Conjecture: Existence of Nash Equilibria in
Modern Internet Congestion Control
Ayush Mishra

Jingzhi Zhang

Melodies Sim

National University of Singapore
Singapore
ayush@comp.nus.edu.sg

Wuhan University
China2017301500155@whu.edu.cn

National University of Singapore
Singapore
melodies@comp.nus.edu.sg

Sean Ng

Raj Joshi

Ben Leong

National University of Singapore
Singaporeseannpn@comp.nus.edu.sg

National University of Singapore
Singaporerajjoshi@comp.nus.edu.sg

National University of Singapore
Singapore
benleong@comp.nus.edu.sg

ABSTRACTThe Internet’s congestion control landscape is currently in the
midst of an unprecedented paradigm shift. A recent measurement
study found that BBR, a congestion control algorithm introduced
by Google in 2016, has seen rapid adoption and is deployed at morethan 20% of the Alexa Top 20,000 websites. Encouraging early deployment results from Google, Dropbox and Spotify suggest thatBBR could potentially replace traditional loss-based congestion control algorithms like CUBIC. In this paper, we study the interactions
between CUBIC and BBR and show that the underlying interactions can be modeled as a normal form game. Our game-theoretic
analysis and testbed measurements suggest that while BBR seems
to achieve somewhat better performance than CUBIC on the Internet today, this advantage will decrease as the proportion of BBR
flows increases. The distribution of congestion control algorithms
on the Internet would likely reach a Nash Equilibrium, where noflow has the incentive to switch from CUBIC to BBR, or vice versa.
We also found that the distribution of CUBIC and BBR flows inthis Nash Equilibrium will be dependent mainly on the size of thebottleneck buffer, and marginally on the RTT distribution of theflows. Our results suggest that the future Internet will likely bemore heterogeneous and that buffer sizing will continue to have asignificant impact on Internet congestion control.

CCS CONCEPTS• Networks → Transport protocols; Public Internet.

KEYWORDS
congestion control; measurement study
ACM Reference Format:
Ayush Mishra, Jingzhi Zhang, Melodies Sim, Sean Ng, Raj Joshi, and Ben
Leong. 2021. Conjecture: Existence of Nash Equilibria inThis work is licensed under a Creative Commons Attribution International
4.0 License.
APNet 2021, June 24–25, 2021, Shenzhen, China, China© 2021 Copyright held by the owner/author(s).
ACM ISBN 978-1-4503-8587-9/21/06.
https://doi.org/10.1145/3469393.3469397

Modern Internet Congestion Control. In 5th Asia-Pacific Workshop on Networking (APNet 2021) (APNet 2021), June 24–25, 2021, Shenzhen, China, China.
ACM, New York, NY, USA, 6 pages. https://doi.org/10.1145/3469393.3469397

1

INTRODUCTION

Today, BBR [2] and CUBIC [6] are the two dominant congestion
control algorithms on the Internet. The apparent performance benefits and promising deployment results of BBR (and its variant
BRRv2) from Google [3], Spotify [5], and Dropbox [9] are likely
going to make website operators evaluate if they should adopt the
new congestion control algorithm. A recent measurement study
found that more than 20% of the Alexa Top 20,000 websites [8]have already adopted BBR to serve HTTP traffic [15]. Since mostof these early adopters are websites that stream video [15], it isplausible that BBR is already contributing to more than 40% of thedownstream traffic on the Internet [20]. This significant change in
the Internet’s congestion control landscape is notable because BBR
represents a paradigm shift in congestion control. While traditional
congestion control algorithms like CUBIC relied on packet losses
to infer congestion, BBR uses bottleneck bandwidth and RTT estimates. By doing so, BBR avoids filling deep buffers to keep latency
low and is more resilient to random packet losses.
There have been many studies on the interactions between CUBIC and BBR. However, unlike previous studies [7, 22, 23] which
are mainly focused on flow-level fairness, we attempt to answer thefollowing question: given the aggressive adoption of BBR in recent
times, how do we expect the Internet to evolve? Would it be reasonable
to expect the whole Internet to switch to BBR, and/or its variants, inthe near future?
Most recent deployments have cited better throughput as a key
reason to switch from CUBIC to BBR [3, 5, 9]. This is not surprisingand it seems perfectly plausible that a website would decide to
switch from one TCP variant to the other if the switch will result inhigher throughput. Hence, we can think of the Internet abstractly
as a system of n flows and model it as a normal form game. In gametheory, a normal form game is a standard representation of a gamewhere the players have a preference of outcome, which in our caseis higher throughput.
Our analysis of such a normal form game suggests that given
the characteristics of the interactions between BBR and CUBIC,
there must exist a Nash Equilibrium [16] (NE). A Nash Equilibriumin a game is a strategy distribution between the players where noMishra et al.
BBR's throughput (normalized)

BBR's throughput (normalized)

APNet 2021, June 24–25, 2021, Shenzhen, China, China

3.5
80 ms BBR - 40 ms CUBIC
40 ms BBR - 80 ms CUBIC
40 ms BBR - 40 ms CUBIC

3
2.5
2
1.5
1
0.5
0
0

1

2

3

4

5

6

7

8

9

10

Buff size (normalized by BDP of smallest RTT flow)

player has anything to gain by changing only their strategy. Thisessentially means that a set of n flows will eventually converge toa distribution of congestion control algorithms, where no flow will
have any incentive to switch from CUBIC to BBR, or vice versa
because doing so will not lead to an increase in throughput. We
verify the existence of such Nash Equilibria and measure the effect of network parameters such as bottleneck link capacity, buffer
size, number of flows, and RTT using a set of testbed experiments.
Our results suggest that the distribution of congestion control algorithms at the NE depends primarily on the size of the bottleneckbuffer and marginally on the RTT distribution of the flows.

4
2
1
0.5

1 BDP Buffer
5 BDP Buffer
10 BDP Buffer

0.25
0.125
0

10 20 30 40 50 60 70 80 90% of BBR flows at the bottleneck

Figure 2: BBR’s throughput vs. % of BBR flows.
aggregate bandwidthof all BBR flows

Figure 1: BBR throughput vs. bottleneck buffer size.

8Lsmall number of BBR
flows getting a large
share of bandwidthlinear function
sub-linear function

0100% of BBR flows at the bottleneck

Figure 3: Sub-linear increase in total BBR bandwidth.
Observation 2. When a single BBR flow competes with a single
CUBIC flow at a bottleneck, there must exist some threshold bottleneck buffer size Tfair such that when the bottleneck buffer size
Buf f < Tfair , the BBR flow gets higher throughput than the CUBIC flow and when Buf f > Tfair , the CUBIC flow gets higher
throughput than BBR.

CUBIC and BBR behave very differently when they are in deep
buffers. Since CUBIC is loss-based, it will continue to fill a buffer tillit experiences a packet loss. On the other hand, BBR caps its cwnd,
which prevents it from filling deep buffers. While this behavior isintended to allow BBR to keep the queuing delay low, a competing
CUBIC flow will be significantly more aggressive (throughput-wise)
and take up a much larger share of the bandwidth. On the other
hand, in shallow buffers, CUBIC’s throughput will be limited because the cwnd will keep reducing because of the frequent packet
drops. Therefore, we make the following general observation which
has been verified in previous studies [22, 23].

Besides buffer size, BBR’s throughput also depends on the number of BBR and CUBIC flows at the bottleneck. We simultaneously
run 20 CUBIC/BBR flows passing through a 100 Mbps bottleneck. In
each iteration, we change the percentage of flows that ran BBR and
CUBIC. We plot the per-flow throughput for BBR normalized by
the average per-flow throughput of the competing CUBIC flows in
Figure 2. All flows had an RTT of 40 ms and ran for 2 minutes. Our
results show that as the percentage of BBR flows in the bottleneck
link increase, BBR flows get lower throughput. This behavior isconsistent when repeated with different buffer sizes, and the average throughput for the BBR flows can drop below that for the
CUBIC flows if the buffer is large and the number of BBR flows
is larger than a threshold value. We therefore make the following
observation:

Observation 1. When competing at the bottleneck where thebuffer is deep, CUBIC tends to have higher throughput than BBR;
the converse is true when the buffer is shallow.

Observation 3. As the percentage of BBR flows at the bottleneck
increases, the per-flow average throughput of BBR flows at thatbottleneck decreases.

We investigate the impact of RTT with a simple experiment
where a CUBIC and a BBR flow share a 20 Mbps bottleneck withvarying buffer sizes. Both flows were run simultaneously for two
minutes and we plot the steady-state throughput for BBR in Figure 1,
normalized against the throughput of the CUBIC flow. As expected,
we see that the throughput for BBR is a smooth function of thebuffer size. We also see that in each case there exists some buffer
size where CUBIC and BBR get an equal share of the bottleneckbandwidth i.e. normalized throughput equals 1. Based on this result,
we can make the next observation:

Earlier measurement studies had found that a small number of
BBR flows can get a disproportionately large amount of bottleneckbandwidth when competing with CUBIC flows [22, 23]. As shown in
Figure 3, the aggregate bandwidth of BBR flows at a fixed-capacity
bottleneck link will increase as the number of flows increases, so
the result is a sub-linear increase in total throughput and thus the
per-flow throughput would decrease with more BBR flows.
Both BBR and CUBIC congestion control algorithms are also
known to be RTT-unfair [10, 22]. CUBIC, being cwnd-based, ishampered in a long RTT path as it can ramp up its cwnd less often

2

KNOWN INTERACTIONS BETWEEN BBR
AND CUBIC

Conjecture: Existence of Nash Equilibria in
Modern Internet Congestion Control

APNet 2021, June 24–25, 2021, Shenzhen, China, China

Table 1: Possible scenarios in a 2-flow system.

than another CUBIC flow that has a short RTT path. BBR on theother hand tends to favor flows with longer RTTs as compared tothose with shorter RTTs. This is because when BBR becomes cwndlimited, it places twice the BDP (Bandwidth Delay Product) amountof packets in flight. We summarize these findings in Observation 4.
Observation 4. When two BBR flows compete at a bottleneck,the flow with a longer RTT will get higher bandwidth than theflow with a shorter RTT. When two CUBIC flows compete at abottleneck, the flow with a shorter RTT will get higher bandwidththan the flow with a longer RTT.

3

MODELLING THE INTERNET AS A
NORMAL FORM GAMEIn this section, we will formulate the problem of flows choosing
between CUBIC and BBR as a normal form game and analyze itusing the observations made in §2.
To investigate the behavior for a set of n flows, we model eachflow as an independent agent that (i) can decide on whether the flow
should run BBR or CUBIC; and (ii) knows its current throughput andalso the throughput if it were to switch to the alternative algorithm,
if all other flows do not change. We assume that websites candetermine (ii) by A/B testing between CUBIC and BBR, and that all
the flows would prefer to adopt the TCP variant that allows it toachieve higher throughput. While senders generally care about a
combination of network parameters, the weights given to each ofthese parameters may be varied across sites. However, improved
throughput was cited as a key factor by all major companies [3, 5, 9]that have reported making a switch from CUBIC to BBR, so forsimplicity, we consider only throughput in our analysis. With theseassumptions, we effectively have a normal form game, where thewebsites/senders can maximize throughput via 2 mutually exclusive
strategies: (i) BBR; or (ii) CUBIC. In this section, we argue that a
Nash Equilibrium should exist.

3.1

Two-flow game

Consider a simple network with two senders S 1 and S 2 sharing acommon bottleneck link. Each sender can pick either CUBIC or
BBR. We denote the RTTs of these two flows as RTT1 and RTT2
respectively. We will assume, without loss of generality, that RTT1 >RTT2 . We will also assume that the bottleneck queue is a simple
FIFO queue and does not deploy any other queuing disciplines or
AQMs. With two senders and two congestion control algorithms,
there can be a total of four possible scenarios or combinations. We
enumerate these possible scenarios in Table 1.
Consider the case where the two flows S 1 and S 2 pick different
algorithms, i.e. scenarios 2 and 3. From Observation 2, we knowthat for scenario 2, there exists a threshold buffer size T2 such thatwhen the bottleneck buffer size Buf f > T2 , the CUBIC flow will get
more bandwidth (CUBIC wins), and when Buf f < T2 , the BBR flow
will get more bandwidth (BBR wins). Similarly, for scenario 3, there
exists a buffer size T3 such that when Buf f > T3 , the CUBIC flow
wins whereas when Buf f < T3 , the BBR flow wins. In summary, T2
and T3 are the threshold buffer sizes below which BBR wins against
CUBIC in scenarios 2 and 3 respectively.

Scenario

S1

S2

1
2
3
4

CUBIC
CUBIC
BBR
BBR

CUBIC
BBR
CUBIC
BBR

The BBR flow in scenario 3 has a higher RTT than the BBR flowin scenario 2. From Observation 3 we know that a BBR flow with ahigher RTT is effectively more aggressive. This means that BBR will
be more aggressive in scenario 3 than in scenario 2 and so CUBIC
will need a deeper buffer in scenario 3 to tie with BBR. Therefore
we conclude that T3 > T2 . This relationship can also be seen in
Figure 1.
On this note, we consider 3 different regimes as illustrated inTable 2, where for each scenario we mark the outcome for a senderthat gets the higher bandwidth as ‘W’ (winner) and the outcome
for the other sender as ‘L’ (loser). A sender choosing either CUBIC
or BBR as their congestion control algorithm is denoted with theletter C and B, respectively, and the winning congestion control
variant in each case has also been highlighted for easy analysis. In
particular, we see that:(1) When Buf f < T2 : S 1 will definitely pick BBR regardless ofwhat S 2 decides to pick. In economic terms, B is a dominant
strategy for S 1 . S 2 will pick the algorithm that achieves thehigher throughput and we get the Nash Equilibrium (B, ∗).
(2) When T2 < Buf f < T3 : B is still a dominant strategy for S 1
and we again have the Nash Equilibrium (B, ∗).
(3) When T3 < Buf f : C is now a dominant strategy for S 2 andwe end up with a Nash Equilibrium (∗, C).
By enumerating all possible scenarios, we have effectively shownthat unique Nash Equilibria must exist for a 2-flow system. We
verified our predictions with experiments where two flows compete
at a 20 Mbps bottleneck with RTT1 = 80 ms and RTT2 = 40 ms. For
buffer sizes less than twice the BDP of the smaller flow, the NE wasat (B,B) and for all other buffer sizes up to 10 BDP the NE was at(B,C).

3.2

Conjecture for n flows

While we can model a system with n flows using the methodology
in §3.1, enumerating all possible scenarios is not a feasible approachto prove the existence of Nash Equilibria. However, some forms ofnormal form games have been studied extensively and proven tohave a pure strategy Nash Equilibrium [14, 19]. These games show
unique properties like continuity and boundedness of pay-offs, andmeasurability of outcome, that can be easily extended to our setting.
It remains as future work to check if it is possible to model an n-flow
system as one of these normal form games so that we can apply
known results.
The complexity for a n-flow system arises from the exponentiation of the number of scenarios and also the fact that flows canhave different RTTs, which makes it intractable to enumerate all
possibilities. However, there are two key observations from ourexperiments that inspire us to propose Conjecture 1 below: (1) As
the proportion of BBR flows increases, the gain in throughput will
decrease (cf. Observation 3 in §2). When the proportion of BBR

APNet 2021, June 24–25, 2021, Shenzhen, China, China

Mishra et al.

Table 2: Outcomes in a two-flow game. ( RTT1 > RTT2 , winning strategies are highlighted)
Buff < T2
Scenario
1
2
3
4
Nash Equilibrium

T2 < Buff < T3

OutcomeStrategies

OutcomeStrategies

Outcome

S1
C
C
B
B

S1
L
L
W
W

S1
C
C
B
B

S1
L
W
W
W

S1
C
C
B
B

S1
L
W
L
W

S2
C
B
C
B

S2
W
W
L
L

S2
C
B
C
B

(B, ∗)

flows increases beyond a threshold, the average throughput of the
BBR flows tends to drop below that of the remaining CUBIC flows(see Figure 1). (2) When the buffer is shallow, BBR is a dominant
strategy for flows with large RTTs; similarly, when the buffer isdeep, CUBIC is a dominant strategy for flows with short RTTs.
Conjecture 1. In a system with n flows that can choose to run
either CUBIC or BBR, there will be at least one Nash Equilibrium.

4

T3 < Buff

Strategies

EVALUATIONIn this section, we present the results of our testbed experiments
that: (i) suggest that NE exists for n flows (for n > 2), and (ii) provide
some insights into how RTT and buffer size affect the NE. Theseexperiments serve as validation of Conjecture 1 in a limited state
space and do not attempt to replicate all the network interactions
possible on the Internet (since that would be impossible).
Testbed setup. Our testbed setup consists of two servers connectedby a 1 Gbps link. We use Mahimahi [17] to spawn individual iperf
senders with different RTTs. We use nfqueue [18] to create a bottleneck with a specified link capacity and buffer size.
Methodology. For each network setting, we run 3 trials of all the
2n possible combinations of the n senders running either CUBIC
or BBR. In each trial, the senders send data for 2 minutes and werecord their average per-flow throughput. To identify the NE, weenumerate all the combinations and check if there is any combination such that no individual flow in that combination can achieve
higher throughput if it switches to the other TCP variant (all otherflows remaining fixed). Since the number of combinations increases
exponentially with n, we have only been able to do this exercise
for up to a maximum of 12 flows. It is clear that we will not be able
to prove Conjecture 1 with experiments. However, in this section,
our goal is to verify that the Conjecture holds for some n > 2 inpractice and that there are no obvious counter-examples. In all our
experiments, we found a NE as our Conjecture suggests.
Nash Equilibria Found. In all cases we found exactly one NE. All
these NE share a common property that allows us to specify it
succinctly with a single number: the proportion of CUBIC flows.
In particular, we observe that in any NE combination of n flows,
if we sort the flows by their RTTs, then it is always possible towrite the NE combination where the first m flows pick CUBIC, andthe remaining (n − m) flows pick BBR. For example, in a 6-flow
experiment, where flows have an RTT distribution of (20 ms, 20 ms,
50 ms, 50 ms, 80 ms, 80 ms), if the NE exists when 50% of the flows
pick CUBIC the NE combination will look like (C, C, C, B, B, B).

(B, ∗)

S2
W
L
L
L

S2
C
B
C
B

S2
W
L
W
L

(∗, C)

That is, all the 20 ms flows and one 50 ms flow picking CUBIC andall the 80 ms flows and one 50 ms flow picking BBR. This indicatesthat smaller RTT flows prefer CUBIC over BBR at the NE. We use
this property to represent a NE combination with a single number– the percentage of flows running CUBIC in the NE combination(m/n×100). The CUBIC flows are always the flows with the smallest
RTTs in the combination. This is not surprising, since smaller RTT
flows perform worse than larger RTT flows when running BBR andvice versa for CUBIC.

4.1

Effect of buffer size

In Figure 4, we present the NE when 6, 9, and 12 flows compete at
3 different bottleneck links with speeds of 20, 50 and 100 Mbps. In
each experiment, the flows are divided into 3 groups with RTTs of
20 ms, 50 ms and 80 ms, respectively. We let these flows compete in
the common bottleneck with different buffer sizes and record thecongestion control algorithm distribution at the NE in each case. In
general, when the buffers are deeper, more flows would pick CUBIC
over BBR. At very large buffer sizes, the NE generally occurs neara fifty-fifty split between CUBIC and BBR.

4.2

Effect of other network parameters

Our experimental results in Figure 4 also suggest that link capacity
and the number of flows have little to no effect on the distributionof CUBIC and BBR in the NE. It is plausible that because we size
the bottleneck buffer based on the BDP of the smallest RTT flow(smallest RTT multiplied by the bottleneck link capacity), any differences caused by an increase in link capacity are normalized by
the proportional increase in the buffer size. While our experimentssuggest that the number of flows does not have much effect on
the distribution of congestion control algorithms in the NE combination, this could also be an artifact of our limited number ofexperiments.
In Figure 5, we plot the results for two different RTT distributions. We found that when the flows have larger RTTs, the NE has
a marginally lower percentage of flows picking CUBIC. In otherwords, the RTT distribution does have a marginal impact on theNE.

5

DISCUSSION AND FUTURE WORK

Our experimental results suggest that with realistically sized (atleast 1 BDP) buffers, the Nash Equilibrium for n flows will be a mixof both CUBIC and BBR flows, and not one where all the flows are
BBR. While recent studies [7, 13] have shown BBR to have betterlink utilization and lower and less variable delay, and BBR’s recent

100

RTT 3 = 80 ms

20 Mbps
50 Mbps
100 Mbps80

RTT 2 = 50 ms

60
40

RTT 1 = 20 ms

20
0
0

1

2

3

4

5

6

7

8

9

100
80

RTT 2 = 50 ms

60
40

RTT 1 = 20 ms

20
0

10

0

Buff size (normalized by BDP of smallest RTT flow)

RTT 3 = 80 ms

20 Mbps
50 Mbps
100 Mbps

1

2

3

4

5

6

7

8

9

% of CUBIC flows at the NE

APNet 2021, June 24–25, 2021, Shenzhen, China, China% of CUBIC flows at the NE% of CUBIC flows at the NE

Conjecture: Existence of Nash Equilibria in
Modern Internet Congestion Control

RTT 3 = 80 ms

20 Mbps
50 Mbps
100 Mbps80

RTT 2 = 50 ms

60
40

RTT 1 = 20 ms

20
0

10

0

Buff size (normalized by BDP of smallest RTT flow)(a) 6 flows

100

1

2

3

4

5

6

7

8

9

10

Buff size (normalized by BDP of smallest RTT flow)(b) 9 flows(c) 12 flows

RTT 320-50-80 ms RTTs
40-80-120 ms RTTs

80

RTT 2

60
40

RTT 1

20
0
0

1

2

3

4

5

6

7

8

9

10

Buff size (normalized by BDP of smallest RTT flow)(a) 6 flows, 50 Mbps

100

RTT 3

20-50-80 ms RTTs
40-80-120 ms RTTs

80

RTT 2

60
40

RTT 1

20
0
0

1

2

3

4

5

6

7

8

9

10

Buff size (normalized by BDP of smallest RTT flow)(b) 9 flows, 50 Mbps

% of CUBIC flows at the NE

100% of CUBIC flows at the NE% of CUBIC flows at the NE

Figure 4: The effect of link capacity and number of flows on the Nash Equilibrium. RTTs 20 ms, 50 ms and 80 ms.
100

RTT 3

20-50-80 ms RTTs
40-80-120 ms RTTs

80

RTT 2

60
40

RTT 1

20
0
0

1

2

3

4

5

6

7

8

9

10

Buff size (normalized by BDP of smallest RTT flow)(c) 12 flows, 50 Mbps

Figure 5: The effect of the RTT distribution on the Nash Equilibrium.
deployments [3, 5, 9] have shown good performance, the corollary
is that we would expect the benefits of deploying BBR to be reduced
as more flows switch from CUBIC to BBR.
The future of Internet congestion control. Notwithstanding
the rapid adoption of BBR in recent years [15], our results suggestthat CUBIC is not about to disappear or to be completely replacedanytime soon. Instead, the Internet is likely to become more heterogeneous. This means that existing networks and middleware willneed to work well with both traditional loss-based TCP, as well aswith BBR and its variants.
The best is yet to come. We note that there is already work being done on BBRv2 [4], which is a less aggressive version of BBR.
HTTP/3.0 is going to adopt QUIC [1, 11] as its transport stack. With
QUIC, the entry barrier to developing and deploying newer congestion control algorithms is even lower. It remains to be seen how thenature of the NE will change in such a heterogeneous environment.
Even if our Conjecture in §3.2 is true for n flows where they can beeither CUBIC or BBR, it is not clear if it would hold for any n flows
if there are no restrictions on the choice of the congestion control
algorithm. It remains as future work to investigate if and how a
NE will exist in the presence of BBRv2 and any future QUIC-basedcongestion control algorithms.
The Internet may not follow economic game theory. We haveapplied concepts from game theory to model the evolution of Internet congestion control. It has also provided us with a principled way
to analyze and reason about the interactions between CUBIC and
BBR. However, it is also clear that the Internet does not necessarily
follow economic game theory perfectly. For example, it is quite
clear that CUBIC is a dominant strategy when competing with New
Reno, i.e. one always achieves higher throughput since CUBIC ismuch more aggressive than New Reno. Despite this, there are still asmall number of websites on the Internet today that continue to run
New Reno [15]. This phenomenon could potentially be attributed
to the indifference of some websites to network performance or
the influence of some other extraneous factors. Such phenomenon
contradicts what game theory would predict.
Uniqueness of the Nash Equilibria. In all the experiments, wefound exactly one NE. We were tempted to conjecture in §3.2 that
not only does a NE exist, but that it would be unique. We resisted
doing so because of the large space of possibilities when there are alarge number of flows and many different RTTs. A Nash Equilibrium
is like a local minimum. It remains as an open question whether an
n-flow system could have multiple Nash Equilibria, and if so, what
determines whether there are multiple NE or a unique one.
Impact of workloads, AQMs, and multi-hop paths Admittedly,
the analysis done in this paper does not represent the conditions at
an Internet scale. It remains to be seen how the normal form game
would be affected by flow sizes and lengths, and how the interaction
between CUBIC and BBR will change when the bottlenecks moveor deploy AQMs like RED and CoDel.

6

RELATED WORKTo the best of our knowledge, there have not been any previous
game-theoretic studies on the interactions between CUBIC and
BBR. However, there have been several measurement studies that
try to measure and model the interactions of BBR with itself andwith traditional loss-based TCP like CUBIC. These works focus on
flow-level fairness and general network metrics like throughput,
utilization, and delay.
Scholz et al. [21] and Miyazawa et al. [24] showed that sharing
a bottleneck with loss-based protocols such as CUBIC causes BBR
to measure an over-inflated RTT as loss-based protocols tend toAPNet 2021, June 24–25, 2021, Shenzhen, China, China

fill up buffers. This drastically increases BBR’s congestion window.
The net result is that the throughput for BBR and CUBIC willfluctuate cyclically. While one can näively expect CUBIC to become
a dominant strategy for very large buffers, the RTT over-estimation
associated with BBR can explain why some BBR flows are still able
to remain competitive and the NE does not end up being all flows
running CUBIC.
Similar phenomena were also reported by Hock et al. [7] who
showed that BBR’s over-estimation of RTT is aggravated when
multiple BBR flows share a bottleneck and over-estimate their fairshare of the bandwidth. They attribute this to the implementationof BBR where each flow will measure the maximum available bandwidth over 10 RTTs. This results in the sum of derived throughput
to be always greater than the actual maximum bandwidth of thebottleneck, causing persistent queues to build up at the bottleneck.
More recently, Ware et al. proposed a model to analyze how
BBR can be detrimental towards CUBIC when the two of themshare a bottleneck [23]. While their results assume that the network has very deep bottleneck buffers, their finding that BBR gets
cwnd-limited when competing with other flows explains the RTTunfairness observed between BBR flows.

7

CONCLUSION

In this paper, we investigate the potential evolution of Internet
congestion control by modeling it as a normal form game. Throughanalysis and testbed experiments, we show that if all flows were
either CUBIC or BBR, Nash Equilibria (NE) exists for a small numberof flows and we conjecture that a Nash Equilibrium will exist in
general for larger systems like the Internet. Our results suggestthat under realistic network settings, this Nash Equilibrium is likely
a heterogeneous mix of CUBIC and BBR flows. Furthermore, therelative proportions of CUBIC and BBR flows that make up theNE will depend primarily on the size of the bottleneck buffer andmarginally on the RTT distribution of the flows, with smaller RTT
flows generally picking CUBIC.
The existence and the nature of these Nash Equilibria will impact
how we can expect the Internet’s congestion control landscape to
evolve over the next few years and how it affects the performance
benefits associated with the adoption of new congestion control
algorithms like BBR. In particular, our findings effectively debunk
the commonly held notion that switching to BBR would improvethroughput [12]. We also suspect that it is unlikely that the entire
Internet will switch from CUBIC to BBR and/or its variants in thenear future.

ACKNOWLEDGMENTS
This work was supported by the Singapore Ministry of Education
grant T1 251RES1917.

REFERENCES[1] Ed. M. Bishop. 2021. Hypertext Transfer Protocol Version 3 (HTTP/3). (2021).
https://tools.ietf.org/html/draft-ietf-quic-http-34[2] Neal Cardwell, Yuchung Cheng, C Stephen Gunn, Soheil Hassas Yeganeh, and
Van Jacobson. 2017. BBR: congestion-based congestion control. Commun. ACM
60, 2 (2017), 58–66.
[3] Neal Cardwell, Yuchung Cheng, C. Stephen Gunn, Soheil Hassas Yeganeh, Van
Jacobson, and Amin Vahdat. 2017. TCP BBR congestion control comes to GCP –your Internet just got faster. (2017). https://tinyurl.com/yc7bd9jk

Mishra et al.

[4] Neal Cardwell, Yuchung Cheng, Soheil Hassas Yeganeh, Ian Swett, Victor Vasiliev,
Priyaranjan Jha, Yousuk Seung, Matt Mathis, and Van Jacobson. 2019. BBR v2
A Model-based Congestion Control. (March 2019). https://datatracker.ietf.org/
meeting/104/materials/slides-104-iccrg-an-update-on-bbr-00[5] Erik Carlsson and Eirini Kakogianni. 2018. Smoother Streaming with BBR. (2018).
https://tinyurl.com/yyt5tbhd[6] Sangtae Ha, Injong Rhee, and Lisong Xu. 2008. CUBIC: A New TCP-friendly
High-speed TCP Variant. SIGOPS Operating Systems Review 42, 5 (2008), 64–74.
[7] Mario Hock, Roland Bless, and Martina Zitterbart. 2017. Experimental Evaluationof BBR Congestion Control. In Proceedings of ICNP.
[8] Alexa Internet Inc. 2018. The Top 500 websites on the Internet. (2018). https://www.alexa.com/topsites[9] Alexey Ivanov. 2019. Evaluating BBRv2 on the Dropbox Edge Network. (2019).
https://tinyurl.com/yyrs68pp[10] Tomoki Kozu, Yuria Akiyama, and Saneyasu Yamaguchi. 2013. Improving RTT
Fairness on CUBIC TCP. In Proceedings of the International Symposium on Computing and Networking.
[11] Adam Langley, Alistair Riddoch, Alyssa Wilk, Antonio Vicente, Charles Krasic,
Dan Zhang, Fan Yang, Fedor Kouranov, Ian Swett, Janardhan Iyengar, Jeff Bailey,
Jeremy Dorfman, Jim Roskind, Joanna Kulik, Patrik Westin, Raman Tenneti,
Robbie Shade, Ryan Hamilton, Victor Vasiliev, Wan-Teh Chang, and Zhongyi Shi.
2017. The QUIC Transport Protocol: Design and Internet-Scale Deployment. In
Proceedings of SIGCOMM.
[12] Frederic Lardinois. 2017. Google Cloud gets a new networking algorithm that
boosts internet throughput. (2017). https://tinyurl.com/yjswop2z[13] Feng Li, Jae Won Chung, Xiaoxiao Jiang, and Mark Claypool. 2018. TCP CUBIC
versus BBR on the Highway. In Proceedings of PAM.
[14] Andreu Mas-Colell. 1984. On a theorem of Schmeidler. Journal of Mathematical
Economics 13, 3 (1984), 201–206.
[15] Ayush Mishra, Xiangpeng Sun, Atishya Jain, Sameer Pande, Raj Joshi, and Ben
Leong. 2019. The Great Internet TCP Congestion Control Census. In Proceedingsof SIGMETRICS.
[16] John F. Nash. 1950. Equilibrium points in n-person games. Proceedings of the
National Academy of Sciences (1950).
[17] Ravi Netravali, Anirudh Sivaraman, Somak Das, Ameesh Goyal, Keith Winstein,
James Mickens, and Hari Balakrishnan. 2015. Mahimahi: Accurate Record-andReplay for HTTP. In Proceedings of ATC.
[18] Netfilter Organization. 2019. libnetfilter_queue. (2019). https://bit.ly/2HimY17[19] Kali P. Rath. 1992. A Direct Proof of the Existence of Pure Strategy Equilibria inGames with a Continuum of Players. Economic Theory 2, 3 (1992), 427–433.
[20] Canada Sandvine Inc. Waterloo, ON. 2018. The 2018 Global Internet Phenomena
Report. (2018). https://www.sandvine.com/phenomena[21] Dominik Scholz, Benedikt Jaeger, Lukas Schwaighofer, Daniel Raumer, Fabien
Geyer, and Georg Carle. 2018. Towards a Deeper Understanding of TCP BBR
Congestion Control. In Proceedings of IFIP.
[22] Y. Tao, J. Jiang, S. Ma, L. Wang, W. Wang, and B. Li. 2018. Unraveling the RTTfairness Problem for BBR: A Queueing Model. In Proceedings of GLOBECOM.
[23] Ranysha Ware, Matthew K. Mukerjee, Srinivasan Seshan, and Justine Sherry. 2019.
Modeling BBR’s Interactions with Loss-Based Congestion Control. In Proceedingsof IMC.
[24] Kouto Miyazawa; Kanon Sasaki; N. Oda; S. Yamaguchi. 2018. Cycle and Divergence of Performance on TCP BBR. In Proceeding of CloudNet.