Content uploaded by John F. Kolen
Author content
All content in this area was uploaded by John F. Kolen on Jul 13, 2018
Content may be subject to copyright.
EOMM: An Engagement Optimized Matchmaking
Framework
Zhengxing Chen ∗
Northeastern University
czxttkl@gmail.com
Su Xue
Electronic Arts, Inc.
sxue@ea.com
John Kolen
Electronic Arts, Inc.
jkolen@ea.com
Navid Aghdaie
Electronic Arts, Inc.
naghdaie@ea.com
Kazi A. Zaman
Electronic Arts, Inc.
kzaman@ea.com
Yizhou Sun
University of California, Los
Angeles
yzsun@cs.ucla.edu
Magy Seif El-Nasr
Northeastern University
m.seifel-nasr@neu.edu
ABSTRACT
Matchmaking connects multiple players to participate in online
player-versus-player games. Current matchmaking systems depend
on a single core strategy: create fair games at all times. These
systems pair similarly skilled players on the assumption that a fair
game is best player experience. We will demonstrate, however,
that this intuitive assumption sometimes fails and that matchmak-
ing based on fairness is not optimal for engagement.
In this paper, we propose an Engagement Optimized Matchmak-
ing (EOMM) framework that maximizes overall player engage-
ment. We prove that equal-skill based matchmaking is a special
case of EOMM on a highly simplified assumption that rarely holds
in reality. Our simulation on real data from a popular game made
by Electronic Arts,Inc. (EA) supports our theoretical results, show-
ing significant improvement in enhancing player engagement com-
pared to existing matchmaking methods.
Keywords
matchmaking, player engagement, video games
1. INTRODUCTION
Player-versus-Player (PvP) is a mode of video game in which
multiple players directly engage in competition or combat. PvP
games, which cover many popular genres, such as multiplayer on-
line battle arena (MOBA), first-person shooting (FPS), and e-Sports,
have increased worldwide popularity in recent years. For example,
League of Legends, one of the most played MOBA games, has 90
million summoner names registered, 27 million unique daily play-
ers and 7.5 million concurrent users [30, 41]. As data released
∗This work was done when Zhengxing Chen was an intern student
at Electronic Arts,Inc.
©2017 International World Wide Web Conference Committee
(IW3C2), published under Creative Commons CC BY 4.0 License.
WWW 2017, April 3–7, 2017, Perth, Australia.
ACM 978-1-4503-4913-0/17/04.
http://dx.doi.org/10.1145/3038912.3052559
.
by [39] shows, e-Sports is estimated to have 188 million viewer-
ship and 748 million dollar worth market in 2015 and the numbers
are expected to grow continuously.
Matchmaking is the process that connects players to form PvP
matches. In practice, a matchmaking system takes practical lim-
itations, such as players’ geo-location and network latency, into
consideration. For example, cross-ocean pairing is not good for
player experience. Beyond technical constraints, the strategy var-
ious matchmaking systems employ is creating fair games. This
strategy relies on the assumption that matching closely skilled play-
ers tend to create competitive games which are desired by play-
ers [21]. In order to establish player skills, numerous models have
been studied, such as Elo [16], Glicko [20] and TrueSkill [23].
Are fairly matched games always beneficial for player experi-
ence? This fundamental, yet intuitive, assumption is worthy of deep
investigation. We can challenge it with a few examples. Consider a
cautious player who cares about protecting his rank among friends,
and a risk taker who enjoys difficult matches. Pairing them with
the similarly skilled opponents will affect these players very differ-
ently. Even for the same player, their expectation on the coming
match when they just lost three games in a row can be very differ-
ent from that when they recently performed well. In Table 1, we
show an example that churn risks vary drastically upon players’ re-
cent match outcomes in a popular PvP game made by EA. These
facts lead to two key insights: (1) the effectiveness of matchmaking
needs to be measured quantitatively; and (2) matchmaking should
depend on dynamic and individual player states.
In this paper, we propose a new matchmaking framework, En-
gagement Optimized Matchmaking (EOMM). By formulating
matchmaking into an optimization problem, we pair players in or-
der to maximize the overall player engagement, or equivalently,
minimize the overall player disengagement. First, we measure a
player’s disengagement by their churn risk after each matchmak-
ing decision. Here churn refers to no gameplay within a period of
time, such as a week. Second, we model all players who wait in
the matchmaking pool as a complete graph, where each player is
a node, and an edge between two players is their sum churn risks
if paired. The churn risk depends on individual player states at the
moment of matchmaking. Last, we can achieve engagement opti-
mized matchmaking efficiently by solving a minimum weight per-
fect matching (MWPM) problem that finds non-overlapping pairs
with the minimal sum of edge weights on a complete graph.
Table 1: An example about the impact of player states on their
engagement. Data is from a popular PvP game made by EA.
Average churn risks vary drastically upon players’ recent three
match outcomes ((W)in,(L)ose or (D)raw). Churn risk is mea-
sured by the ratio of the players who stop playing within a pe-
riod time (7 days in this table) after a match. The churn risk of
some states with repeated losses (5.1%) is almost twice as much
as those of other "safer" states (2.6%-2.7%).
Last 3 Outcomes Churn Risk
DLW |LLW |LDW |DDD 2.6% - 2.7%
... ...
WWW 3.7%
... ...
DLL |LWL |LDL 4.6% - 4.7%
WWL 4.9%
LLL 5.1%
EOMM provides a solid theoretical framework for matchmaking
analysis. With it, we prove that equal-skill based matchmaking is
a special case of EOMM on a simplified and often inapplicable
assumption about player states. The generic EOMM instead proves
to be optimal across a wide range of contexts.
For system development, EOMM is both flexible and computa-
tionally feasible. The optimization objective can be tuned for var-
ious interests, e.g., in-game time, or even spending. Furthermore,
EOMM consists of three components: a skill model, a churn pre-
diction model and a graph matching model. All can be efficiently
implemented and independently upgraded. We built a simulated
system based on real data of a popular game made by Electronic
Arts, Inc. (EA), showing significant improvement in enhancing
player engagement by EOMM against equal-skill based and other
matchmaking methods.
In sum, this paper contains the following contributions. First, we
propose an engagement optimized matchmaking framework, i.e.,
EOMM, which solves matchmaking as an optimization problem of
maximizing the overall player engagement. Second, we provide
theoretical analysis about the optimality of EOMM and the condi-
tions of the applicability of existing matchmaking methods. Last,
we build a simulated system using real game data to show signif-
icant advantages of EOMM in retaining players over the existing
matchmaking methods.
The rest of this paper is organized as follows. After reviewing the
related work, we will present the formulation of matchmaking as
an optimization problem on a graph. Then we describe theoretical
findings comparing EOMM and other matchmaking methods. We
then show the case study applying EOMM on real data. Finally, we
will conclude with a discussion of the results and future directions.
2. PREVIOUS WORK
2.1 Skill Modeling
The motivation behind skill rating is to rank players and to enable
skill-based matchmaking. Dating back to 1952, the Bradley-Terry
model [7] was developed to deal with repeated pairwise compar-
isons among a group of subjects. In the Bradley-Terry model, a
player iis assumed to have a fixed, positive skill scalar, ri, and the
winning probability of player iagainst player jis the ratio of player
i’s skill in the sum of skills of both players. In its original form, the
Bradley-Terry model estimates player skills only after observing all
pairwise comparisons. While feasible for small groups of players,
requiring O(n2)matches is prohibitive for large player pools. One
can show that the Bradley-Terry model is equivalent to a logistic
regression model [2] in which each coefficient wicorresponds to
log(ri).
The Elo system [16] addresses the relative skill ratings in player-
versus-player games, such as chess, with a probabilistic model. Elo
captures player performance, pi, as a random variable following a
one dimension Gaussian distribution with a mean, ri, and a fixed
variance, β2, shared by all players. In the Elo system, rigets up-
dated depending on the extent of agreement between expected out-
comes and real outcomes. For example, a low skill player beating a
high skill player yields a large update in adjusting their skill means
closer. Unlike the original Bradley-Terry model, rican be updated
at an ongoing basis, i.e., as soon as after every match of player i.
The Glicko system [20], a Bayesian ranking rating system, was
later introduced. Besides mean player skill, ri, it also models the
belief about a player’s skill as RDi(rating deviation). As they
play an increasingly number of games, the belief about their skills
become stronger hence RDidecreases. However, RDiincreases
when a player ceases to play for long time. To achieve high effi-
ciency, Glicko uses an approximation Bayesian algorithm to update
riand RDi.
Neither the Bradley-Terry model, the Elo system or the Glicko
system was initially applicable to team-oriented games until works
such as [23, 24, 29] to generalize these models. For example,
TrueSkill system [23] extends the Elo system to games with flex-
ible numbers of players and teams.
Researchers have proposed more advanced skill models to cap-
ture player skills in multiple facets. The works in [8, 38] model
player skills in multi-dimensions such as offensive and defensive
abilities. Delalleau et al. [10] proposed a neural network based
skill model which learns latent skill embeddings of players and is
claimed to outperform TrueSkill in a team based game. There are
skill models proposed for specific game genres, such as [11] for
chess, [9, 40] for MOBA games and for [3] RTS games.
In our paper, we will compare EOMM with skill based match-
making methods which leverage skill models. Skill models can also
facilitate EOMM in the decision of player assignment.
2.2 Matchmaking Strategies
There has been much research ensuring physical criteria of match-
making services such as network connection quality [1, 27, 28].
Besides physical criteria, matchmaking can be seen as a player
modeling technique [45] that extracts player information and deliv-
ers adaptive gaming experience [33]. A fair amount of matchmak-
ing systems assume that skill balanced games are good for engage-
ment [21] and hence resort to skill rating algorithms to identifying
similarly skilled opponents. My´
slak and Deja [32] suggests addi-
tional information about player preferences in in-game avatar roles
can further improve fairness-based matchmaking systems. A few
researchers have explored methods to improve player engagement
through matchmaking. Delalleau et al. [10] proposed to train a
neural network based architecture which predicts player enjoyment
based on their historical statistics. They measured enjoyment by
directly asking players for feedback after each match. Whether or
not this method can effectively collect sufficient feedbacks has not
been demonstrated with real data. Jiménez-Rodrıguez et al. [25]
proposed that matchmaking could be based on preferred roles by
players. They argue that a fun match should have players act in
roles with perceivably joyful role distribution. However, it is still
a conceptual, heuristic-based method without experiment showing
that such matchmaking system indeed improves concrete engage-
ment metrics. To our best knowledge, we have not seen any exist-
ing matchmaking method that formally treats matchmaking as an
optimization problem to maximize player engagement.
2.3 Player Engagement Prediction
Player engagement can be seen as an objective measurement of
user experience in games [6]. Player engagement can be embodied
by many specific metrics, such as time or money spent in the game,
the number of matches played within a time window, or churn risk.
We define churn risk as the proportion of total players stopping
playing the game over a period of time.
Churn prediction has been applied within various disciplines for
decades, such as telecommunications [17], online advertisements
[46] and insurance [31]. Video games have also sparked a num-
ber of churn analysis studies. For instance, Weber et al. [43] built
a regression model to predict the number of games played. They
also used the model to aid game design by identifying the most in-
fluential features on player retention. Hadiji et al. [22] established
the fundamental in churn prediction in free-to-play (F2P) games
by suggesting definitions of various churn behaviors, proposing
universal behavioral telemetry, and comparing different machine
learning models across five commercial F2P games. Runge et al.
[37] not only trained a churn prediction model for a casual social
game but also showed how the game can leverage the model to in-
crease the effectiveness of promotions to players. In EOMM, we
employ churn prediction as an important building component in
engagement optimization.
2.4 Graph Matching
In a graph G= (V, E ), a matching is a set of pairwise non-
adjacent edges [44]; that is, no two edges share a common vertex.
Aperfect matching is a matching with every vertex in Gincident on
exactly one edge in the matching. In a weighted graph G, a mini-
mum weight matching (MWM) is the matching with the lowest sum
of edge weights. A minimum weight perfect matching (MWPM) is
the perfect matching with the lowest sum of edge weights.
As will be shown in Section 3, the EOMM framework converts
the problem of determining optimal match assignment to the prob-
lem of seeking MWPM on a weighted graph. MWM/MWPM have
broad applications in other fields, including creating pairs follow-
ing specific rules in chess tournaments [34], schdeduling training
sessions among NASA shuttle cockpit simulators [4] and transmit-
ting images over networks [36]. In a similar spirit, Ólafsson [34]
leverages MWPM algorithm to determine opponents. Their goal,
however, was to create matches maximally adhering specific rules
of chess tournament, which is different than ours to optimize for
player engagement.
The first attempt to solve MWPM is the polynomial time blos-
som algorithm proposed by Edmond [14, 15] in 1965. Since then,
researchers have steadily improved upon this algorithm. We will
compare and discuss those improved methods later when we intro-
duce EOMM.
3. ENGAGEMENT OPTIMIZED
MATCHMAKING
In this section, we will introduce the EOMM framework which
formulates matchmaking as an optimization problem. In contrast
with the existing matchmaking methods that heuristically pair sim-
ilarly skilled co-players, EOMM aims to match players in an opti-
mal way that maximizes overall player engagement. Here we will
describe the details of match assignment for 1-vs-1 games. We will
discuss how EOMM can be extended to the matches with more
players in the final section.
Complete graph 𝒢
Player pool 𝒫
Engagement Optimized
Matchmaking (EOMM)
Figure 1: Model matchmaking on a complete graph. Each node
represents a player, and every edge is associated with the sum
engagement metric of two players if paired. EOMM amounts
to finding an optimal pair assignment on G.
3.1 Optimization Objective
In practice, matchmaking is applied to a pool of players,
P={p1,· · · , pN}, who are waiting to start 1-vs-1 matches. We
assume Nto be an even number such that all players can be paired.
The objective of EOMM is to maximize the overall player engage-
ment, or equivalently, minimize the overall player disengagement.
We use churn risk as a concrete metric of disengagement. The
term “churn” is used by convention, which actually represents a sta-
tus of disengagement, i.e., a player not playing any games within
a subsequent time frame, not necessarily a permanent churn. We
denote the churn risk of player piafter matchmaking with player
pjas ci,j , which is a function of both players’ states, i.e., ci,j =
Pr(pichurns|si,sj) = c(si,sj). A player state is a collection
of features that profile an individual player, including but not lim-
ited to install date, skill, play frequency, performance and etc. We
will elaborate on learning ci,j in the subsequent sections. Note that
ci,j 6=cj,i since two players in a paired match may be impacted
differently. We use a list of player tuples, M={(pi, pj)}, to
denote a matchmaking result, i.e., a pair assignment, in which all
players in Pare paired once and only once. Defining the overall
player disengagement as the sum of individual churn risks, EOMM
seeks for an optimal pair assignment M∗such that:
M∗= arg min
MX
(pi,pj)∈M
c(si,sj) + c(sj,si)(1)
We construct a graph, G, to model this environment (see Fig-
ure 1). Each player piis a node of the graph, who has a player
state, si, before matchmaking. The edge between two players pi
and pjis associated with a weight ci,j +cj,i, which is the expected
sum disengagement metric if they are paired. Note that Gis a com-
plete graph in that all pairs of players can be possibly connected.
Once all ci,j are computed, finding M∗in Eqn. 1 is converted to
aminimum weight perfect matching problem, i.e., finding a pair
assignment with the minimal sum weights of edges on graph G.
3.2 Predicting Churn Risks
We learn the function ci,j =c(si,sj)as a churn prediction
problem. In its original form, the churn risk ci,j of player piaf-
ter matchmaking depends on the states from both the player and
their opponent. Unfortunately, the well-established churn predic-
tion studies cannot be employed here because they only use fea-
tures of players themselves without considering those of opponents.
Also, naively feeding both player states as input will double the
feature dimension, which makes the prediction unintelligible and
harder since much more training data is needed.
One way to simplify the prediction of ci,j is to base it only on
player pi’s own state, si, and the resulting match outcome, oi,j ,
from the view of pi. This works because the opponent’s state, sj,
such as skill, play history and style, does not directly interact with
player pi’s churn risk ci,j. It, however, influences the upcoming
match outcome, which is directly perceivable by player piand thus
affects pi’s churn. Once the match outcome oi,j is known, ci,j be-
comes conditionally independent to the opponent’s state, sj. For-
mally, this property is represented as:
Pr(pichurns|si,sj, oi,j ) = Pr(pichurns|si, oi,j ),(2)
which can be written in a concise form:
c(si,sj, oi,j ) = c(si, oi,j )(3)
In this paper, we assume that game outcomes are sampled from a
finite set, O, such as Win,Lose and Draw. For example, oi,j =W
means that piwins over pj, while oj,i =Lrepresents the same
outcome from the view of pj. To predict game outcomes, we em-
ploy the standard skill models [16, 20] that are widely adopted in
the video game industry. These models use both players’ skills,
which are a proxy of their entire player states, as the input for the
prediction. We denote player pi’s skill representation as µi, which
is, for example, Elo score [16] or Glicko mean and RD [20]. Note
that µiis part of player state si. As a result, we have:
Pr(oi,j |si,sj)≈Pr(oi,j |µi,µj),(4)
Putting them together, we can efficiently predict the churn risks
of paired players in Eqn. 1:
c(si,sj) + c(sj,si)(5)
=X
oi,j ∈O
Pr(oi,j |si,sj) (c(si,sj, oi,j ) + c(sj,si, oj,i)) (6)
≈X
oi,j ∈O
Pr(oi,j |µi,µj) (c(si, oi,j ) + c(sj, oj,i)) ,(7)
where the first equality is a marginalization on game outcome, oi,j.
In the approximate equality, the conditional independence of ci,j
on sjgiven oi,j (Eqn. 3) and the game outcome prediction (Eqn. 4)
are used.
Now c(si, oi,j)can be efficiently learned based on any preferred
churn prediction model. The input features are the updated player
state based on the predicted game outcome of the hypothetical
matchmaking, i.e., supdate
i←siand oi,j . We can decompose
the original player state as si= [oK
i,ˆ
si], where oK
iis a vector
of the latest Kgame outcomes (for example, oK
i=LW LDL
when K= 5), and ˆ
sirepresent the rest of features in si. If piis
hypothetically matched with pj,siwill be updated as:
supdate
i←siand oi,j (8)
=[oK
i,ˆ
si]and oi,j (9)
=[oK+1
i,ˆ
supdate
i](10)
We use ˆ
supdate
ito indicate that non-game-outcome features are
also updated after the new match. For example, the total number of
games played increments by one.
3.3 Finding the Optimal Pair Assignment
Given the predicted churn risks of each pair of players, i.e., the
weight of every edge in G, EOMM reduces to a minimum weight
perfect matching (MWPM) problem. The goal is to find a pair
assignment, M∗, on a complete graph, G, which has the minimal
sum weights of edges.
For a graph with Nnode, the brute-force way is to exhaustedly
compare all N
N/2/2N
2possible pair assignments and find the best
one, but the time complexity is too high to be feasible in practical
systems. Fortunately, many polynomial time algorithms exist for
the MWPM problem. For example, several algorithms can solve
the problem in the worst time complexity O(N3)[18, 26]. If en-
gagement measurements are pure integers, there exists a slightly
faster algorithm [19] with running time O(N23
4log K)where K
is the largest magnitude of an edge weight. There also exist greedy
algorithms, such as [12] and [13], with faster running time to find
suboptimal solutions. Moreover, MWPM can be solved in parallel
as proposed by [35].
4. THEORETICAL FINDINGS
Besides generating optimal matchmaking assignments, EOMM
provides a framework to conduct theoretical analysis on other match-
making related problems. We use this framework to compare
EOMM with other matchmaking strategies under different hypo-
thetical situations to obtain many insights. Without loss of gener-
ality, we focus our discussion on 1-vs-1 games with possible game
outcomes sampled from Win,Lose and Draw.
Using the same notation in Section 3, we investigate a pair of
players pi, pj∈ P, i 6=j. When c(si,sj) = c(oi,j ), i.e., a
player’s churn risk only depends on the game outcome of the up-
coming match, regardless of all other states. This simplification
for Eqn. 8, where supdate
ionly considers oi,j but ignores si, has
interesting implications.
•If c(W in) + c(Lose)>2·c(Draw), i.e., the sum churn
risk of two matched players in a tied game is lower than that
in a non-tied game. Under this circumstance, the equal-skill
based matchmaking is equivalent to EOMM, as both strive
to form matches with Draw outcomes as many as possible.
This explains the intuition and popularity behind equal-skill
matchmaking. But we should be very aware of its conditional
applicability, while EOMM is instead always optimal.
•If c(W in) + c(Lose)<2·c(Draw), equal-skill based
matchmaking is actually worst among all matchmaking
schemes, as its goal to create close matches contrarily mini-
mizes the overall player engagement. Although this situation
contradicts with the common intuition that fair matches are
good, it is possible for a real game. Therefore validating the
assumptions with real game data is critical before applying
an equal-skill based matchmaking algorithm.
When c(si,sj) = c(si), i.e., a player’s churn risk is determined
by his state before matchmaking, then it does not matter whom
they will play. In this case, EOMM can do no better than a ran-
dom matchmaking. Random matchmaking, from this perspective,
is not as trivial as we thought. It is a relative safe and stable base-
line choice in lack of prior information. While equal-skill based
method can perform the worst under certain conditions, random
matchmaking will never fall into the worst case.
The analysis above shows that the existing matchmaking meth-
ods, such as equal-skill based and random matching, arise within
the EOMM framework on different conditions. Practitioners can
safely apply EOMM while gathering more information about their
game and players.
Figure 2: Predicted win probability vs. real win probability.
Real win probability is the ratio of matches, with similar pre-
dicted winning probabilities, whose outcomes are real “Wins”.
5. CASE STUDY
To test the proposed matchmaking framework, we ran simulation
which is configured based on the real data from a popular PvP game
made by Electronic Arts Inc.(EA). In the simulation, we compared
different matchmaking methods applied to the same player popu-
lation. In the end, EOMM retained significantly higher number of
players than other matchmaking methods.
5.1 Data Collection
We collected 1-vs-1 matches from a popular game made by EA.
There are three possible match outcomes, namely Win,Lose and
Draw. In total, we collected 36.9 million matches played by 1.68
million unique players in the first half of 2016.
5.2 Preparation
To create a realistic environment for simulation, the following
models and functions are needed. We compute them based on real
game data.
Player Skills We need to establish a distribution of player skills
for the population we simulate on. The distribution is learned from
real game data. We sorted the collected real matches temporally
and applied Glicko [20] to compute each player’s final skill. For
each player i, the skill vector is represented by mean riand vari-
ance RDi, i.e. µi= (ri, RDi). In simulation, we assume that the
game and player skills are stationary. The population’s skill distri-
bution is constant, where each player’s skill does not change any
more over time.
While Glicko scores can be used to estimate the winning prob-
ability of player iover player j,Pr(i > j|µi,µj), they cannot
provide the probability of draws. We defined a set of rules to allow
the estimation of win/lose/draw probabilities from Glicko scores:
Pr∗(i=j) = 20% (11)
Pr∗(i > j) = 80% ·Pr(i > j|µi,µj)
Pr(i > j|µi,µj) + Pr(j > i|µj,µi)(12)
Pr∗(i < j) = 1 −Pr∗(i=j)−Pr∗(i > j)(13)
Figure 3: Predicted churn risk vs. real churn risk. Real churn
risk is the ratio of matches, with similar predicted churn risks,
which are indeed the last match before churn.
Basically, the draw probability (Eqn. 11) is set to 20% regardless
of skill gaps. This is based on our findings that 1) draw outcomes
only have −0.05 correlation with the difference of skill means in
the collected game data; 2) around 20% matches are draws re-
gardless of skill gaps. The win/lose probabilities are normalized
such that the probabilities of win, lose and draw sum up to 1. Fig-
ure 2 shows that the predicted win probabilities using Glicko scores
based on our rules are well aligned with the real match outcomes.
Churn Prediction Model We trained a logistic regression model
for predicting whether a player will be an eight-hour churner after
a match. The input features describe the upcoming match and the
player’s 10 most recent matches. A player is labeled as an eight-
hour churner if they do not play any 1-vs-1 match within the next
eight hours after playing this match. As discussed in Section 3,
the term of “churn” is used by convention. It represents “stopping
playing” within a period of time, which is a metric of disengage-
ment.
We use Eqn. 7 to estimate c(si,sj)+c(sj,si). The model takes
as input the player’s state sibefore matchmaking along with the
upcoming match outcome oi,j .
Specifically, the input features consist of:
•Each of the player’s 10 most recent matches: win/lose/draw
status, time passage since the previous match, time passage
to the upcoming match, and goal difference against his op-
ponent
•Upcoming match: one-hot encoding of the upcoming match’s
outcome win/lose/draw
•Other: the number of 1-vs-1 matches played in the last eight
hours, one day, one week and one month.
We use 5-fold cross validation and grid search to determine the
proper L2regularization strength when training the model. The
predicted probabilities are well aligned with the real churn prob-
abilities, in particular when churn risk is less than 0.8, as shown
in Figure 3. While the performance of the predictive model still
has room to improve, the flexibility of EOMM allows one to easily
refine or replace the model if better ones are found.
Player States In simulation, each player’s state is sampled from
a collection of states, which are established based on real players’
states in the collected data. We first randomly sample a subset of
matches. Both players’ states in those matches are gathered to cre-
ate this collection. A player state contains the needed features for
churn prediction, as well as the player’s skill score.
5.3 Simulation Procedure
In the simulation, we compared EOMM with three matchmak-
ing mechanisms: random matchmaking (RandomMM), which ran-
domly pairs available players in the waiting pool, skill-based match-
making (SkillMM), which pairs every two consecutive players after
sorting them by skills, and worst matchmaking (WorstMM), which
does the opposite of EOMM by minimizing the objective func-
tion of EOMM. SkillMM always seeks “fair games”. We added
WorstMM as a validation.
All methods are applied on the same population (waiting pool),
where the same player skill distribution, churn model and player
state distribution as described in Section 5.2 are used. EOMM fol-
lows Eqn. 7 to estimate churn risk c(si,sj) + c(sj,si). We used
the perfect matching algorithm [18, 26] implemented by an open-
source library [42].
For each matchmaking method M, the procedure within each
round of simulation is as follows:
1. Create a waiting pool of Pplayers, whose player states are
sampled from the player state collection.
2. Use Mto determine the pair assignment (matchmaking).
3. Simulate match outcomes according to the win/lose/draw prob-
ability predicted by the skill model
4. For each player, simulate if he will churn according to the
predicted churn probability by churn model.
5. Record the number of retained players.
In experiments, we tested P= 100,200,300,400 and 500. For
each setting of P, we repeated the simulation by 10,000 rounds
of matchmaking. We compare different matchmaking methods by
the average number of their retained players per round, i.e., the
players who continue playing in the next eight hours. In order to
test statistical significance, we conducted Welch’s t-test between
every pair of the matchmaking algorithms.
5.4 Results and Discussion
The results are shown in Table 2. All pairwise differences of re-
tained players are statistically significant (p-value <0.01) except
EOMM vs. RandomMM (when P= 100) and SkillMM vs. Ran-
domMM (when P= 400). In all other scenarios, EOMM outper-
forms the other three matchmaking methods. The results prove the
applicability of EOMM to act as an engagement optimizer. When
P= 100, EOMM does not retain a significantly higher number
of players than RandomMM, and even retains fewer players than
SkillMM. It is possibly because that when Pis small, the random-
ness has higher impact, and also, the room for arranging opponents
is smaller. More rounds of simulations might be needed to show
significance in this case.
The improvement of EOMM over SkillMM, the most common
matchmaking method, in terms of the average number of retained
players are 0.3%,0.9%,1.1%, and 0.6% when P= 200,300,400
and 500 respectively. On average EOMM retains 0.7% more player
compared with SkillMM after one round of matchmaking. No-
tably the benefit of retention will accumulate over time in a con-
stant population. For players who play 20 rounds of matchmaking
Table 2: Average number of retained players per round of
matchmaking simulation. 10,000 rounds of matchmaking were
simulated.
Method P=100 P=200 P=300 P=400 P=500
WorstMM 51.50 103.39 154.57 206.65 258.21
SkillMM 52.52 103.96 156.05 207.43 259.24
RandomMM 51.81 103.97 156.09 207.09 259.65
EOMM 51.90 104.24 157.50 209.37 261.19
games within eight hours, there will be 15% more players retained
(1.00720 ≈1.15) by EOMM over those by SkillMM. The more
rounds of matchmaking are conducted, the more significant is the
accumulative advantage of EOMM in engagement.
We did not find a consistent climb in retention boost as Pin-
creased. This may suggest that when the player pool reaches certain
size, the choices of opponents are enough to rescue those players
on the edge of churn. Beyond this size, a larger player pool may not
bring in significantly extra benefits in engagement maximization.
As a validation, WorstMM consistently retains the fewest play-
ers in the pools of all sizes. This result verifies the optimum of
EOMM from the opposite side. It is also interesting to note that
SkillMM does not consistently outperform RandomMM, which is
aligned with our discussion in the theoretical findings in Section 4,
that is, balanced matches are not always optimal for engagement.
6. CONCLUSION
The paper presents a novel framework to achieve engagement
optimized matchmaking (EOMM). It formulates matchmaking as
a problem of maximizing the player engagement, and solves the
optimization efficiently. EOMM employs three components, a skill
model, an engagement predictive model and a minimum weight
perfect matching algorithm, each of which can be tailored flexibly
for specific applications. We ran simulations whose configurations
were based on real data from an online PvP game. The results
show that EOMM significantly outperforms all other methods in
the number of retained players. EOMM also provides a theoretical
framework to analyze various matchmaking algorithms.
EOMM provides a measurable and flexible matchmaking frame-
work. It has well-defined quantitative objectives that can be mon-
itored, evaluated and improved. Within the EOMM framework,
the core building components, skill model, churn model and graph
pairing model, are uncoupled so that they can be tuned and replaced
independently. Moreover, we can even change the objective func-
tion to other core game metrics of interest, such as play time, re-
tention, or spending. EOMM allows one to easily plug in different
types of predictive models to achieve the optimization.
So far we have discussed EOMM in 1-vs-1 game scenarios. This
framework also applies to PvP games that involve teams of players,
where every component needs to be extended with additional care.
The skill model can be simply applied to a team by adding up skills
for all team members [23]. For churn prediction, we can use the
same idea that one player’s churn risk is conditionally independent
with other players’ states given that their influence on the player’s
own state, such as the game outcome, is known. Last, the minimum
weight perfect matching algorithms for pairs are no longer applica-
ble. Instead of a pair assignment, we seek a grouping assignment
on a complete graph. A related area to investigate is perfect match-
ing in hypergraphs [5], where an edge can connect more than two
vertices. Furthermore, EOMM is not even limited to games. In
broad applications, such as friend connection in a social network
and 1-on-1 tutoring in online education, EOMM’s formulation and
optimization techniques still apply.
In the future, we expect EOMM equipped with more advanced
models, such as skill model and churn model, can have higher op-
timal bound. We will explore EOMM performance in more realis-
tic situations, where practical restrictions are applied, such as net-
work connectivity, regions and friend/black lists. More restrictions
would result in fewer edges in the constructed graph of EOMM.
Last, we will explore EOMM in multi-player games with more than
two players involved and efficient algorithms analogous to perfect
matching algorithms within hypergraphs.
7. ACKNOWLEDGMENTS
This work is partially supported by NSF Career #1453800.
8. REFERENCES
[1] S. Agarwal and J. R. Lorch. Matchmaking for online games
and other latency-sensitive P2P systems. In ACM SIGCOMM
Computer Communication Review, volume 39, pages
315–326. ACM, 2009.
[2] A. Agresti and M. Kateri. Categorical data analysis.
Springer, 2011.
[3] T. Avontuur, P. Spronck, and M. Van Zaanen. Player Skill
Modeling in Starcraft II. In AAAI Conference on Artificial
Intelligence and Interactive Digital Entertainment (AIIDE),
2013.
[4] C. E. Bell. Weighted matching with vertex weights: An
application to scheduling training sessions in NASA space
shuttle cockpit simulators. European Journal of Operational
Research, 73(3):443–449, 1994.
[5] C. Berge. Hypergraphs: combinatorics of finite sets,
volume 45. Elsevier, 1984.
[6] R. Bernhaupt. User experience evaluation in entertainment.
In Evaluating User Experience in Games, pages 3–7.
Springer, 2010.
[7] R. A. Bradley and M. E. Terry. Rank analysis of incomplete
block designs: I. The method of paired comparisons.
Biometrika, 39(3/4):324–345, 1952.
[8] S. Chen and T. Joachims. Modeling intransitivity in matchup
and comparison data. In Proceedings of the Ninth ACM
International Conference on Web Search and Data Mining,
pages 227–236. ACM, 2016.
[9] Z. Chen, Y. Sun, M. Seif El-Nasr, and T.-H. D. Nguyen.
Player skill decomposition in multiplayer online battle
arenas. In Meaningful Play, 2016.
[10] O. Delalleau, E. Contal, E. Thibodeau-Laufer, R. C. Ferrari,
Y. Bengio, and F. Zhang. Beyond Skill Rating: Advanced
Matchmaking in Ghost Recon Online. IEEE Transactions on
Computational Intelligence and AI in Games, 4(3):167–177,
sep 2012.
[11] G. Di Fatta, G. M. Haworth, and K. W. Regan. Skill rating by
bayesian inference. In IEEE Symposium on Computational
Intelligence and Data Mining (CIDM), pages 89–94. IEEE,
2009.
[12] D. E. Drake and S. Hougardy. A simple approximation
algorithm for the weighted matching problem. Information
Processing Letters, 85(4):211–213, 2003.
[13] R. Duan and S. Pettie. Linear-time approximation for
maximum weight matching. Journal of the ACM (JACM),
61(1):1, 2014.
[14] J. Edmonds. Maximum matching and a polyhedron with 0,
l-vertices. Journal of Research of the National Bureau of
Standards, 69(1965):125–130, 1965.
[15] J. Edmonds. Paths, trees, and flowers. Canadian Journal of
Mathematics, 17(3):449–467, 1965.
[16] A. E. Elo. The rating of chessplayers, past and present. Arco
Pub., 1978.
[17] J. Ferreira, M. B. Vellasco, M. A. C. Pacheco, R. Carlos, and
H. Barbosa. Data mining techniques on the evaluation of
wireless churn. In European Symposium on Artificial Neural
Networks, Computational Intelligence and Machine
Learning (ESANN), pages 483–488, 2004.
[18] H. N. Gabow. Implementation of algorithms for maximum
matching on nonbipartite graphs. In PhD Thesis. Stanford
University, 1974.
[19] H. N. Gabow. A scaling algorithm for weighted matching on
general graphs. In 26th Annual Symposium on Foundations
of Computer Science, pages 90–100. IEEE, 1985.
[20] M. E. Glickman. Parameter estimation in large dynamic
paired comparison experiments. Applied Statistics, pages
377–394, 1999.
[21] T. Graepel and R. Herbrich. Ranking and matchmaking.
Game Developer Magazine, 25:34, 2006.
[22] F. Hadiji, R. Sifa, A. Drachen, C. Thurau, K. Kersting, and
C. Bauckhage. Predicting player churn in the wild. In IEEE
Conference on Computational Intelligence and Games
(CIG), pages 1–8. IEEE, 2014.
[23] R. Herbrich, T. Minka, and T. Graepel. Trueskill™: A
bayesian skill rating system. In Advances in Neural
Information Processing Systems, pages 569–576, 2006.
[24] T.-K. Huang, C.-J. Lin, and R. C. Weng. A generalized
Bradley-Terry model: From group competition to individual
skill. In Advances in Neural Information Processing Systems,
pages 601–608, 2004.
[25] J. Jiménez-Rodrıguez, G. Jiménez-Dıaz, and B. Dıaz-Agudo.
Matchmaking and case-based recommendations. In 19th
International Conference on Case Based Reasoning, 2011.
[26] E. L. Lawler. Combinatorial optimization: networks and
matroids. Courier Corporation, 2001.
[27] Y. Lee, S. Agarwal, C. Butcher, and J. Padhye. Measurement
and estimation of network QoS among peer Xbox 360 game
players. In International Conference on Passive and Active
Network Measurement, pages 41–50. Springer, 2008.
[28] J. Manweiler, S. Agarwal, M. Zhang, R. Roy Choudhury,
and P. Bahl. Switchboard: a matchmaking system for
multiplayer mobile games. In Proceedings of the 9th
International Conference on Mobile Systems, Applications,
and Services, pages 71–84. ACM, 2011.
[29] J. E. Menke and T. R. Martinez. A Bradley–Terry artificial
neural network model for individual ratings in group
competitions. Neural Computing and Applications,
17(2):175–186, 2008.
[30] M. Minotti. Comparing MOBAs: League of Legends vs.
Dota 2 vs. Smite vs. Heroes of the Storm.
http://venturebeat.com/2015/07/15/compa
ring-mobas- league-of- legends- vs-dota- 2- v
s-smite- vs-heroes- of- the-storm/. Online;
accessed May, 2016.
[31] K. Morik and H. Köpcke. Analysing customer churn in
insurance data–a case study. In European Conference on
Principles of Data Mining and Knowledge Discovery, pages
325–336. Springer, 2004.
[32] M. My´
slak and D. Deja. Developing game-structure sensitive
matchmaking system for massive-multiplayer online games.
In Social Informatics, pages 200–208. Springer, 2014.
[33] T.-H. D. Nguyen, Z. Chen, and M. S. El-Nasr.
Analytics-based AI Techniques for Better Gaming
Experience, volume 2 of Game AI Pro. CRC Press, Boca
Raton, Florida, 2015.
[34] S. Ólafsson. Weighted matching in chess tournaments.
Journal of the Operational Research Society, 41(1):17–24,
1990.
[35] C. N. Osiakwan and S. G. Akl. The maximum weight perfect
matching problem for complete weighted graphs is in PC. In
Proceedings of the Second IEEE Symposium on Parallel and
Distributed Processing, pages 880–887. IEEE, 1990.
[36] E. A. Riskin, R. Ladner, R.-Y. Wang, and L. E. Atlas. Index
assignment for progressive transmission of full-search vector
quantization. IEEE Transactions on Image Processing,
3(3):307–312, 1994.
[37] J. Runge, P. Gao, F. Garcin, and B. Faltings. Churn
prediction for high-value players in casual social games. In
IEEE Conference on Computational Intelligence and Games,
pages 1–8. IEEE, 2014.
[38] M. Stanescu. Rating systems with multiple factors. In Master
Thesis. School of Informatics, University of Edinburgh,
2011.
[39] SuperData. eSports market brief: US accounts for almost
half of total viewership. https://www.superdatares
earch.com/blog/esports-brief/. Online;
accessed Mar, 2016.
[40] M. Suznjevic, M. Matijasevic, and J. Konfic. Application
context based algorithm for player skill evaluation in MOBA
games. In International Workshop on Network and Systems
Support for Games (NetGames), pages 1–6. IEEE, 2015.
[41] P. Tassi. Riot’s ’League of Legends’ reveals astonishing 27
million daily players, 67 million monthly.
http://www.forbes.com/sites/insertcoin/
2014/01/27/riots-league- of-legends- revea
ls-astonishing- 27-million- daily- players
-67- million-monthly/#26ff8e543511. Online;
accessed May, 2016.
[42] J. van Rantwijk. Maximum Weighted Matching. http:
//jorisvr.nl/article/maximum-matching.
Online; accessed May, 2016.
[43] B. G. Weber, M. John, M. Mateas, and A. Jhala. Modeling
player retention in Madden NFL 11. In Innovative
Applications of Artificial Intelligence (IAAI), 2011.
[44] D. B. West. Introduction to graph theory. Prentice Hall,
2001.
[45] G. Yannakakis, P. Spronck, D. Loiacono, and E. André.
Player modeling. In Artificial and Computational
Intelligence in Games, pages 45–59. 2013.
[46] S. Yoon, J. Koehler, and A. Ghobarah. Prediction of
advertiser churn for Google AdWords. In the Joint Statistical
Meetings (JSM) Proceedings, 2010.
