A Game Generative Network Framework with Its Application to Relationship Inference

A Game Generative Network Framework with Its Application to Relationship
 Inference
 Jie Huang1 , Fanghua Ye2 , Xu Chen3
 1
 University of Illinois at Urbana-Champaign
 2
 University College London
 3
 Sun Yat-sen University
 jeffhj@illinois.edu, smartyfh@outlook.com, chenxu35@mail.sysu.edu.cn

 Abstract
arXiv:2001.03758v2 [cs.SI] 30 Mar 2021

 relationships, either explicit or implicit. Thus it is possible
 for us to generate a network for a game process and mine the
 A game process is a system where the decisions relationships between agents on the network. Although there
 of one agent can influence the decisions of other are a large number of studies on social network mining, none
 agents. In the real world, social influences and re- can be directly applied to social agents with game behaviors.
 lationships between agents may influence the deci- Generating networks for game processes will no doubt open
 sion makings of agents with game behaviors. And a door for mining hidden information of agents with game
 in turn, this also gives us the opportunity to mine behaviors.
 such information from agents by the observed in- In addition, relationships between agents in game pro-
 teractions of them in a game process. In this paper, cesses are usually more complex than common relationships
 we propose a Game Generative Network (GGN) modeled by traditional networks. In addition to positive rela-
 framework that utilizes the deviation between the tionships, there may also be some negative relationships be-
 real game outcome and the ideal game model to tween agents in a game system, which greatly influence the
 generate networks for game processes, which opens strategies taken by agents. In a game process, positive and
 a door for understanding agents with game behav- negative relationships reflect in many different ways. For in-
 iors by network mining approaches. We illustrate stance, a positive relationship may represent friendship, coop-
 how to apply GGNs to infer the hidden relation- eration, trust or win-win and a negative relationship may rep-
 ships between agents with game behaviors and con- resent hostility, competition, distrust or loss-loss [Guha et al.,
 duct experiments on team games as a concrete ex- 2004; Brzozowski et al., 2008; Pang et al., 2008]. Compared
 ample. Experimental results demonstrate that our to traditional networks that reduce the relationships between
 proposed framework can reveal the hidden relation- objects as simple pairwise links, signed networks [Tang et
 ships of agents in such games. al., 2016] in which the weights of edges can be positive and
 negative are better to represent such positive and negative re-
 1 Introduction lationships between agents in game processes.
 In the real world, there are a variety of systems with coop- Based on the above considerations, we propose a general
 eration and competition behaviors among agents. Such be- Game Generative Network (GGN) framework, which gener-
 haviors are usually modeled as games where the decisions ates signed networks for game processes based on the de-
 of one agent can influence the decisions of other agents [Os- viation between the real game outcome and the ideal game
 borne and Rubinstein, 1994; Myerson, 2013], and the deci- model. Our assumption for such games is that if there are
 sions of agents may also be affected by the relationships be- some relationships between agents, the relationships will af-
 tween them [Chen et al., 2014]. However, due to the com- fect the strategies that agents take. In general, GGNs serve
 plexity of relationships between agents – difficult to obtain as the intermediaries between game processes and network
 and model, most game-theoretic models do not consider re- mining approaches. In this paper, we mainly focus on the re-
 lationships between agents comprehensively or even ignore lationship inference task. In spite of the generality of GGNs
 such information. If we want to better understand agents with with various potential applications, we first give a simple il-
 game behaviors, it is necessary for us to mine the social influ- lustration of the Prisoners’ Dilemma [Rapoport et al., 1965],
 ences and relationships between them. Also, by incorporating and then introduce a game generative network mining ap-
 such information, we can build game models that better char- proach and apply it to infer the relationships between social
 acterize agents’ real game behaviors. agents in team games.
 On the other hand, the pattern of connections and relation- The main contributions of this paper are summarized as
 ships between objects in a system can be represented as a follows:
 network [Newman, 2003], where the objects correspond to • We appear to be the first to mine positive and negative
 nodes and the relationships between objects correspond to relationships between social agents with game behaviors
 edges. A game is a system of multiple agents with special from game processes.

 1

• We propose a Game Generative Network framework, to build the edges of the network, the weight of each edge is
 which opens a door for analyzing hidden relationships formulated as follows:
 of agents with game behaviors via network mining ap- (t)
 proaches. w(ei,j ) = uj (s0 ) − uj (s), (2)
 • We study team game and apply our framework to it. where uj (s) is the utility of agent pj with strategy list s, and
 Experimental results demonstrate the feasibility of our (t)
 ei,j is the directed edge from pi to pj built at time t. Re-
 GGN framework on revealing relationships between peating the above process for T times, we will end up with
 agents in such games. a directed signed network containing the interaction informa-
 The code and data are available at https://github.com/jeffhj/ tion of agents.
GGN.
 2.3 General Game Generative Network
2 Game Generative Network Framework
 For the special case in the previous section, it is natural to
2.1 Preliminaries generate a network implying the relationship information of
Game Theory agents. However, in most cases, we cannot get the whole
Game theory is a tool to analyze decision making in multi- game process. This is to say, in a real game, we may only
agent interactions. Based on different analyzing methods, get the final outcome, i.e., the real strategies taken by agents.
there are different types of game models designed for dif- Besides, such dynamic games are not universal. So a more
ferent specific games. A game denoted as Γ usually includes general game generative network framework is necessary.
three parts: the set of agents P, the strategy space S, and Most existing game-theoretic models assume that all the
the utility function u : S → R. If the number of agents in agents are in selfish behaviors, where each agent aims at max-
a game is n, we call it n-agent or n-player game. A (pure) imizing her own utility. Such assumptions ignore social influ-
Nash equilibrium in an n-agent game is a list of strategies ences and relationships between agents. Besides, sometimes
s∗ = {s∗1 , s∗2 , . . . , s∗n } such that relationships among agents will cause a deviation in the util-
 ity of each agent. On one hand, most game models either do
 s∗i = argmax ui (s∗1 , s∗2 , . . . , si , . . . , s∗n ), (1) not consider relationships between agents or do not consider
 si
 such information comprehensively. One the other hand, such
where si is the strategy taken by agent pi . In other words, information of agents is also difficult to obtain and hard to
Nash equilibrium is a stable strategy list that neither agent characterize by game models.
can increase her payoff by taking another strategy, thus no As discussed above, an ideal game model is based on the
agent will try to change her strategy. assumption that each agent is selfish and usually does not
 consider the complete relationships between agents, but the
Signed Network real outcome does not follow such assumptions and reflects
A signed network is defined as a graph G(V, E, w), where V some unobserved relationships. For instance, from an idea
is the node set, E is the edge set, and w : E → R is a weight game model, we may find that both agent p1 and agent p2 can
mapping function associated with each edge and the weight achieve a higher payoff by reducing the other’s payoff, but
can be either positive or negative. the real outcome shows that they did not do so (we will use
 Prisoners’ Dilemma as a concrete example in the following
2.2 A Special Case: Dynamic Game section). This may be because there is a positive relationship
We first study a special type of game named dynamic game. between them, and such a relationship will be reflected in the
Dynamic game is a kind of game in which decisions of agents deviation between the real game outcome and the ideal game
are made at various times with some of the earlier decisions model. Therefore, in order to generate networks for game
being public knowledge when the later decisions are being processes, one practical approach is to use the deviation be-
made. Here we consider the n-agent dynamic game with tween the real game outcome and the ideal game model.
agent set P = {p1 , p2 , . . . , pn }, where each agent has multi- Assume the real strategy list is sr =
ple opportunities to change her strategies and the payoff up- {sr1 , sr2 , . . . , sri , . . . , srn }. For agent pi , we can predict
dates after each strategy changes. the strategy s∗i taken by pi from an ideal game model.
 In order to involve the interactions between agents, we By replacing sri with s∗i , we get a new strategy list
build a network with node set V = P. The original strategy s0 = {sr1 , sr2 , . . . , s∗i , . . . , srn }. And then we use the utility
list of each agent at time t − 1 is s = {s1 , s2 , . . . , si , . . . , sn }. difference before and after the conversion of each agent to
At time t, agent pi changes her strategy from si to s0i , which build the edges of the network, the weight of each edge is
converts s to s0 = {s1 , s2 , . . . , s0i , . . . , sn }. Since the strate- formulated as follows:
gies made by agents in dynamic games are based on some w(ei,j ) = uj (sr ) − uj (s0 ). (3)
public knowledge, agents are aware of others’ strategies and
can make decisions according to such knowledge. Thus re- There are two ways to choose s∗i : one way is to use the
lationships between agents may be reflected dynamically in strategy in a pure strategy Nash equilibrium. If there are mul-
such interactions. Based on this observation, we use the util- tiple pure strategy Nash equilibria, we can also build multiple
ity difference before and after the conversion of each agent networks. But the concern is that the Nash equilibrium may

 2

For example, suppose that the real outcome of the Prison-
 2 4
 -1 1 2 ers’ Dilemma game happens as follows: p1 chooses to keep
 Q S quiet and p2 chooses to squeal. Based on the framework of
 1 Q -2, -2 -5, 0 GGN, we can generate a network of p1 and p2 in Fig. 1 (upper
 right). For example, w(e1,2 ) = u2 ({Q, S}) − u2 ({S, S}) =
 S 0, -5 -4, -4 3 0 − (−4) = 4. From the network, we can conclude that p1
 -2 1 2 -2
 has a positive relationship to p2 , that is p1 is doing favor to-
 wards p2 . Another possible situation is that both p1 and p2
 Figure 1: The illustration of Prisoners’ Dilemma. keep quiet. From the GGN illustrated in Fig. 1 (lower right),
 we can conclude that there is a strong relationship between
 p1 and p2 so that they believe in each other. These exam-
 ples illustrate that GGNs can be useful to reveal relationships
be hard to find and pure strategy Nash equilibrium does not between agents.
exist in some cases. The second way is to choose the strategy
that maximizes the utility of pi as follows: 3.2 Game Generative Network Mining
 For small GGNs (e.g., n ≤ 10), we can conclude the rela-
 s∗i = argmax ui (sr1 , sr2 , . . . , s# r
 i , . . . , sn ). (4) tionships between agents by observation or simple statistical
 s#
 i analysis. For big GGNs, it is necessary to apply an auto-
 Based on the above framework, we can generate signed matic network mining method. From our definition above,
networks for most game processes, namely Game Generative a GGN is essentially a signed network. With such cogni-
Networks (GGNs). tion, the problem of mining for GGNs is indeed the prob-
 lem of mining for signed networks [Leskovec et al., 2010a;
 Kunegis et al., 2009; Tang et al., 2016]. Since the focus of
3 Relationship Inference on Game Generative this paper is on exploring the capabilities of GGN, we design
 Network a relatively simple and intuitive method to mine for GGN and
To show how to apply GGNs to solve downstream tasks, we observe the power of GGN by using this method.
study relationship inference with GGNs. Relationship infer- The most direct relationship between agents in GGNs is
ence aims to infer the relationships between agents. And the first-order proximity. For each pair linked by an edge
specifically, in this paper, we aim at judging whether the rela- eu,v with weight wuv , if wuv > 0, eu,v represents a positive
tionship between two agents in a game is positive or negative. relationship from u to v; if wuv < 0, eu,v represents a nega-
In some games, the relationships between agents may influ- tive relationship from u to v; otherwise, there is no first-order
ence agents’ decision makings. For instance, some positive proximity between u and v represented by eu,v . Since not
relationships (e.g., friendship, cooperation, trust, and win- all the edges generated are meaningful, we should filter out
win) and negative relationships (e.g., hostility, competition, some edges based on specific cases. For instance, given a re-
distrust, or loss-loss) may be involved in game processes, and lationship network of agents, we have prior knowledge about
the utilities of agents are affected by such relationships. But whether two agents know each other, thus the edges beyond
most of the time, we have no knowledge about relationships the edges in the relationship network, including self-loops,
between agents or the signs of such relationships. Thus re- should be filtered out.
lationship inference on social agents with game behaviors is Besides, the first-order proximity cannot represent all rela-
quite interesting and useful if we want to further understand tionships between agents, and because the behaviors of agents
agents’ behaviors and the relationships between them. are not fully reflected in a game, some edges may be missed.
 Based on the multiplicative transitivity [Kunegis et al., 2009]
3.1 Prisoners’ Dilemma of signed networks, if there is an edge with weight wuv be-
Here we give a simple illustration of relationship inference tween u and v, and there is an edge with weight wvx be-
with GGNs. We study the famous problem of Prisoners’ tween v and x, then there may be a hidden third edge with
Dilemma [Rapoport et al., 1965]. The problem is defined as weight wuv · wvx between u and x. Multiplicative transitivity
follows: two criminals (p1 and p2 ) are imprisoned and they is proved by the fact that two agents connected by an even
cannot talk to each other. The police offer each prisoner a number of negative edges can be considered balance [Hage
bargain – keep quiet (Q) or to squeal (S). The payoff table of and Harary, 1983].
these two criminals is shown in Fig. 1 (left). For instance, if Based on the multiplicative transitivity, we introduce the
p1 chooses strategy S and p2 chooses strategy Q, p1 will be exponential kernel to evaluate the kth-order relationships be-
released and p2 will be imprisoned for 5 years. tween agents. The kth-order exponential kernel is defined as
 In the ideal game model where the relationship between the weighted sum of matrix powers, where the weight decays
these two agents is ignored, Nash equilibrium of Prisoners’ with the inverse factorial:
 k
Dilemma is that both criminals choose to squeal thus both of X 1 i
them will be imprisoned for 4 years. But if we consider the expk (A) = A. (5)
 i!
relationship between p1 and p2 , the real strategies they take i=0
may not be consistent with the Nash equilibrium of an ideal The sign of the (i, j)-th entry of the exponential kernel in-
model. dicates the positive or negative relationship from i to j. The

 3

Real
 Outcome
 Game Mining
 Process
 (Black Box)

 Ideal Game Ideal
 Model Outcome
 Social Individuals Game Generative Network Relationship Inference

Figure 2: The process of relationship inference with GGN, where GGN serves as an intermediary that combines game processes and network
mining approaches. The color of positive edges in the GGN is red and the color of negative edges is blue.

greater the absolute value, the stronger the relationship. Com- Algorithm 1: Team Game(G)
bining with the generative process of GGN, the process of
relationship inference with GGN is illustrated in Fig. 2. Initialize each node to a singleton team;
 while not converge do
 Random shuffle the nodes;
4 Game Generative Network for Team Game for i from 1 to n do
In spite of the generality of GGNs that can apply to a vari- si = argmaxs∗i ui ({s1 , s2 , . . . , s∗i , . . . , sn });
ety of games, for an empirical study on a specific game, we Node i joins the new team by taking strategy
will show how to apply GGNs to the team game for rela- si ;
tionship inference as a concrete example. The goal of team
game is to divide n agents into small groups. The utility of
each agent in team games is affected by the relationships be- However, when most of the relationships of agents are pos-
tween agents. Relationships between agents in team games itive, such gain function will lead to a trivial solution where
have a broad definition. For instance, a positive or negative all the agents form a single team. To solve this problem, we
relationship can represent a friend or a foe relation between consider that the number of members in each group should
two agents. In this case, everyone wants to team up with her be as balanced as possible, which is consistent with the fact
best friends and tries not to team up with those she dislikes. that a team with a large size will weaken the relationships. To
On the other hand, relationships in team games can evaluate this end, we define the loss function as li (s) = c|t(i)|, where
whether the cooperation between two agents will bring posi- |t(i)| is the size of the team which agent i belongs to and c
tive (win-win) or negative (loss-loss) effects. And in this case, is a parameter to balance the gain and the loss. Finally, the
everyone wants to team up with a group of agents that make utility of agent i with strategy list s is calculated as follows:
her obtain the biggest profits.
 n
 X
4.1 Team Game Definition ui (s) = gi (s) − li (s) = Aij δ(i, j) − c|t(i)|. (7)
 j=1,j6=i
In this section, we design an ideal game model for the team
game. Consider an undirected signed network G(V, E, w)
with the node (agent) set V, the edge set E and the weight Existence of Nash Equilibria We will prove that the team
w(e) associated with each edge e. In our settings here, team game we designed is a finite exact potential game which al-
game aims to find a set of teams and each node has one and ways possesses pure strategy Nash equilibria [Monderer and
only one team, and we use t(i) to denote the team which node Shapley, 1996]. Firstly, let us recall the definition of the ex-
i belongs to. The strategy of agent i is to quit the current team act potential game. A game is an exact potential game if
t(i) and join a new team t0 (i). there exists an associated potential function Φ(·) defined on
 From the above analysis, we can conclude that each agent the strategy profiles that satisfies Φ(s0i , s−i ) − Φ(si , s−i ) =
wants to team with those agents who have positive relation- ui (s0i , s−i ) − ui (si , s−i ) for every strategy profile s−i of all
ships with her, thus we define the gain function as follows: agents except agent i and every strategy si of agent i. In an
 exact potential game that contains a finite number of strategy
 n
 X profiles, Nash equilibria always exist. And every better re-
 gi (s) = Aij δ(i, j), (6) sponse in which each agent sequentially changes her strategy
 j=1,j6=i to improve her own utility, will finally converge to a Nash
where n is the number of nodes, Aij is the (i, j)-th entry equilibrium [Monderer and Shapley, 1996].
of the weighted adjacent matrix A of the signed network Theorem 1. The team game with utility function ui (·) is
G(V, E, w) above, and δ(i, j) is an indicator function which an exact potential game with potential function Φ(s) =
 P n 1
is equal to 1 when t(i) = t(j) and 0 otherwise. i=1 2 (gi (s) − li (s)).

 4

1.0
 Datasets Nodes Edges + edges - edges Slashdot1
 0.9 Slashdot2
 Slashdot1 3,869 93,498 77,052 16,446

 Average accuracy
 Epinions1
 Slashdot2 3,872 27,298 20,134 7,164 0.8 Epinions2
 Epinions1 6,605 182,674 158,170 24,504
 Epinions2 6,591 36,992 32,832 4,160 0.7

 0.6

 Table 1: Statistics of the datasets.
 0.5

 29 27 Random GGN
 24 26 331418
 25 32 22 15
 31 28 8 30
 2
 19 13 20
 17
 1
 0 7 3
 9 25 Figure 4: Average accuracy of relationship inference in the team
 11 21 12
 6 45 10 game.
 16

 23
 29
 but we do not know which kind of relationship (e.g., posi-
 27 26 tive or negative) between them, we assume that the network
 23
 24 G we get of n agents is unsigned and unweighted, where an
 edge between u and v means that u and v know each other.
 (a) (b) Given the network of the agents and the results of their de-
 cision makings, we can build a network in the framework of
Figure 3: The network structure (a) and the GGN (b) of case study. GGN.
In the GGN, the color of positive edges is red and the color of nega- Based on the above considerations, we design the experi-
tive edges is blue. ments as follows: For a real network G 0 which is signed and
 weighted, we stimulate the team game on G 0 and generate
Proof. teaming results which can be considered as the real strategies
 the agents take. We use the teaming results and the ideal team
 Φ(s0i , s−i ) − Φ(si , s−i ) game model on G (without G 0 ) to generate GGN and calcu-
 n n late the kth-order exponential kernel of the GGN to infer the
 1 X X
 relationships. Since G 0 is the real network, the weighted adja-
 = ( Aij δ̂(i, j) − Aij δ(i, j))
 2 cency matrix of G 0 represents the real relationships between
 j=1,j6=i j=1,j6=i
 c agents, which can serve as the ground-truth to evaluate the
 − ((|t0 (i)| + 1)2 + (|t(i)| − 1)2 − |t0 (i)|2 − |t(i)|2 ) performance with respect to relationship inference.
 2
 We extract four datasets from Slashdot and Epinions
 =gi (s0i , s−i ) − gi (si , s−i ) − c(|t0 (i)| + 1) + c|t(i)| [Massa and Avesani, 2005; Kunegis et al., 2009; Leskovec
 =gi (s0i , s−i ) − gi (si , s−i ) − li (s0i , s−i ) + li (si , s−i ) et al., 2010b] (two from each one). Slashdot is a technology
 =ui (s0i , s−i ) − ui (si , s−i ). news website that lets users tag other users as friends and foes
 and Epinions is a product review website where users build
 links that indicate trust or distrust about other users. In order
 to make the team game converge, we build the network with
 The team game is a game with finite agents and finite strat- undirected edges. The statistics of these datasets are listed in
egy space, so the team game with utility function ui (s) is a Table 1.
finite exact potential game, thus possesses pure strategy Nash
equilibria. Case Study
 Based on the finite improvement property of potential We use Zachary’s Karate Club network [Zachary, 1977] as a
game, we propose an algorithm for computing the Nash equi- toy example in our experiments. Zachary’s Karate Club is a
librium of the ideal team game model, which is shown in well-known social network of a university karate club with
Algorithm 1. We firstly initialize each node to a singleton 34 nodes and 78 edges. We assume that node 23 has negative
team and then repeat the following process until the game relationships with all her neighbors and all other relationships
converges: random shuffle the nodes, and then let each node are positive, which leads to the outcome that no one wants
quit the current team and join the team which maximizes her to team with node 23. We stimulate the game process and
utility in the current state. generate the GGN. The structure of the real network and the
 generated GGN are illustrated in Fig. 3.
4.2 Experiments From the snapshot of the GGN in Fig. 3(b), we observe that
 node 25, 27, 29 have first-order negative relationships to node
In this section, we will show how to apply GGNs to conduct 23, and node 24, 26 have second-order negative relationships
relationship inference in team games. to node 23, which is consistent with the real network.
Experimental Setup Relationship Inference for Team Game
Based on the fact that most of the time we can only get the In this section, we evaluate the performance of mining on
information that there is a relationship between two agents, GGN in the relationship inference task. Since the number

 5

1.0
 0.4 5 Related Work

 Predictive percentage
Average accuracy

 0.9
 0.3
 In general, our work is related to game theory [Osborne and
 0.8
 0.2

 Slashdot1
 Rubinstein, 1994; Myerson, 2013] and social network anal-
 0.7 0.1 Slashdot2
 Epinions1
 ysis [Wasserman and Faust, 1994; Scott, 2017]. There are a
 Epinions2
 0.6
 2 4 6 8 10
 0.0
 2 4 6 8 10
 lot of studies on social network discovery, such as link anal-
 k k
 ysis [Getoor and Diehl, 2005; Liben-Nowell and Kleinberg,
Figure 5: Results of relationship inference in the team game when 2007] and community detection [Fortunato, 2010]. Although
c = 0.2. there are some research [Chen et al., 2010] applying game-
 theoretic models to network mining, no one, in turn, applies
 network mining to game systems. In addition, some work
 1.0
 0.4
 conducts relationship inference tasks on various social net-
 Predictive percentage

 works [Diehl et al., 2007] or studies the cooperation and
 Average accuracy

 0.9
 0.3

 0.8
 0.2
 competition among agents in game theory [Colman, 2003;
 0.7 0.1
 Slashdot1
 Slashdot2
 Barash, 2004], but none can infer the hidden relationships
 0.6 0.0
 Epinions1
 Epinions2 between agents with game behaviors. Owing to the frame-
 2 4
 k
 6 8 10 2 4
 k
 6 8 10
 work of GGN we designed, our work has a strong con-
 nection to signed network mining [Leskovec et al., 2010b;
Figure 6: Results of relationship inference in the team game when Leskovec et al., 2010a; Kunegis et al., 2009; Yang et al.,
c = 0.3. 2012; Tang et al., 2016]. From the perspective of economics,
 our framework is related to revealed preference [Samuelson,
 1948], which aims at inferring the preferences of individuals
of positive edges and negative edges is imbalanced in our with the observed choices.
datasets and for ease of understanding, we use average ac- Recently, Serrino et al. train a multi-agent reinforce-
curacy (the average accuracy of each class) as the evaluation ment learning agent to find friend and foe in The Resistance:
metric. We set c = 0.2 for the team game process and k = 2 Avalon, a hidden role game played by several players. How-
for the kth-order exponential kernel in our network mining ever, their model is just designed for similar hidden role
approach. Here we should mention that our task is brand new games with several players, and can not be applied to infer
and there is no baseline that uses the game outcome, i.e., the the relationships between a large number of social individu-
strategies taken by agents, as input to infer the relationships als with game behaviors.
between agents, so only random guessing can be compared in
our experiments. 6 Discussion and Conclusion
 From the results shown in Fig. 4, we have several observa- In this paper, we propose a novel game generative network
tions. First, the prediction based on the GGN framework far framework to build networks for game systems, which is a
outperforms random guessing, which indicates the feasibil- combination of game theory and network mining. In general,
ity of our model for mining the hidden relationships among GGNs sever as the intermediaries between game processes
agents with game behaviors. Second, comparing with Slash- and network mining approaches, which provides a new way
dot1 and Epinions1, Slashdot2 and Epinions2 are sparser, and to model and mine relationships between social agents with
the average accuracies of Slashdot2 and Epinions2 are higher game behaviors. To show how to apply GGNs to reveal the
than those of Slashdot1 and Epinions1 respectively. This is relationships of agents, we first give an illustration on Pris-
because, in the team game, the complexity of strategies taken oners’ Dilemma, and then introduce the team game as a con-
by users is related to the complexity of their relationships, a crete example and conduct experiments on it. Indeed, there
sparser network means that users’ strategies are easier to un- are many other potential applications of game generative net-
derstand, which is compatible with our results. works, such as mining the hidden features of agents by using
 To observe the results in different settings of team game, advanced technologies, e.g., graph neural networks (GNNs).
we also conduct experiments on team games with different c For future work, we plan to further improve the game gener-
values by different kth-order exponential kernels. The results ative network framework and apply it to more real applica-
are shown in Fig. 5 and Fig. 6, where predictive percentage tions.
is the proportion of non-zero values in the predicted edges.
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