer that transmits during split slots to succeed at the time step τ , then pτ is alsothe probability for a player that stays quiet during split slots to succeed at the timestep τ + 1 (since validation slots occur immediately after split slots). Let X t beany transmission history, and suppose that t +1 is a split slot that i participates in(otherwise i’s optimal strategy is to behave according to the protocol). Let A(X t) =t + c∑τ≤t Xiτ be the actual cost incurred by i until time t, and let T (X t),Q(X t) bethe total expected cost of player i if she always transmits, stays quiet, respectively,in all future split slots she participates in.

Then, the above discussion implies that Q(X t) = T (X t)+ 1. This could cre-ate strong incentive to deviate in the low-probability event that very small groupswere generated very early in the protocol. Line 24 in the code of Protocol MUL-TIPLICITY, handles such a situation—either the actual cost already incurred by i isA(X t)≥ log logn, or i belongs in a group of size at least logn, implying an expectedfuture cost at least log logn (i.e., the expected number of splits required until sheobtains a unique id). Therefore, in all cases T (X t)≥ log logn, and we obtain

Q(X t)

T (X t)=

T (X t)+1T (X t)

= 1+1

T (X t)≤ 1+

1loglogn

= 1+o(1)

Q(X t) is an upper bound on the total expected cost of the protocol: Stayingquiet in split slots is always worse than transmitting; according to the protocol aplayer stays quiet only in half (in expectation) of the future split slots she partic-ipates in. Thus, at any time during the execution of the protocol, player i willnot benefit by deviating from the protocol by more than o(1) (with respect to thetotal number of players n), which means that Protocol MULTIPLICITY is a o(1)-equilibrium.

Protocol PERFECT with non-negligible transmission cost Going back to theperfect information protocol, we can now easily adapt protocol PERFECT for thecase that c is non-negligible. Note that the main issue that arises in this case isthe fact that, unlike in the case of negligible c, a player now has an incentive notto transmit during a split in order to avoid the transmission costs. All we have to

19

do to address this, is to add a validation slot after each split (all players that donot transmit during the split, must transmit during the validation). The punishmentprotocol is used again to force players to transmit in exactly one of the split andvalidation slots of each round. As in Protocol MULTIPLICITY, this way a playercannot decrease her transmission costs by staying quiet in a split. The analysisbecomes similar to the analysis of Protocol MULTIPLICITY and we obtain a o(1)-equilibrium.

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Protocol 4.1 MULTIPLICITYRequire: Code for player i, at the beginning of round k+1:

G1,k,G2,k, . . . ,GMk,k groups formed by the end of round k in order of in-creasing id

indexi index of the group of player i in the sorted sequenceof groups at the end of round k

xk total #players that transmitted in all split slots ofround k.

1: for j in increasing order if xk is even, but in decreasing order if xk is odd, do2: if j 6= indexi then3: Stay quiet in current split slot s Group G j,k performs a split4: nT, j := #players that transmitted in slot s5: if nT, j 6= |G j,k| then The validation only happens if not all players

of G j,k transmitted6: Stay quiet in current validation slot s′ Group G j,k validates the

number of its members that stayed quiet in the previous slot7: nQ, j := #players that transmitted in slot s′

8: else nT, j = |G j,k| holds9: nQ, j := 0

10: else Now is the turn of i’s group11: Transmit in current split slot s with probability 1/212: nT,indexi := #players that transmitted in slot s13: if nT,indexi 6= |Gindexi,k| then The validation only happens if not all

players of group Gindexi,k transmitted14: if i transmitted in s then15: Stay quiet in current validation slot s′

16: else17: Transmit in current validation slot s′

18: nQ,indexi := #players that transmitted in slot s′

19: else nT,indexi = |Gindexi,k| holds20: nQ,indexi := 021: if nT, j +nQ, j 6= |G j,k| then A player tried to “cheat”22: Switch to the punishment protocol23: if total time since begining of the game < log logn then24: if 1 < nT, j < log logn or 1 < nQ, j < log logn then one of the sub-

groups is too small25: Cancel the split of G j,k

[End of round k+1]

23