The present study considered the network games with n players, in which each player must adopt one of two strategies 0 and 1.
$$V=\left\{\text{1,2},\dots ,n\right\} and \forall i\in V ; {A}_{i}=\left\{\text{0,1}\right\}$$
1
In this case, the total number of possible combinations of players' strategies is equal to \({2}^{n}\). Assuming each of these combinations as a hypothetical Nash equilibrium, the network \(G\left(V,E\right)\)structures were specified, that the players should connect in the network \(G\left(V,E\right)\)format. The set of each player’s neighbors was determined by the set of the network edges (\(E\)). In other words:
$$\forall i\in V ; { N}_{i}\left(G\right)=\left\{ j \right|(i,j)\in E\}$$
2
In network games, player's decisions aren't affected by the strategy of the whole players, rather, the effectiveness of the decisions is determined by the network edges, and as a result, each player's decisions are affected by just the strategy of player's neighbors. and when the vertices are considered similar, only the number of neighbors, who have adopted each of the possible strategies, affects the strategy selection of each vertex. Therefore, in this framework, in the Nash equilibrium, the strategy of both vertices is the same if the number and strategies of their neighbors are equivalent. Based on this, for each of the three games; the majority game, the best-shot public goods game, and the minority game, we have obtained the set of rules and restrictions that the network between the players must have, corresponding to the existing Nash equilibrium, which is described below:
Majority game
In the majority game, each player prefers to choose the strategy that most of its neighbors have chosen. In other words, for each player assuming that the number of the members of the set, who have selected strategy 1, is greater than the number of the members who have selected strategy 0, will benefit more, if player adopts strategy 1 instead of strategy 0 and vice versa. If and, respectively, represent the strategy of player and the combination of the strategies of the set of neighboring vertices of the player then
$$\left\{\begin{array}{c}{u}_{i}\left(1,{\varvec{a}}_{{N}_{i}\left(G\right)}\right)>{u}_{i}\left(0,{\varvec{a}}_{{N}_{i}\left(G\right)}\right) if \frac{{\sum }_{j\in {N}_{i}\left(G\right)}{\varvec{a}}_{j}}{\left|{N}_{i}\left(G\right)\right|}>\frac{1}{2}\\ {u}_{i}\left(1,{\varvec{a}}_{{N}_{i}\left(G\right)}\right)<{u}_{i}\left(0,{\varvec{a}}_{{N}_{i}\left(G\right)}\right) if \frac{{\sum }_{j\in {N}_{i}\left(G\right)}{\varvec{a}}_{j}}{\left|{N}_{i}\left(G\right)\right|}<\frac{1}{2}\end{array}\right.$$
3
Assuming in a hypothetical Nash equilibrium which represents the macroscopic state of the players' strategy, \({n}_{0}\)and \({n}_{1}\) players will adopt strategy 0 and strategy 1, respectively, \({(n}_{0}+{n}_{1}=n\) ). if vertex \(i\) adopts strategy 1 in this equilibrium, then the following relation is met in the corresponding network:
$${|N}_{i}\left(G\right)\cap {V}_{1}| >{ |N}_{i}\left(G\right)\cap {V}_{0}|$$
4
$$\text{t}\text{h}\text{a}\text{t} {V}_{0}=\{v\in V; {\varvec{a}}_{v}=0\} \text{a}\text{n}\text{d} {V}_{1}=\{v\in V; {\varvec{a}}_{v}=1\} .$$
In other words, the number of the edges of vertex \(i\) to the set of vertices \({V}_{1}\)must always be more than the number of the edges of vertex \(i\) to the set of vertices \({V}_{0}\). also, for every vertex like \(j\) which adopts strategy 0, the following relation should be met.
$${|N}_{j}\left(G\right)\cap {V}_{1}|<{ |N}_{j}\left(G\right)\cap {V}_{0}|$$
5
If these conditions do not obtain even for one vertex of the network, the assumed Nash equilibrium will not be created in the network. Therefore, in the basic state, we can consider a network that for each vertex \(i\) adopts strategy 1 and for each vertex \(j\) with strategy 0:
= \({|N}_{j}\left(G\right)\cap {V}_{1}|=0\) \({ |N}_{i}\left(G\right)\cap {V}_{0}|\) & \({ |N}_{i}\left(G\right)\cap {V}_{1}| >0\) > 0 & \({ |N}_{j}\left(G\right)\cap {V}_{0}|\)(6)
and every time an edge in the network causes the numbers corresponding to \({|N}_{i}\left(G\right)\cap {V}_{0}|\) and/or\({ |N}_{j}\left(G\right)\cap {V}_{1}|\)to increase, edges should be added to the network so that the numbers corresponding to\({ |N}_{i}\left(G\right)\cap {V}_{1}|\) and\({ |N}_{j}\left(G\right)\cap {V}_{0}|\) are also increased in such a way that the above inequalities (relations 4 and 5) are always maintained.
Best-shot public goods game
In the Nash equilibrium of the best-shot public goods game, if all the neighbors of vertex adopt strategy 0 (), then the vertex must adopt strategy 1; if at least one of the neighbors of vertex adopts strategy 1 (), then the vertex must adopt strategy 0. In other words, if, the following relationship holds
$${u}_{i}\left(\varvec{a},G\right)=\left\{\begin{array}{c}1-c if {\varvec{a}}_{i}=1\\ 1 if {\varvec{a}}_{i}=0 and \exists j\in {N}_{i}\left(G\right) s.t. {\varvec{a}}_{j}=1\\ 0 if {\varvec{a}}_{i}=0 and \forall j\in {N}_{i}\left(G\right) {\varvec{a}}_{j}=0\end{array}\right.$$
7
Therefore, there should be no edges between the vertices of the set \({V}_{1}\), and each vertex of the set \({V}_{0} \text{s}\text{h}\text{o}\text{u}\text{l}\text{d}\)have at least one neighbor in the set \({V}_{1}\). The presence or absence of an edge between the vertices of the set \({V}_{0}\) does not make a difference.
Minority game
The following relation holds in the minority game
$$\left\{\begin{array}{c}{u}_{i}\left(1,{\mathbf{a}}_{{N}_{i}\left(G\right)}\right)<{u}_{i}\left(0,{\varvec{a}}_{{N}_{i}\left(G\right)}\right) if \frac{{\sum }_{j\in {N}_{i}\left(G\right)}{\varvec{a}}_{j}}{\left|{N}_{i}\left(G\right)\right|}>\frac{1}{2}\\ {u}_{i}\left(1,{\varvec{a}}_{{N}_{i}\left(G\right)}\right)>{u}_{i}\left(0,{\varvec{a}}_{{N}_{i}\left(G\right)}\right) if \frac{{\sum }_{j\in {N}_{i}\left(G\right)}{\varvec{a}}_{j}}{\left|{N}_{i}\left(G\right)\right|}<\frac{1}{2}\end{array}\right.$$
8
So, for each vertex i with strategy 1 and each vertex j with strategy 0:
\({ |N}_{i}\left(G\right)\cap {V}_{1}|<{|N}_{i}\left(G\right)\cap {V}_{0}|\) & \({ |N}_{j}\left(G\right)\cap {V}_{1}|>{ |N}_{j}\left(G\right)\cap {V}_{0}|\).(9)
Therefore, in the basic state, a network can be considered that;
\({ |N}_{i}\left(G\right)\cap {V}_{1}\left| ={ |N}_{j}\left(G\right)\cap {V}_{0}\right|=0\) & \({|N}_{i}\left(G\right)\cap {V}_{0}|>0\) & \({ |N}_{j}\left(G\right)\cap {V}_{1}|>0\)(10)
and every time an edge in the network causes the numbers corresponding to \({|N}_{i}\left(G\right)\cap {V}_{1}|\) and/or\({ |N}_{j}\left(G\right)\cap {V}_{0}|\)to increase, edges should be added to the network that the numbers corresponding to\({ |N}_{i}\left(G\right)\cap {V}_{0}|\) and \({|N}_{j}\left(G\right)\cap {V}_{1}|\) are also increased in such a way that the above inequalities (relation 9) are always maintained.
In Fig. 1, a Nash equilibrium for the 6 –player game was considered in such a way that 3 players {1,3,4} can adopt strategy 1 and 3 players {2,5,6} can adopt strategy 0. For each of the above 3 games, conditions and restrictions governing the network between the players were shown along with an example of acceptable networks.
The number of acceptable networks
Based on the proposed conditions for the acceptable networks, it is obvious that these networks are not unique in each game corresponding to the given Nash equilibrium and are in fact a function of the number of players and the combination of strategies. To obtain the total number of acceptable networks corresponding to the given Nash equilibrium, we provided the following mathematical formulas. In the best-shot public goods game, Theorem 1 specifies the number of networks for any desired number of players who adopt strategies 0 and 1, and in the majority and minority games, Theorems 2 and 3, respectively, determine the number of these networks under certain conditions. The theorems will be proven in the Materials and Methods section.
Theorem 1
The number of networks with n nodes that and (i.e., ) in the best-shot public goods game is obtained from the following equation:
$${F}_{1}\left({n}_{0},{n}_{1}\right)={2}^{\frac{{n}_{0}\left({n}_{0}-1\right)}{2}}\times {\sum }_{i=0}^{{n}_{0}}{\left(-1\right)}^{i}\times \left(\genfrac{}{}{0pt}{}{{n}_{0}}{i}\right)\times {2}^{\left({n}_{0}-i\right)\times {n}_{1}}$$
11
Theorem 2
If be the number of networks that exist in the Nash equilibrium of the majority game, then the following propositions are available:
a) | \({F}_{2}\left({n}_{0},0\right)={\sum }_{i=0}^{{n}_{0}}{(-1)}^{i}\times \left(\genfrac{}{}{0pt}{}{{n}_{0}}{i}\right)\times {2}^{\left(\genfrac{}{}{0pt}{}{{n}_{0}-i}{2}\right)}\) | (12) |
b) | \({F}_{2}\left({n}_{0},{n}_{1}\right)={F}_{2}({n}_{1},{n}_{0})\) | (13) |
c) | \({F}_{2}\left({n}_{0},2\right)={F}_{2}\left({n}_{0},1\right)={F}_{2}({n}_{0},0)\) | (14) |
Theorem 3
If be the number of networks that exist in the Nash equilibrium of the minority game, then the following propositions are available:
a) | \({F}_{3}\left({n}_{0},0\right)={F}_{3}\left({n}_{0},1\right)=1\) | (15) |
b) | \({F}_{3}\left({n}_{0},{n}_{1}\right)={F}_{3}({n}_{1},{n}_{0})\) | (16) |
We also obtained the number of networks corresponding to the considered Nash equilibrium for the 2, 5 and 8–player games in which each of the 0 and 1 strategies is chosen by the different combinations in these games. The results were shown in Fig. 2.
The majority and minority games’ graphs are symmetric. However, the difference lies in the fact that the largest number of networks in the minority game corresponds to the situation where half of the players adopt strategy 1 and half of the players adopt strategy 0, whereas in the majority game, the largest number of networks corresponds to the situation where all the players adopt the same strategy. The best-shot public goods game’s graph is not symmetric and its largest number of networks corresponds to the situation where few players adopt strategy 1.
The probability of occurrence of acceptable networks
As shown in Fig. 2, the number of acceptable networks corresponding to the given Nash equilibrium is very large. The distribution of acceptable networks and the probability of their occurrence were investigated below. The results obtained according to the number of networks edges for the 5–player games were presented in Fig. 3. To calculate the probability of occurrence of each of the acceptable networks, the space of all possible networks was considered and an edge was randomly added/removed in such a way that it can converge to the first acceptable network. In this simulation, the number of players who adopt strategy 1 is equal to zero or five in the majority game, one in the best-shot public goods game, and two or three in the minority game. The possible networks can have from 0 to 10 edges and the total number of possible networks is equal to 1024. Figure 3 displays the distribution of acceptable networks according to the number of edges and the number of networks that have converged to the first acceptable network of the desired Nash equilibrium. We also calculated the ratio of the number of converged networks to the number of acceptable networks. The results indicated that although the distribution of acceptable networks follows a normal distribution, their occurrence probability is non-uniform, and the probability of occurrence increases with density. In other words, a greater number of networks, from the total space of possible networks, have converged to the acceptable dense networks.