936 ieee/acm transactions on networking, vol...

936 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 3, JUNE 2009

An Analysis of Generalized Slotted-Aloha ProtocolsRichard T. B. Ma, Student Member, IEEE, Vishal Misra, Member, IEEE, and Dan Rubenstein, Member, IEEE

Abstract—Aloha and its slotted variation are commonly de-ployed Medium Access Control (MAC) protocols in environmentswhere multiple transmitting devices compete for a medium, yetmay have difficulty sensing each other’s presence (the ‘‘hiddenterminal problem’’). Competing 802.11 gateways, as well as mostmodern digital cellular systems, like GSM, are examples. Thispaper models and evaluates the throughput that can be achieved ina system where nodes compete for bandwidth using a generalizedversion of slotted-Aloha protocols. The protocol is implementedas a two-state system, where the probability that a node transmitsin a given slot depends on whether the node’s prior transmissionattempt was successful. Using Markov Models, we evaluate thechannel utilization and fairness of this class of protocols fora variety of node objectives, including maximizing aggregatethroughput of the channel, each node selfishly maximizing its ownthroughput, and attacker nodes attempting to jam the channel. Ifall nodes are selfish and strategically attempt to maximize theirown throughput, a situation similar to the traditional Prisoner’sDilemma arises. Our results reveal that under heavy loads, agreedy strategy reduces the utilization, and that attackers cannotdo much better than attacking during randomly selected slots.

Index Terms—MAC protocols, Markovian decision, Prisoner’sDilemma, short-term fairness, slotted-Aloha, Stackelberg Game.

I. INTRODUCTION

I N MANY communication networks, the communicationmedium is often shared by multiple users who must com-

pete for access. In Ethernet [1], nodes use CSMA/CD [2], [3] asa MAC protocol. In order to reduce the probability of collisions,each node implements CSMA/CD, sensing the medium to en-sure the medium is available prior to transmitting. However, forwireless ad hoc networks or sensor networks, carrier sensingmay not be effective. This is because nodes may not be able tosense one another’s presence, yet their transmissions may stillinterfere. Ad hoc networks, sensor networks, and competing“hotspot” 802.11 gateways are examples where this so-called“hidden terminal problem” occurs.

The Aloha protocol [4] is a fully decentralized medium accesscontrol protocol that does not perform carrier sensing. The sub-sequent slotted-Aloha protocol [5] was introduced to improvethe utilization of the shared medium by synchronizing the trans-mission of devices within time-slots. Today, various forms ofslotted-Aloha protocols are widely used in most of the current

Manuscript received February 25, 2007; revised October 30, 2007 and Feb-ruary 25, 2008; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Ed-itor A. Kumar. First published June 20, 2008; current version published June17, 2009.

R. T. B. Ma and D. Rubenstein are with the Department of ElectricalEngineering, Columbia University, New York, NY 10027-6623USA (e-mail:[email protected]; [email protected]).

V. Misra is with the Department of Computer Science, Columbia University,New York, NY 10023-6367 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNET.2008.925633

digital cellular networks, such as the Global System for Mobilecommunications (GSM).1

In this work, we consider a generalization of theslotted-Aloha protocol. Like slotted-Aloha, the decision totransmit within a slot has a random component. However, intraditional slotted-Aloha, the user continues transmission insubsequent slots until a collision occurs. In our generalizedversion, the user may cease transmitting with some fixed(non-zero) probability. We model a system of users im-plementing this generalized protocol with tunable parametersvia Markov Models that allow us to measure the rate at whichnodes attempt to transmit packets (cost), and their rates of suc-cess (throughput). In parts, we impose budget constraints thatrestrict the nodes’ costs, such that the fraction of slots withinwhich a node attempts transmissions is bounded. In practice,these additional constraints may be due to energy constraints,or bandwidth limitations.

This generalized version of slotted-Aloha is worth studyingfor several reasons. First, it is derived from a protocol that iscommonly used today. Second, introducing additional states tocapture backlog or number of successive collisions more closelyemulates binary exponential backoff protocols, e.g., the 802.11family of MAC protocols; however, this generalized version ofslotted-Aloha retains the simplicity and elegance of the originalAloha approach. Third, we will show that the generalized ver-sions can outperform the original version, both in terms of ag-gregate throughput, as well as the ability to cope with malicioususers. Fourth, by using this generalized slotted-Aloha protocol,we provide a framework to study the user behaviors in cooper-ative, competitive and adversarial environments.

We begin by exploring an environment where users co-operate and set the protocol parameters to maximize the totalsystem throughput while sharing the bandwidth evenly. We findthat the throughput is bounded by and that toachieve this utilization, users who gain access to the channelmust transmit over a large number of consecutive slots. We thenexplore how throughput decreases as “short-term fairness” ismore strictly enforced, reducing the expected number of con-secutive slots.

Next, we consider selfish users who wish to maximize theirown throughput, perhaps at the expense of the nodes againstwhom they compete for medium access. We fist identify a Nashequilibrium for short-sighted selfish users where the aggregatethroughput is zero. We then formulate a Stackelberg game,where a leader node strategically chooses its parameters and afollower node subsequently modifies its parameters in responseto short-sightedly maximize its own throughput. Under thismodel, we find that performance of the protocol depends onnodes’ budgets, and takes on three distinct types of behavior.When nodes’ budgets are low, the aggressive strategy is op-timal. When nodes’ budgets fall within a medium range, all

1In the GSM network, the control channels of the TDM channels use slotted-Aloha.

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MA et al.: AN ANALYSIS OF GENERALIZED SLOTTED-ALOHA PROTOCOLS 937

nodes achieve the same throughput in a unique equilibrium,but the throughput is less than what would be obtained in a co-operative setting. When nodes’ budgets fall within the highestrange, the leader node obtains much higher throughput thanthe follower node. We further model two strategic selfish nodesusing a simultaneous move game, where multiple equilibriaexhibit that the Prisoner’s Dilemma can occur. We develop anadditional enhancement to the protocol that can be implementedby cooperative nodes which will encourage selfish users to tunetheir protocol parameters to match those of a cooperative node,maximizing the aggregate throughput.

Last, we consider an attacking node that, with a limitedbudget, seeks to minimize the throughput of the other nodes inthe system. We show that when the attacker’s budget is small,selecting random slots (i.e., via a Bernoulli process) is optimal.When the budget is large, the optimal strategy is to mimic agreedy user. Our analysis provides insights on the limits ofsuccess a jammer can have in disrupting a slotted-Aloha likenetwork.

We summarize the main contributions of this paper asfollows.

1) We formulate different user behaviors under a generalizedslotted-Aloha protocol where users make transmission de-cisions using a two-state system.

2) We identify throughput bounds for a system of cooperativeusers and explore the trade-off between user throughputand short-term fairness.

3) Under selfish behavior of the users, we identify a Prisoner’sDilemma phenomenon and propose methods to detect andprevent nodes from acting selfishly without regard for othernodes’ throughput.

4) Under adversarial behavior of one user, we measure themaximum possible deterioration of the system and try tounderstand the behavior of an attacker.

We organize our paper as follows. In Section II, we reviewrelated work. In Section III, we motivate the protocol and con-struct a Markov Model for the generalized slotted-Aloha pro-tocol. In Section IV, we measure the system throughput in a co-operative environment where users want to maximize the totalthroughput of the system. In Sections V and VI, we evaluateboth the aggregate and individual user throughput where selfishusers exist in the system. We formulate a Stackelberg game [6]and identify a Prisoner’s Dilemma situation in Section V, and inSection VI present strategies that cooperative nodes can imple-ment to detect and prevent selfish user behaviors. In Section VII,we explore a system in which an attacker tries to minimize thethroughput of the remaining nodes. Section VIII concludes.

II. RELATED WORK

The Aloha protocol and its slotted version have been studiedfor decades. Because slotted-Aloha exhibits an instability as thenumber of transmitting nodes increases [7]–[12], early researchfocused on stabilization [12], [13]. For instance, Rivest [13] pro-posed a pseudo-Bayesian algorithm, which utilizes feedback toestimate the number of current backlogged nodes in the system.Subsequent studies utilized dynamic controls [9], [14] to sta-bilize the systems. However, today’s networks avoid stabiliza-tion challenges by implementing an admission control proce-dure that limits the number of simultaneous users in the system.

In this work, we focus on the performance of stable slotted-Aloha type systems, where only a finite number of users will ac-cess the shared medium simultaneously. Early work on slottedAloha with finite number of users can be found in [7]. Notonly does our work evaluate the throughput bounds for a finiteslotted-Aloha type system, but it also considers the performanceof individual users under different types of user behavior. Manyprior works [15], [16] show that users always have incentive notto follow the designed protocol (i.e., not backoff) in order toachieve higher throughput. Consequently, game-theoretic anal-ysis can be applied [6], [17].

Recent work using Game Theory to analyze user behavior inMAC protocols and wireless ad hoc networks can be found in[18]–[21]. More specifically, game-theoretical analysis of theAloha protocols can be found in [22]–[27].

MakKenzie and Wicker’s work [24], [25] discussed the sta-bility of slotted-Aloha with selfish user behavior and perfect in-formation. Our work is different in three ways. First, we focuson performance (attainable throughput) instead of stability. Interms of data backlog at the users, we consider scenarios ofelastic transfers, where users always have data to send and uti-lize whatever bandwidth is available, and hence classical sta-bility results do not apply to our analysis. Second, we assumethat nodes do not know the number of transmitting nodes a par-ticular slot a priori, and know only whether or not their trans-mission succeeded after the fact. Third, we consider cooperativeand attacking strategies in addition to selfish strategies.

Jin and Kesidis’s work [26] discusses the equilibria of a non-cooperative game for Aloha protocols. In their non-cooperativegame formulation, each user only uses one transmitting proba-bility (i.e., always in a backlogged state). Moreover, utility func-tions and payments are specified for each user. Our work, onthe other hand, formulates generalized slotted-Aloha protocolthat considers the Markovian decisions depending on whetherthe most recent transmission is a success (in a Free State) or afailure (in a Backlogged State). Also, we do not impose pay-ments.

Altman et al. [27] analyze slotted-Aloha systems as both co-operative and non-cooperative games with partial information.Their work assumes that there are a finite number of buffer-less sources. The arrival of packets to each source follows aBernoulli process. As in typical slotted-Aloha, users only con-trol the backlog probability in both systems. In our work, we as-sume saturated arrivals (elastic transfer) where each user alwayshas packets to transmit. Our users’ strategies are more broad,because users are permitted to choose a non-zero probability toback off even their previous transmissions are successful. In ad-dition, we analyze an adversarial game where an attacker wantsto minimize other users’ throughput.

III. PROTOCOL DESCRIPTION AND MODEL

In this section, we describe a generalized slotted-AlohaMAC protocol and construct a Markov Model from which itsthroughput can be analyzed. We first overview the originalslotted-Aloha protocol:

1) Time is divided into slots, and each node can attempt tosend one packet in each time-slot.



2) If a node has a new packet to send, it attempts transmissionduring the next time-slot.

3) If a node successfully transmits one packet, it can transmita new packet in the next time-slot.

4) If a node detects a collision, it retransmits the old packetin each subsequent time-slot with a pre-determined prob-ability, , until the packet is successfully transmitted. Re-turning to step 3 after a successful transmission.

The slotted-Aloha protocol described above can be imple-mented as a two-state system, where the state maintains theoutcome of the previously attempted transmission. A node isin its Free State if the most recent transmission from that nodeis a success; otherwise, the node is in its Backlogged State.In the Free State, a node transmits during the next slot withprobability 1, and in the Backlogged State, it transmits duringthe next slot with probability . Our generalization of the aboveprotocol is to allow a node to vary the probability with which anode transmits a packet when it resides within the Free State.Later, we will see that this generalization enables us to modelselfish as well as malicious behaviors of users.

Our evaluation will consider a network of contentiousnodes, which always have a backlog of data packets to transfer.We assume that data packets are fragmented into lengths whichcan be transmitted within a time-slot, and that nodes are able tocoordinate slot transmission times and can estimate the numberof nodes with which they compete for bandwidth. How-ever, because nodes’ transmissions may interfere but cannotbe recovered, methods to prevent slot contention that requireexplicit communication and coordination among the competingmembers (e.g., TDMA, RTS-CTS) cannot be used. In practice,many real systems implement admission control mechanismswhich constraint the number of users in the system. In practice,a system can often rely on a centralized authority to broadcastthe number of competing users (e.g., this information can beembedded in beacon messages).

Each node can tune its protocol using two parameters: ,the transmitting probability in the Free State, and , the trans-mitting probability in the Backlogged State. Given and thetransmitting probabilities for each of the nodes in each of thestates, we can compute the following performance metrics:

• , the throughput of node , which is the fraction of slotswithin which successfully completes a transmission andis the only device to attempt transmission.

• , the cost for node , which is the fraction of slots withinwhich attempts transmission (regardless of whether thetransmission fails or succeeds).

Each node’s decision to transmit within a particular slot de-pends only on the outcome of its previous attempt (success orfailure), and does not depend on the state of other nodes. Hence,this protocol can be easily implemented in a distributed manner.Moreover, each node’s decision is Markovian, as it depends onlyon its previous attempt’s outcome.

Fig. 1 shows the state transition diagram for a two-nodesystem with node and . and represent that node

is in a Free State and a Backlogged State, respectively. Asystem for nodes is easily modeled by as a Markov Modelwhere the chain would consist of states. By numbering the

Fig. 1. Two-node Markov chain.

states to be 1,2,3,4, thetransition matrix for a two-node Markov Model is

where .If , the Markov Model is positive-recurrent.

The steady state distribution is the following:

where

(1)The corresponding throughput and cost of node are

(2)

(3)

Nodes may have physical limitations (e.g. power consump-tion constraints or application throughput constraints) that maybound its cost function. We bound allowed cost by a budget, ,such that a node’s parameters must produce a cost .

When we consider cooperative nodes that seek to maximizethroughput, we are also interested in system fairness: all nodesshould get an equal share of the aggregate throughput. In ad-dition, we assume that it is undesirable for any one node to“capture” the medium for an extended number of slots—a long-term capture can be thought of as unfair over a short duration.Koksal’s work [28] gives an analysis of the short-term fairnessof MAC protocols. It provides some insight into why MAC pro-tocols exhibit bad short-term fairness using two different fair-ness indexes. In this paper, we measure short-term fairness viaa more fundamental quantity defined as the following.

Definition 1: Let be the number of consecutive slots fol-lowing an initially successful transmission over which node



successfully transmits packets (i.e., if there are successful con-secutive transmissions, then ). The system is saidto be -short-term fair to all nodes if for allnodes .

Remark: By definition, if a system is -short-term fair, it isalso -short-term fair for any . If short-term fairnessis enforced, then a node cannot “capture” the channel for anexcessive amount of time, i.e., that the medium is shared moreevenly on smaller time scales.

IV. COOPERATIVE PERFORMANCE ANALYSIS

In this section, we assume that nodes cooperate to fairly (i.e.,equally) share the available bandwidth and to maximize the ag-gregate system throughput. By doing so, each node achieves themaximum throughput possible in a fair allocation when limitedto protocols that cannot sense the wireless medium. Clearly, if itwere permissible to bias the allocation toward one of the nodes,the system could achieve full utilization by allowing only onenode to always transmit. If a centralized scheduler or carriersensing mechanism were permitted, we could also make fairshare of the medium with almost 100% utilization. Here, weseek an unbiased and distributed solution for all nodes such thatnodes will achieve the same average throughputs.

Our goal in this section is to answer the following questions:1) What are the values of and for each node that

maximize the total throughput of the system?2) What is this maximum achievable throughput of the

system?3) What is the short-term fairness of the optimal allocation,

and how can that short-term fairness be improved?Theorem 1: For two homogeneous nodes with

and , .Proof: Substitute all the transmitting probabilities with

and into (1) and (2), we have

and

where

When , andas . By symmetry, as .

Next, we want to show for all . It isequivalent to show the following:

Fig. 2. � -node Markov Chain with {� � �, � � �}.

We define .Two boundary conditions are and

. Since is a cubic function of , it is sufficientto show that the local maximum is less than zero, so as to provethat for any , . At the local maximum,

Using the above condition, it is equivalent to show

The maximum of the above function is

The denominator is negative, while the numerator is positivebecause

Finally, because the local maximum , we concludethat for all .

Theorem 1 upper-bounds the maximum fair throughput at2/3, which is achieved in the limit as both nodes choose { ,

}. This solution is intuitive: collisions are less likely tooccur in a carrier-sense free environment when nodes are veryunlikely to start transmitting, but hold the medium until a sub-sequent collision.

Theorem 2: For homogeneous nodes with , ,the total throughput approaches .

Proof: Consider in each time-slot, the whole system is incertain state. We aggregate all the system states into the fol-lowing two states. One state is the “Busy” state where only oneof the nodes is transmitting in the time-slot. The other state is the“Idle or Collision” state where no node or more than one nodeare transmitting in the time-slot. The state transition diagram isshown in Fig. 2.

We define the transition probabilities asand . indicates the probability that allof the backlogged nodes do not transmit. indicates the



Fig. 3. Aggregate throughput for fixed � .

probability that only one of the nodes transmits. The systemutilization becomes

Therefore, as .Intuitively, when the number of nodes increases in the system,

with higher probability, the channel is jammed with more thanone node transmitting at the same time. Consequently, theaggregate system throughput decreases. However, Theorem 2shows that the throughput does not drop to zero: even when thenumber of nodes tends to infinity, the aggregate throughput re-mains larger than one half. Note that this result differs from thetraditional performance bound of slotted-Aloha becauseour generalized model permits the capture of the resource. Thisallows a node to use the channel for very long but bounded

intervals (given a fixed and non-zero value of ) of slots whileall other nodes back off. An alternative analysis of this capturephenomenon can be found in [7].

Both Theorem 1 and Theorem 2 focus on the class of solu-tions where and . For , we proved in The-orem 1 that the optimal throughput is achieved at

. Nevertheless, the maximum throughput is also achieved atfor . The formal proof can be found in

our technical report [29]. Here, we present some evidence whichshows the optimality of . We start with thefollowing observation.

Theorem 3: For homogeneous nodes, the solutionsand both achieve the

throughput .Proof: When , each node tries to

transmit at each time-slot independently with probability .Therefore, the throughput is just the probability that if and onlyif one of the nodes transmits:

When and , we can adopt the MarkovChain in Fig. 2. We have the transition probabilities



Fig. 4. Aggregate throughput.

and. The throughput is

Theorem 3 provides a reference point to divide the solutionspace into groups. Fig. 3 plots the throughput for systems of

,5,10 and 20. In each subplot, varies along the -axis,and different curves plot different values for . For any , weuse the curve (from Theorem 3, we know the exactvalue when or ) as a reference to dividesolutions into two groups: and . We plotthe curve in solid lines. We compare the solutionwithin each group and across groups, and make the followingobservations:

• When the value of increases from , the throughputdecreases. The curve of is above all curves of

.• When the value of decreases from , the throughput

curve becomes lower in the region , butthe maximum of each curve is at . This maximumincreases when decreases.

In Fig. 4, we plot the surface of throughput for systems withvarious number of nodes. In each subplot, varies along the

-axis (on the right); varies along the -axis (on the left).By increasing of the number of nodes in system, the surfacesbend down dramatically in the region of small and large(left corner). The high throughput is reached in the region oflarge and small (right corner). In particular, the maximumis achieved at .

Although the solution might maximizethroughput, it is not short-term fair, in which a single transmittergains exclusive access of the medium for a long time. As

, we have . We now consider how to enforceshort-term fairness.

Theorem 4: For homogeneous nodes with and, the system is -short-term fair2 to

all nodes.Proof: Because is a geometric random variable with

parameter , we have

Since , we have . Bydefinition, the system is -short-term fair to all nodes.

2See Definition 1.



Fig. 5. Throughput under different fairness conditions.

Theorem 4 quantifies how to select to achieve a certainshort-term fairness. In particular, in order to achieve -short-term fairness, we can choose the following value of :

(4)

The total throughput becomes a function of :

(5)

(6)

Fig. 5 plots the total throughput under different short-termfairness constraints as the number of nodes, , variesalong the -axis. This figure shows the tradeoff betweenshort-term fairness and throughput. However, without sac-rificing much throughput, the system achieves desirableshort-term fairness. For example, if we want the system tobe 8-short-term fair, which means each node can successfullytransmit no more than 8 consecutive slots on average whenit captures the channel, we can achieve a total throughputclose to 1/2 even for large . In fact, when , thetotal throughput does not collapse to zero. We will discuss thethroughput limits in the next theorem.

Lemma 1: For any constant , if, then is monotonically decreasing with

.Proof: Let and . We have

First, is strictly decreasing in , because. We define .

The last inequality holds because of the following. We define. is a strictly concave function, because

and . Sinceand , we see attains its maximum value

0 at . holds under our context; therefore, the lastinequality holds.

Finally, , and consequently,.

Theorem 5: Under any short-term fairness condition, the total throughput, , is lower-bounded by

.Proof: We choose the value of in (4) to satisfy the short-

term fairness condition. Accordingly, from (6), we have

The right-hand side is of the form 0/0 as . By L’hos-pital’s rule

On the other hand,

Thus, by Lemma 1, is monotonically decreasing in (as wellas in ). Therefore, is lower-bounded by

.Theorem 5 provides the lower bounds for the curves in Fig. 5.

We draw the limits of the throughput in dotted lines under eachthroughput curve. We see, for to be reasonably large (e.g.

), the throughput lower limit is close to 1/2. Indeed, whentends to infinity, this limit also tends to 1/2. In practice, a

fairness requirement can be achieved by choosing a suitablefor all users. Like the information of the number of users (N),

we suggest that the value can be broadcasted in the beaconmessages by the system infrastructure.

V. COMPETITIVE PERFORMANCE ANALYSIS

In the previous section, we identified the lower bounds of theobtainable throughput among cooperative nodes, additionallyconsidering short-term fairness constraints. In this section, weassume that each node is autonomous and sets its protocolparameters to maximize its own throughput. We start froma short-sighted selfish strategy that maximizes one’s ownthroughput given any fixed parameters of other nodes. We showthat this strategy induces a zero-throughput Nash equilibriumin the system. After that, we construct a Stackelberg game [6]by formulating a constrained optimization problem for eachnode. In particular, the leader player in the Stackelberg gamestrategically sets the parameters by taking the other node’s



reaction into account. Last, we form a simultaneous move gamewhere nodes all strategically set parameters, and reveal that aPrisoner’s Dilemma [6] phenomenon can occur.

A. Short-Sighted Selfish Behavior

Suppose nodes are originally cooperative and useand to achieve the maximum -short-term fair aggregate throughput. In this system, each node ob-tains throughput

If one node deviates from this cooperative solution and setsinstead, its throughput increases to

Its throughput now equals the probability that no other node istransmitting in each time-slot. Comparing the above two equal-ities, we have

Hence, by unilaterally changing to be 1, a selfish node canusually increase its throughput at least times (if ).This selfish behavior sacrifices the throughput of all other nodes,because they can no longer obtain any throughput. In fact, givenall other nodes’ parameters fixed, to set is alwaysthe best strategy to maximize one’s own throughput. However,unfortunately, this implies a Nash equilibrium of the systemwith zero aggregate throughput.

B. Stackelberg Game

As shown above, multiple short-sighted selfish users candrive the system throughput to zero. Here, we extend thisselfish behavior into a more sophisticated strategy which takesother nodes’ short-sighted selfish behavior into account. Weformulate a Stackelberg game [6], which enables one of thenodes (the leader) to anticipate the short-sighted selfish be-havior of the other node (the follower) and to choose its optimalparameters. Stackelberg games have been applied to differentareas of networking protocols (e.g. routing strategies [30], [31])in order to achieve efficient equilibria.

In this Stackelberg game, we consider a network that con-sists only of two selfish nodes and , each of which wantsto maximize its own throughput. In addition, we assume thateach has budget constraints and , respec-tively. and are the costs of both nodes as defined in (3).

are two budget constants that physically restrictthe average number of packets the node can transmit in eachtime-slot. We impose these budget constraints in order to modelthe nodes in a wireless ad hoc network or a sensor network. Be-cause nodes in these networks are very sensitive to power con-sumption, and transmitting packets consumes a lot of batterypower. Consequently, the behavior of nodes may largely dependon their budget constraints.

We formally describe the Stackelberg game as follows.Players: The leader node and the follower node .Strategy: for ; for .

Payoff: and for and , respectively.Game rule: Given any ’s strategy , chooses thebest response accordingly.

Follower’s Problem: In the Stackelberg competition, the fol-lower is modeled as a short-sighted selfish node that alwaystunes its parameters, responding to the other node’s parameters,to maximize its own throughput. Formally, for any given , thefollower node solves

Leader’s Problem: In the Stackelberg competition, the leaderapplies the strategy that takes the follower’s behavior into

account. It maximizes its own throughput by anticipating theshort-sighted strategy of the follower. Formally, the leader node

solves

In practice, users might not know one anothers’ strategies.The Stackelberg game assumes that the follower will adjust itsparameters to maximize its throughput given any fixed param-eters of the leader, , and will maintain fixed parameters solong as does as well. Both the leader and the follower are notnecessarily very different; however, the leader is more strategi-cally sophisticated and the follower is simply a throughput max-imizer.

To solve this Stackelberg game mathematically, we first solvethe follower’s problem for every possible strategy taken by node

. Thus, we obtain the best response strategy of as a functionof node ’s strategy. After that, the leader decides its optimalstrategy according to node ’s best response strategy. This pro-cedure is often referred to as backwards induction3 [17]. Thecorresponding game solution is often referred to as a Stackel-berg equilibrium. We derive numerical results based on (2) and(3).

C. Three Stackelberg Equilibrium Regions

We solve the above Stackelberg game for nodes who have thesame budget constraints, i.e., .

Fig. 6 and Fig. 7 show the throughput and costs of bothplayers in the Stackelberg equilibrium. The x-axis representsthe budget constraint for both players. The change in thethroughput as a function of the budget behaves differently inthree different solution regions:

1) When the budget is less than 1/3, both players achievethe same throughput, and they both use up their budgets.Within this region, an increase of budget improves theachieved throughput. The throughput is mainly limited bythe budget constraints rather than the competition betweenthese two players.

2) When the budget is between 1/3 and 2/3, both playersagain achieve similar throughput and use up their budgets.

3Backward induction is actually a more general procedure to identify the Sub-game Perfect Nash Equilibria in any finite dynamic game with perfect informa-tion.



Fig. 6. Throughput in the Stackelberg equilibrium.

Fig. 7. Cost in the Stackelberg equilibrium.

However, an increase of budget deteriorates each player’sthroughput. In this region, competition between these twoplayers further reduces the throughput as well.

3) When the budget is more than 2/3, the leader can select pa-rameters that give it a larger fraction of the throughput. Asthe budget increases, this unfair allocation of throughputexacerbates and the follower, still wishing to maximize itsown throughput, actually becomes less aggressive and usesa partial budget. In this region, the Stackelberg game bene-fits the leader by sacrificing the throughput of the follower.

Fig. 8 plots values of and , revealing the strategies ofboth players in Stackelberg equilibrium. In the first two solutionregions, both players use similar strategies. As a result, it doesnot matter (to a node) whether it is the leader or the follower,because both players achieve similar throughput. In particular,when the budgets are close to 1/3, the strategies selected by theplayers are similar to what would be selected by cooperativeplayers, and the aggregate throughput approaches 2/3. As thebudgets are further increased, the nodes’ additional contentionon the medium and the rate of interference become significant.

When the budgets exceed 2/3, the leader’s strategy changesdramatically. Instead of setting , it sets . This

Fig. 8. Strategies in Stackelberg equilibrium. (a) Leader’s strategy. (b) Fol-lower’s strategy.

implies that if a transmission fails in a slot, it will attempt toretransmit during the next slot. This makes sense intuitively be-cause the follower, attempting to maximize its own throughputwith its confined budget, must back off with high probabilityafter a collision, and the "safest" time for the follower to transmitwill be following a previous successful transmission. Becausethe leader uses , the follower can only successfullytransmit when the leader is in the Free State. Therefore, the fol-lower must not fully use its budget, since it has to reduce thecollision probability that leads the leader to a Backlogged State.

Note that at around a budget of 0.55, where the leader starts toset , the follower’s strategy begins to mimic the leader’sstrategy to maximize its own throughput. When the budget infurther increased, the follower cannot use the same aggressivestrategy to maximize its own throughput.

D. Simultaneous Move Game and Prisoner’s Dilemma

The Stackelberg game assumes that the leader is strategicallymore sophisticated that the follower. To relax this bias, we nowassume that both nodes, not being able to anticipate the otherparty’s strategy, will decide their strategies simultaneously. Thismodels situations in wireless network where users do not knowother users’ backoff rates, which can be any strategy of and



, in advance. We focus on three representative budget sce-narios and the corresponding strategies (from Fig. 8) that wouldbe played in the Stackelberg game:

• Low budget region: . Strategy.

• Medium budget region: . Strategy.

• High budget region: . Strategy, and

.Strategy in the lower budget region is similar to the strategyplayed by the nodes in a cooperative environment. Strategy ,more aggressive than , is played by both the leader and thefollower in the medium budget region. Finally, and arethe strategies of the leader and the follower in the high budgetregion.

We consider two simultaneous move games, where twonodes must choose their parameters to maximize their in-dividual throughput without knowing what their opponentchooses to do. Each game models a budget scenario wherenodes are confined to use some of representative strategiesabove.

To be “Cooperative” or to be “Greedy”?: First, we considera game in which two nodes, each with a budget of 0.5, must de-cide whether they will perform a cooperative-type of strategy orbehave in a greedy manner (i.e., should the node set its param-eters according to or ?) We evaluate (2) for both nodes,and list the throughput in the following table:

Here, we see a typical Prisoner’s Dilemma [6]. Althoughfrom a global perspective, both players know the best solutionis , from any hypothetical local point, strategyshould always be played. This is because, for any fixed strategyby the opponent, choosing is always better than choosing

. Strategy is called the dominating strategy [6] for bothplayers and the solution is the unique Nash equilib-rium [6] solution of this game.

To be “Aggressive” or to be “Mild”?: In the second game,we assume that nodes have budgets in the third region. As in theStackelberg game, the leader player is better off by playing anaggressive strategy; however, now the nodes must also decidewhether to choose the leader-type of strategy or the follower-type of strategy, without knowing the other player’s responsein advance. Notice that there is no actual leader or follower inthe simultaneous move game. Here, nodes are restricted to thestrategies and used by the leader and the follower in theStackelberg game.

Here, a node’s best strategy is not clear. A node is always betteroff choosing the opposing strategy of its competitor. Choosing

the follower strategy is more conservative. A throughput of atleast 0.1233 is ensured, but the throughput can be at most 0.25.If the leader strategy is chosen, throughput of 0.3595 is possible,but throughput of 0 is also a possible outcome. Interestingly,this game has two symmetric Nash equilibrium solutions, i.e.,

and .From the results of the above two simultaneous move games,

we can further explain the three solution regions of the Stack-elberg game in Fig. 6. When the budget is between 1/3 and 2/3,both nodes are afford to use classes of cooperative (e.g. ) orcompetitive (e.g. ) strategies. The uniqueness of the Nashequilibrium solution implies that nodes would similar strate-gies regardless it is the leader player or the follow player. Asthe budget increases, it is affordable for the nodes to use anystrategy. From the symmetric Nash equilibria in the simulta-neous move game, the leader player can always take advantageto choose a favorable equilibrium in the Stackelberg game.

VI. SELFISH BEHAVIOR DETECTION AND PREVENTION

In the previous section, we used non-cooperative games oftwo nodes to show that selfish behavior of nodes deteriorates theoverall throughput obtained across the transmission medium, aswell as that of the individual nodes. In this section, we discusshow cooperative nodes can identify and prevent selfish behaviorin a general -node system.

A. Transmitting is a Dominating Strategy

Consider any node at any time-slot . If it attempts totransmit, the probability of success is

where is the probability ( or depending on ’s state)that node transmits in that time-slot. Without any budget con-straint, node can achieve the highest throughput by transmit-ting a packet during every time-slot. However, if node trans-mits a packet in every time-slot, other nodes transmission at-tempts will always fail. Over time, this phenomenon can beeasily observed. Here, we consider how cooperative nodes canalter their parameters if their perceived success rates are toosmall in such a way that selfish nodes become “encouraged” toset their parameters to mimic the behavior of cooperative nodes.

B. Selfish Behavior Detection

Theorem 6: For an -short-term fair cooperative en-vironment, where each node uses and

, the success rate defined by for any nodeis lower-bounded by .

Proof:

where is equal to the total average cost for all nodes.Suppose all nodes are in backlogged state. Let be thenumber of nodes that decide to transmit in a time-slot. Then



, where is a binomialrandom variable with parameters and .

Since and, we obtain

By Lemma 1, is monotonically decreasing in . When, is the maximum for . Substituting

with , we have

Theorem 6 provides a guideline for how cooperative nodescan, in a distributed fashion, detect the existence of any selfishnode. A fraction of at least

of a cooperative node’s transmissions should be suc-cessful. For instance, when equals 8, this average successrate lower-bound is 42/43. When is larger, the success rateis even higher. Notice that this success rate is different from thethroughput that a node achieves. With the increase of the numberof nodes, each node’s throughput decreases however the successrate lower-bound remains the same. In practice, each node canmeasure this quantity to infer if there is any selfish node in thesystem.

Fig. 9. Cooperative � , selfish preventive � and its approximation.

C. Selfish Behavior Prevention

In order to prevent selfish behavior in the system, each co-operative node can implement a new strategy when they de-tect selfish nodes. The new strategy uses a that re-duces the throughput of the selfish node to a level below what itwould have achieved if were used by all cooperative nodes.Knowing that such a reduction will occur, selfish nodes have thenecessary incentive to remain cooperative.

Suppose after all cooperative nodes activate the new strategy, the selfish node obtains throughput . has to be less than

, which is the fair share throughput gained in a cooperativeenvironment:

From Theorem 2, we know that is lower-bounded by 1/2.Hence, we can substitute in 1/2 for when calculating the lower-bound of as an approximation.

Fig. 9 shows the cooperative strategy and the selfish pre-vention strategy . We see is a good approxi-mation for the lower-bound for .

VII. ADVERSARIAL MODEL ANALYSIS

All previous scenarios assume that each node, whethercooperative or selfish, is interested in maximizing its ownthroughput. In this section, we consider an attacking nodewhose goal is to use its restricted budget to minimize thethroughput of the other nodes in the system, i.e., to cause asmany of its packets to collide with what would otherwise besuccessful transmissions. We first discuss how much damage anadversary node can inflict if it uses a random (stateless) attack.Next, we formulate this attack model as another Stackelberggame.

A. Pure Random (Stateless) Attack

If an attacking node is able to transmit a packet in every time-slot, it can clearly jam all transmissions. We assume that theadversary node has a budget , allowing it to transmitin at most a fraction of the slots. This budget represents as the



highest frequency of transmission under which an attack cannotbe detected.

Definition 2: An adversary node is said to use a -purerandom attack if it transmits a packet in each time-slot inde-pendently with probability .

By Definition 2, an adversary node with a budget can usea -pure random attack for any . We can imagine that a

-pure random attack for a communication channel is identicalto a lossy channel where a packet is lost with probability .

Theorem 7: Suppose there are two nodes and in thesystem. If node is an adversary node that uses a -pure randomattack, then regardless of the strategy of player , ’s throughput

is equal to .Proof: Substitute and with in the corresponding

throughput for as in (2). We have

Since the corresponding cost function for as in (3), we have

Therefore, .Theorem 7 formalizes the intuitive result that a -pure random

attack reduces the capacity to be of the original capacity.Interestingly and counter-intuitively, if we have more than onecooperative node, the damage caused by a -pure random attackis often larger than a factor of .

Theorem 8: Suppose originally there are homogeneousnodes that use and in the system. Theyachieve an aggregate throughput . If an adversary node joinsthe system and uses -pure random attack, then the aggregatethroughput of the cooperative node is less than .

Proof: Before the adversary node comes into the system,we can model the system as in Fig. 2. The transition probabili-ties are and . Afterthe adversary node comes, we define the corresponding transi-tion probabilities to be and . Because a successful packetfrom a normal node happens only if the adversary node does nottransmit, we have and . From(5), we obtain

The new throughput is

Therefore,

The last inequality holds if .An explanation of this result is that as more nodes participate

in the cooperative process, the expected number of slots betweentransmissions in the Backlogged State grows at a faster rate thanthe expected number of slots between transmissions in the FreeState. A random seeding of losses forces more nodes to spendincreased time in the Backlogged State, and as a result, eachnode attempts fewer transmissions over time, yet still loses afraction of the attempts to the random loss process.

B. Adversary Stackelberg Game

Now, we compute the reduction in throughput caused by anadversary node when it maximizes its attack power under atwo-state system. As in Section V-B, we model this system as aStackelberg game. The difference between the previous modeland this model is that we assume the leader node is the at-tacker and its sole objective is to minimize the throughput ofnode . Because the leader has an advantage over the follower,making the adversary node the leader maximizes its potentialfor damage. We still assume that node and have budget con-straints: and , respectively. The adversaryStackelberg game can be formally described as follows.

Player: The leader node and the follower node .Strategy: for ; for .Payoff: and for and , respectively.Game rule: Given any ’s strategy , chooses thebest response accordingly.Follower’s Problem:For any given , the follower node solves:

Leader’s Problem:The leader node solves:

C. Two Stackelberg Equilibrium Regions

By backward induction, we solve the above adversary Stack-elberg game for nodes who have the same budget constraints,i.e., .

Fig. 10 plots the throughput of the follower (non-attacking)node when chooses the optimal attacking strategy of theStackelberg game. It also plots the curve , whichgives the throughput of node when the attacker uses a -purerandom attack with . Fig. 11 shows the costs incurred byboth players. We identify two regions in the Stackelberg equi-librium solutions:

1) When the budget is less than 2/3, both players use up theirbudgets. Player achieves identical throughput when at-tacked by the adversarial leader player and by a -purerandom attacker.



Fig. 10. Throughput in (adversary) Stackelberg equilibrium.

Fig. 11. Cost in (adversary) Stackelberg equilibrium.

2) When the budget is larger than 2/3, player achievesslightly but observably lower throughput when attackedby the adversarial leader player than attacked by the -pureattacker.

Intuitively, the attacking node will always use up its budget toattack. But surprisingly, a strategic, two-state attack cannot dobetter than pure random attack if the adversary node does nothave a budget larger than 2/3. When the budget is larger than2/3, the two-state attack is only slightly more effective.

D. Random Attack Versus Strategic Attack

We show the strategy solutions of both players in Fig. 12. Wefind that the strategies played in the two budget regions are quitedifferent.

Not surprisingly, when the budget is less than 2/3, the at-tacking node uses the pure random strategy .Theorem explains why the throughput is so close to curve

in the lower budget region. It turns out that playerhas multiple strategies to maximize its throughput, but all of

these strategies use up the budget . Therefore, although thestrategies played by node seem to be irregular, node al-ways gains a throughput that is close to . Noticethat when the budget is very small (e.g. ), the prob-ability of a collision is extremely small. Therefore, there aremultiple optimal strategies that can maximize throughput. Of

Fig. 12. Strategies in Stackelberg equilibrium. (a) Leader’s strategy; (b) fol-lower’s strategy.

course, should be one of the optimal strategiesthat maximizes throughput.

After comparing the strategies played by both nodes in thelarger budget region with those used by two non-cooperative,non-attacking nodes in Fig. 8, we notice that they are strikinglysimilar. This means that an adversary node chooses a strategyvery similar to what is chosen by a node who wishes to selfishlymaximize its own throughput. Of course, node would there-fore use the same response strategy.

In conclusion, if bandwidth requirements/capabilities are low,an attacker cannot do much better than attacking at randompoints in time. If the bandwidth requirements and capabilitiesare high, then an attacker behaves similarly to a node seeking tomaximize its own throughput.

VIII. CONCLUSION

In this paper, we generalize the slotted-Aloha protocol toa two-state protocol and construct its Markov Model. Wefind that if all nodes cooperate in an effort to maximize theaggregate throughput, an aggregate throughput of at least onehalf can be achieved regardless of the number ofnodes competing for bandwidth. If all nodes are selfish andattempt to maximize their own individual throughput, both theaggregate throughput and system fairness will be compromised.In a Stackelberg game, a leader node can achieve much higherthroughput than a follower node with large budget limits. In



a simultaneous move game, where nodes both strategicallychoose parameters, a situation similar to the traditional Pris-oner’s Dilemma arises. Finally, we showed that adversarynodes with limited budgets can do little better than a randomattack, and nodes with large budgets should behave like theirselfish counterparts.

The generalized slotted-Aloha provides a framework toanalyze different behavior of autonomous nodes in system.Some analytical observations from different behavioral mod-elings provide guidelines for building robust and efficientmedia access protocols, from which systems can obtain higheraggregate throughput, as well as the ability to cope with selfishand malicious users.

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Richard T. B. Ma (S’07) received the B.Sc. (first-class honors) degree in computer science in July 2002and the M.Phil. degree in computer science and engi-neering in July 2004, both from the Chinese Univer-sity of Hong Kong. He is currently working towardthe Ph.D. degree at Columbia University, New York,NY.

His research interests are in network economicsand game theory.

Vishal Misra (S’98–M’99) received the B.Tech. de-gree in from IIT Mumbai in 1992 and the Ph.D. de-gree from the University of Massachusetts, Amherst,in 2000, both in electrical engineering.

He is an Associate Professor of computer scienceand Director of the Distributed Network AnalysisLaboratory at Columbia University, New York,NY. His current research interests include wirelessnetwork protocol design, network and distributedchannel coding, multipath and robust routing proto-cols and scheduling mechanisms.

Prof. Misra has served on the TPCs of all major networking and performanceevaluation conferences, and as the TPC Chair of ACM Sigmetrics 2008. He hasreceived the NSF Career award, the DoE Career award, and IBM faculty awards.

Dan Rubenstein (S’97–A’00–M’03) received theB.S. degree in mathematics from the MassachusettsInstitute of Technology (MIT), Cambridge, MA, theM.A. degree in mathematics from the University ofCalifornia at Los Angeles (UCLA), and the Ph.D.degree in computer science from the University ofMassachusetts, Amherst.

He is now an Associate Professor of electrical en-gineering and computer science at Columbia Univer-sity, New York, NY. He is generally interested in thedesign and performance of networked systems, and

also has interests in the security and integrity of such systems, as well as thegeneral design of distributed communication algorithms.

Prof. Rubenstein has received a National Science Foundation CAREERAward, an IBM Faculty Award, an IEEE ICNP Best Paper Award, and (as astudent) an ACM SIGMETRICS Best Student Paper Award.


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