CC BY
4Contributions to Game Theory and Management, XII, 316-324
Two-Stage Network Formation Game with Heterogeneous Players and Private Information*
Ping Sun1'2 and Elena Parilina1'2
1 St. Petersburg State University,
7/9 Universitetskaya nab., Saint Petersburg, 199034, Russia 2 School of Mathematics and Statistics and Institute of Applied Mathematics of Shandong, Qingdao University, Qingdao 266071, PR China E-mail: [email protected], [email protected]
Abstract We consider a two-stage network formation game with heterogeneous players and private information. The player set consists of a leader and a finite number of other common players, which are divided into two types, passive and positive players. At the first stage, the leader suggests a connected communication network for all players to join. While it is assumed that the link information which every common player receives from the leader is private. Based on the private information, every player chooses the action, accept or reject, at the second stage. A network is formed finally. We show the existence of subgame perfect Nash equilibrium in the game. The result is illustrated by an example.
Keywords: heterogeneous players, private information, Myerson value, subgame perfect Nash equilibrium.
1. Introduction
In recent years, network games, network stability, network formation as well as issues about communication networks have been widely studied. Jackson and Wolinsky (1996) initially proposed the concept of pairwise stability to characterize stable network in which the rule of network formation is called JW rule. Bala and Goyal (2000) mainly studied the Nash equilibrium network and its dynamic formation process, showing that the Nash network has special structures, such as the star or the wheel. Avrachenkov et al. (2011) addressed network formation issue using cooperative game theory and solve the cooperative network formation game with the Nash bargaining solution concept.
An important extensive research in network formation is to introduce heterogeneity. There are various types of heterogeneity, such as heterogeneous players, heterogeneous costs of forming links, heterogeneous information delivering qualities of links, etc. Heterogeneous players were introduced in (Larrosa and Tohme, 2003) where the payoff of each player is not only associated with the number of links in those paths, the end of which is such player, but also related with the values of himself, and the values of players are various. Galeotti et al. (2006) introduced heterogeneous costs of forming links as well as heterogeneous values in the two-way-flow model, and proved that centrality and short average distances between players are robust features of equilibrium networks.
* This work was supported by the Shandong Province "Double-Hundred Talent Plan" (No. WST2017009).
Besides, Petrosyan and Sedakov (2014) considered the multistage network games with perfect information in which players can change the network structure at each stage, and proposed a method for finding optimal behavior for players in games of this type. The endogenous dynamic formation of the network was introduced in (Aumann and Myerson, 2003) where an auxiliary linking game which consists of pairs of players being offered to form links while the offers are made one by one according to some chosen order of feasible links was constructed. And the linking game was with perfect information.
In practice, heterogeneous people are fairly common among the community, for instance, female and male, individuals with various education backgrounds, like bachelor, master, doctor, people from different countries, and so on. And it is also reasonable that various people have different standards and face various cases although in the same community, such as different levels of salary, getting different information about the community, etc. In this paper, we consider the game of incomplete information, and to simplify the complicated case brought by incomplete information, heterogeneous players are introduced simultaneously.
The paper is organized as follows. In Section 2 some basic definitions and notations are briefly introduced. And in Section 3, the model of two-stage network formation game with heterogeneous players and private information is introduced. Then the two-stage game introduced in Section 3 in extensive form is described in Section 4. Section 5 contains the theorem about the existence of subgame perfect Nash equilibrium in the game as well as a corresponding example.
2. Basic Definitions and Notations
Let the set of players be N = {1,... , n}, |N| = n > 3. Suppose there is a player called the leader of other players referred as player 1. A cooperative game with transferable utility is a pair (N, v), where v : 2N ^ IR is a characteristic function that assigns to every coalition of players S C N its worth v(S), with v(0) = 0. For
v
game (N, v). A singleton solution of game (N, v) is a function £ : G ^ IRN, where G is the set of games (N, v), and £ = (£i,..., £n) is a vector of payoffs to players in
v
A communication structure on N is specified by a graph (N, r), where r C rC = {ij I i, j € N, i = j} is a collection of unordered pairs of nodes. And similarly, we write r when we refer to a graph (N, r). In graph r, a sequence of different nodes (ii,..., ifc), k > 2 is a path from h to ik, if for all h = 1,..., k — 1, ihih+1 € r. We say two nodes are connected, if there exists a path from one node to another, and graph r is connected, if any two nodes are connected in graph r. Here we denote the set of all connected graph on N by G(N).
Given the characteristic function v(S), S C N and graph r, determine the new characteristic function using the following approach:
vr(S)= £ v(T), (1)
t es/r
where S/r = {{i | i, j are connected in S by r} | j € S}.
Vector £r = (£1(r),..., £n(r)) is defined as a payoff vector in cooperative game with the given graph r. For instance, given v(S), S C N mid r, if the Myerson
value (Myerson, 1977) Y(r) is chosen as the cooperative solution concept, then
for all i € N, where Shi(vr) is the i-th component of the Shapley value (Shapley, 1953) of player i in game (N,vr).
3. The Model
3.1. Two-Stage Network Formation Game with Private Information
Two-stage network formation game with private information takes places as follows.
Stage 1. The leader chooses a network (graph) r from his strategy set U (e.g. he starts a joint project), where U is a given set of connected graphs without loops on N. The cooperative game v showing the power of any coalition S (in the project) is given and known for all players.
After choosing network r, the leader informs players 2,.. .,n about the links that the player will have in r. The information is private, which means that if player 1 chooses network r, then player i will get information from player 1 that he will have the set of links r(i) = {ij | ij € r} in the network. Therefore, we have the game with imperfect information.
2, ... , n
U
which is {accept, reject}. By accepting the network, player j € {22,... ,n} joins the network. Otherwise, he starts playing as individual player. If he accepts the network, he pays a fee of Oi(r) which is a function of network r. We assume that the network is formed only if all players accept the network simultaneously. Otherwise, the network is not formed and all players act as individual players. If r is formed, player i gets a payoff of &i(r) — Oi(r). We notice that in the following part, we denote the action 'accept' by a, and action 'reject' by r.
3.2. Heterogeneous Players: Passive and Positive
With private information, players are not sure about the network structure which
is suggested by player 1 at the first stage. Consequently, every player needs to guess
the structure of the network based on the private information, thus choosing the
action according to the payoff which is strongly related to the network structure.
According to the private information which players N\{1} get, it is easily shown
that the set of all the networks player i expects to be formed is {ri € G(N) | r(i) C
ri C r(i) UA}, denoted by PIi5 where A = {jk | jk € rN,j = i,k = i,j = k}. Thus
player i is able to choose his action based on the payoff (ri) — 0i(ri), ri € PIi.
Obviously &i(ri) — 0i(rl) may be different for different ri € PIi. We call player
i a passive player, if he chooses action based on the payoff min {&i,(ri) — 0i(r^^
r iePii
N P i
positive player, if he chooses the action according to payoff max {&i(ri) — 0i(r^^
r iePii
and we denote the set of positive players in N by Q. And we assume P U Q = N \{1} holds. Thus, the payoff of player i € N \ {1} in the described two-stage game if at the end of stage 2 the network r is formed is
Ki(r)= I{i € P}^ min {Zi(n - Oiir)} + I{i € Q}^ max {&(!«) - 0^)} (3)
& (r )= Yi (r ) = Shi(vr),
(2)
r *eFii
r *eFii
where
I{i€ S}={i;i €S (4) While if r is not formed, then player i's payoff in the two-stage game is v{i}.
4. Two-Stage Game as a Game in Extensive Form
The described two-stage game with private information can be regarded as an extensive-form game ^ with player set N on a game tree denoted by Z. Let X = X1 U • • • U Xn U Xn+1 be the finite set of vertices, with X1 = {x0} being the only personal vertex of player 1, Xj being the set of personal vertices of player i € N \{1} and Xn+1 = {x : Fx = 0} being the set of terminal vertices at which the game ends and players get their payoffs. For any x € X, Fx is the set of those vertices which can be realized immediately after the vertex x has been realized, and FX = F(Fx), Fk = F(Fxfc-1). By the construction, FX0 = X2 and |J Fx = Xi+1 for i = 2,..., n.
xeXi
Thus we have |X11 = 1 |Xj| = 2i-2|U1|, for i € N\{1}, |Xn+1| = 2n-1|U1|. Specifically, we denote the vertex to which the game process moves after player 1 suggesting network rk by xpk € X2.
In terms of different private information players get, the set of personal vertices of player i € N\{1} is partitioned into subsets Xj, which is referred to as information sets of player i. Specifically, Xj = {x,y | x € Fx-2, y € Fx-2, rk(i) = rg(i)}. For any player i € N \ {1} and x € Xj, player i does not know the vertex itself, but knows that this vertex is in a certain information set Xj c Xj.
i€N
strategy of player 1 is a rule u1 assigning an action from the set U1 to the only personal vertex x0. And the strategy of player i € N \ {1}isa rule uj assigning an action from the action set {a, r} to any information set Xj c Xj. And a strategy profile u = (u1,...,un) can uniquely define a terminal vertex, in particular, the terminal vertex which is achieved by the strategy profile u = (u1,..., un), where u1(x0) = rk, is denoted by xrk,(u2Subg^e which begins at vertex xrk is denoted by Z(xrk), and ujk,i € N \ {1} denotes the truncation of strategy uj to subgame Z(xrk). In other words, ujk is a rule assigning an action from {a, r} to any information set Xj c Xj, Fx-2 C Xj'.
Given the characteristic function v(S), S C N, payoff to player i € N \ {1} at the terminal vertex xFk,(«2,...,«n) is defined as
K.(u) = H. (x ( )) = ( v({i}); 3uj(Xj) = r; j C Xj, Kj(u) = Hj (xrk,(„2,... , ^ Kj(rfc), otherwise. (5)
And for player 1, it is defined as
is , \ u ( \ ( v({1}), 3uj(Xl)= r, Fx-2 C Xl,
K1(u) = H1 (xrk ,(„2,..,„J ^ £m) — №)/ orkherwis:. (6)
The game proceeds as follows. At vertex x0, player 1 chooses a network rk € U1, then the game process moves to information sets X| , ..., X^ simultaneously, where X22' = {xrj | Tj(2) = A(2)} Fx-2 C Xj', i = 3,..., n, and players N \ {1} {a, r}
with Ui(Xi ) being the choice of player i. Fin ally, the game terminates at vertex xpk (U2 u ) ■> and player i e N gets his payoff Hi (xrk (U2 u )) defined by (5) and (6).'
5. Main Result and Example
Theorem 1. The extensive-form, game on game tree Z admits a subgame perfect Nash equilibrium, (SPNE').
Proof. Consider the families of subgame Z(xrk), Pk e U1? there are only two kinds of payoff vectors among all the terminal vetices. The first case is when all players N \{1} choose act ion a for the corresponding information sets. And the other case is when there exists at least one player choosing action r. Thus, it is easily known that (u*k,..., u*nk), where u*k(Xf) = r, i e N \ {1} is the Nash equilibrium of subgame Z(xrk), rk e U1 because any player can not change the payoff in subgame Z(xrk) by deviating from choice r. Therefore, strategy profiles (PJ , u*,..., u^), where u* = (u*,... ,u*Ui|) l = 1,..., |U]J, i e N \ {1} are all subgame perfect Nash equilibria of the game. The theorem is proved. □
Example 1. Let the set of players be N = {1, 2, 3}. The values of characteristic function are v({1}) = 1, v({2}) = v({3}) = 1/2 v({1,2}) = 3 v({1,3}) = 2, v({2,3}) = 3/2, v({1,2,3}) = 5. The strategy set of the leader (player 1) is Ui = {A,/2,13,14}, where Pi = {12,13} F2 = {12,13,23} r3 = {12,23}, r4 = {13, 23}.
Here we use Myerson value (Myerson, 1977) as a singleton solution, and 9i (r) = c|r(i)| ^s defined as the cost for player i to hold links in network r, where |r(i)| is the number of links in r(i) c being the holding cost per link.
Fig. 1 to Fig. 4 show the game trees and SPNE with c = 1/2, and different cases: 1) Q = {2}, P = {3}; 2) Q = {2, 3}; 3) P = {2}, Q = {3}; 4) P = {2, 3} respectively. Fig. 5 to Fig. 8 show the game trees and SPNE with c = 1/4 and cases: 1) — 4) respectively. And the colored links in figures show the SPNE (not unique) in the game.
Fig. 1. Two-stage game with c = 1/2, Q = {2}, P = {3}.
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Fig. 2. Two-stage game with c = 1/2, Q = {2, 3}.
Fig. 3. Two-stage game with c = 1/2, P = {2}, Q = {3}.
Fig. 4. Two-stage game with c = 1/2, P = {2, 3}.
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Fig. 5. Two-stage game with c = 1/4,Q = {2}, P = {3}.
Fig. 6. Two-stage game with c = 1/4, Q = {2, 3}.
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Fig. 7. Two-stage game with c = 1/4, P = {2}, Q = {3}.
E.g., in Fig. 1 in subgame Z(xri), with payoff vectors shown at the terminal vertices, it is easily seen that both the strategy profiles (a, a) and (r, r) are the Nash equilibria. Then in subgame Z(xr2), both the strategy profiles (a, r) and (r, r) are
the Nash equilibria. And in subgame Z(xr3), both the strategy profiles (a, a) and (r, r) are the Nash equilibria. In subgame Z(xr4), both the strategy profiles (a, r) and (r, r) are the Nash equilibria. Finally, in game Z with x0 as the initial vertex, we can get that all the strategy profiles shown in the Table 1 are the SPNE of the game.
Table 1. All SPNE and corresponding networks in the game described in Fig. 1.
Strategy Profiles \Players ^— Player 1 Player 2 Player 3
A A (a, a, a) (a, r, a)
A (a, a, r) (a, r, a)
A (r, a, a) (r, r, a)
A (r, a, r) (r, r, a)
ri ri (a, r, r) (a, r, r)
ri (a, r, a) (a, r, r)
0 ri (r, r, r) (r, r, r)
r2 (r, r, r) (r, r, r)
r (r, r, r) (r, r, r)
A (r, r, r) (r, r, r)
ri (r, r, a) (r, r, r)
r2 (r, r, a) (r, r, r)
r (r, r, a) (r, r, r)
A (r, r, a) (r, r, r)
We can also analyze the set of SPNE for games from Fig. 2 to Fig. 8 respectively. And in fact, the sets of SPNE in games from Fig. 1 to Fig. 4 are not the same. For instance, strategy profile (r3, u2,«3) where «2 = (r, a, a), «3 = (r, a, a) is a SPNE in the game shown in Fig. 3. While it is not a SPNE in the game described in Fig. 4. Thus, we can conclude that the types of players can have an effect on SPNE. While it is also easily seen that, for games which are shown from Fig. 5 to Fig. 8, the sets of SPNE are the same. In other word, the types of players do not affect the the set of SPNE in those cases.
References
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Larrosa, J. and F. Tohme (2003). Network formation with heterogenous agents. Economics Working Paper Archive Econ-WPA, Microeconomics series, No. 0301002.
Galeotti, A., S. Goyal and J. Kamphorst (2006). Network formation with heterogeneous player. Games and Economic Behavior, 54(2), 353-372.
Petrosyan, L. A. and A. Sedakov (2014). Multistage network games with perfect information. Automation and Remote Control, 75(8), 1532-1540.
Aumann, R.J. and R. Myerson (2003). Endogenous formation of links between players and of coalitions: An application of the Shapley value. Networks and Groups. Springer, Berlin, Heidelberg, 207-220.
Myerson, R. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225-229.
Shapley, L. S. (1953). A value for n-person games. Contributions to the Theory of Games, 2(28), 307-317.
